Reverse engineering
smooth muscle molecular mechanics
Lennart Hilbert
Doctor of Philosophy
Department of Physiology
McGill University
Montréal,Québec
2014-4-15
A thesis submitted to McGill University in partial fulfillment of the
requirements of the degree of Doctor of Philosophy
c
Lennart
Hilbert, 2014
DEDICATION
Research can not exist without a material and societal foundation. This
thesis is dedicated to a society that believes that sacrificing a part of their
productivity for academic research ultimately supports its well-being and
progress.
On a personal level, I am deeply thankful for being welcomed in
Canada, in Québec, and in Montréal as a new home; and for not being
forgotten in my “old home” either. Most importantly, there are my partner
Carolyn, her family who has shown true hospitality, and my parents and
brother.
Inspired by daring fellow students and trusted by my artistic collaborators, during my graduate studies I joined artistic expression and scientific
research, two integral parts of my life.
In moments of vanity I believe that my path is guided by an unerring
gut feeling. In fact, I owe much to my teachers and mentors, who fostered
my educational and academic transitions.
ii
ACKNOWLEDGEMENTS
First and fore-most, both my thesis supervisors have contributed
extensively to this work and my development. Great teachers teach not
through instruction, but by their actions and way of being. Michael C.
Mackey embodies the curiosity, perseverance, willingness to challenge
and collaborate that has set humans so far apart. Also, anyone who has
positively contributed to as many persons’ lives as Michael must have done
a couple of things right. Anne-Marie Lauzon is a leader, in the best sense of
the word. Trusting her group, accepting the limited control we have over the
results of our work, and always supporting those in her responsibility, she
allows people to grow, gives room for real team work, and is always ready
to be surprised by a truly unexpected finding. Thus, while not necessarily
always by their conscious intent, working with and for both of them has
profoundly affected my perspectives on work and life.
I was lucky to work in an experimental group and a theoretical group
simultaneously. My experimental colleagues might, at times, not have
felt as blessed by this than I did, polluting the lab meeting with always
a little too much mathematical “enlightenment”. I certainly benefited,
but also hope to not remain the sole beneficiary after all. I am endlessly
grateful for Nic Roman and Genevieve Bates’s patience in face of my hands
trying to handle something different than a computer keyboard and a pen,
and my brain trying to dissect an experiment where some down-to-earth
troubleshooting would be in order. I have learned much from Gen and Nic,
sharing equipment, worries, and successes over the past years.
In Anne-Marie’s laboratory, I worked with bright and highly motivated
undergraduates – who made me wonder what I was still doing when I was
iii
at that stage of my studies. Jenna Blumenthal recorded motility data,
priming the actin and tropomyosin isoform investigation included in this
thesis. Shivaram Cumarasamy worked hand in hand with me, recording
the motility assay data included in the first article included in this thesis.
Stéphanie Néron got used to running phosphorylation quantification assays
very quickly, which we analyzed and fitted together. Zsombor Balassy, after
fighting his way through cataloguing our entire stock of chemicals, is at the
time of writing working with me as the first “pure” modelling student in
Anne-Marie Lauzon’s laboratory.
The laboratory being the lively and enjoyable place it is, I want to
also thank Rachel Bullimore, Gijs Ijpma, Linda Kachmar, Jack Kinsalla,
Oleg Matusovsky, Nedjma Zitouni for all support in the lab and the the
good times shared. The same applies for the students of James Martin’s
laboratory, Mike O’Sullivan – my common Physiologist in the MeakinsChristie outpost, and the wider Meakins-Christie community.
Lastly, acknowledgements go to Apolinary Sobieszek, who fills our
freezer with more purified proteins than we can ever hope to investigate.
In Michael Mackey’s theoretical research group and its environment, I
have seen the beginnings, progressions, and completions of several graduate
degrees and visitors’ research projects. On my way, I was happy to have the
academic input and comradeship of Raluca Apostu, Bartek Borek, Grace
Brooks, Morgan Craig, Gabriel Provencher Langlois, Thomas Quail, Shahed
Riaz, T. K. Shajahan, Frédéric Simard, Romain Yvinec, and Daniel Zysman.
It would not have been the same without their company and guidance.
Two visiting faculty members, Moises Santillan and Jinzhi Lei, have
taken distinct influence on my research. Moises introduced me to molecular
iv
motors. Discussions with and lectures by Jinzhi were essential to the
development of the model of the actin-myosin interface.
My research was monitored by an enthusiastic and constructively
critical committee, consisting of Leon Glass, Dilson Rassier, Maurice
Chacron, Anmar Khadra, and my supervisors – and on one occasion Michael
Guevara, who seemed to be the only one unaware that he was not on the
committee but nevertheless had several great comments.
My appreciation also goes to Dilson Rassier and his laboratory, especially Fabio Minozzo. Dilson has allowed me to talk to everyone in his lab
for one whole day and many smaller visits after. From this openness, an
effective collaboration and close friendship with Fabio came into existence.
David Albrecht, who is a close friend from the early days with the
Studienstiftung, has also become a valuable and esteemed collaborator. The
investment of his time and knowledge has made essential contributions to
my first publication and my understanding of the Molecular Biology of the
SOS response.
An international travel award of the Faculty of Medicine allowed me
to visit Josh Baker and his laboratory at the University of Nevada, Reno.
Their hospitality made the stay very enjoyable and let me gain insight into
the methods and approaches another lab uses to address questions similar to
the ones addressed in this thesis and Anne-Marie Lauzon’s laboratory.
At the Centre for Applied Mathematics, we started a Student chapter,
first Thomas Quail and me, and were later joined by Frédéric Simard and
Morgan Craig. Considering everyone’s busy schedules, diverse interests, and
different places we work in – we have made some great things happen, and
some of it is still underway. One immediate result of the student chapter is
v
a collaboration between Thomas Quail, Greg Stacey, and myself – I finally
revoked my decision to “never get my hands dirty in Neuroscience”.
The graduate experience in the Department of Physiology has been a
good one. We have an active and diverse body of students. Our graduate
coordinators always kept us sorted – Christine Pamplin and Rosie Vasile.
John White as our graduate adviser has given me great advice when the
“going got tough”. Lastly, our chair John Orlowski inspires much of the
fascination and readiness to enter ever new territories and fields, which is
the life blood of Physiology.
Anne-Marie Lauzon and Michael Mackey gave critical comments on the
thesis draft. Anne-Marie Lauzon and Nedjma Zitouni translated the thesis
abstract into French.
Financial support as well as the interdisciplinary experience of summer
academies of the Studienstiftung des Deutschen Volkes (the German people’s
foundation for studies) were instrumental in my transition from Physics
into Physiology and Biology and from Germany to Canada. My research
was supported by funding from the Natural Sciences and Engineering
Research Council of Canada (NSERC), the National Institutes of Health
(NIH), and the Canadian Institutes of Health Research (CIHR). I was
directly supported by the J. P. Collip Fellowship, a Travel Award, and
several Graduate Excellence Awards from the Faculty of Medicine, Graduate
Research Awards from the Research Institute of the McGill University
Health Centre, a two-year salary award from the Canadian Institutes
for Health Research (CIHR) funded McGill Systems Biology Graduate
Training Program, a Graduate Research Award from the Centre for Applied
Mathematics in Bioscience and Medicine, and GREAT travel awards from
the Department of Physiology.
vi
ABSTRACT
The interaction of filaments of muscle myosin motor proteins with
actin filaments is central to contraction of skeletal, cardiac, and smooth
muscle. In smooth muscle, however, the myosin filament structure, the
arrangement of actin and myosin into filament-filament interfaces, and the
ultrastructural arrangement into contractile units differ fundamentally from
skeletal and cardiac muscle. Cell activation and contraction alter the length
and ultrastructural arrangement of actin-myosin filament-filament interfaces.
Many smooth muscle contractile proteins and their isoforms directly affect
actin-myosin interaction and are also changed in disease.
In this thesis, we utilized the in vitro motility assay as an effective
readout of the mechanochemistry of a group of myosins coupled via an actin
filament. Using newly developed video analysis software and mathematical
models, we inferred the influence of actin-myosin interface length. We
also determined the effect of actin and tropomyosin isoforms on individual
myosin mechanochemistry and myosin-myosin coupling. We provide new
insights to molecular mechanics and regulation of actin-myosin interfaces of
varied length, which are relevant to the unique molecular and ultrastructural
force generation mechanisms in smooth muscle.
Our first study investigates the emergence of coordinated myosin group
mechanochemistry with an increasing number of coupled myosins (N ,
assumed proportional to actin length). Actin was observed in two discrete
states, the arrested and the actively sliding state. These states’ existence
depended on N . At intermediate N , slow switching between the two states
occured. Stochastic simulation of individual myosin binding sites on actin
that are coupled via a rigid actin backbone and have strain dependent
vii
mechanical stepping rates reproduced the experimental data. In the model,
the arrested and the motile state corresponded to persistent arrest vs.
coordinated cycling of actin-myosin on the level of the mechanically coupled
myosin group. A simplified model identified N dependent bifurcations and a
bistable region at intermediate values of N .
Our second study assesses the effect that combinations of different
contractile actin isoforms and smooth muscle tropomyosin isoforms on
molecular mechanics of smooth muscle myosin. We developed a combination
of bootstrap statistics and principal component analysis, which we used
to identify molecular mechanics differences between actin isoforms and
between tropomyosin isoforms. In the absence of tropomyosin, no differences
were found. Differences were specific to each actin-tropomyosin condition.
Using our stochastic simulation, we inferred changes in individual myosin
mechanochemistry and myosin to myosin coupling that could explain these
differences.
viii
ABRÉGÉ
L’interaction entre les filaments de myosine et les filaments d’actine
est essentielle dans la contraction des muscles squelettiques, cardiaques
et lisses. Par contre, dans les muscles lisses, la structure des filaments de
myosine, l’interface des filaments de myosine et d’actine et l’arrangement
en unité contractile diffèrent fondamentalement des muscles squelettiques
et cardiaques. L’activation des cellules musculaires lisses et leur contraction
altèrent la longueur et l’ultrastructure des filaments de myosine et d’actine.
Plusieurs protéines contractiles des muscles lisses et leurs isoformes affectent
directement les interactions entre la myosine et l’actine; elles sont également
altérées dans certaines maladies.
Dans cette thése, nous avons validé l’utilisation du test de motilité
in vitro en tant que lecteur de la mécano-chimie d’un groupe de myosines
couplées par un filament d’actine. En utilisant un logiciel d’analyse de
vidéo et des modèles mathématiques que nous avons récemment développés,
nous avons déduit l’influence de la longueur de l’interface entre myosines
et les filaments d’actine. Nous avons également étudié l’effet des isoformes
de l’actine et de la tropomyosine sur la mécano-chimie des molécules de
myosine et sur les interactions d’une myosine à l’autre. Nous apportons
des nouvelles perspectives sur la mécanique moléculaire et la régulation des
interfaces actine-myosine à différentes longueurs, applicables aux mécanismes
moléculaires et ultrastructurelles de la génération de force, spécifiques aux
muscles lisses.
Notre première étude a mis en évidence la mécano-chimie d’un groupe
de myosines coordonnées, à un nombre croissant de myosines couplées (N,
étant proportionnel à la longueur d’actine). L’actine est observée dans
ix
deux états distincts, l’état immobilisé et l’état propulsé. Ces états existent
dépendamment de N. Pour un N Intermédiaire, un lent changement entre
les deux états a lieu. Une simulation stochastique des sites de liaison de
myosines individuelles sur l’actine a été réalisée. Les myosines ont été
couplées via une structure rigide d’actine avec un taux d’achèvement des
pas mécaniques dépendant de la tension; ceci a reproduit nos données
expérimentales. Dans ce modèle, l’état immobilisé et mobile correspondent
à un arrêt persistant vs un cyclage coordonné de l’actine-myosine au niveau
du groupe de myosines couplées mécaniquement. Un modèle simplifié a
identifié des bifurcations dépendants de N et une région bistable à des N
intermédiaires.
Notre seconde étude a établi l’effet des combinaisons des différents isoformes d’actine et de tropomyosine sur la mécanique moléculaire du muscle
lisse. Nous avons développé une combinaison des statistiques «bootstrap»
et d’analyse du composant principal, permettant ainsi l’identification des
différences mécaniques moléculaires entre les isoformes d’actine et les isoformes de tropomyosine. En absence de tropomyosine, aucune différence n’a
été observée. Les différences etainent spécifiques à chaque condition actinetropomyosine. Notre simulation stochastique, nous a permis de déduire des
changements dans la mécano-chimie des myosines individuelles ainsi que
dans le couplage myosine-myosine. Ceci pourrait expliquer les différences
observées.
x
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . .
iii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
ABRÉGÉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
1
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
1.2
Author Contributions . . . . . . . . . . . . . . . . . . . . .
Original Contributions to the Field . . . . . . . . . . . . .
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2
2.1
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Published Article One . . . . . . . . . . . . . . . . . . . . . . . .
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2.2
2.3
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3.1
3.2
3.3
3.4
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Research Questions . . . . . . . . . . . . . . .
2.1.1 Mechanical Plasticity . . . . . . . . . .
2.1.2 Dynamic Phenotype Change . . . . . .
2.1.3 Phenotype in Physiology and Disease .
2.1.4 Specific Questions . . . . . . . . . . . .
Detailed Background . . . . . . . . . . . . . .
2.2.1 Overview . . . . . . . . . . . . . . . . .
2.2.2 Cellular Contractile Structures . . . . .
2.2.3 Mechanochemistry of Actin and Myosin
2.2.4 Plasticity of Cellular Ultrastructures . .
2.2.5 Plasticity in Physiology and Disease . .
Research Outline . . . . . . . . . . . . . . . .
Article Information . . . .
Main Article . . . . . . . .
Supplementary Material .
Thesis Revision Comments
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Published Article Two . . . . . . . . . . . . . . . . . . . . . . . . 113
4.1
4.2
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Article Information . . . . . . . . . . . . . . . . . . . . . . 113
Main Article . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Supplementary Methods . . . . . . . . . . . . . . . . . . . 134
xi
5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.1
5.2
Conclusions . . . . . . . . . . . . . . . . . .
Future Directions . . . . . . . . . . . . . . .
5.2.1 Active State of Myosin Group Action
5.2.2 Classification of Regulatory Proteins .
5.2.3 Challenges in Plasticity Theory . . . .
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xii
Chapter 1
Preface
In accordance with McGill thesis regulations, the different authors’ contributions to the work presented in this thesis will be explained, followed by
a list detailing original scholarship and distinct contributions to knowledge.
1.1
Author Contributions
For the first article, the experiments were conceived and designed by
Lennart Hilbert and Anne-Marie Lauzon. The experiments were performed
by Shivaram Cumarasamy and Nedjma Zitouni. The data were analyzed
by Lennart Hilbert and Shivaram Cumarasamy. The mathematical model
was developed by Lennart Hilbert and Michael Mackey. The numerical
simulations were developed, programmed, and analyzed by Lennart Hilbert.
The paper was written by Lennart Hilbert.
For the second article, in accordance with what is stated in the published article, the author contributions are the following. The experiments
were conceived and designed by Lennart Hilbert, Jenna Blumenthal, and
Anne-Marie Lauzon. The experiments were performed by Lennart Hilbert
and Nedjma Zitouni. The data were analyzed by Lennart Hilbert. Reagents
were contributed by Apolinary Sobieszek. Preliminary experiments that
preceded final data collection were performed by Genevieve Bates and Jenna
Blumenthal. The video analysis software was designed by Lennart Hilbert.
The video analysis software development was instructed and tested by
Genevieve Bates and Horia Roman. The mathematical model was developed
by Lennart Hilbert and Michael Mackey. The numerical simulations were
1
developed, programmed, and analyzed by Lennart Hilbert. The paper was
written by Lennart Hilbert.
1.2
Original Contributions to the Field
1. I have developed a new video analysis software for the sliding actin
filament in vitro motility assay. This software provides significantly
faster processing of the raw data than using manual or semi-automated
methods. It allows fully automated processing of the raw data, thus
reducing analysis workload. Its detailed analysis features and the
possibility to integrate further analysis steps and statistics allow other
researchers to write their own MatLab scripts to execute custom
tailored analysis of formerly inaccessible features of actin motility.
2. I have discovered fundamental discrepancies between the Uyeda et al.
mathematical model [173] commonly used to interpret myosin driven
in vitro actin sliding and actual experimental measurements.
3. I have discovered that myosin driven in vitro actin sliding is fully
arrested below ≈ 0.3 μm actin length, switches between arrest and
continuous sliding between 0.3 μm and 1.0 μm, and is continuous above
1.0 μm.
4. I have discovered that the transition into continuous motility depends
on the number of myosin motors that can effectively interact with an
individual actin filament, and not on the actin length per se.
5. I have discovered that increasing the number of myosins that can
simultaneously interact with an individual actin filament increases
the sliding velocity of actin, even when there is always enough myosin
available to drive actin.
6. Based on experimentally measured individual myosin mechanochemical
mechanisms and rigid mechanical coupling via the commonly propelled
2
actin filament, I have developed a mechanically realistic mathematical
model that explains or predicts all aforementioned features of in vitro
actin sliding.
7. I have measured the smooth muscle myosin driven in vitro actin
sliding for different combinations of isoforms of contractile actin and
smooth muscle tropomyosin.
8. I have created a new combination of bootstrapping statistics and
principal component analysis, which allowed me to identify significant
differences in molecular mechanics from complex in vitro actin sliding
data consisting of four actin length resolved features of actin motility.
9. I have retraced these differences to distinct, isoform combination
specific features of actin sliding.
10. I have executed simulations of the mathematical model of smooth
muscle myosin driven in vitro actin sliding on the Consortium Laval,
Université du Québec, McGill and Eastern Quebec (CLUMEQ) high
performance computing infrastructure to assess the influence of all
parameters of the mathematical model on measurements made from
the in vitro motility assay of smooth muscle myosin.
11. Based on this assessment of the model parameters’ influence, I
have proposed changes in the individual myosin chemistry and the
intermyosin mechanical coupling that explain the in vitro actin
sliding differences observed for the different actin-tropomyosin isoform
combinations.
3
Chapter 2
Introduction
2.1
Research Questions
2.1.1
Mechanical Plasticity
Smooth muscle lines many hollow organs, giving dynamic control over
tissue mechanical properties, cross sectional areas, and hollow volumes [134].
It is a regulatory and structural element of “containers”, “ducts”, and “valves”
in the human body. The evolution of smooth muscle has lead to its unique
ability to alter its working range as well as its contractile characteristics
dynamically; this ability distinguishes it from skeletal and cardiac muscle
and allows efficient and effective fulfilment of its physiological function
[134, 69, 65]. These cellular properties are constituted by equally unique
dynamic plasticity in the mechanisms of force generation: the length of actin
and myosin filaments, their association into filament-filament interfaces, and
the ultrastructural organization into contractile units adjust in response to,
and in temporal coordination with, active and passive mechanical changes.
2.1.2
Dynamic Phenotype Change
Other than their mechanical plasticity, smooth muscle cells also show
a unique phenotypic plasticity: (1) cells can alter between the contractile
and the proliferative (non-contractile) phenotypes, (2) expression levels
of contractile proteins and their isoforms can change, and (3) cells of
contractile phenotype can show differences in their contractile apparatus’s
response to mechanical cues [69, 65, 54].
4
2.1.3
Phenotype in Physiology and Disease
The contractile plasticity and phenotypic plasticity are at the core of
smooth muscle physiological function. At the same time they are believed to
be a major source of smooth muscle associated diseases.
Permanently contracted smooth muscle contributes to blood vessel wall
mechanical properties and adjustment of vessel diameter. Smooth muscle is
involved as a mechanical effector and in many pathways in vascular disease,
such as hypertension, ischemia, and heart failure [48]. Much attention has
been paid to hypertrophy, hyperplasia, and dedifferentiation into immature
smooth muscle cells as causes of the observed mechanical changes, as well
as to the underlying pathogenic factors and pathways [103]. More recently,
experimental evidence from animal disease models also suggests that the
contractile phenotype (tonic vs. phasic smooth muscle) can be altered and
thereby contribute to the observed pathologies [48].
Bands of smooth muscle wrap around the airways, and changes in this
smooth muscle contribute to airway narrowing and hyperresponsiveness in
asthma. Much attention has been given to the general increase in airways
smooth muscle mass, leading to thickening, increased contractility, and
increased risk of airway obstruction [34]. The underlying mechanisms are
an increase in size (hypertrophy) or number (hyperplasia by proliferation,
switching between contractile and non-contractile phenotype, cell migration)
of contractile airway smooth muscle cells [82]. However, as for other smooth
muscle contractility related pathologies, alternative mechanisms of smooth
muscle hyperresponsiveness have been discussed. These include mechanical
plasticity in response to mechanical cues [82, 164], and changes in contractile
protein expression [107] and in the ultrastructures formed by these proteins
[65].
5
Smooth muscle constitutes the uterine contractile structure, the
myometrium. Alterations in the myometrium smooth muscle can lead
to different problems in reproductive health [1]. While not actually a
pathological state, large alterations in smooth muscle contractility must
occur during pregnancy before term vs. during term [167]
Smooth muscle in the wall of the urinary bladder maintains structural
integrity and drives voiding across a volume range from full bladder to the
expulsion of 95 % of full volume. Pathological and age related alterations
in bladder smooth muscle contribute to decreased capability to build
up pressure needed for voiding. Experimental findings indicate that,
for example, partial bladder outlet obstruction can lead to changes in
contractile protein expression and molecular and ultrastructural contractile
properties in the smooth muscle cells, which lead to loss of capability to hold
and void a regular volume of urine in a controlled manner [186].
Intestinal smooth muscle goes through cycles of contraction and
relaxation, constituting peristalsis and segmentation. In rats and guinea
pigs used as animal models of intestinal obstruction, increased tension
development, increased responsiveness to contraction stimulation, decreased
unloaded shortening velocity, a shift in the expression of smooth muscle
myosin isoforms, and an altered sensitivity of unloaded shortening velocity
to ADP concentrations was found [12, 13, 116]. Mechanically detected
hypertrophy and hyperresponsiveness to agonists disappeared two weeks
after the removal of the obstruction [13].
For all these pathologies, it is a common theme that the observed
smooth muscle phenotype changes are induced by various stimuli, comprising inflammatory signals, neural control, as well as mechanical cues [117].
A promising avenue towards understanding these pathological changes is
6
the comparison to differences occurring in different parts of the body and
throughout normal ageing [186, 106, 166, 105]. It is not unthinkable that
many diseases are caused by dynamic maladjustment of the smooth muscle
phenotype [69]. The molecular and cellular differences underlying the phenotype changes are therefore relevant for understanding disease mechanisms
as well as for answering fundamental questions regarding smooth muscle
contraction.
2.1.4
Specific Questions
The above review of disease related changes in smooth muscle phenotype shows changes at various levels. Expression levels and ultrastructural
organization of contractile proteins, cell-cell interactions, mechanical protocols, animal model results, and regulatory pathways are all represented and
related to each other. Limiting our perspective to individual cells, still many
molecular mechanisms and levels of cellular organization interact to determine active and passive mechanical properties. When considering primarily
active properties, one essential molecular interaction must be pointed out:
the mechanical action of myosin protein motors on actin filaments. Even
though a multitude of regulators, pathways, and levels of organization are in
play, ultimately contraction is effected by active stepping of myosin motors
that transiently bind to passive actin filaments. Taking this interaction as
an “effective read-out”, the relevance of various influences can be compared
at the level of a single active molecular mechanical process.
The goal of this thesis is to develop, understand, and apply new quantitative methods and models for the in vitro motility assay, which monitors
the propulsion of actin by myosin, which will deliver this “effective read-out”.
7
Importantly, this approach establishes an experimental and theoretical connection between (1) individual actin-myosin interactions, (2) the physiologically relevant coordinated mechanical action of groups of myosins working
on the same actin filament, and (3) the regulation of the mechanochemistry
of groups of myosin by actin filament binding proteins. Connecting these different aspects and levels of smooth muscle molecular mechanics gives insight
into the following challenges in understanding smooth muscle molecular
mechanics, which relate to smooth muscle mechanical phenotypes.
1. The ratios between fast and slow smooth muscle myosin isoforms
change between smooth muscle mechanical phenotypes. Single
molecule insights into the isoforms’ actin-myosin kinetic differences, as
well as measurements from in vitro motility assays with actin filaments
and tissue mechanics experiments exist. However, currently there is
no coherent way to extrapolate from the single molecule findings to
situations where myosins are mechanically coupled into groups of 50 or
more motor proteins working on an individual actin filament.
2. The ratios of contractile actin isoforms and smooth muscle tropomyosin
isoforms change between smooth muscle mechanical phenotypes. Any
experimental insight into the isoform specific differences in molecular
mechanical function is lacking. The first reason is that actin and
tropomyosin isoforms likely have a functional synergy in molecular
mechanics, which has not yet been investigated. The second reason is
that the molecular mechanical process that these isoforms would effectively alter is the interaction of many myosin motor proteins with one
actin filament. The assessment of the alterations of the kinetics of individual myosin-actin interactions in the background of a mechanically
coupled group of myosins is not currently accessible in experiment.
8
3. It is widely believed that the smooth muscle phenotype also alters
the plasticity of the smooth muscle cell cytoskeleton and contractile
apparatus in response to mechanical protocols. Amongst other aspects,
this plasticity affects the length of actin filaments and the length of
actin-myosin interaction interfaces. While this should logically affect
the number of myosin proteins capable of acting at any individual
actin-myosin interface, it is unclear how the number of mechanically
coupled myosin motor proteins affects their mechanical performance as
a group.
2.2
Detailed Background
2.2.1
Overview
Compared to differentiated skeletal and cardiac muscle cells (both
striated muscles), smooth muscle cells have a unique capacity to dynamically
change their phenotype. The phenotype can be categorized according to
several perspectives, amongst them physiological, embryologic, molecular,
and anatomic [48]. In the beginning, three specific phenotypic categorizations will be discussed, each of which have been shown to undergo dynamic,
reversible changes: (1) cells can be “synthetic” (also called “immature” or
“proliferative” phenotype) or “contractile” (also called “mature” phenotype,
in which cells exhibit contractile protein structures), (2) contractile phenotype cells can differ in the expression levels of contractile proteins and their
isoforms, thus modulating their mechanical properties (3) contractile phenotype cells can differ in the way they exhibit mechanical plasticity of their
contractile apparatus in response to mechanical protocols [69]. Functional
differences between the phenotypes are manifested in smooth muscle cell
electrophysiology, contractile behavior, pharmacology, and ultrastructure
[69]. They occur between organs, between localized regions of organs, or
9
even individual smooth muscle cells [69]. Molecular and biochemical counterparts of these differences can be found at the level of contractile proteins,
ion channels, and proteins that more directly regulate contraction [69].
Expression levels that are typical of a specific phenotype can result from
stimuli such as mechanical protocols, cyctokines, and extracellular matrix
composition [69].
Synthetic and contractile smooth muscle phenotype. When
smooth muscle cells are cultured, a loss of their contractile capacity is observed; they undergo modulation into a non-contractile, synthetic phenotype
[68]. Conversely, prolonged serum deprivation of cultured cells leads to differentiation of a subgroup into the contractile phenotype [68], which exhibits
characteristic gene expression profiles upon differentiation [121]. Similar
changes between a proliferating, a synthetic, and a contractile phenotype are
associated with pathological airway and vascular remodelling [147, 82, 151].
Shifts between the proliferative, synthetic, and contractile phenotype are associated with specific pathologies, e.g. chronic asthma and airway narrowing,
which stem from more permanent changes in the airway wall [86, 147, 151].
These differences do not immediately contribute to pathologies that manifest
via a response to a trigger stimulus, such as airway hyperresponsiveness.
Such stimulus triggered pathologies relate more to the mechanical properties
of the smooth muscle cells. To focus on these pathologies, our review will
be limited on the foundations and mechanisms of change in the mechanical properties of contractile smooth muscle cells. These properties will be
interchangeably referred to as the “contractile phenotype” or “mechanical
phenotype” of muscle cells.
10
Figure 2–1: Contractile ultrastructure of the smooth muscle cell A)
A network of dense bodies (gray circle) is connected by contractile bundles
(black lines). Upon shortening of the contractile bundles, the network contracts and the contraction is transmitted to the cell membrane via dense
bodies attached to the cell membrane (also called attachment plaques). Arrows indicate direction of contraction. B) Contractile bundles between dense
bodies consist of actin thin filaments (black) attached to the dense bodies
and myosin thick filaments (red) that can interconnect actin filaments from
both dense bodies. C) Shortening of the bundles occurs by the mechanical
action of individual myosins in the thick filament on actin thin filaments
from both dense bodies. The actin filaments emanate from the dense bodies
with a specific polarity, as indicated by the arrowheads in the thin filaments.
Upon myosin activation, the two actin filaments move towards each other.
2.2.2
Cellular Contractile Structures
Smooth muscle cells have a distinctly different architecture than
skeletal and cardiac muscle cells. Where the latter display static arrays
of axially and laterally aligned actin-myosin filaments, the former have a
less clearly structured actin and myosin filament organization and specific
ultrastructural components and arrangements to transmit force from actinmyosin interfaces to the cellular level.
Smooth muscle cell contractile structure and its constituents
To put the molecular and ultrastructures of smooth muscle into
perspective, some general parameters describing smooth muscle structure
and function will be reviewed [52]. The length of smooth muscle cells
is relatively small – mammalian visceral muscle cells are 2.3 to 3.5 μm
long, while skeletal muscle cells can span the whole length of a muscle.
11
Smooth muscle cells do, however, possess a wide range of contraction, in
situ as well as in vitro. Shortening up to a quarter of slack length can be
observed, and is fully reversible. Muscle cells do not slide past each other,
the relative geometry of the cells in the tissue is maintained. Cells do not
undergo irreversible structural changes. Compared to skeletal muscle cells,
a similar force (isometric) of 200 to 400 mN/mm2 is achieved. Differently
from skeletal muscle, reaching isometric force can take several seconds, the
force is supported by a markedly lower relative content of myosin (molar
ratio of actin to myosin ranging between 1.5 and 3.85, compared to ≈ 0.36
in skeletal muscle fibers), and can be maintained for for several hours.
In transverse sections of smooth muscle cells, different dominant
molecular structures associated with contractile function can be identified:
actin (thin) filaments, myosin (thick) filaments, intermediate filaments, and
dense bodies [162, 52, 153]. The thin filaments are mostly organized into a
regular, lattice like order, with occasional association into rosettes or orbits
of several thin filaments bundled around thick filaments [52, 153]. Thick
filaments are more rare, but also exhibit a regular spacing in transverse
sections. Intermediate filaments consists of desmin, vimentin, or pre-keratin
and are found in the proximity of dense bodies [162, 153], and appear as
linking elements between actin and the dense bodies [52, 29, 154, 37]. The
ratio of myosin filament count to actin filament count is approximately 12:1
to 15:1 [52, 162, 153].
Within smooth muscle cells, cytoplasmic dense bodies that are held in
place by cytoskeletal structures order the contractile elements, consisting of
myosin thick filaments and actin thin filaments (Fig. 2–1 A) [37, 52, 17, 5].
Attachment plaques, or dense bands, anchor the actin and myosin filament
containing contractile elements to the cell wall for force transmission,
12
probably with involvement of the proteins α-actinin, filamin, and vinculin
[168, 52, 5]. In intact smooth muscle cells, the dense bodies exhibit clear
geometric, “plait like” order [154]. When adhered to a cover slip surface, the
cytoskeletal ordering elements lose integrity, and surface bound contraction
obviates a separation into contractile bands resembling of skeletal and
cardiac muscle sarcomeres [154]. This supports the common notion that
the contractile unit of smooth muscle are interfaces between actin thin and
myosin thick filaments, which exist in between adjacent dense bodies (Fig.
2–1 B). Structures that would support this notion have been recreated in
vitro in a reconstitution assay of different muscle type myosins and actin
[169].
Actin and Myosin filaments.
A question that has attracted much attention is that of the length
of smooth muscle actin thin and myosin thick filaments. These lengths
are highly relevant to understanding the basic contractile elements, the
contractile ultrastructure, and the macroscopic contractile properties of
smooth muscle cells. In general, thin filaments clearly exceed thick filaments
in length, and both filaments exhibit broad and potentially dynamic
variation in lengths. Using supercontracted cell fragments, actin filament
lengths of 3 μm to 10 μm were estimated to be present in intact smooth
muscle cells. In the same study, the myosin filament length was estimated
to be 1.6 ± 0.3 μm [154]. Using serial transverse sections, myosin filament
length was initially estimated as ≈ 2.2 μm [5], while more recent findings
demonstrated an exponential distribution of myosin filament lengths with
a characteristic length of 0.116 to 0.182 μm, dependent on the source of
the smooth muscle and the muscle’s contractile state [115]. In an in vitro
reconstitution assay where thick and thin filaments spontaneously self
13
organized into contractile bundles, smooth muscle myosin thick filament
lengths could be reproducibly set to specified lengths between 0.2 and
0.9 μm [169]. Setting the length required changes in the ionic strength and
application sequence and timing of the myosin polymerization buffers, and
affected the contractile bundles’ maximal contraction force.
Ultrastructure
While it is clear that actin-myosin interactions drive smooth muscle
contraction, the ultrastructure that contains the actin-myosin interfaces
has been more elusive than the well known sarcomere in striated muscle.
Actin filaments, which are connected to and emanate from the dense
bodies, exhibit a regular polarity in myosin binding. Actin filaments in
longitudinal smooth muscle cell sections that were decorated with the
myosin subfragment 1 (S1) exhibited arrow like patterns that consistently
pointed away from the dense body they originated from [17]. Because the
actin filaments emanate with the same outbound polarity from dense bodies
and are organized longitudinally, at a given myosin filamentous structure
actin filaments with opposing polarity will make contact (Fig. 2–1 C).
One can therefore expect a myosin filamentous structure that supports
this mode of contact. Indeed, in vitro reconstruction of myosin filaments
under near physiological conditions yielded side polar filaments, which are
in good agreement with other, more in situ observations [29, 153]. From
the perspective of the long range of contraction in smooth muscle (relative
to striated muscle), side polar myosin filaments which can slide along
the full extent of much longer actin filaments make sense [29, 30]. More
specific statements about the detailed structure of the basic contractile
unit were made based on combinations of tracheal smooth muscle strips,
electron microscopy of dense body-myosin thick filament organization, and
14
modelling of force-length relationships under different length adaptation
protocols [77]. This study coherently explains all their observations with
a model where actin and myosin filaments organize into sarcomere like
structures, which are established along the whole distance between adjacent
dense bodies. In an ultrarapid freeze study, these claims could largely be
substantiated [83]. Considering that the distance between dense bodies
changes as the cell length is adjusted, an interesting observation can be
made about the sarcomere like structures when the smooth muscle strips
are rapidly stretched (Fig. 2–1 C) [77]. Immediately following the stretch,
myosin content is reduced in the proximity of dense bodies, indicating the
creation of an actin-myosin overlap free zone. After allowing sufficient time
for adaptation, these zones are filled with myosin to prestretch levels. This
dynamic restructuring is an ultrastructural manifestation of the unique
plasticity of smooth muscle in response to mechanical stimuli.
2.2.3
Mechanochemistry of Actin and Myosin
The interaction of smooth muscle myosin motor proteins with filaments
of actin is the central driver of macroscopic contraction of smooth muscle
cells. Due to its central nature and high relevance to this thesis, experimental findings and mathematical models of this interaction will be discussed in
more detail.
The actin-myosin interaction poses a complex challenge to our understanding. While several myosin motor proteins are simultaneously exerting
their mechanical effect on an actin filament, each individual myosin progresses through its own ATP hydrolysis cycle. This cycle energetically
drives the individual myosins mechanical action, but is also dependent on
mechanical loading from the actin filament that the myosin is attached to.
This somewhat overwhelming complexity can be addressed in a stepwise
15
Figure 2–2: Adaptation of the actin-myosin interface to rapid
stretch. A) At rest, myosin thick filaments associate with actin along
the whole distance between dense bodies. B) During a rapid stretch, the
length of actin and myosin filaments remains the same, leading to an absence of myosin in the proximity of the dense bodies. C) After sufficient
time has passed, the length of actin and myosin thick filaments adjusts to
the increased rest length.
16
approach. First the basic kinetic schemes for the interaction of a single
myosin protein motor with an actin filament will be discussed. These foundations of muscle molecular mechanics have been investigated primarily in
skeletal muscle. Given fundamental differences in how muscle activation
causes contraction in skeletal vs. smooth muscle, the regulation of smooth
muscle contraction at the molecular level will also be discussed. Next, the
experimental knowledge and theoretical models regarding the action of large
groups of myosin in the contraction of muscle tissue will be introduced.
Lastly, selected in vitro experiments and theoretical models concerning protein motor cooperativity arising from mechanical coupling via a shared cargo
will be introduced. This will demonstrate a lack of experimental results and
sparsity of theoretical predictions for cooperativity of myosin motor groups
of 10 to 100 coupled motors.
General Actin-Myosin Kinetic Schemes
The sliding filament model. Interference microscopy experiments
with frog skeletal muscle and rabbit skeletal muscle fibrils suggested the sliding filament model of muscle contraction, which has gained wide acceptance.
Activation by adenosine triphosphate leads to the relative sliding of actin
thin filaments and myosin thick filaments, which are organized into a parallel array inside the sarcomeres. Thin and thick filaments do not shorten, but
anchoring to structural elements leads to overall shortening of the sarcomere
(similar to relative sliding in smooth muscle, Fig. 2–1 C) [88, 90].
Crossbridge theory of muscle contraction. Another major step
was the proposition of a specific type of myosin-actin “crossbridge” [91]. Numerous observations of skeletal muscle actin and myosin filament structures
using electron microscopy suggested myosin molecules as linkers protruding
17
!
Figure 2–3: Huxley-Simmons Crossbridge Model of Step Length
Change Experiments. A) Length change protocol and tension response.
B) Crossbridge configurations at different time points, corresponding to T0 ,
T1 , and T2 , from top to bottom. C) Two conformations used in the Huxley
and Simmons crossbridge model. D) Relative tensions from Huxley and
Simmon’s model with their parameters, y0 = 8 nm, h = 8 nm, 1/α = 2 nm
[89].
out of the thick filament and attaching to actin with a catalytically and
mechanically active globular head group [91].
Following up on myosin crossbridges as the supposed effectors of
relative filament sliding and tension development, a combination of length
step experiments and mathematical modelling effectively laid the foundation
for most mechanistic models of actin-myosin interactions still being used
today [89]. In the experiments, frog muscle fibers were allowed to contract
isometrically to plateau force (T0 ), and then submitted to a step change
in length (Fig. 2–3 A). The immediate, linear response to length steps
was taken as a measure for an elastic part of actin-myosin links (T1 ). A
second rapid partial force recovery (T2 ) was taken as an indicator for
conformational changes in myosins bound to actin (Fig. 2–3 A). This
18
characterization of the molecular linkers connecting and working in between
actin and myosin filaments firmly established the “crossbridge theory” of
muscle contraction (Fig. 2–3 B).
Mathematical model of crossbridge mechanics. A relatively
simple mathematical model was used to explain these observations [89].
Many of the basic concepts and assumptions of this model of myosin as a
mechanochemical crossbridge remain at the basis of models currently used
to describe the mechanochemical interaction of muscle myosin with actin
(not to be confused with the initial sliding filament model [87]). A detailed
derivation will serve as a background for the discussions in the chapters
representing published articles. The derivation is taken directly from Huxley
and Simmon’s article [89]. An initial assumption is that many myosins work
at the same time, so that an average can be made and stochastic effects
can be neglected. The population of actin-bound myosins (crossbridges)
was further assumed to be distributed into two distinct states, defined by
the conformation of the force generating part of myosin (Fig. 2–3 C). The
fraction of crossbridges in the less strained state is 1 − n (state one), the
fraction in the more strained state is n (state two). The overall relative
displacement of actin and myosin filaments is y, with y = 0 referring to the
relative position at the isometric plateau.
The tension development in the linear elastic parts of the crossbridges
was modelled by a displacement away from a reference point y0 , leading to
an elastic response with stiffness K. y0 refers to the crossbridge head part
orientation being mid way between the two confirmations. The distance
between the two states was h, allowing the calculation of tension developed
19
in the two states at the isometric plateau,
h
h
F10 = K(y0 − ), F20 = K(y0 + ).
2
2
(2.1)
Considering also y = 0, the tension in the two states is
h
h
F1 = K(y + y0 − ), F2 = K(y + y0 + ).
2
2
(2.2)
The average tension developed by a crossbridge could be calculated as
φ = F1 (1 − n) + F2 n.
(2.3)
At the isometric plateau y = 0 as well as n0 = 1/2 as will be seen later, so
φ0 = Ky0 .
(2.4)
n changes in time according to
ṅ = +k + (1 − n) − k − n,
(2.5)
where k + is the rate for going from state 1 to state 2, k − the rate for
the reverse transition. It is assumed that the ratio of these rates depend
exponentially on the relative potential difference between the two states,
k+
= e(B1 −B2 )/kB T ,
k−
(2.6)
where B1 and B2 are the potential levels of the first and second state,
respectively. Replacing them with chemical potentials E1 , E2 and the work
W that is exerted during a transition from state one to state two, the
effective potential difference governs the ratio of forward and reverse rate,
k+
= e(E1 −E2 −W )/kB T .
k−
20
(2.7)
Earlier, the assumption was made that n = 1/2 at the isometric plateau. At
a steady state of Eq. 2.5, n = k − /(k − + k + ), which is equivalent to k + = k− ,
or E1 − E2 = W . Thus, the chemical potential difference between the two
states is exactly equal to the mechanical work for the transition from state
one to state two. In other words, the tension at the isometric plateau is
defined by the energy available for a myosin mechanical transition.
At the steady state, the work W associated with a transition from state
one to state two is
W = hφ0 = h
F1 + F2
= Kh(y + y0 ),
2
(2.8)
so the chemical energy difference can be inferred,
E1 − E2 = Khy0 .
(2.9)
Inserting Eq. 2.8 and 2.9 into Eq. 2.7, the following relation between the
two transition rates was found:
k+ = k− e
−y kKhT .
(2.10)
B
Inserting into the steady state (n = k − /(k − + k + )) gives
1
n=
2
yKh
1 − tanh
2kB T
.
(2.11)
Finally the immediate elastic tension response (φ1 ) and the quick
recovery tension (φ2 ) after the application of a step of length y were
calculated. Immediately after the step, it can be assumed that n has not yet
changed, i.e. n = 1/2. From Eq. 2.3 it is found that
φ1 =
F1 + F2
= K(y + y0 ).
2
21
(2.12)
Assuming that the quick force recovery corresponds to the point where
n attains a new steady state that is adjusted to the imposed y = 0, Eq. 2.3
and 2.11 give
yKh
h
φ2 = K y + y0 − tanh
2
2kB T
(2.13)
Comparison of φ1,2 to the experimentally measured T1,2 demonstrated an
excellent agreement between theory and experiment (Fig. 2–3 D and Huxley
and Simmons [89]).
Chemistry of the actin-myosin interaction. Beyond a purely
mechanical understanding, it is necessary to explain how myosin converts
chemical free energy stored in ATP into mechanical work. In solution
assays with actin and myosin (extracted from skeletal muscle), ATPase
(enzymatic hydrolysis of ATP into ADP and mono-phosphate) activity
and the association of myosin and actin can be measured. A paradoxical
dependence on ATP concentration is seen: an addition of ATP induces
enzymatic activity and a simultaneous decrease in the association of actin
and myosin. Considering that major ATPase activity is dependent on actin
activation of myosin, it seems surprising that a dissociation of actin and
myosin leads to an increase in ATPase activity.
The explanation of this observation has become known as the LymnTaylor scheme, which describes actin-myosin interaction kinetics as a cyclic
reaction (Fig. 2–4) [119]. The detailed steps in the hydrolysis of ATP as
well as their relative rates were addressed in reaction kinetics experiments,
which measured the different rate constants from phosphate liberation
time courses. Myosin undergoes a cyclic interaction with actin, where one
cycle hydrolyzes one ATP molecule (Fig. 2–4). Starting with myosin with
an empty catalytic site, the binding of an ATP leads to rapid dissociation
22
Figure 2–4: Lymn-Taylor scheme of ATP hydrolysis by myosin. ATP
bound to myosin is hydrolyzed, myosin attaches to actin and releases the
hydrolysis products ADP and phosphate, and upon binding of unhydrolyzed
ATP myosin detaches from actin and the hydrolysis cycle can start over
[119].
from actin, followed by hydrolysis at the rate observed in solution without
actin. Free myosin molecules, with the hydrolysis products ADP and P
bound, reattach to actin. When bound to actin, the hydrolysis products are
released, and myosin can receive another ATP molecule to start over the
kinetic cycle. This cycle forms the chemical complement to the mechanical
actin-myosin cycle brought forward in the mechanical crossbridge theory.
The Lymn-Taylor scheme has been extended to account for additional
findings on the possible chemical transitions and a separation of actin
bound myosins into a weakly and a strongly bound state. A first refinement
step was the differentiation into weak vs. strong binding in the different
bound states of myosin and actin. The experimental finding that the
actin-myosin cycle could also progress through hydrolysis when myosin is
bound to actin required a generalization of the Lymn-Taylor scheme. This
generalized scheme, proposed in qualitative terms by Greene and Eisenberg,
states that the actin-myosin cycle can proceed in myosin alone but also
in myosin bound to actin, necessitating that before the release of ATP
hydrolysis products, myosin only weakly attaches to actin so that hydrolysis
23
can still occur in this weakly bound state [44]. From the perspective of
the mechanochemistry of a group of myosins interacting with actin, an
interesting suggestion was made in the Greene Eisenberg scheme. Strain
dependent detachment rates for myosin were introduced in a way that
effectively lead to a directed displacement of actin by myosin. The GreeneEisenberg scheme, proposed mostly in verbal and pictorial form, rests upon
an earlier quantitative treatment by Hill, Eisenberg, and Chen [46, 45].
Later, an even more generalized model of the relation between mechanical and chemical transitions was proposed, again in verbal and pictorial
form, by Geeves, Goody, and Gutfreund [53]. The main difference in this
model was that the actin-myosin complex could be in a strong or a weak
binding state irrespective of the nucleotide binding state of myosin. This
allowed an indirect coupling between the generation of tension in the actinmyosin complex and the release of hydrolysis products and the binding of
ATP.
Mathematical models of crossbridge mechanochemistry.
The degree of qualitative inference and verbal reasoning in the chemistry
oriented studies is surprising, formal and quantitative evaluations are
relatively rare. Much of the progress in theoretical understanding was
made in incremental steps over consecutive review articles, rather than
research articles specifically developing mathematical models. One case
where a full mechanochemical model was in fact fitted to and evaluated
in detail against experimental data is in the work of Hill, Eisenberg, and
Chen [46, 45]. Nevertheless, even in this work the actual formalisms used
are described separately in book form [81] or publications explaining the
formalism at excess of 100 pages [79, 80]. Those works combine much
of the prior formalisms to model the molecular mechanics leading to
24
Figure 2–5: Kinetic scheme for the example Hill crossbridge model.
tissue level measurements in skeletal muscle with a chemical kinetic and
thermodynamical perspective. Given its comprehensive quantitative
treatment of “classical” crossbridge theory, some time will be spent reviewing
the central formalisms and concepts used therein. The review essentially
follows an example given by Hill [81].
The model follows a large ensemble of independently cycling myosin
crossbridges. These crossbridges are not directly coupled to each other, they
are only affected by motion of the actin filament they are attaching to, the
movement of which is given as a model input. It therefore makes sense to
start the model development at the level of the individual crossbridge. The
crossbridges are assumed to follow a simplified kinetic scheme, consisting
of three discrete chemical states (i = 1, 2, 3) (Fig. 2–5). These states are
the unbound state (i = 1), the attached state before the mechanical power
stroke (i = 2), and the attached state after the mechanical power stroke
(i = 3). Associated with each of these states is an energy level (Gi ). For the
states in which myosin is attached to actin, a mechanical component enters
the Gi . Dependent on the relative distance between a reference point on
myosin and the attachment point on actin (x), the elastic element of myosin
is stretched and contributes a potential energy stored in the elastic element.
25
G1 and G2 effectively become dependent on x. From here, the mechanical
force (Fi (x)) created by one crossbridge in state i can be calculated as a
function of x,
Fi =
∂Gi
∂x
=
p,T
d
Gi (x|p, T = const.)
dx
(2.14)
This implicitly assumes that every individual crossbridge is always at
thermodynamic equilibrium. It needs to be pointed out that Hill’s treatment
includes the mechanical potential within the Gi . Thus, when taking the
derivative of Gi with respect to x – which is a mechanical coordinate – only
the mechanical potential will change. In this way, as far as the derivative
with respect to x is concerned, Gi does not undergo chemical alterations.
In this context it turns into a purely mechanical potential, whose derivative
describes a mechanical force Fi .
Certain relationships between the transition rate constants can be
established when taking a thermodynamical perspective. Defining αi,j as the
transition rate from state i to state j (i, j = 1, 2, 3), one can write
G2 (x) − G1
α1,2 (x)
= exp −
α2,1 (x)
kB T
α2,3 (x)
G3 (x) + μP − G2 (x)
= exp −
α3,2 (x)
kB T
α3,1 (x)
G1 + μD − (G3 (x) + μT )
= exp −
α1,3 (x)
kB T
(2.15)
(2.16)
(2.17)
Here, it is taken into consideration that all transition rates can depend on
x because for every transition at least the source or the target state energy
level has a dependence on x. μP refers to the chemical potential of a single
phosphate, the corresponding potentials for ATP and ADP are μT and μD .
The chemical potentials are assigned according to binding and unbinding
26
events associated with the kinetic transitions,
μT = μ0T + kB T ln[ATP], μD = μ0D + kB T ln[ADP], μP = μ0P + kB T ln[Pi ],
(2.18)
where 0 superscripts refer to standard chemical potentials. Taking the
reaction rate products of the forward ( + ) and the reverse ( − ) reaction
rates, one can calculate the free energy release (XT ) for one forward cycle
completion with one ATP hydrolysis,
+
α1,2 α2,3 α3,1
,
XT = μT − μD − μP = kB T ln − = kB T ln
α2,1 α3,2 α1,3
(2.19)
which describes a relationship between the reaction rates that must be
obeyed for any given value of x. In this section of the thesis, no actual rate
expressions will be assigned. When adjusting the rates to represent a specific
system or experiment, the above relationship must be obeyed by the rate
expressions.
To assign functional expressions to the Gi , x = 0 is defined as the point
where a post power stroke myosin is in a relaxed configuration,
(2.20)
F3 (x = 0) = 0.
The energy levels are then defined as
G1 /kB T = 20, G2 /kB T = 16 + (x − 80 Å)2 /(2σ 2 ),
2
(2.21)
G3 /kB T = μP + x2 /(2σ 2 ), σ 2 = 200 Å , XT /kB T = 23,
containing the chemical free energy levels, free energy from hydrolysis
product release (in case of G3 ), and mechanical contributions (G2 and G3 ).
For consistency, it should be pointed out that in fact not only three energy
levels per myosin exist. There are in fact infinitely many sets of the three
basic energy levels, each of them with a constant offset of rXT to the Gi ,
27
with r = 0, ±1, ±2, . . . . Whenever a transition from i = 3 to i = 1 occurs,
r is decreased by 1, whenever a transition from i = 1 to i = 3 occurs, r is
increased by 1. These constant offsets do not affect the Fi , and the influence
of the chemical potentials that make up the XT is already contained in the
conditions on the αi,j . The offsets are therefore omitted from the Eqs 2.21.
To progress from the dynamics of a single crossbridge to those of an
entire ensemble, the time evolution of the probability distribution at the
level of subensembles is employed: pi (x, t) is the probability density, at time
t, to find a given crossbridge in the infinitesimally small interval [x, x + dt]
in state i. When integrated over an interval [a, b] on the x coordinate, the
probability Pi (a, b) that a given crossbridge is both in state i and within the
interval x ∈ [a, b]. The P time evolution is
dPi (a, b)
=
dt
b
a
∂pi
dx,
∂t
which can be assigned more specifically as
b dPi (a, b)
∂pi
=
αj,i pj (x, t) −
αi,j pi (x, t) + v
dx.
dt
∂x
a
j=i
j=i
(2.22)
(2.23)
The summation terms represent chemical transitions from other states j
into state i and from state i into other states j, respectively. The last term
represents changes due to relative sliding of actin and myosin with velocity
v. v > 0 represents sliding towards lower x values. For Eq. 2.22 and 2.23 to
be true for all possible combinations of a and b, the following must hold:
∂pi
∂pi =
.
αj,i pj (x, t) −
αi,j pi (x, t) + v
∂t
∂x
j=i
j=i
28
(2.24)
To model isotonic shortening at a constant velocity v, v = const. and a
local steady state assumption, (∂pi /∂t)x = 0, are applied, resulting in
−v
∂pi =
αj,i pj (x, t) −
αi,j pi (x, t).
∂x
j=i
j=i
(2.25)
The pi (x) solving this differential equation can then be used to calculate
the average force F generated by the myosin ensemble. The special case of
isometric plateau force can be calculated in the same way, but additionally
setting v = 0. The force generated by all myosins at a specific x is
F (x) =
pi (x)Fi (x),
(2.26)
i
which needs to be averaged in x,
F =
x
F (x∗ )dx∗
, x ∈ R.
dx∗
x
(2.27)
The x integration range needs to be approximated for numerical or analytical tractability. By solving for different v and calculating the corresponding
F , force-velocity curves can be calculated from this model.
The chemical state diagram in this example corresponds to the chemical
states used in the studies contained in this thesis. However, in this thesis
a unidirectional cycle of chemical transitions is used, making a coherent
thermodynamical definition of the cycle kinetics impossible. While the
nucleotide binding and unbinding events were the same as in the transitions
proposed here, in our work a secondary mechanical step is associated with
the transition from state i = 3 to i = 1 (not present in this specific example,
but included in some of the aforementioned, more detailed studies).
Also, it is important to point out several assumptions by which the
class of models represented by this example – one could call them “classical”
29
crossbridge models – differs from later models and simulations. First, crossbridges are treated by a continuum approach, instead of following individual
crossbridges. This treatment is justified by the large number of myosins
that work together in a sarcomere, and the ordered contractile structure of
skeletal muscle. However, for situations where few myosins interact with
actin – such as in an in vitro motility assay – this approximation might be
inappropriate. An example would be a situation where a long waiting time
occurs until a first myosin undergoes a crucial mechanical step, which will
cause other myosins to step shortly after. This kind of threshold behavior
can only be captured in a formalism that explicitly treats individual myosin
counts. A formalism that treats only averages as the dynamic variable will,
in this example, understimate the relevance of such a threshold. Second,
while the unbinding rates of myosin from actin depend on the strain on
the molecule, this strain is only imposed as a tension that builds due to
an interaction between the muscle and an external load generating object.
The possibility of internal loads that stem from stretch or compression of
the crossbridge elastic elements can, by definition of the model formalism,
not affect the attachment rates of other myosins. Such a mechanical “feedback” or “coupling” between different myosins working on the same passive
structure can potentially lead to dynamic phenomena typical of nonlinear,
feedback regulated systems. These phenomena can play an important role in
force and length transients, but can not be treated with this formalism.
The second limitation was later overcome, by inverting the requirement
of a specific v to calculate the resulting F [179]. The opposite way of constructing a force velocity curve is to set a specific F value (external load),
and calculate the resulting v. Effectively, this leads to a force equilibration
of the myosins attached to actin with each other as well as the external
30
load. Allowing the myosin strain distributions to feed back on their own
dynamics in this way also lead to the emergence of hysteresis, bistability,
and oscillations in v [179].
A second, ADP release associated mechanical step. More
sophisticated mathematical models of the “classical” crossbridge theory have
generally assumed that a minor mechanical step must occur in association
with the release of ADP from myosin. Experimental evidence of this step
was challenging to come by. Interestingly, related experimental works
discuss that the existence of a secondary mechanical step was seen as a
novel idea – not as an investigation of a postulation already contained in
the mathematical models based on the Lymn-Taylor scheme. Interestingly,
this incremental discovery process by successive experiments has lead to
results that are largely in agreement with the “classical” models – with
the exception that the role of individual myosin motors and the coupling
between them is receiving greater attention than in the classical models
which treated individual myosins as noninteracting particles of an ensemble.
In a cryoelectron microscopy study of smooth muscle myosin, a ≈ 35 Å
displacement step associated with ADP release was found, which makes
up a sizeable fraction of the ≈ 100 Å overall displacement resulting from a
whole actin-myosin interaction cycle [182]. This study also indicated that
small changes in the nucleotide binding cleft of myosin can translate into
mechanically effective displacements, due to a rigid lever arm formed by the
molecular structure of myosin. Further studies using electron paramagnetic
resonance spectroscopy with probes ligated to the regulatory light chain
also demonstrated an orientation change in the myosin neck region upon
ADP release, however, only in smooth muscle myosin but not in skeletal
muscle myosin [58]. The limitation to smooth muscle myosin, after all, was
31
corrected only after single molecule experiments that directly demonstrated
the occurrence of a secondary mechanical step of skeletal muscle myosin that
follows the main power stroke and is associated with ADP release [23].
Along with the discovery and discussion of the secondary, ADP release
associated mechanical step emerged the discussion of its strain dependence.
Comparison of solution kinetics of the actin-myosin ATP activity for
different muscle myosin types demonstrated that ADP release from smooth
muscle myosin is energetically much less favourable than from skeletal
or cardiac muscle myosin [31]. A thermodynamic estimation suggested
that ADP release would likely require mechanical work, and in conclusion
would be dependent on external mechanical loads. A follow up study
using electron paramagnetic resonance spectroscopy demonstrated that the
ADP release associated mechanical step is also present in skinned smooth
muscle, and also proposed a strain dependence of the ADP release step,
recognizing that this was already assumed in the classical crossbridge models
[59]. Following these initial pieces of evidence, more definitive studies of
the strain dependence of myosin mechanical steps were executed, and
theoretical formulations of the strain dependence were developed, which will
be discussed in the following section.
Strain Dependent Actin-Myosin Kinetics
Experimental findings. Taking literally the description of myosin
as a “molecular motor” leads to the question how the motor performs
under load. As explained above, actin-myosin interaction kinetics are often
described as a series of discrete mechanochemical transitions. Accordingly,
it seems appropriate to probe the rate and step length of these transitions
for their dependence on an external mechanical load. A load dependent
mechanical step was first measured for smooth muscle myosin in a loaded
32
tri-bead assay, where higher loads decreased the stepping rate, and negative
loads increased the stepping rate [174]. This study measured the second,
slower mechanical step of smooth muscle myosin, due to the interest in the
ADP detachment step in smooth muscle explained above, but presumably
also because the time to detect of myosin to actin binding events based on
a reduction of trap oscillation amplitude and subsequent mounting of load
force was not fast enough. A second study with a smooth muscle mutant
that affects the translation of nucleotide binding into larger conformational
changes, demonstrated the dependence of strain dependence on an intact
nucleotide binding site [94].
The issue of loading delay was recently addressed, for the first time
giving access to the load dependence of the main myosin power stroke, and
a load dependent mechanical step in skeletal muscle myosin [24]. Using
a microscope stage that moves at a constant velocity, binding events are
detected by deflections of the laser trapped beads, which is more rapid
than the detection of a reduction of fluctuation amplitude. Due to the
ongoing motion of the stage, loading begins the moment that a myosin
attaches, the load is set by stopping the stage motion when the trapped
bead displacement corresponding to the desired loading force has been
attained.
Mathematical descriptions of strain-dependent transition
rates. A mathematical description of the changes in the rate of mechanical
transitions under different mechanical loads will explain these processes in
more detail, and serve as major building block for models of mechanically
coupled myosin motors. To estimate the rate of the mechanical step (main
power stroke or second mechanical step, respectively), it is assumed that its
rate is equivalent to the rate of the associated chemical reaction (Pi release
33
or ADP release, respectively). Further, it is assumed that the reaction rate
(kn , n referring to a specific reaction) obeys Arrhenius’s law,
kn = k̃0 e−ΔG
‡ /k
BT
,
(2.28)
where kB is Boltzmann’s constant, T is the absolute temperature in K, ΔG‡
is the activation barrier or transition state of the reaction, and k̃0 = const.
quantifies a basic rate constant. Rescaling into units of kB T , one gets
‡
kn = k̃0 e−ΔG ,
(2.29)
‡
where ΔG‡ = ΔG /(kB T ). Next, ΔG‡ , the effective activation barrier
is specified. ΔG‡ consists of a constant term that is given by the specific
nature and properties of the chemical reaction (ΔG‡0 ) minus energy that is
dissipated into heat during approach to the reaction’s transition state (Q),
ΔG = ΔG‡0 − Q.
(2.30)
Due to conservation of energy, the heat release is the difference between
chemical free energy that is released during approach to the reaction barrier
(−ΔG) and mechanical work executed during while reaching the reaction
barrier (W ‡ ),
Q = −ΔG − W ‡ .
(2.31)
Assuming ΔG‡0 and ΔG to be constant, the reaction rate can be written as a
function of W ‡ ,
‡
kn = k 0 e−W ,
where k 0 = k̃ 0 exp(−ΔG‡ + ΔG).
34
(2.32)
W ‡ of a specific transition can now be approximated in different
ways, dependent on the mechanical setting. Generally speaking, work is
determined for a change in system state q from q1 to q2 as the integral of the
force along the path q(s). q(s) is parametrized by x ∈ [x1 , x2 ] ∈ R, so that
W =−
F (q)dq = −
x2
F (q(x))
x1
dq
dx.
dx
(2.33)
Making a linear spring approximation of the force F depending on a
onedimensional displacement allows the modelling of two different experimental settings, which will demonstrate the limitations of the linear
approximation. The linear spring approximation is
F (x) = −κx.
(2.34)
It is now possible to address specific experimental situations. In case
of a loaded laser trap assay, the trap focus is adjusted to rapidly build a
load F = −κtrap x1 on a myosin attached to actin, κtrap is the stiffness of the
laser trap, x1 the displacement of the trap from the equilibrium position.
F < 0 (retarding load) implies x1 > 0. In loaded laser trap experiments,
κtrap ≈ 0.02 pN/nm can be assumed, while |F | ≈ 1 pN are loaded onto the
myosin [174], leading to |x1 | ≈ 50 nm. Assuming that the loaded mechanical
step is d = 2 nm long [174], x would increase to x2 = x1 + d = 0.52 nm,
leading to a force increase of < 10% during the mechanical step. Here,
the approximation F = −κx1 = const. is justifiable. The question now is
how to translate the observable mechanical step d and the applied force F
into the work W ‡ done to reach the transition state. The chemical reaction
that supplies the chemical free energy −ΔG and the mechanical work W ‡
occur in different parts of the myosin molecule. This means that F could
be communicated via a lever like molecular structure, changing the length
35
of the reaction coordinate and potentially applying a scaling factor to F .
Lastly, the transition state of the reaction does not have to coincide with
the completion of the mechanical step, meaning that d is not necessarily the
length that determines W ‡ . Clearly, several formulations and assumptions
are possible and potentially valid here. To avoid additional complexity, the
full length of the observable mechanical step (d ≈ 2 nm) and the force of
the laser trap (F ) will be used. This gives W ‡ = −dκtrap x1 . Inserting this
W ‡ into Eq. 2.28 leads to what is often referred to as Bell’s equation, and
constitutes the simplest approximation of the load dependence of chemical
transition rates [177],
kn = kn0 e−F d ,
(2.35)
Again, it is important that in Bell’s equation F is not necessarily the force
effected by the molecular motor or imposed by an external load, but needs
to be considered with respect to the chosen reaction coordinate. Further,
the stiffness in calculating F can be due to the mechanical element that
myosin works on, the elastic tail of myosin, or internal elasticity between
different parts of the myosin proteins. The pragmatic choice of using the
stiffness of the external elastic element is therefore a strong simplification.
Finally, d also does not necessarily need to be the step length measured
in an experiment, but refers to the distance to a transition state, which is
defined along the reaction coordinate.
The constant force approximation is invalid when addressing a typical
in vitro motility assay. Given that myosins that attach to actin bind in an
unstrained configuration (possibly with an unbiased prestrain due to the
relative position of the myosin attachment point on the cover slip and the
binding site on actin that the myosin binds to), x1 ≈ 0 can be assumed.
36
This requires to use an x2 dependent F term, see Eq. 2.34, resulting in
W‡ = −
σκ
(x2 − x1 )2 ,
2
(2.36)
where σκ is the compound stiffness of all other myosins attached to the actin
filament (that is, excluding the myosin undergoing the mechanical step).
Considering the in vitro motility assay situation in a more complete
perspective, however, presents an even more complicated picture. A condition that is used in most models that follow individual motors that undergo
discrete steps is the attainment of “force equilibrium” between all motors
working on the same cargo. This implies that the sum of the forces exerted
by all motors (m) bound to the cargo is 0,
Fm = 0.
(2.37)
m
It is proposed in this thesis that the force equilibrium must also be
considered during conformational changes leading to the occurrence of
chemical reactions. Here, different time scales need to be considered. The
fastest changes happen when a chemical reaction actually occurs, meaning
that a transition state for a chemical reaction has been crossed and the
according chemical and mechanical changes are executed. The rapid changes
during such a transition are significantly faster than the attainment of force
equilibrium. The other time scale is the waiting time between chemical
reactions. During this time, the actomyosin system and its constituent
proteins undergo small, random confirmational alterations that are biased
by the energy function of the overall actomyosin system. By definition, no
chemical changes occur during these phases, so this energy function should
be defined only by changes in the mechanical work stored in the actomyosin
system. Based on these considerations, the force equilibrium assumption was
37
extended to apply during the conformational changes that ultimately lead to
the occurrence of the chemical reaction accompanying the mechanical step.
Again, to avoid additional complexity, viscous effects due to motion of actin
and myosin were neglected – the motion ranges during the conformational
fluctuations are typically below 4 nm. Conformational fluctuations relevant
to actin propulsion also occur in the longitudinal direction of actin, for
which motion over several μm in split seconds can be readiliy observed when
actin is not bound to by myosin [173]. This indicates that viscous effects can
be neglected on the relevant length scales.
Given the assumption that the force equilibrium is maintained while
transition states are approached,
Fm = 0 must hold at all times. One
consequence of this is that any force developed in a myosin approaching a
transition state towards a mechanical step forward must offset the opposed
forces developed by other myosins. Mounting such a force also implies that
mechanical work is exerted not only upon the not stepping myosins, but also
on the elastic elements of the myosin approaching the transition state. In
such a situation, a stringent application of the linear force approximation
(Eq. 2.34) is not obvious any longer, and alternative descriptions that
satisfy
Fm = 0 for all myosins become necessary. These are developed in
the articles included in this thesis.
Smooth Muscle Myosin Light Chain Regulation
Most of the fundamental concepts of muscle physiology have been
derived from skeletal muscle. One major difference between skeletal and
smooth muscle is the regulation of contraction by Ca2+ at the molecular
level. In skeletal muscle, myosin binding sites on actin are covered by
tropomyosin, which can be displaced from its blocking position by troponin
in reaction to Ca2+ signalling. Differently, the main Ca2+ regulatory
38
mechanism in smooth muscle occurs at the level of individual myosins.
Activation of the 20 kD regulatory light chain (RLC), located in the neck of
the each myosin head, regulates the molecule’s ATPase activity, capability to
bind to actin, and its polymerization behavior.
Experimental findings. The actin activated ATPase activity of
smooth muscle myosin was found to be proportional to the phosphorylation
level of a specific Serine residue of the RLC. The phosphorylation is effected
by myosin light chain kinase (MLCK), which itself depends on Ca2+ [159,
4, 134]. In more detail, the RLC is phosphorylated by fast acting MLCK
whose activity depends on active calmodulin, which is in turn activated by
Ca2+ from extracellular sources or the sarcoplasmic reticulum [4]. Inside
smooth muscle cells, dephosphorylation of the RLC via myosin light chain
phosphatase (MLCP) has been shown [97]. MLCP dephosphorylation
activity towards the RLC is regulated by extracellular signals; the main
access points for extracellular signals are integrins, G coupled protein
receptors, and nitric oxide induced signalling, which drive interleaved
regulatory pathways ultimately affecting MLCP activity [97]. Thus, the
amount of active crossbridges is regulated by Ca2+ , additional extracellular
signals, and external load [134, 97].
Structurally, the activation and inactivation is matched by an unfolding
or folding of the myosin long tail domain, respectively [171, 22]. As the
myosin tail region folds around the myosin heavy chain head region, binding
to actin and access to the ATP hydrolysis site is hindered.
In vitro motility assays with varying phosphorylated fractions of smooth
muscle myosin confirmed the gradual regulation of actin propulsion velocity
[181]. Further assays confirmed the reversible activation and deactivation by
MLCK/calmodulin-Ca2+ and MLCP, respectively [136].
39
In the context of regulation of contraction by myosin light chain
phosphorylation, the phenomenon of “latch” in smooth muscle should be
mentioned. Interestingly, during continued activation, smooth muscle myosin
regulatory light chain phosphorylation decreases, while the muscle’s force
bearing capacity remains at high levels [35, 134]. This “latching in” during
sustained contraction effectively allows prolonged contraction without the
continuous expense of chemical free energy, which is well suited to the force
maintaining function of smooth muscle.
Increases in myosin light chain kinase and myosin phosphorylation were
seen in animal models of airway hyperresponsiveness [99, 93, 55]. Bronchial
smooth muscle cells from asthmatic subjects showed increased capacity
and velocity of shortening, and an increase in myosin light chain kinase
expression [120].
The Hai-Murphy model of smooth muscle contraction regulation. To build a mathematical model of the specific regulatory mechanism
of smooth muscle, an initial step was to simplify the formalisms and
concepts developed for skeletal muscle molecular mechanics. The most
prominent example, the Hai-Murphy model [67], is a system of four ordinary
differential equations, disregarding much of the complexity of the mathematical models developed for skeletal muscle. For example, the model contains
no variable that explicitly corresponds to the extent of shortening and no
strain dependence is considered, only the chemical state of the muscle is
explicitly treated. This leads to a simple and clear formalism, but also neglects important additional aspects that could contribute to the phenomena
explained by this model.
The model construction begins with the cyclic binding and unbinding
of phosphorylated myosin to and from actin. Two states are considered,
40
detached phosphorylated myosin (Mp) and actin attached phosphorylated
myosin (AMp). One binding and one unbinding transition can occur, and
the fraction of myosin in the attached state determines the force developed
by the activated myosin. In the actual model, a ratio of 4:1 of the binding
over the unbinding rate was used, leading to a fraction of 0.8 of the total
activated myosin being attached in the steady state. The measured isometric
force was assumed proportional to the bound fraction, and 0.8 corresponded
to the plateau force. In addition, two corresponding nonphosphorylated
states were used (M and AM), and phosphorylation and dephosphorylation
reactions could convert between Mp and M and between AMP and AM. The
scheme is completed by the crucial assumptions that only phosphorylated
myosin can bind to actin, and that unphosphorylated myosin unbinds
from actin at a much slower rate than phosphorylated myosin. Using these
assumptions, a set of ordinary differential equations can be constructed:
d[M]
= −K1 [M] + K2 [Mp] + K7 [AM]
dt
d[Mp]
= K4 [AMp] + K1 [M] − (K2 + K3 )[Mp]
dt
d[AMp]
= K3 [Mp] + K6 [AM] − (K4 + K5 )[AMp]
dt
d[AM]
= K5 [AMp] − (K7 + K6 )[AM]
dt
(2.38)
(2.39)
(2.40)
(2.41)
The transition rates representing MLCK phosphorylation of the
regulatory light chain were assumed to be independent of myosin being
bound to actin, K1 = K6 . The MLCP dephosphorylation rates were also
assumed equal, K2 = K5 . While the notation of concentrations ([X]) was
used, these quantities were liberally used to represent probabilities. The
differential equations fulfil the conservation of total myosin,
[M] + [Mp] + [AM] + [AMp] = const.,
41
(2.42)
which together with initial condition [M] = 1.0 and [Mp] = [AM] = [AMp] =
0 leads to the concentrations being equal to the probability of finding a
given myosin in the respective state.
From this model, the phosphorylated fraction (pP , part of total myosin)
and the normalized active stress level (T , normalized by maximal activation
plateau stress level) are
pP = [AMp] + [Mp], T = [AMp] + [AM].
(2.43)
The model is externally controlled by temporal modulation of
K1 = K6 = f (t). A stepwise constant driving function f (t) was designed that mimicked the regulation resulting from a Ca2+ transient. For
0 to 5 seconds, f = 0.55 s−1 , after 5 seconds that f was lowered to 0.3 s−1
(K2 = K5 = 0.5 s−1 , K4 = 0.1 s−1 , K3 = 4K4 ). The resulting T and pP
time courses quantitatively fit experimental results, where an initial increase
in phosphorylation was followed by a decrease in phosphorylation and an
even later, but persistent increase in active stress. This persistent tension
development in spite of lowered phosphorylation (which is often seen as
the primary marker of myosin activation) is called the latch state. The Hai
Murphy model is a clear hypothesis on how an active myosin cycle combined
with a unidirectional, slow detachment upon dephosphorylation could be
the source of this phenomenon. Assessment of the model for a broad range
of phosphorylation levels (K1 = K6 ), clearly demonstrated the possiblity
of tension maintenance without the maintenance of high phosphorylation
values. For increasing K1 = K6 , first the steady state tension increases to
maximal level. Steady state phosphorylation only increases for increasing
K1 = K6 after two further orders of magnitude. This indicates an efficient mechanism of tension generation that persists in spite of low levels of
42
phosphorylation, leading to arrest of unphosphorylated myosin in the actin
attached state, and consequently a low ATP consumption rate.
In a later review paper by Murphy, a number of alternative mechanisms
and explanations are discussed, and generally dismissed for reasons of
being either incoherent with experimental knowledge, not sufficiently well
known from experiments or in the cellular environment, or as proposing
superfluous detail [134]. Surprisingly, an obvious simplification made in
the Hai-Murphy model in comparison with crossbridge mechanics models
developed in skeletal muscle – namely the complete removal of detailed
crossbridge mechanics – is not addressed. However, an important criticism
can be raised, which stems immediately from this specific simplification.
Slow cycling crossbridges may not only arise from dephosphorylation,
but also regular strain dependence already leads to cycling arrest. Such
strain dependence was neglected in the Hai-Murphy model. In a model
with tight coupling of ATP hydrolysis to mechanical transitions and strain
dependent mechanical transitions, at isometric plateau force, no effective
cycling occurs. Thus, no ATP would be consumed, without any need for
dephosphorylation of the myosin. This aspect is even more relevant given
more recent indications that in smooth muscle myosin the release of ADP –
which precedes ATP binding and myosin detachment in all muscle myosins
– is slowed down by strain and much less energetically favourable than in
skeletal muscle myosin [135, 8, 174].
Kinetics of Coupled Protein Motors
The explanation of a tissue level phenomenon, such as muscle contraction, by a molecular level interaction, such as the actin-myosin
mechanochemical cycle, spans many spatial scales and layers of biological
organization. The total number of microscopic interactions that contribute
43
Figure 2–6: Mechanical coupling between two myosins’ minor mechanical steps. In this model, myosins binding to actin immediately undergo the main power stroke of length d1 . Thus, a strain on actin connecting
the two myosins builds only when the second myosin has attached, x > 0.
After the second myosin has bound, the strain x is increased or decreased,
dependent on which myosin undergoes the minor power stroke of length d2 .
This difference leads to different kinetic rates for the secondary power stroke
for each individual myosin. Scheme adapted from Jackson and Baker [92].
to the macroscopically observable behavior is large. The first transition of
scales is the mechanical coupling of several myosin motors, which is due to
force transmission through the actin filament they are jointly working on.
This specific transition of scales was initially addressed in vitro by
Baker and coworkers [9, 8]. The single myosin kinetics in a laser trap assay
were compared to actin propulsion velocity in the motility assay, where
several myosins simultaneously propel actin filaments. Given a sufficient
free energy of ATP hydrolysis, actin sliding velocities exceeded predictions
based on single molecule measurements up to twofold. This was explained
by mechanical coupling, which transfers mechanical work from kinetic steps
with a large free energy release (main power stroke) to transitions without a
large free energy release (minor mechanical step that is associated with ADP
release, and precedes myosin detachment from actin).
This energy transfer can be explained using a simple though effective
mechanical argument brought forward by Jackson and Baker [92]. The
44
general result is that two myosin motors that sequentially bind to actin
influence each others rate for ADP release. It is assumed that both myosins
undergo the weak to strong binding transition immediately upon attaching
to actin, and also execute the associated main power stroke of length d1
(Fig. 2–6). The initial binding occurs in an unstrained configuration, and
the main power stroke takes the myosin away from this unstrained position
by the length d1 . The first myosin that binds to actin can freely move the
actin, and is therefore still unstrained after its initial power stroke. The
second myosin now binds without strain. Assuming that the actin is a
Hookean spring between the two myosin heads (spring constant κ), the force
upon attachment is
F = κx = 0, x = 0.
(2.44)
Here x is the deviation of the actin link between the two myosin heads away
from the unstrained position. After loose attachment, the second myosin
immediately undergoes a d1 displacement. This changes x to x = d1 , which
is associated with an increase in the mechanical potential,
μmech =
κ
(d1 )2 .
2
(2.45)
μmech = 0 is taken as a reference point from the unstrained position
at x = 0. Only considering the interaction of two myosins attached to
actin and only kinetic steps in forward direction of the actin-myosin cycle,
two mechanical events are possible from here: (1) the initially attached
myosin undergoes the secondary, minor power stroke of length d2 , making
x = d1 − d2 and μmech = κ/2 · (d1 − d2 )2 , or (2) the second myosin
undergoes the secondary, minor power stroke, making x = d1 + d2 and
μmech = κ/2 · (d1 + d2 )2 . One can thus write down the mechanical potential
45
differences for the first and the second possibility,
Δμ1mech =
κ
κ
((d1 + d2 )2 − (d1 )2 ), Δμ2mech = ((d1 − d2 )2 − (d1 )2 ).
2
2
(2.46)
From single molecule experiments, it is known that d1 > d2 > 0. This
implies
Δμ1mech < 0 and Δμ2mech > 0.
(2.47)
This means that due to the mechanical coupling via actin, a further mechanical step of the second myosin will increase the work in the system, thus
slowing down this specific transition. On the other hand, a further mechanical step of the first attached myosin will lower the work in the system, which
will be dissipated as heat and accelerates this particular transition. These
mechanical terms, which are added to the individual myosins’ chemical
potential changes associated with the possible steps, stem from transfer of
mechanical work between the second and the first myosin via actin. In a
situation with saturated ATP, myosin detachment will practically instantaneously follow the mechanical step. In a situation where many myosins
propel an actin filament, this is a probable mechanism of coordination and
acceleration of myosin stepping and an increased observable actin sliding
velocity.
Mathematical Models of Coupled Myosin Motors
An interesting model that implicitly addresses the interaction of a
group of myosins has been developed for in vitro motility mixture assays
[181, 73]. In a mixture assay, varying fractions of two types of myosin are
deposited on the motility surface together, and the resulting actin sliding
velocities are interpreted in terms of the mechanical interaction of the two
myosin types.
46
Figure 2–7: Mixture assay sliding velocities from mathematical
model. This model calculates the sliding velocity V of actin filaments propelled by a mixture of fast and slow myosins in an in vitro motility assay.
A) The velocity-force curves of the two myosins can be used to calculate
the sliding velocities for specific mixtures (red dotted lines, for 75%, 50%,
and 25% fast myosin, from top to bottom). Model parameters: af = 0.65,
bf = 1, Pf0 = 1, as = 0.7, bs = 0.1, Ps0 = 1. B) Depending on the shape
of the velocity-force curves, a convex or a concave shape can be observed,
indicating a “stronger” fast or slow myosin, respectively. Red boxes show the
specifically labelled V from panel A. Changed model parameters for dashed
curve: af = 1.25, as = 0.15. Model and evaluation according to Warshaw et
al. [181]
47
For both the faster and the slower myosin (indicated by f and s,
respectively), a Hill-type relationship between velocity and force was
assumed;
Vf = bf [{(Pf0 + af )/(Pf + af )} − 1],
(2.48)
Vs = bs [{(Ps0 + as )/(Ps + as )} − 1],
(2.49)
where P0 is the crossbridge stall force and a and b are free model parameters. Due to coupling via a rigid actin filament, all myosins must propel
actin at the same velocity,
Va = Vf = Vs ,
(2.50)
and the mean crossbridge force is determined by the two crossbridge
populations contributions,
P = kPf + (1 − k)Ps ,
(2.51)
where k is the fraction of fast myosin. In the absence of external forces
(P = 0), one gets
kPf = (k − 1)Ps .
(2.52)
This condition makes clear that Pf and Ps must have opposite sign, which
means that the fast myosins experience a slowing load, which is matched
by an accelerating load on the slow myosins. Equating Eqs. 2.48 and
2.49, using Eq. 2.50, and finally replacing Pf using Eq. 2.52, the following
quadratic equation can be solved
A1 (Ps )2 + A2 Ps + A3 = 0,
48
(2.53)
where
A1 = (bs − bf )(k − 1)/k,
A2 = (bs − bf )(af + as (k − 1)/k) + bf (Pf0 + af ) − bs (Ps0 + as )(k − 1)/k,
A3 = (bs − bf )af as + bf (Pf0 as + as af ) − bs af (Ps0 + as ).
Using this Ps , Va = Vs can be calculated for different parameter values (Fig.
2–7 A). The model evaluation can also be understood graphically: when
both myosins’ velocity-force curves are plotted together, a horizontal line
determining the velocity is shifted up or down till the ratio of the distances
from P = 0 to the two curves is k/(1 − k) (Fig. 2–7 A). This model implies
that a concave Va vs. k curve is found when the slow myosin is “stronger”,
i.e. an increase in Ps decreases Vs more slowly (relative to Vs at Ps = 0)
than Pf decreases Vf (Fig 2–7 B). A convex curve indicates that the fast
myosin is “stronger” (Fig. 2–7 B).
Another relevant model for the in vitro motility assay has been developed by Uyeda et al. to understand actin length dependent changes in
sliding velocities [173, 72]. This model is based on the simple assumption
that an actin filament in a motility assay slides at maximum sliding velocity
νmax whenever at least one myosin is bound to it.
The duty cycle (f ) is the fraction of an actin-myosin interaction cycle
that is spent attached to actin,
f=
katt
τon
=
,
τof f + τon
katt + kdet
(2.54)
where τon and τof f are the average life times of myosin in the actin attached
or the unbound state, respectively, and katt and kdet are the average rates
of myosin attachment to and detachment from actin, respectively. The
probability of having at least one myosin bound to actin is the complement
49
of the probability to not have any myosin bound to actin,
P (n > 0) = 1 − P (n = 0), P (n = 0|N ) = (1 − f )N ,
(2.55)
where n is the number of bound myosins and N the number of total myosins
that are in reach for binding to an actin filament. The sliding velocity of
actin at a given length then is
ν = νmax P (n > 0|N = L/C) = νmax 1 − (1 − f )L/C ,
(2.56)
where C is a constant determined by the density of myosin on the motility
surface and the distance across which a myosin can bind to an actin. C
relates the number of myosins that can bind to an actin filament to its
length (L). The main advantage of this model is its conceptual simplicity
and small number of parameters, which has lead to its widespread use for estimating myosin duty cycles from low myosin concentration motility assays.
There is, however, a clear disagreement between the model assumption of
continuous sliding due to individual myosin interaction and the practically
instantaneous completion of myosin mechanical steps on any measurable
time scale [24]. A comparison to the experimental findings in this thesis, will
demonstrate that none of the observations made from our L resolved actin
sliding analysis are explained by this model. Further, our simulation results
from a mechanistically detailed model question the general validity of the
individual myosin duty cycle as a concept to understand myosins working in
a mechanically coupled group.
A more detailed approach to modelling the in vitro motility assay
are stochastic simulations following individual myosin extension, force
development, and chemical states. Within this thesis, this approach is
central to the mathematical modelling. There are, to my knowledge, two
50
other groups of researchers – Duke and Vilfan and Walcott and coworkers
– who have also published work using similar models of the interaction of
actin and myosin filaments [39, 40, 175, 178, 33].
Walcott, in close coordination with experimental work, has developed a
combination of chemical transition simulations using the Gillespie algorithm
and mechanical equilibrations between myosins working on the same actin
filament. Two studies detailing this approach and comparing simulation and
in vitro motility assay and laser trapping data have been published [178, 33].
The basic Gillespie-mechanical model formalism and evaluation of model
results are equivalent to what is developed and used in this thesis, with
crucial differences in the mechanochemical scheme and the treatment of the
strain dependence of myosin mechanical steps.
A series of articles by Duke and Vilfan progressed beyond two limitations of the sliding filament model [39, 40, 175]. Following the stochastic
reactions and mechanical transitions of individual myosin crossbridges, the
relevance of stochastic effects was considered. Also, the myosins were not
treated as independent instances of identical particles, but mechanical coupling between the myosins lead to phenomena typical of nonlinear dynamical
systems, which are also observed in experiment and are not explicable by
“classical” crossbridge theory models. In the first paper, introducing the
stochastic model, the following formalism is used [39]. A simplified chemical
scheme for the interaction of individual myosins with actin is devised; the
transition rates associated with mechanical transitions are dynamically
altered dependent on the current sliding distance and the individual location
of the myosin relative to the attachment point on actin; reactions are simulated as instantaneous, single events, after which the elastic elements of all
attached myosins relax towards a force equilibrium. This model will now be
51
discussed in detail. While the original article left parts of the used algorithm
and model unclear, they were revisited by Walcott and Sun, allowing a
coherent description of the approach [179].
N = 100 myosin motors are simulated, representative of one thick
and thin filament pair. Each of these myosins can be in one of three kinetic
states. Myosins can be not bound by actin (state 1), bound to actin while
having an ADP and phosphate bound to them (state 2), or be bound to
actin while having only ADP bound to them (state 3). When going from 2
to 3, the major power stroke of length d associated with phosphate release
occurs, when going from state 3 to 1, a minor power stroke of length δ
associated with ADP release occurs.
To specify the kinetic transitions of the individual myosins, the rates
and their load dependence need to be specified. Between state 1 and 2,
constant rates are assumed,
k1,2 = kbind = const., k2,1 = kunbind = const.
(2.57)
For the mechanical transitions, it is important to also consider the effect of
load on the kinetic rates. The first mechanical transition, the main power
stroke and its reversal, was assumed to occur at a significantly higher rate
than all other reactions. Therefore, a fast equilibration between pre and post
power stroke myosins was assumed, which was updated whenever any of
the other reactions had occurred. In accordance with the thermodynamical
formulation of a fast detailed equilibrium between states 2 and 3, the
fraction of myosins in the pre power stroke state is
ppre =
1
1 + k2,3 /k3,2
52
=
1
,
1 + K23
(2.58)
where
K23 = exp
−ΔG − W
kB T
(2.59)
and ΔG is the free energy change and W the mechanical work done during
the main power stroke. Conversely, the post power stroke fraction of myosin
is
ppost =
K23
,
1 + K23
(2.60)
Duke specifies
d+x
W =
x
1
Kx∗ dx∗ = Kd(2x + d),
2
(2.61)
which is the elastic work due to moving from the current elongation of the
elastic element x of the crossbridge to the elongation after the power stroke,
x + d. Between states 3 and 1, a unidirectional transition associated with
ADP release and subsequent ATP binding occurs. Its rate is
0
exp −WADP , WADP =
k3,1 = kADP
K
δ(d + x),
2
(2.62)
which apparently was based on the assumption that δ 2 dδ,
x+d+δ
WADP =
x+d
K(x∗ )dx∗ =
K
(2(d + x)δ + δ 2 ) ≈ Kδ(d + x).
2
(2.63)
Due to the fast equilibrium between pre and post power stroke myosin,
these were combined into a single kinetic state, the system was reduced to
two states, attached and detached myosin. The overall detachment rate for
a given attached myosin now is the probability to be in the pre power stroke
fraction times the rate of detachment out of the pre power stroke state plus
the probability to be in the post power stroke fraction times the rate of
53
detachment out of the post power stroke
g(x) = ppre k2,1 + ppost k3,1 ,
(2.64)
while the overall attachment rate is simply f (x) = k1,2 .
The slow transitions were then implemented in a Gillespie algorithm.
Whenever a slow attachment or detachment transition occurred, the position
of the actin filament was updated, thereby shifting the individual filaments’
elongations x. To update the filament position, simultaneously a force
equilibrium between all attached myosins and the external load needs to be
fulfilled and the fast equilibrium between ppre and ppost must be obeyed. A
filament position that fulfills both conditions needs to be found numerically
[179]. This update sets the new attachment and detachment rates, which is
followed by another reaction step in the Gillespie algorithm.
Simulations of this model qualitatively reproduced force-velocity
curves and the dependence of efficiency on external force. In simulations
where N was set to a value of 150, oscillations in relative sliding velocity
occurred, which resulted from synchronized stepping of the myosins in
response to step reductions in external load. These simulations explained
the experimentally observed oscillations in isotonic shortening velocity
transients after step shortening, and averaging over several thousand of pairs
of simulated filament pairs successfully predicted average observations from
these experiments.
2.2.4
Plasticity of Cellular Ultrastructures
The mathematical models of Duke and Walcott (see 2.2.3) demonstrate
how mechanical coupling between myosin motors at the microscopic level
can contribute to mechanical properties at the macroscopic level of muscle fibre and tissue. It is still relatively unclear how the specific cellular
54
ultrastucture of smooth muscle – differing fundamentally from the skeletal
muscle sarcomeric organization – relays the molecular mechanics of actin
and myosin to cellular and ultimately tissue contraction. Observations
of mechanical plasticity at the cell and tissue level will be reviewed here,
alongside with existent hypotheses explaining these in terms of changes of
the contractile ultrastructure. This will obviate that mechanical coupling
between smooth muscle myosins can potentially affect many of the considered phenomena – in spite of this aspect mostly not being considered in
the literature discussing mechanical plasticity. This will become especially
prominent for the hypotheses regarding dynamic alterations of myosin filament length, which in the cellular context is a limiting factor for the number
of myosin motors that can act as a mechanically coupled unit. Therefore,
this discussion will provide the cellular and tissue level context in which the
molecular level findings made in this thesis can be used to better understand
observations and to refine mechanistic hypotheses.
As discussed in 2.2.2, the length of smooth muscle “sarcomeres”
seems to adjust dynamically to the distance between dense bodies [77].
This finding leads to a more recent branch of smooth muscle research,
which assesses the dynamic response of the cell’s contractile apparatus to
mechanical and regulatory stimuli [149, 54]. Two major determinants of
these dynamic responses are dynamic changes in the actin-myosin filament
arrangement that makes up the contractile apparatus and alterations in the
cytoskeleton that determines cell structure and relays forces developed by
actin-myosin interactions [149, 54]. Several associated processes are related
to the topic of this thesis. The actin and myosin ultrastructure is altered as
actin and myosin rearrange in response to altered cell geometry: polymerize
upon cell contraction; and depolymerize in the absence of stress, strain, and
55
Figure 2–8: Length Adaptation of Contractile Ultrastructure. When
a smooth muscle cell is adapted to a new length, the cell length at which the
greatest force can be developed adjusts to this length. This is supposedly
due to an underlying restructuring of the ultrastructure that drives cell contraction. Nonpolymerized elements of the contractile structure associated
into functional units that are added in series to the existent ultrastructure.
activation of the muscle (evanescence) [149, 54]. Beyond alterations in the
contractile apparatus’s actin-myosin ultrastructure, rearrangements of the
cytoskeleton and influence from actin associated regulatory proteins play an
important role [66, 126, 54].
Response to Mechanical Stimuli
Striated muscle has a short length range over which near maximal force
can develop. In contrast, smooth muscle, upon instantaneous measurement,
has a broader force plateau [149]. The plateau is widened even further when
the muscle is given time to adapt to an adjusted length. This phenomenon
is called length adaptation, referring to the adjustment of the optimal length
for force development when the muscle is set to an altered length. Length
adaptation is supposedly due to rearrangements of the contractile apparatus
and cytoskeleton to a new rest length.
A similar adjustment occurs when smooth muscle strips were preactivated to maintain a basal tone [18]. Activation via Acetylcholine in the
tissue bath adapted the basal tone over a time course of approximately 50
minutes, leading to increased force development and rate of shortening upon
additional activation by electrical field stimulation via electrodes [18].
56
While the force development is in fact constant over a wide range of
lengths (after adaptation to new length), the velocity of shortening varies
largely with length. A model that can explain this change pattern is a
contractile apparatus that contains myosin filaments in series and in parallel,
where additional myosin filaments are added in series as new contractile
elements upon cell lengthening (Fig. 2–8). The observed rapidity of changes
would be accounted for by a rapid equilibration between soluble, monomeric
myosin molecules and filamentous myosins in the contractile units. Forcelength relationships, ATP consumption, muscle power output, and myosin
filament content in rapidly stretched trachealis muscle supports this model
[77, 101, 140, 101].
Response to Activation and Phosphorylation
Especially for the myosin thick filaments, not only direct mechanical
stimuli, but also regulatory influences and cytoplasmic conditions significantly affect the equilibrium between filamentous and nonfilamentous
myosin, as well as the length of myosin filaments [130, 169].
In solutions that are close to physiological conditions, smooth muscle
myosin is found in an equilibrium between two states: (1) a “closed”
monomeric state, where the molecule’s tail domain is attached to and
blocks its catalytic and motor domain, and (2) the “open” filamentous state,
where myosin associates into side polar filaments [149]. This equilibrium
can be shifted towards the open, filamentous form by increased myosin
monomer concentration, increased ionic strength, decreased levels of ATP,
and increased phosphorylation of the regulatory light chain [149, 169].
Also, in different phases of development or in hypertrophy, the relative
content of the carboxy terminal (myosin tail tip region where myosins make
tail to tail contacts that form myosin filaments) isoforms SM1 and SM2
57
of smooth muscle myosin differ. In vitro SM2 leads to less stable myosin
filaments, which, if applicable in vivo, would lead to greater plasticity of the
contractile apparatus [149].
The question remains what the relevance of the different molecular
mechanisms that shift the equilibrium between monomers and filaments,
and thereby the predominance of myosin in its filamentous form, mean
with respect to contractile apparatus reorganization inside of living cells.
Experiments with human airway smooth muscle cells have shown (1) that
two dynamically interconvertible pools of smooth muscle myosin exists, one
monomeric and inactive pool, and a filamentous, contractile pool, and (2)
that application of a peptide that destabilizes the monomeric conformation
leads to dynamic establishment of more and longer myosin filaments [130].
Further, in smooth muscle from rat, swine, and dog, thick filament
content increased markedly upon contractile activation, suggesting that
myosin thick filaments are recruited from a large pool of myosin monomers.
While the general cytoplasmic conditions, especially the ATP concentration,
would bias myosin towards a monomeric state, rapid phosphorylation of the
regulatory light chain following activation of the muscle can oppositely bias
myosin towards the filamentous form, which would explain that drastic and
rapid shift [149].
Following the line of smooth muscle activation induced alterations in
the contractile apparatus ultrastructure, it was found that smooth muscle
stimulation leads to inverse changes in maximal force and velocity that can
be developed by the muscle [150]. This relationship was fully explained
by a model where smooth muscle stimulation leads to a “series to parallel”
restructuring in the thick filaments (Fig 2–9). Upon activation and following
approach to an isometric contraction plateau, myosin supposedly forms
58
Figure 2–9: Series to parallel transition upon activation and contraction. In activated, isometrically contracting smooth muscle cells, the
contractile units supposedly undergo dynamic restructuring. The increasing number of myosins that are arranged in parallel with increasing force
development allow can jointly bear higher maximal contraction forces. The
reduced number of contractile units arranged in series lead to a lower unloaded contraction velocity. Figure adapted from Seow et al. [150].
longer filaments, which contain more myosins in parallel. In combination,
these myosins can develop a greater force, as the total force is shared
amongst the myosin heads. On the other hand, the unloaded sliding velocity
of the myosin filaments does not change upon lengthening. Under the
assumption that the same number of myosins is available for filament
formation, longer filaments mean fewer filaments organized in series along
the cell’s longitudinal axis, which would result in slower contraction of the
cell under unloaded conditions [150].
Another important factor in the polymerization into myosin thick
filaments in vivo are thin filament associated proteins. Caldesmon, and
potentially telokin and the 38k protein might explain the gap between
monomer prone cytosolic and filament prone in vitro conditions [149]. As
templates for myosin filament polymerization, they can locally switch the
dissociation constant towards a polymerization. This localized effect at the
same time favors a targeted and oriented polymerization of the side polar
myosin thick filaments so as to polymerize the contractile apparatus in a
proper form for contraction at a high rate [149].
59
Contractile Apparatus and Cytoskeleton Restructuring
When investigated in more detail, the polymerization and depolymerization of myosin thick filaments seems to be controlled by an integrated
interaction between mechanical destabilization, myosin depolymerization in
non-activated muscle, and polymerization by phosphorylation upon activation of the muscle cell [102, 149, 158]. Mechanical destabilization is affected
by externally applied length changes of muscle cells, and leads to the depolymerization of parts of the myosin thick filaments. Ensuing activation
and contraction leads to a repolymerization of the myosin thick filaments
[102] with myosin thick filaments added or removed in series to adjust to a
new rest length [158]. Such depolymerization and repolymerization cycles
are believed to underlie the contractile apparatus’s long range of adaptation
[102].
Regarding actin thin filaments, a clear response to mechanical stimuli
and activation of smooth muscle has been shown in numerous publications
[66]. The response to mechanical stimuli follows a different pattern than
that of myosin [76]. Different mechanical stimuli increase the density of
actin filaments. Imposed length oscillations lead to a small but significant
increase in actin filament density, activation and subsequent isometric
contraction lead to filament density increase, and so did 1.5-fold extension
from in situ length. For length increase and contraction, the increases
are additive, length oscillations have not been combined with the other
conditions. At in situ length, relaxed and contracted cells both showed
≈ 10% more actin filaments in the proximity of dense bodies compared
to the spatial average over the whole cell. The difference disappeared in
relaxed as well as contracted elongated cells. The result of impairment
of these polymerization processes has unequivocally been shown to be a
60
reduction of shortening and tension development [66]. Specific inhibition
has also demonstrated the necessity of actin polymerization for maximal
force production as well as length dependent differences in maximal force,
and demonstrated that the mechanism is based on shifting between actin
globular and filamentous form [128].
An important question that remains is the mechanism that drives actin
polymerization upon stimulation of smooth muscle cells. The answer could
lie within the cytoskeletal structures, which have the potential to connect
external mechanical stimuli, intracellular signalling, and regulation of
polymerization [66]. Smooth muscle has three major cytoskeletal structures,
actin stress fibers, actin microtubules, and intermediate filaments [126].
Importantly, these components exhibit differences between different cells
[126, 54] and are regulated in response to mechanical and other stimuli [66].
This is a relatively recent view to which the findings of this thesis do not
directly relate – the interested reader will find detailed information in review
articles [126, 66, 54]
A relatively recent study found that the dense bodies also exhibit a cell
wide organization, which furthermore exhibits strain dependent plasticity
[187]. Dense bodies are aligned along cables parallel to the cell’s long axis,
which stiffen and straighten upon cell stretch, and adapt their lengths on a
slower time scale when the cell is set to a shorter length [187].
Actin Associated Regulatory Proteins
As discussed before, proteins associated with actin and the thin
filaments appear to play important roles in providing templates and
environments for myosin to polymerize. More often, however, the thin
filament associated proteins are referred to as thin filament regulatory
proteins. This refers to the influence that they exert on the basic mechanical
61
interaction of actin and myosin, providing an additional means to modify
the actin-myosin interaction cycle’s kinetics beyond the influence of the
phosphorylation state of smooth muscle myosin [4].
The regulatory proteins associated with actin in smooth muscle
are tropomyosin, caldesmon, calponin, and transgelin (SM22) [133].
Tropomyosin forms elongated dimers, which bind to actin and influence
the access of myosin to actin binding sites [133, 110]. Caldesmon is a long
monomer that aligns with actin, can bind to actin and myosin, which
regulates their interactions, and this regulation is dependent on various
phosphorylation sites [133, 110]. Its capacity to influence the tropomyosin
regulation of myosin access to actin is suggestive of caldesmon serving the
purpose of an additional regulation pathway of smooth muscle contraction
[133, 110]. The role of calponin is less clear, it is suggested to directly inhibit the rate of actin-myosin interactions or affect the signalling in smooth
muscle activation [133]. For SM22, the functional relevance is even unclearer, but existent data indicate a potential binding to actin and a possible
involvement in cytoskeletal rearrangement [133].
In this thesis, work specifically on the interaction of actin and smooth
muscle tropomyosin is presented, and an in depth discussion of this regulation is given in the introduction of the according article included in this
thesis.
2.2.5
Plasticity in Physiology and Disease
Given that this thesis’s findings of mechanical coupling at the level of
groups of muscle myosin motors are relevant to mechanical plasticity at the
cell and tissue level, the question of relevance at the systemic level, i.e. in
physiology and disease, arises. The following discussion will demonstrate
that the reviewed modes of smooth muscle plasticity are believed to underlie
62
physiology and disease of smooth muscle – as investigated extensively in, for
example, airway smooth muscle.
Mechanical Property Changes and Ultrastructural Plasticity
Force modulations during oscillation cycles were observed when length
oscillations were superimposed onto tracheal smooth muscle at isometric
force. The force modulations could not be explained purely by actin-myosin
interactions, but rather seem to stem from cytoskeletal plasticity. These
investigations indicate that cytoskeletal plasticity contributes to force
modulations with frequency and amplitude of the superimposed oscillations,
as seen in tidal breathing [152].
In an effort to connect restructuring of the basic contractile element
formed by actin and myosin filaments to tissue level mechanics, a mathematical model was constructed to reproduce the time courses of isotonic
shortening at different isotonic loads. The time courses display two characteristic features, active shortening and an adjustment of the muscle’s
rest length. These two observations were explained by a combination of
two mechanisms, actin-myosin interactions and a viscoelastic element,
respectively [165].
Hyperresponsiveness and excessive narrowing of airways, which are
symptoms for many asthmatics, are prevented by deep inspirations in
healthy individuals. Deep inspirations prevent these symptoms for the
following 20 to 30 minutes, and also when the symptoms are currently
occurring. Deep inspirations extend narrowed airways and give a “bronchoprotection” from narrowing, and healthy individuals who suppress their
deep inspirations voluntarily lose this protection. In asthmatics, the bronchoprotective effects of deep inspirations are absent, or even reversed, which
contributes to the symptom of excessive airway narrowing in asthma. Out
63
of the different mechanisms that are discussed for these observations, length
induced adaptation of smooth muscle resting length via plasticity of the
contractile apparatus seems the most likely [180].
In vascular smooth muscle, the length dependent plasticity is less well
investigated. However, the length-force relationship has also been found
to change with time after activation, and the force production is affected
by imposed length oscillations. At small amplitude, imposed oscillations
increase force development, for larger amplitude oscillations the force is
lowered. [148]
When cultured smooth muscle cells are prestrained, an increase in
their force response to Ca2+ (maximally produced force) and the expression
of myosin light chain kinase can be observed [157]. When myosin was
permanently activated by thiophosphorylation and cells were strained
for such short time that contractile protein expression levels did not yet
change, a change in the cytoskeletal organization and a force increase were
still observed [3, 157]. Prestrained cells that are allowed to contract freely
exhibit increased total shortening and shortening velocity [156]. Prestrain
also induces an increase in cell stiffness.
Tracheal smooth muscle strips from guinea pigs that were immunosensitized with ovalbumin, but not guinea pigs with induced neurogenic
inflammation, showed increased shortening capacity and velocity but not
isometric plateau force, indicating a connection between specific immune
sensitizations and restructuring in the contractile apparatus of airway
smooth muscle [132].
One way of interpreting the plasticity observed in airway smooth muscle
that has recently gained much attention is the soft glass transition theory.
This theory states that mechanical changes in response to mechanical
64
loading or external activation lead to a fluidization of the cytoskeleton
and contractile apparatus. This fluidization is similar to that in a soft
glass, and is taken as an explanation for smooth muscle’s adaptability to
significant length changes. This explanation captures much of the observed
dynamical changes and specific material properties of smooth muscle tissue,
but remains entirely nonspecific with respect to underlying molecular
mechanisms [64].
Molecular to Systems Level
Even the limited and selected review in this thesis obviates a broad
range of mechanisms, levels of biological organization, and signalling defining
smooth muscle contraction and its contribution to disease. As our knowledge
on these different aspects grows, it becomes increasingly important to
integrate them into coherent, mechanistically justified explanations of
observed phenomena and symptoms.
One way to integrate findings on smooth muscle plasticity is to assess
the causal connection of molecular level results to physiologically and clinically observed phenomena. This could be seen as a “vertical” integration,
across different scales of biological organization. One of the most investigated connections is the shift between two isoforms of the myosin heavy
chain, which is found in asthmatic bronchi.
The first clear confirmation of the functional relevance of myosin heavy
chain sequence alterations was found in myosin purifications from turkey
and chicken aorta and gizzard [96]. A sequence difference resulting in the
inclusion or exclusion of seven amino acids near the ATP binding domain
of the myosin head lead to 2.5-fold differences in the in vitro motility assay
unloaded sliding velocity [96]. The so called (+)insert isoform (SM-B) was
found preferentially in the gizzard purfication, the (-)insert isoform (SM-A)
65
preferentially in the aorta purification, further suggesting that the molecular
differences contribute to the phasic or tonic contractile properties of these
tissues, respectively [96].
The (+) and (-)insert molecular mechanics were then found to differ
only in one specific kinetic step of the actin-myosin interaction cycle [104].
The release of ADP from and the binding of ATP to myosin’s catalytic
pocket are slowed down in the (-)insert, while the mechanical power stroke
length as well as isometric force of myosin groups and unitary force did
not differ between the isoforms [104]. In loaded in vitro motility assays,
myosin purified from tissues containing mostly the (-)insert isform shows
a higher reduction of force generating capacity upon addition of MgADP
than myosin purified from tissues containing mostly the (+)insert isform
[109]. The unbinding force for detachment of unphosphorylated myosins of
both isoforms from actin in single molecule assays does not differ, indicating
a bond strength that is independent of the myosin isoform [109]. From a
tissue level perspective, the relative (+)insert mRNA content (normalized by
total muscle myosin mRNA content) correlates positively with contraction
velocity across various animal models, normal tissue, and also remodelled
tissue [117].
In human tissue, the expression level of (+)insert myosin heavy
chain correlated strongly with the physiological requirements for phasic
contraction across different organs, as assessed by tissue shortening rates
[106]. Asthmatic bronchoscopies showed significant increases in the relative
(+)insert content, which aligns with hyperresponsiveness observed in
many asthmatic patients [108]. The sequence of the human seven amino
acid insert was found to correspond to that in rat smooth muscle myosin,
suggesting in vitro motility assays of rat smooth muscle myosin from
66
different organs as a quantification of molecular level unloaded mechanics
[106]. In vitro sliding velocities were elevated in organs with increased
(+)insert content (normalized to total muscle myosin content) [106].
Working with a (+)insert knockout mouse, all tissues displayed a
decrease in the maximal shortening velocity under no load, which was
paralleled by a reduction in (+)insert expression [117]. In these tissues,
the force generation did not appear to be affected by the knockout, which
is in agreement with the molecule level finding that only myosin kinetics,
but not the force generating capacity are affected by the (+)insert [117].
Interestingly, the (+)insert knockout did also lead to expression level
changes in other contractile proteins and isoforms, such as a calponin
increase, a caldesmon decrease and a regulatory light chain LC17 increase,
which points to a more global contractile phenotype alteration [117].
Global Shift Profiles
As seen for myosin heavy chain isoforms, expression differences in one
contractile proteins rarely occur alone in vivo, making it necessary to take
into consideration several affected molecular mechanisms simultaneously
[166]. This could be seen as a “horizontal” integration of changes in the
different molecular constituents of smooth muscle contractile properties.
The myosin heavy chain isoform, actin isoform, transgelin (SM22), and
tropomyosin isoform levels differ in asthmatic bronchoscopies, indicating
a global expression profile shift [108, 107]. Similarly, myosin heavy chain
isoform, actin isoform, and calponin isoform differences are found in vascular
smooth muscle between different tissues, ages, and in disease states [48]. In
urinary bladder, partial outlet obstruction leads to expression differences
in myosin isoforms, caldesmon isoforms, tropomyosin isoforms, calponin
67
isoforms, and proteins associated with tension transfer from the contractile
apparatus to the extracellular domain [186].
Attempts to integrate these findings appear to be rare. One possibility
would be mathematical models that rebuild the different mechanisms contributing to cell mechanical properties – contractile apparatus, cytoskeleton,
membrane bound plaques, thin and thick filament polymerization dynamics
– in a whole cell model [21]. Given that the molecular functional differences
of the different isoforms were known, such a model could explain or predict
the expected impact on cellular mechanical properties.
Complex Interactions between Altered Proteins
Additional complexity comes from the spatial proximity of actin
regulatory proteins, actin, and myosin. This is due to association with
similar binding regions on actin, and that the regulatory proteins exert their
effect on the same process, actin-myosin interaction [110]. It is therefore
likely that a functional interaction not only directly between the regulatory
proteins, actin, and myosin exists, but also functional interactions between
the different thin filament associated regulatory proteins [110]. Given that
expression differences occur on the scale of global expression profiles, this
means that a simple addition of effects is likely to give misleading results.
Instead, changes in cellular mechanical properties are likely caused by
effects resulting also from interactions between several proteins and protein
isoforms, all of which can exhibit expression level changes.
In the case of caldesmon, an interaction with tropomyosin’s effect on
actin-myosin interaction is observed [133]. Smooth muscle tropomyosin can
accelerate the interaction kinetics of actin and myosin. When caldesmon
is bound in addition, this effect is inverted, and the kinetics are slowed
down. This regulatory effect, again, is inverted when caldesmon is either
68
phosphorylated at two specific sites that inhibit full binding to actin or
bound by Ca2+ activated calmodulin, leading to a potentiation of the initial
acceleration of kinetics by tropomyosin [124, 136]. Further, an in vitro
association between caldesmon and calponin has been observed. Beyond
this, the knowledge regarding functional interactions between different
regulatory proteins in terms of molecular mechanics is sparse. These
examples do, however, indicate that such interactions can play an important
role.
2.3
Research Outline
The central molecular mechanism in our review has been the interaction
of myosin with actin. When the contractile properties of smooth muscle cells
are concerned, it represents the ultimate effector of measurable changes.
The understanding of the effect of changes in the molecular constituents of
smooth muscle contraction, their expression levels or isoform specifications,
and different ultrastructural configurations depend crucially on an accurate
understanding of this effector.
Current experimental techniques are insufficient for the understanding
of the interaction of actin and myosin molecules in the context of this effector. In any of the physiologically relevant situations, groups consisting of
varying numbers of myosin motors act on actin filaments simultaneously, influencing each others kinetic rates in the completion of their mechanochemical cycle. It is possible to measure the simultaneous mechanochemistry
of small groups of myosin motors working on actin. For larger groups, as
would occur in living cells, the resolution of individual mechanochemical
events is currently not possible, even less so the assessment of the detailed
mechanochemistry of the individual myosins in the context of the coupled
group.
69
The in vitro motility assay, while lacking this detailed resolution, can
rapidly provide large data sets at the macroscopic level of the actin filament
sliding motion. Actin filaments occur at different lengths, thus giving access
to different numbers of myosin working as a mechanically coupled group.
Actin regulatory proteins can be added to the assay to assess their influence
on the myosin group mechanochemistry. The inference of the individual
myosin kinetics in the context of the mechanically coupled group – the
microscopic level of the investigated system – requires a reverse engineering
starting from the experimentally observable actin sliding motion – the
macroscopic level.
Main Goal: To understand how the macroscopic process of actin
propulsion by groups of myosin motor proteins emerges from the microscopic
chemical kinetics of individual myosins interacting with actin and the
mechanical coupling of these myosins via the commonly propelled actin
filament.
Hypothesis: The individual myosins’ chemical steps that are coupled
to mechanical transitions of the myosins are slowed down or accelerated
dependent on the mechanical work they exert on the elastic network of
myosins simultaneously bound to actin, thereby giving rise to coordinated
group kinetics of myosins bound to the same actin filament.
Specific Goal 1: To develop a video analysis software that allows to
process data in the necessary amount and to request detailed features of
actin sliding motion, resolved by actin filament length.
Specific Goal 2: To develop a mathematical model that explains the
emergence of the observable actin sliding motion based on known strain
dependent myosin-actin interaction kinetics and rigid coupling via the
commonly propelled actin filament.
70
Specific Goal 3: To assess the impact of the isoforms of contractile
actin and smooth muscle tropomyosin on the in vitro propulsion of actin by
smooth muscle myosin, and infer the underlying changes in chemical kinetics
and mechanical coupling at the molecular level.
71
Chapter 3
Published Article One
This article addresses the specific goals 1 and 2 as outlined in 2.3 (p.
70): a video analysis software for the in vitro motility is developed and
applied, and a mathematical model of the interaction of intermediate size
groups of mechanically coupled myosin motor proteins is developed.
3.1
Article Information
Title: The Kinetics of Mechanically Coupled Myosins Exhibit Group
Size Dependent Regimes Journal Information: Biophysical Journal, Volume
105, 1466–1474 (2013)
Authors: Lennart Hilbert1,2,3 , Shivaram Cumarasamy 3 , Nedjma B
Zitouni 3 , Michael C. Mackey
1,4,2
, Anne-Marie Lauzon
5,1,3,∗
; (1) Dept.
Physiology, McGill University, Montréal, Québec, Canada; (2) Centre
for Applied Mathematics in Bioscience and Medicine, McGill University,
Montréal, Québec, Canada; (3) Meakins-Christie Laboratories, McGill
University, Montréal, Québec, Canada; (4) Depts. Mathematics and Physics,
McGill University, Montréal, Québec, Canada; (5) Depts. Medicine and
Biomedical Engineering, McGill University, Montréal, Québec, Canada;
∗
3.2
Corresponding author, [email protected];
Main Article
Abstract
Naturally occurring groups of muscle myosin behave differently from
individual myosins or small groups commonly assayed in vitro. Here,
we investigate the emergence of myosin group behaviour with increasing
myosin group size. Assuming the number of myosin binding sites (N ) is
72
proportional to actin length (L) (N = L/35.5 nm), we resolve in vitro
motility of actin propelled by skeletal muscle myosin for L = 0.2 −
3 μm. Three distinct regimes were found: L < 0.3 μm, sliding arrest;
0.3 μm ≤ L < 1 μm, alternation between arrest and continuous sliding;
L > 1 μm, continuous sliding. We theoretically investigated the myosin
group kinetics with mechanical coupling via actin. We find rapid actin
sliding steps driven by power stroke cascades supported by post power
stroke myosins, and phases without actin sliding caused by pre power
stroke myosin build up. The three regimes are explained: N = 8, rare
cascades; N = 15, cascade bursts; N = 35, continuous cascading. Two
saddle-node bifurcations occur for increasing N (mono→bi→monostability),
with steady states corresponding to arrest and continuous cascading. The
experimentally measured dependence of actin sliding statistics on L and
myosin concentration is correctly predicted.
Keywords: Molecular Motor Cooperativity; Muscle Molecular Mechanics; Emergent Behaviour; Mechanical Coupling; Collective Phenomena;
Protein Motors
73
Introduction
It is widely believed that the relative sliding of actin and myosin
filaments, driven by groups of myosin protein motors pulling on actin, is
the basic mechanism of muscle contraction. However, the most detailed
knowledge of this mechanism is gained from laser trap studies, which
investigate the in vitro molecular mechanics of single or a few myosin
molecules’ interactions with a single actin filament. It has been found that
the effect larger groups of myosin exert on a single actin filament can differ
significantly, e.g. the actin sliding velocity in in vitro motility assays exceeds
expectations from single myosin studies up to twofold [9, 8]. It was deduced
that this augmentation results from cooperativity between individual
myosins that is established by mechanical coupling via the jointly propelled
actin filament [9, 8, 92, 178].
The interaction of a single myosin with actin can be described as a
mechanochemical cycle that couples hydrolysis of adenosine triphosphate
(ATP) to myosin conformational changes that propel actin [89, 46]. Two
mechanical steps occur during one mechanochemical cycle: the main power
stroke transition (power stroke step size d = 4 nm) and a minor mechanical
transition (predetachment step size d = 2 nm) that precedes myosin
detachment from actin [23]. Accordingly, actin’s myosin binding sites can be
in one of three distinct kinetic states at any given time: bound by myosin
before its fast main power stroke (pre power stroke – Pre), bound by myosin
after its main power stroke (post power stroke – Post), or not bound by
myosin (unbound). The rate of the detachment step is strain dependent
in smooth muscle myosin: a retarding load slows down detachment and
a supporting load accelerates it [174, 94]. The same strain sensitivity of
detachment has been hypothesized to be present in skeletal muscle myosin
74
[135, 131]. The reasoning is that work has to be exerted when mechanical
transitions occur under load, which alters the reaction’s energetics and in
turn the reaction rate. By this logic, the rates of both the main power stroke
and the predetachment step in skeletal muscle myosin should be strain
dependent.
Strain dependent detachment kinetics, in combination with mechanical
coupling via a jointly propelled actin filament, were proposed as a mechanism of accelerated actin sliding [8, 92, 178]. Such coupling can, however,
give rise to additional macroscopic behaviours [63, 41]. For actomyosin interactions, complex behaviour was observed near loads constituting isometric
force (Fmax ) [63, 39, 40, 175, 143, 139]. In the two headed myosin V cellular
cargo transport motor, strain sensitive protein motor kinetics have been
identified as the crucial mechanism underlying coordinated stepping of the
two heads[10, 141, 28, 118]. There, single motors act as loads on other single
motors’ strain dependent kinetics, and no external loads are required to
elicit the behaviour that emerges for a group of motors. Further, motility
assays using different ratios of slow and fast myosin have shown that interactions between myosin motors working on the same actin filament lead to
nonlinear interactions between fast and slow myosin kinetics [73, 181].
In this study, we use an in vitro motility assay of actin filaments propelled by skeletal muscle myosin to investigate how the group behaviour
exhibited by mechanically coupled myosins emerges as the number of coupled myosins increases. Given the proportionality of the number of myosin
binding sites to actin length (L) [163, 23], we employ newly developed
software to analyze motility assay videos and resolve the results by L.
Further, we develop three mathematical models of the mechanochemistry
of the interaction of myosin with myosin binding sites on actin. A detailed
75
stochastic mechanical model is used to explain the most striking experimental observations and justify a simplified continuous model and its stochastic
implementation for further explanation of our experimental observations.
Methods
Protein purification
Skeletal muscle myosin was purified from chicken pectoralis muscle as in
Sobieszek’s protocol [161] with a modified extraction buffer ([KCl]=0.5M).
Actin filaments were purified from chicken pectoralis acetone powder [137]
and stored at 4◦ C. Actin was fluorescently labelled by incubation with
tetramethylrhodamine isothiocyanate labelled phalloidin (Sigma-Aldrich)
[181].
In vitro motility assay
Nonfunctional myosin heads were removed by ultracentrifugation
(TLA-42.2 rotor in Optima L-90K ultracentrifuge, Beckman Coulter,
Fullerton, CA) and flow through chambers and buffers were prepared
and used as previously described by Léguilette et al. [109]. The oxygen
scavenger contained 0.16 mg/ml glucose oxidase, 0.045 mg/ml catalase, 5.75
mg/ml glucose. Before incubation in the flow through chamber, myosin at
a concentration of 0.5 mg/ml was diluted by addition of myosin buffer, see
table 1.
Video recording
Actin motility was visualized using an inverted microscope (IX70,
Olympus), recorded with an image intensified CCD camera (KP-E500,
Hitachi, 30 fps), and digitized with a custom built computer (Norbec
Communication, Montreal, QC).
76
Video analysis
Custom written MatLab software was used for analysis of actin filament motion. Contrast was enhanced by averaging every 3 consecutive
video frames, giving an effective time resolution of 100 ms. Image frames
were thresholded into binary black and white images. Filament images
were extracted using the MatLab Image Processing Toolbox’s Connected
Components function and tracked based on similarity of area and centroid.
The frame to frame velocity (Vf 2f ) was calculated by dividing the distance
between a filament’s centroid position (center of mass from gray scale pixel
values) in two consecutive frames by the time resolution of Δt = 0.1 s. Vf 2f
was used where a fixed observation time across all measured velocities was
desirable. The trace velocity (ν) was determined by tracking the length of
the path travelled by the leading tip of a filament, and dividing by the time
of observation of the filament (T ). The trace velocity avoids the centroid
method’s bias to underestimate the velocity for longer filaments due to the
centroid “cutting corners” during filament turning [184], and was used where
unbiased, quantitative accuracy for longer filaments was desirable. The trace
velocity as well as the filament length were determined using a transformation of images of elongated objects into rectangles of equal perimeter and
area, with the longer edge representing the object length [145].
Statistical analysis
95% bootstrap confidence intervals (2500 resamples) were used. Statistical significance (p < 0.05) is indicated by confidence intervals not
overlapping 0.
Experimental Results
Below L ≈ 0.3 μm, all analyzed actin filaments are in a nonmotile
state: there is no motion other than random fluctuations around a fixed
77
1/4
1/5
0.5
V
f2f
[μm/s]
0
6
4
prob. dens.
p
1
0
V
f2f
0.25
0.1
5
[μm/s]
0.05
0.3
0.35
0.6
1
fmot
[μm/s]
f2f
0
4
2
0.4
0.5
0
0.2
0.5 1.5 2.5
0
ν [μm/s]
4
0
1/4
1/5
2
0
0.2
0.15
0.1
0
V
0.2
2
6
1/7
1/9
0.5
1
1.5
μm]
2
1/7
1/9
2.5
Figure 3–1: Experimental detection of the three regimes of actin
sliding. A) (Main panel) Histograms of frame to frame velocities of actin
sliding (Vf 2f ) determined across a range of actin filament length (L) for
[My] = 0.125 mg/ml. (Inset) Empirical Vf 2f probability distributions in
specific L ranges (dashed red, dotted blue, and solid black lines represent
L ∈ [0, 0.26], [0.29, 0.31], and [0.325, 0.4] μm, respectively), indicating that
below L ≈ 0.3 μm only a single mode of Vf 2f is visible. Small top panel:
The Dip Test p value (p < 0.05 suggests unimodality) further indicates that
only a single mode exists in the Vf 2f distribution below L ≈ 0.3 μm[74].
This single mode observed below L
≈
0.3 μm corresponds to actin that
is fully arrested to the motility surface (Fig. 3–7). B) Main panel as in A,
except for a different L range. (Solid lines) Means of a two Gaussian mixture model that was fitted to the Vf 2f distribution. Inset: Up to L ≈ 1 μm
the fraction of sliding motion associated with the motile state (motile fraction, fmot ) linearly increases up to fmot
≈
1, indicating a characteristic
length at which continuous actin sliding is established. C) Two Gaussian
mixture model means as shown in B, but determined for the four different [My] used in our experiments. Lowering [My] reduces the maximally
attained velocity of the motile state. Sliding window parameters: (A)
Lmin = 0.175 μm, Lmax = 0.4 μm, window width 0.019 μm, 60 windows,
(B and C) Lmin = 0 μm, Lmax = 3.0 μm, window width 0.3 μm, 150 windows.
78
Dilution1 Final [My]
n Flow through chambers
1/4
0.125 mg/ml 16
1/5
0.100 mg/ml 16
1/7
0.071 mg/ml 8
1/9
0.056 mg/ml 8
Table 3–1: Overview of recorded motility data. 1 Fraction of 0.5 mg/ml
myosin preparation in assay buffer.
position on the motility surface (Fig. 3–1 A, B). Above L ≈ 0.3 μm,
filaments switched between the nonmotile state and a motile state of
directed motion. For increasing L beyond 0.3 μm, increasingly less filaments
move at intermediate velocities (Fig. 3–1 A, B.) The velocity of the motile
state increased with actin length (Fig. 3–1 C). Below L ≈ 1 μm the motile
fraction (fmot ) linearly ascends to a plateau value close to 1, and above
L ≈ 1 μm remains at this plateau value. This indicates that L ≈ 1 μm
is a qualitative separating point between actin that is sufficiently long to
support continuous, uninterrupted sliding motion, and actin that is too short
to do so. In summary, there are three actin length dependent regimes of
actin sliding behaviour: below L ≈ 0.3 μm, actin filaments are persistently
arrested to the motility surface (nonmotile state); between L ≈ 0.3 μm
and L ≈ 1 μm, actin filaments discretely switch between the nonmotile
and the motile state; above L ≈ 1 μm, the motile state is observed almost
exclusively.
Lowering the myosin concentration ([My]) lowered the velocity of the
motile state and led to a slower increase of this velocity with increasing
L (Fig. 3–1 D). The “critical length” (Lc , the length above which fmot
has attained a plateau value) was statistically significantly increased for
[My] = 0.056 mg/ml when compared to [My] = 0.125 mg/ml (Fig. 3–2 A-C).
At a given L, a lower [My] can be assumed to result in a lower number
of myosins being simultaneously bound to the actin filament. This result
79
0
4
2
0
0
1/4
1/5
1/7
2
4
0.5
2
μm]
2
0
3
1
0.5
0.5
0
1
μm]
2
1/4
1/5
1/7
1/9
Myosin dilution
0.3
0
−0.5
0
3
0
−0.5
1
0.5
δν Slope
c
ΔL [μm]
1
ν Slope
2
1.5
0
1.5
1/9
0
1
0.5
δν
fmot
1
1.5
δν
ν [μm/s]
0.5
2
6
ν [μm/s]
fmot
1
0.2
0
1
μm]
2
3
0.1
0
−0.1
1/4
1/5
1/7
1/9
1/4
1/5
1/7
1/9
Myosin dilution
Figure 3–2: Quantitative features of actin sliding. A) fmot , quantified
as fraction of Vf 2f above threshold velocity Vthr = 1.0 μms−1 . Blue dots –
single filament fmot , thick black solid line – sliding window average Fmot ,
thin blue solid line – best fit (nonlinear least squares) of α(1 − e−β(L−γ) ).
α, β, and γ are free fitting parameters for adjustment to the sliding window
average. Gray area – 95% bootstrap confidence interval. The critical length
(Lc , red vertical line, dashed lines – 95% bootstrap confidence interval) was
determined as the point where the fitted fmot (L) function reached 0.95 · α,
indicating that the fmot plateau identified in Fig. 3–1 C, inset, was attained.
B) As in A but for all [My], only sliding window average is shown, [My] see
inset. C) ΔLc differences (relative to [My] = 0.125 mg/ml), values are ΔLc
from the full data set for a given [My] with 95% bootstrap confidence intervals. D) ν, quantified as mean of trace velocities, lines and symbols as A
except black solid line – linear fit to all shown single filament ν values above
Lc . E) As D for all [My], only sliding window average is shown, [My] see
inset in B. F) Slopes of linear fit to ν for L ≥ Lc . G) δν, lines and symbols
as A except black solid line – linear fit to δν values above Lc . H) As G for
all [My], only sliding window average is shown (see inset in B for values of
[My]). I) Slopes of linear fit to δν for L ≥ Lc . Sliding window parameters
for all length resolved plots: Lmin = 0.15 μm, Lmax = 3.0 μm, window width
0.10 μm, 200 windows. Note that differences from 0 were assumed statistically significant (p < 0.05) where confidence intervals did not overlap with
0.
80
indicates that the number of simultaneously bound myosin motors, and
not primarily L, is the parameter determining the transition from mixed to
persistently motile actin sliding [51].
As discussed earlier, above Lc , actin has attained a plateau in fmot .
Therefore, above Lc , the mean actin sliding velocity is not influenced
by the amount of time actin spends in a stopped state. However, for
[My] = 0.125 mg/ml, we observed a statistically significant increase in
the mean sliding velocity ν even above Lc (Fig. 3–2 D). In prior studies,
ν increases with L were explained by phases in which no myosin is bound
to actin and therefore actin is not propelled [173, 72]. Here, however, actin
is consistently propelled for L > Lc , but ν still increases. This indicates
that above Lc , there are always myosins attached to actin, while a further
increase of the number of myosins simultaneously attached to actin leads
to a further ν increase. At lower [My], this group effect is lost, possibly
indicating that a minimal [My] is necessary for this intermyosin effect (Fig.
3–2 E, F).
Although the stop and go motion in the intermediate regime is visible
at time scales of hundreds of milliseconds to seconds, the common molecular
level explanation – the temporary unavailability of myosin to propel actin
– should take place on a much smaller time scale of tens of milliseconds
[173, 72]. We investigated this discrepancy based on fluctuations in the
velocity. The coefficient of variation (standard deviation σν over mean
μν ) gives a nondimensional measure of fluctuation strength. Further, for
a sufficiently large sample of Nν independent positive measurements the
Law of Large Numbers states σν /μν ∝ Nν −1/2 . To scale out Nν influence,
√
we defined the “scaled deviation” δν = (σν /μν ) Nν . We assume that the
measured velocities result from underlying actin-myosin interactions: if
81
myosin binding sites are independent, then Nν ∝ L; if the average rate of
actin-myosin interactions is stationary over the time of observation T , then
Nν ∝ T . In this hypothetical situation, we would expect
δν =
σν LT /(1 μm s) = const.
μν
(3.1)
across all L. In contrast, an increase in δν indicates a decrease in the
degree of independence (stronger correlation between interaction events)
or a slowing down of kinetics. A decrease in δν indicates an increased
independence (decreased correlation between interaction events) or an
acceleration of kinetics.
The value δν exhibits a nontrivial L and [My] dependence. For all [My],
starting from lowest L, we observed an initial δν increase up to a maximum
close to L ≈ 0.3 μm (Fig. 3–2 G, H). This coincides closely with the L
value at which the motile state first emerges (Fig. 3–1 A, B). Then, for
[My] = 0.0125, 0.100, 0.071 mg/ml, but not for [My] = 0.052 mg/ml, δν
drops to a minimum at L ≈ 1 μm ( Fig. 3–2 G, H). This minimum coincides
closely with the attainment of consistent sliding motion (Fig. 3–1 B, inset).
All features seemed sufficiently clear to not require statistical testing, except
for the increase in δν following the minimum at L ≈ 1 μm, which upon
testing does reach statistical significance for the [My] exhibiting a minimum
(Fig. 3–2 I).
Considering Vf 2f , fmot , ν, and δν observations, we suggest the following
interpretation of the three regimes of actin sliding:
1. Below L ≈ 0.3 μm, actin sliding is fully arrested, while myosins are still
acting as crosslinkers preventing actin from floating off of the motility
surface. This indicates that the completion of full actin-myosin
interaction cycles is slowed down or fully stopped. All fluctuations
82
in ν result from random fluctuations around the fixed position of
the actin filaments. For increasing L, a second, motile state occurs,
which introduces slow jumping between full arrest and a discrete, but
very slow, motile state. This slow jumping leads to the observation of
several consequent instantaneous velocities in either the arrested or the
motile state – highly correlated, consecutive sliding velocities, which
lead to an increase in δν.
2. Between L ≈ 0.3 μm and L ≈ 1 μm, the velocity of the motile state as
well as the fraction of overall observation time that filaments spend in
the motile state increase with L. Thus, the rate at which actin-myosin
interaction cycles are completed while in the motile state, as well
as the time spent in this motile state, increase with L. Both factors
accelerate the gross rate of completion of actin-myosin interactions,
which reduces δν.
3. Above L ≈ 1 μm, almost all of the observation time is spent in the
motile state, and either an increase or no statistically significant
change in the velocity of the motile state is observed – these observations by themselves would warrant either a further decrease or, more
likely, a constant level of δν. However, an increase of δν is observed.
This indicates that increasing L beyond L ≈ 1 μm no longer increases
the effective “sample size” of actin-myosin interactions that result in
the measured sliding velocities. This would be explicable by a situation where all myosins that are simultaneously bound to an actin
filament are maximally mechanically coupled, and thus effectively
appear as a completely integrated macroscopic system. Such a system’s dynamics would become effectively independent of the system’s
83
constituent parts, and thus not change fundamentally anymore, even
when the number of parts increases.
Mathematical Models
Mechanistic explanation of nonmotile and motile state
We constructed three mathematical models, all resting on the assumption that the number of myosin binding sites on an actin filament (N ) is
proportional to the length of the actin filament (L). We assume a distance
of 35.5 nm between major binding sites [163, 23]. Our first model monitors
stochastic changes in the chemical as well as the mechanical state of the
individual myosin binding sites located on the same actin filament. A unidirectional three state chemical cycle (unbound→pre power stroke→post
power stroke→unbound) was assumed [23, 174]. Chemical reactions take
place at discrete time points, after which the overall actin-myosin system
immediately relaxes to a force equilibrium. Myosins bound to a binding site
on actin were assumed to have an unstrained position, and a linear force
response when being moved away from that position. The rates of both, the
main power stroke and a minor power stroke preceding the detachment, depend on the mechanical work that would have to be exerted on all myosins
currently bound to the actin filament to execute the mechanical transition
[174, 9, 8, 92]. (See also Fig. 3–3 A and Supplementary Information.)
Our stochastic simulation, unlike commonly used Euler type schemes
with equal time steps, iterates from one chemical reaction to the next. This
results in an exact time resolution for all transitions in the system, and
inherently circumvents problems of methods approximating continuous
processes by finite time differences. We observe that actin sliding occurs
in rapid steps, which are interspersed in between longer phases without
actin motion (Fig. 3–3 B-D). For increasing N , the frequency of these steps
84
!"#
Figure 3–3: Description and results of the detailed stochastic mechanical model. A) Model description: Due to its helical form, binding
sites are placed equidistantly along the actin filament (helical symmetry at
35.5 nm) [163, 23], leading to a proportionality of L and the number of binding sites (N ). Each myosin binding site with a myosin bound is assumed
to exhibit a linear force response to displacement from its unstrained position. A unidirectional three state kinetic scheme is used, where the main
power stroke is 4 nm long; myosin detachment is conditional on the prior
occurrence of a second minor power stroke of 2 nm, after which detachment
immediately occurs due to ATP saturation in the motility buffer [23, 174].
For both mechanical transitions, the rate is dependent on the mechanical
work that has to be exerted to adjust the overall actin-myosin mechanical
network to the changes inflicted by this step [174, 9, 8, 92]. The iteration
scheme for simulation is shown in the bottom of the panel. B-J) Example
filament sliding distances (B-D), post power stroke and pre power stroke
binding site counts (E-G), and individual myosins’ displacements from the
unstrained position (H-J) for different N as indicated above panels (values
drawn for every chemical reaction). K-N) Histograms of Vf 2f (sampled at
the time resolution used in experiment, Δt = 0.1 s) for different N as indicated in the panels (bin height is a bin’s percentage of the sum of all bins’
counts).
85
increases and the phases without motion become shorter, which leads to
an average acceleration of actin sliding. The rapid steps in actin sliding
coincide with cascades in the transition of myosin binding site chemical
states (Fig. 3–3 E-G). During phases without motion preceding an actin
sliding step, myosin binding sites accumulate in the pre power stroke state,
until a rapid global transition into the post power stroke state and the
unbound state occurs, which coincides with the rapid step in actin sliding.
In terms of individual myosin strains, in phases preceding cascades few
Pre→Post transitions occur, which slowly build an increased strain on the
remaining pre power stroke state myosins (Fig. 3–3 H-J). The Pre→Post
main power stroke transition appears more frequently for higher N , which
is due to strains being more homogeneously distributed across the range
of strains; for low N , it is more likely that combinations of myosin strains
occur that lead to a persistent block of Pre→Post transitions. (see Movie S1
and Movie S2).
At intermediate N , cascades occur in bursts, which are series of rapid
consecutive cascades, or even overlap for high N (Fig. 3–3 D, G, J and
Movie S3, Movie S4, Movie S5, Movie S6, Movie S7, and Movie S8).
When analyzed at Δt = 0.1 s, which we used for the analysis of our
experiments, the L dependent presence of a nonmotile and a motile state
(Fig. 3–1 A, B) are reproduced (Fig. 3–3 K-N). The reproduction of the
transient bimodal regime by the model in spite of the relatively coarse
Δt = 0.1 is due to the grouping of cascades into bursts at intermediate
N , causing a slow time scale stochastic alternation between nonmotile and
motile phases (see Movie S9, Movie S10, and Movie S11).
86
3 0.8
4 0.4
Vf2f [μm/s]
0.6
2
1
0
25
50
N
75
100
0
3
0.6
2
0.4
1
0.2
0
0.8
ν [μm/s]
1
0
25
50
N
75
100
0
1
0.2
20
40
60
80
100
0
N
Figure 3–4: N and [My] dependent existence of the nonmotile
and the motile configuration of the actin-myosin system. A, B)
The simplified deterministic model exhibits up to three steady states
(solid lines – stable, dashed line – unstable) in npost
+
nunb (A) which
can be linked to sliding velocity predictions for experimentally observed
actin sliding (B). Arrows indicate the direction in which [My] is reduced,
[My] = 0.125, 0.1, 0.072, 0.056 mg/ml were used, corresponding to our experiments. C) Vf 2f distributions from the stochastic implementation of the
simplified deterministic model (sampled at the time resolution used in experiment, Δt = 0.1 s). The two N dependent populations correspond to the
stable steady states.
N dependent saddle node bifurcations
The detailed stochastic mechanical simulations indicated that the
overall rate of the chemical reactions is accelerated by up to two orders
of magnitude for an increased number of actin bound post power stroke
myosins, and decreased up to 2 orders of magnitude for an increased number
of actin bound pre power stroke myosins (Fig. 3–8). Also, the myosin
attachment to unbound binding sites was found as a rate limiting step after
the occurrence of a cascade, so that myosin detachment could be assumed
as instantaneous and the post power stroke state and the unbound state
could be joined into one state (Fig. 3–9). Based on this, we formulated
a simplified, continuous model describing the dynamics of the fraction
n = (npost + nunb )/N of the total binding sites that is not in the pre power
stroke state (for details see SI).
87
The simplified continuous model explains the three regimes of actin
sliding behavior (nonmotile, intermittent motility, continuous motility) as
a progression through two saddle node bifurcations (monostable, bistable,
monostable) for increasing N . The two stable steady states have the
following properties (Fig. 3–4 A, B):
1. A large fraction of the myosin binding sites is in the pre power stroke
state and the sliding velocity is ≈ 0,
2. A small fraction of the myosin binding sites in the pre power stroke
state and the actin sliding velocity is clearly > 0.
These properties suggest that the steady states are identifiable with:
1. The precascade phases in which actin is not sliding, and
2. The cascading phases during which actin slides. This also indicates
that the transitions from monomodal to bimodal behaviour in actin
sliding indeed occur at critical L associated with the saddle node
bifurcations.
The continuous model can further explain the dependence of the critical
L on [My] (Fig. 3–2 C), as the saddle node bifurcations occur at higher N
for lower [My] (Fig. 3–4 B). In addition, the reduced velocity of the motile
population for lower [My] (Fig. 3–1 C) is captured (Fig. 3–4 B).
Prediction of actin sliding statistics
Lastly, we derived a simplified stochastic simulation from our continuous model, which allowed us to sample n and a time courses (for details see
SI). When sampled at Δt = 0.1 s, this simplified model also exhibits the N
dependent bimodality in Vf 2f that is observed in our experiments and the
detailed stochastic mechanical model. Further, this model correctly predicts
the experimentally observed dependence of fmot , ν, and δν on N ∝ L and
[My]:
88
Figure 3–5: Theoretically predicted quantitative features of actin
sliding. Shown are simulation results of the stochastic implementation of
the simplified mathematical model of actin-myosin interactions. 500 filaments were simulated at each N value with 5000 iterations per myosin binding site, of which the first 3500 were removed to prevent bias from initial
conditions. No sliding window averaging was applied, Vf 2f were sampled at
time resolution of Δt = 0.1 s (corresponding to our experiments). For each
N , the mean sliding velocity (A, arithmetic mean of Vf 2f ), the mean motile
fraction (B, fraction of Vf 2f > Vthr ), and the scaled deviation (C, standard
√
deviation of Vf 2f divided by arithmetic mean of Vf 2f and multiplied by N )
were determined. Inset in C demonstrates increasing prominence of the positive slope of δν at high N for higher [My]. The colors referring to [My] are
given in the inset in C for all panels.
89
1. For all [My], fmot reaches a value close to 1; for lower [My], fmot = 1 is
only reached at greater N ∝ L.
2. ν increases more slowly for lower [My]; ν still increases beyond the
point where fmot has reached 95% of its plateau value.
3. δν follows the same qualitative pattern as in the experimentally
determined statistics; the main peak quantitatively matches our
experimental results, while away from the peak the overall values of
δν are smaller than in our experimental results. This is likely due to
additional sources of noise that are present in the experiment (Fig.
3–7). However, the changes in δν for lower [My] (shift of δν peak
location, loss of positive slope at high N ) are predicted.
Discussion
Based on our experimental and theoretical findings, we suggest that the
three regimes of actin sliding are defined by the existence or nonexistence
of two global kinetic states and can be interpreted as follows (see also Fig.
3–6):
Regime 1
Below L ≈ 0.3 μm, actin filaments are permanently arrested to
the motility surface. Nonfunctional myosin was removed from the assay
(ultracentrifugation removal of nonfunctional myosin and blocking of
nonfunctional myosins in the flow through chamber by unlabelled actin),
making permanent surface binding by nonfunctional myosin unlikely.
Further, at least one myosin should be attached to actin at all times: the
methylcellulose used in the motility buffer limits the lateral diffusion of
actin; when no myosin is attached to actin, rapid longitudinal diffusion
would occur [173]. We did not observe such diffusion. Instead, as detailed
by our mathematical models, perfectly functional myosins accumulate in the
90
Figure 3–6: Three distinct, N dependent regimes of actin sliding
and underlying actin-myosin kinetics. Main panel: The red and the
black solid line indicate stable configurations of the overall actin-myosin
network, which are attainable at a specific N . The red line represents the
nonmotile, pre power stroke dominated state. The black line represents the
motile state without pre power stroke build up. For increasing N , occurrence and disappearance of these states akin to saddle node bifurcations is
observed. For intermediate N , a coexistence of both states is found, leading
to the actin-myosin system stochastically alternating between the nonmotile
and the motile state. (Dashed black line: unstable steady state.) Insets:
Example configurations of the actin-myosin system in the nonmotile (left,
N = 8) and the motile regime (right, N = 35). Each line represents the
displacement of a myosin bound to actin away from its unstrained position;
pre power stroke myosins have vertical top parts, post power stroke myosins
have angled top parts spanning 4 nm (For examples of the dynamic behavior
of the individual myosins, see Movie S1 and Movie S2). Small panels on top:
Example traces showing the dynamics of the number of myosin binding sites
in the post power stroke and the unbound state (npost + nunb ) for different N
(detailed stochastic model, N indicated by the topmost tick on the vertical
axis in each panel).
91
pre power stroke state and mutually hinder each other’s progression through
the main power stroke, ultimately leading to a persistent self blockade. The
actin length of L ≈ 0.3 μm is close to the diffraction limit for our fluorescent
label (L ≈ 0.2 μm), so the absolute value should be understood as an upper
boundary for this regime.
Regime 2
Above L ≈ 0.3 μm, actin sliding switches between the two discrete
states of full arrest to the motility surface vs. continuous forward sliding.
It is unlikely that this is due to longitudinal diffusion during phases where
actin is not bound to any myosin, because even shorter actin filaments
with a lower chance of not binding a myosin did not exhibit such diffusion
behavior. Rather, as detailed by our mathematical models, the whole group
of myosins simultaneously bound to actin dynamically switches between two
group states: the nonmotile state and a motile state. The motile state is
driven by bursts of cascades, during which the main power stroke transition
is supported by post power stroke myosins. At L ≈ 1.0 μm, the motile
state fully dominates, and actin slides continuously at almost all times. This
level of continuous sliding is attained in a linear fashion for increasing L,
where continuous motion is established rather sharply at L ≈ 1.0 μm. This
length of full continuous sliding increases for lowered [My], indicating a
critical group size and not a critical actin length is necessary for continuous
performance of the myosin motor group.
Regime 3
Above this critical group size, continuous actin sliding (full sustained
motility) is established, driven by an unbroken sequence of cascades,
meaning a permanent support of the main power stroke by post power
stroke myosins. For [My] = 0.125 mg/ml the sliding velocity keeps increasing
92
with increasing actin length. This indicates that a further increase in
the myosin group size can indeed lead to a further increase in the rate of
detachment of myosin from the actin filament, which is also indicated by our
mathematical models.
Concluding Remarks
Our modelling approach could be extended by:
1. Explicit treatment of the nonbound state in the continuous model
analysis,
2. The inclusion of nonlinear elasticity of myosin [95], or
3. Including possible interactions between the two motor heads of myosin
[2].
Actin length resolved analysis of motility assay data has been used in earlier
studies [173, 72], and the switching between a nonmotile and a motile state
of motility has been studied by Marston and coworkers while characterizing
actin regulatory proteins [123, 50, 49, 15]. To our knowledge, however, both
aspects have not yet been investigated simultaneously before. Thus, existent
mathematical models of actin length dependent changes only deal with
average sliding velocities, but not with a bimodality between a nonmotile
and a motile state [173, 72, 178]. Also, these mathematical models cannot
explain how actin sliding velocity keeps increasing when a plateau in the
motile fraction has already been reached (Fig. 3–2 D-F) [173, 72].
Although it is intuitively clear that a group behaviour must emerge as
the group size is increased, most experimental studies tackle either small
(around 5-8 coupled motors) [118, 63, 38, 14, 144, 114, 98, 11, 51, 113, 32]
or large (100 or more motors) motor groups [178, 63, 39, 40, 175, 143, 139].
Most studies investigate one specific group size rather than the dependence
of group behaviour on group size. Some investigate within a range of up to
93
±3 motors. Effectively the middle ground (10-100 coupled motors) where
myosin group behaviour emerges is left out. Note, however, Badoual et
al. [7], Uyeda et al. [173], Harris and Warshaw [72], Li et al. [112], and
Kaya and Higuchi [95]. Our study demonstrates and conceptualizes the
emergence of a group behaviour, starting from a group size of N ≈ 10
coupled myosin binding sites, which actually seems to be a hindrance to
the group’s performance, and progresses up to N ≈ 100 binding sites, thus
capturing the establishment of full group behaviour.
Debold et al. have executed laser trap (tribead) assays with N ≈ 8
motors being in the binding range of an individual L ≈ 0.5 μm actin
filament at the same time. In contrast to our experiments, the actin filament
was readily and continuously moved at this group size [32]. Compared to
our experiments, a slower binding of myosin to free binding sites can be
expected. Their myosin concentration of 15 μg/ml is almost one tenth what
we used, and the presentation of myosin on a pedestal is a geometry that
brings less myosin coated surface near the actin filament. Indeed, reducing
ka in our detailed stochastic mechanical simulation leads to a significant
reduction of actin sliding arrest (Fig. 3–10). Thus, ka differences might
explain the different observations between our studies. Kaya and Higuchi
investigated the in vitro interaction of actin with myosin-myorod filaments
[95]. Actin sliding in their assays happened in a stepwise fashion, with
unloaded sliding velocities of 0.2 − 0.5 μm/s. Due to the myosin-myorod
preparation, few myosins (N ≈ 5) interact with actin, which again would
suggest a lowered ka , so that sliding arrest is reduced (Fig. 3–10).
Badoual et al. theoretically investigated bidirectional in vitro sliding
behaviour in terms of motor group size, and the predicted behaviour could
be confirmed in an in vitro system using engineered motors working in
94
opposite directions [7]. We detected bimodal motile/nonmotile behaviour
in a unidirectional, unengineered motor system. This indicates that a physiologically relevant skeletal muscle myosin motor cooperativity mechanism
exists. This cooperativity mechanism collapses at unusually small motor
group sizes, thus allowing us to understand its detailed workings.
Acknowledgments
We thank Josh E. Baker, Genevieve Bates, Gijs Ijpma, Del R. Jackson,
Jinzhi Lei, Oleg S. Matusovsky, Horia N. Roman, and Romain Yvinec for
discussions; Marvid Poultry for the procurement of the chicken breasts for
the purification of skeletal muscle actin. We also thank the anonymous
reviewers for their suggestions. Computational Resources: The Colosse super
computer of the CLUMEQ consortium was used for some of the stochastic
simulations. Funding: National Sciences and Engineering Council of Canada
grants 217457 and 224631. The Meakins-Christie Laboratories MUHC-RI
are supported in part by a Center grant from Le Fonds de la Recherche en
Santé du Québec. LH – McGill University Health Centre Research Institute
Graduate Fellowship, Centre for Applied Mathematics in Biosciences and
Medicine Graduate Fellowship.
95
4
0.1
V
2
0
0.05
0.25
0.3
L [μm]
0.35
0.25
0.3
L [μm]
0.15
f2f
[μm/s]
0.2
6 0.35
Figure 3–7: Control experiment to determine baseline noise that affects
nonmotile actin filaments. Frame-to-frame velocities (Vf 2f ) are plotted vs.
filament length L for a regular motility assay (A) and for no addition of
ATP to the motility buffer (B).A) In the regular motility assay, short actin
filaments exhibit a state of arrest to the motility surface, which is affected
by a baseline noise which causes velocities that are slightly elevated from 0
(Vf 2f from three different flow-through chambers). A “baseline velocity” of
0.121 ± 0.098 μms−1 (standard deviation) was determined as the arithmetic
mean over 29708 Vf 2f values from three different flow-through chambers
recorded in the -ATP condition.
3.3
Supplementary Material
Determination of Noise in Centroid Displacement Measurements
In the regular motility assay, short actin filaments exhibit a state
of arrest to the motility surface, which is affected by a baseline noise
which causes velocities that are slightly elevated from 0 (Fig. 3–7, A). A
comparable baseline noise can be observed in the absence of ATP from
the motility buffer. The absence of ATP leads to the establishment of
persistent actin-myosin rigor bonds which permanently arrest filament
sliding motion (Fig. 3–7, B). Here, we still detected a “baseline velocity” of
0.121 ± 0.098 μms−1 resulting from fluctuations in the centroid position.
Mathematical Model Development
Detailed Stochastic Mechanical Simulation
Mechanochemistry of individual myosin binding sites: Each
individual myosin binding site on the actin filament is referred to by an
96
integer index (m = 0, . . . , N ). It is characterized by its chemical state
(cm ∈ {0, 1, 2}, representing the unbound, the pre power stroke, and the
post power stroke state, respectively) and its mechanical displacement from
an unstrained position (xm , in units of μm). From the chemical state, two
properties are calculated. σm indicates if binding site m is bound by a
myosin or not:
⎧
⎪
⎨ 0 if
σm =
⎪
⎩ 1 if
cm = 0,
cm ∈ {1, 2},
(3.2)
where the curly bracket denotes a case dependent function, whose result
describes a free binding site (σm = 0) or an occupied binding site (σm = 1).
When the myosin is bound to binding site m, and has undergone the main
power stroke, the equilibrium position is shifted by a step length Δstep
m ,
calculated as
Δstep
m
⎧
⎪
⎨ 0 nm if
=
⎪
⎩ 4 nm if
cm ∈ {0, 1},
(3.3)
cm = 2.
This formulation reflects that only after the main power stroke (cm = 2), the
main power stroke length (4 nm) alters the equilibrium position. For cm = 2,
either no myosin is bound or no mechanical step has occurred, giving a
0 nm step length. This means that the minor mechanical step preceding
detachment (2 nm, see Capitanio et al. [23]) does not take effect in terms of
Δstep
m .
For simplicity, we consider only forward chemical transitions, leading to
a unidirectional reaction cycle 0 → 1 → 2 → 0. This approximations can
be justified by the releases of free energy in the reactions’ forward direction,
which lead to a forward directed bias of the reaction cycle and its individual
97
reactions. The transition rates are ka (attachment, 0 → 1), kp (main power
stroke, 1 → 2), and kd (detachment, 2 → 0) [23].
Mechanical equilibration along the whole actin filament: xm is
calculated in the context of the whole actin filament’s N binding sites. Actin
filament sliding is monitored by the length travelled by the actin filament (a,
in units of μm). When a binding site goes from the unbound to the bound
state (0 → 1), a filament position at which the myosin now bound to site m
would be unstrained (x0m ) is stochastically drawn from a normal distribution
around the current a:
x0m ∼ a + N (0, w),
(3.4)
where N (0, w) represents a normal distribution (mean μ = 0, standard
deviation σ = w); w determines how far x0m scatters around a.
0
step
Knowing a, x0m , and Δstep
m , we can calculate xm = a − xm − Δm .
Assuming linear elasticity, a force acting towards relaxation of the myosin
strain develops at binding site m:
fm = −σm K(a − x0m − Δstep
m ),
(3.5)
where K is the spring constant. We assume that after each chemical
transition an instantaneous equilibration of all fm occurs:
N
fm = 0,
(3.6)
m=1
so that all forces cancel. This condition is reached by adjustment of a, and
insertion of Eqn. 3.5 gives
a=
σm (x0m + Δstep
m )
,
natt
98
(3.7)
where natt =
σm and the sum is over m = 1, . . . , N . If natt = 0, a is left
unchanged from the last time natt > 0 applied.
Transition rate calculation: The transition rate of each binding site
(rm ) depends on the current chemical state cm , and for cm ∈ {1, 2} also on
the mechanical work ΔWm that must be exerted on the overall actin-myosin
system for the myosin bound to myosin binding site m to execute its next
mechanical step . rm is calculated using a case dependent function for each
of the chemical states,
⎧
⎪
⎪
if
ka
⎪
⎪
⎨
2
1→2
rm =
if
kp0 e− K cf ΔWm
⎪
⎪
⎪
⎪
⎩ k 0 e− K2 cf ΔWm2→0 if
d
cm = 0,
(3.8)
cm = 1,
cm = 2.
2cf /K quantifies the strength of the effect of mechanical coupling on the
reaction rates. The 2/K terms are used alongside cf in the exponent to
later cancel out with terms in the expressions for the ΔWm . The ΔWm are
calculated as
ΔWm = Wmaf ter − Wmbef ore ,
(3.9)
where Wmbef ore is the mechanical work stored in the actin-myosin system
before the mechanical step, and Wmaf ter is the mechanical work stored in
the actin-myosin system after the mechanical step [38]. At any time, the
mechanical work is the sum of the mechanical work stored in each of the
attached myosins (W m , calculated by an integral from the unstrained
position x = 0 to the current position x = a − x0m − Δstep
m ):
W =
m
m
W , W
m
x
=−
fm (x )dx = σm
0
99
2
K
a − x0m − Δstep
.
m
2
(3.10)
Thus, for a set of given a, x0m and cm , the mechanical work stored in the
actin-myosin system is
W =
N
m=1
σm
2
K
a − x0m − Δstep
.
m
2
(3.11)
The notation m is introduced to distinguish between the general summation
over all myosin binding sites (m) and the calculation of ΔWm for a specific
myosin binding site (m).
The task of finding ΔWm for a specific m is now to calculate the Wmaf ter
that would be required for the stepping of a myosin specifically bound to
binding site m. A mechanical step at binding site m would lead to a change
in the overall sliding distance a and and the stepped distance of binding site
1
ter
m, Δstep
For a step size d, the altered aaf
follows from Eqn. 3.7:
m .
m
ter
= abef ore +
aaf
m
d
.
natt
(3.12)
is increased by d. Inserting the altered a and Δstep
Other than a, only Δstep
m
m
into Eqn. 3.11 allows the calculation of the ΔWm ,
d 2
step 2d
K
0
ΔWm = 2
+ (a − xm − Δm ) natt
m
natt
d
+d2 − a + natt
− Δstep
2d + x0m 2d .
m
(3.13)
By inserting this expression in Eqn. 3.8, one can see that all K terms cancel
out, effectively leaving only cf as a coefficient in the exponent. In this way,
the redundant model parameter K was removed. cf now represents the
1
In the case of the minor power stroke preceding myosin detachment from
a binding site, we assume that the initial mechanical step precedes and is
kinetically separate from the chemical unbinding, which instantaneously follows the mechanical step. Thus, chemical state changes are irrelevant in our
calculation of Wmaf ter .
100
effective strength with which the expense of mechanical work for a myosin
step influences the rate of this step to occur.2 The step size d is 4 nm for
cm = 1 and 2 nm for cm = 2 [23].
Stochastic choice of next chemical reaction: After mechanical
equilibration and calculating the rm , the next chemical reaction r and the
time to its occurrence τ are randomly chosen using the Gillespie Algorithm.
Two random numbers (r1 and r2 ) are drawn from a uniform distribution in
the range from 0 to 1. The time to the next reaction is
τ = −log r1
N
rm ,
(3.14)
m=1
and the next chemical reaction occurs at binding site mr determined by
m
r
m
≥ r2 .
mr = min m : m=1
(3.15)
N
r
m
m=1
Iteration scheme: The simulation of an actin filament’s sliding is
initiated with a = 0, x0m = 0, and cm assigned to 0, 1, or 2 randomly with
equal weights. Then, the following sequence (Fig. 3–3, A) is iterated until
a maximum simulation time or a maximum number of kinetic steps per
binding site is reached.
1. Instantaneous Equilibration amongst all Attached Myosins. a is
determined based on the force equilibrium condition (Eqn. 3.7).
2. Calculate Transition Rates for all Binding Sites. The current chemical
configuration (cm ) and mechanical configuration (xm ) are considered
to calculate the forward transition rates (rm ) for all individual binding
sites (Eqn. 3.8).
2
To clarify, Δstep
refers to the the state before the step, not after the
m
step, and therefore does not include d of the potential mechanical transition.
101
Parameter
Value
Myosin binding rate
ka = 825 s−1
Main power stroke zeroth order rate
kp0 = 11, 000 s−1
Detachment zeroth order rate
kd0 = 27, 500 s−1
Mechanical coupling strength
cf = 0.925 nm−2
Standard deviation in attachment strains w = 1.225 nm
Table 3–2: Parameter values used for simulation of the detailed
stochastic mechanical model. Initial parameter values of kp0 and kd0 were
taken from Capitanio et al. [23]. kp0
=
1, 500 s−1 , which Capitanio et al.
identified as the rate limiting step of the skeletal muscle myosin kinetics, was
taken directly, and kd0 = 5, 500 s−1 was calculated for [ATP]=1mM (saturated
ATP used in our motility buffer) based on the formula given by Capitanio et
al. Due to higher temperature in our experiments (22◦ C vs. ≈ 31◦ C), kp0 and
kd0 were increased by a factor of 5.5, which was determined so as to make
the simulation results match the L dependent transition from nonmotile to
motile actin sliding as well as the velocity of the motile population. Other
parameter values were adjusted to match experimental data in the same
respects. For ka , a roughly 10-fold lower value than for kp0 (rate limiting
step for detachment after attachment in single myosin experiments [23])
was chosen to account for the low duty cycle of skeletal muscle myosin [72].
w ≈ 1 nm seems to be an easily imaginable fluctuation in myosin position at
the moment of binding to actin, given that the sizes of mechanical steps are
2 and 4 nm and that a myosin molecule can span several tens of nanometers.
3. Stochastically Draw next Transitioning Binding Site. One binding
site mr is stochastically chosen to undergo a single forward transition
(Eqn. 3.15) after a waiting time τ that also is stochastically determined (Eqn. 3.14). The according chemical state cmr is incremented
to the next chemical state in the reaction cycle, meaning that it goes
from its current cmr to a new value cmr = mod(crm + 1, 3).
Dependent on the desired type of output, the chemical and mechanical state
were then evaluated at each reaction step or at a fixed time resolution (Δt,
necessary to mimic the time resolution of Δt = 0.1 s used in the analysis of
our experiments).
102
Figure 3–8: Dependence of interstep intervals on the chemical state
of the actomyosin network. For each step in the stochastic mechanical
simulation, the waiting time to the next chemical reactions (Δt) was calculated, and the number of pre power stroke myosins (npre , panel A) and post
power stroke myosins (npost , panel B) were evaluated. Simulation times: 0.1
s (black), 0.2 s (gray), 0.3 s (light gray). Standard model parameters (Table
3–2) and N = 15 were used, and the simulation state was evaluated after
each chemical reaction. Note that on the logarithmic scale for Δt roughly
two orders of magnitude are spanned for different npre and npost .
Deterministic Approximation
Approximations Derived from Detailed Stochastic Mechanical
Simulations
Plotting npre and npost versus Δt demonstrates that the rate of the
chemical reactions increases for increasing npost and decreases for increasing
npre , with a change of two orders of magnitude (Fig. 3–8). This dependence
will be used for the following development of a simplified deterministic
model.
From the detailed stochastic mechanical simulation we can extract
the most commonly used kinetic pathways (Fig 3–9). Chemical transitions
predominantly occur away from the center of the triangular plots. This
indicates that ((npre , npost , nunb ) generally follows a three phase cyclic
pattern throughout detachment cascades and reattachment sequences:
1. (≈ N, ≈ 0, ≈ 0) → (≈ 0, ≈ N, ≈ 0)
2. (≈ 0, ≈ N, ≈ 0) → (≈ 0, ≈ 0, ≈ N )
103
Figure 3–9: Main kinetic pathways. A-C) npre , npost , and nunb represent the count of myosin binding sites in the pre power stroke, the post
power stroke, or the unbound state, respectively. For all chemical transitions
between the (npre , npost , nunb ) states, the number of times the transitions
occurred was counted. Gray scales represent these counts, scaled by the
transition with the highest count, which is shown in full black. Simulation
time: 150 s, standard model parameters (Table 3–2) and N = 15 were used.
3. (≈ 0, ≈ 0, ≈ N ) → (≈ N, ≈ 0, ≈ 0)
One implication of these cyclic kinetics is that kd ka during cascades of
myosin unbinding (post power stroke→unbound), which effectively renders
myosin binding the rate limiting step in this moment. Accordingly, we
decided to monitor the binding sites in the post power stroke state and the
unbound state as one population. To do so, we introduce a new variable
x = (npost + nunb )/N .
Model Development
This reduces the kinetics to one dimension. We further assume this
dimension to be continuous. Consequently, we define an ordinary differential
equation model for n, which is amenable to steady state analysis,
dx
= −ka x + kp (1 − x).
dt
(3.16)
Since ka = const., we can develop an expression for the kp dependence on x
to approximate the results of our detailed mechanical simulations.
104
Mechanical strain on myosin accelerates or decelerates its power stroke
transition via a mechanical potential μW (W for work) [174, 94, 92]:
kp = kp0 e−μW .
(3.17)
The units of μW are chosen to account for the coefficient relating μW
to changes in the rate, kp0 is the unstrained main power stroke rate. We
assume a neutral number of Post + Unbound binding sites (N 0 ), at which
the effect of Pre (retarding) and Post (forward pulling) myosins exactly
cancels (μW = 0), so that the main power stroke is neither accelerated nor
decelerated. Assuming that μW changes linearly in x, we write
μW = N x − N 0 .
(3.18)
To have a well controlled range in which the mechanical coupling takes
effect, we rewrite the above expression as a Hill function with an asymptote
for N x → ∞,
μW = I
Nx
1
−
0
Nx + N
2
,
(3.19)
which is 0 at N x0 = N 0 , negative below, positive above, and is bounded
by −I/2 ≤ μW < I/2 for x ∈ [0, 1]. I is the “coupling strength”, relating
mechanical interactions between myosins to the effect on the main power
stroke rate.
We assume that the strength of the mechanical coupling gets diminished along the actin filament by a “longitudinal damping factor” over the
distance between two neighbouring binding sites [173]. Taking a continuum
approximation, over a distance of Nb binding sites, the mechanical coupling
would be decreased by e−Nb /N , where N is the longitudinal damping factor.
Given that our model is nonspatial with respect to the location of myosin
105
binding sites on actin, we make another approximation of the effective
number of myosins Ne that any myosin bound to actin is coupled to:
N
Ne =
pM y e−n/N dn = pM y N 1 − e−N/N ,
(3.20)
0
where N now is the number of overall myosin binding sites present on the
actin filament. Note that N , which is a freely chosen model parameter, now
combines adjustments made for the nonlocal averaging approach as well as
the longitudinal damping of mechanical coupling between myosin binding
sites. pM y is the probability that a given myosin binding site currently is
bound by a myosin: pM y = [My]/([My] + KMy ), where KM y is a half-value
constant we adjusted to our experimental findings (Fig. 3–1 C). We replace
N 0 /N by N 0 /Ne and insert this into Eq. 3.17 to find
kp =
kp0
exp I
x
1
−
x + x0 2
,
(3.21)
with
x0 =
−1
N 0 [My] + KMy 1 − e−N/N
.
[My]
N
(3.22)
This approximate dependence of kp on x (and in effect on npre = N (1−x)) is
in qualitative agreement with the dependence seen in the detailed stochastic
mechanical simulation (Fig. 3–8).
For a given x, the sliding velocity is
ν = d1 · kp N (1 − x)/N = d1 kp (1 − x).
(3.23)
This is the product of a single myosin main power stroke length (d1 = 4 nm)
and the rate at which binding sites with myosin bound in the pre power
stroke state transition into the post power stroke state. It is normalized
106
Parameter
Value
Main power stoke rate
kp0 = 5500−1
Detachment and binding rate
ka = 850−1
Interaction strength
I = 10
Pre/Post balance point
N0 = 10
Myosin binding half value constant KM y = 0.0225 mg/ml
Myosin coupling range
N = 15
Table 3–3: Parameters for the Ordinary Differential Equation model
and its stochastic implementation.
Reaction
Rate
Δx
Δa
Main power stroke
(1 − x)kp +1/N d1
Attachment to actin xka
−1/N 0
Table 3–4: Reaction channels, reaction rates, chemical state changes
(Δx), and actin propulsion length (Δa) used in the stochastic simulation of the simplified deterministic model.
by the number of overall binding sites to distribute the d1 mechanical step
length to an advancement of all binding sites, cf. Eqn. 3.12.
Steady state values x∗ (defined by dx
= 0) were determined by
dt x=x∗
numerically solving x∗ ka = (1 − x∗ )kp . The steady state sliding velocities ν ∗
were determined by inserting n∗ into Eqn. 3.23.
Stochastic Simulation of the ODE Model
Again, the Gillespie Algorithm was used (reaction channels and state
changes shown in Table 3–4) to simulate stochastic a and x time courses.
The reaction rates are the same as in the deterministic model, and state
changes have been added to describe changes in a and discrete transitions
in x (The simplified model has only two reaction channels for x → x + 1/N
and x → x − 1/N ). After simulation, a was sampled at a time resolution of
Δt = 0.1 s (corresponding to the time resolution of our experimental data)
as described above.
107
Figure 3–10: Frame to frame sliding velocity histograms for lowered
attachment rate of myosin to actin. Histograms bin heights represent
the bin’s percentage of the sum of all bins’ counts. ka values are indicated
in the panels. Δt = 0.1 s, maximal number of steps – 10,000 per myosin
binding site, maximal time simulated – 120 s.
Influence of Attachment Rate at N = 8
To investigate the influence of ka on the group kinetics for a small
number of binding sites (N = 8), we executed our detailed stochastic
mechanical simulation for reduced ka values (Fig. 3–10). Over a range
of three orders of magnitude (ka = 0.825 − 825 s−1 ), large qualitative
differences in the distribution of Vf 2f can be observed. For ka < 82.5 s−1 ,
unimodal distributions are observed. For ka = 0.825 s−1 the single mode
is close to Vf 2f = 0, and gradually shifts to Vf 2f > 0 for increasing ka .
A second, Vf 2f = 0 associated mode only emerges for ka = 26.089 s−1 ,
gains considerable weight in the distribution for ka = 82.5 s−1 , and keeps
increasing in weight for further increasing ka .
Description of Movie Files
Movie files are available online in the published article’s Supplementary
Material: http://www.cell.com/biophysj/supplemental/S0006-3495
108
Movie S1 Movie displaying the deviation from the unstrained position
of all attached myosins for an actin filament
xm and head tilting state Δstep
m
with N = 8 myosin binding sites. All myosins’ unstrained positions are
drawn in the bottom center of the video, so that all myosins “anchor points”
superimpose. The very top ends of the lines represent the deviation from
the unstrained position, xm . For pre power stroke myosins, a vertical head
section is drawn in the upper section of the line. For post power stroke
myosins this section bridges a 4 nm displacement, representing the head
tilting of the myosin head region. Between each two consecutive two video
frames, a Δt = 0.01 ms elapses, the video is rendered at 30 frames per
second.
Movie S2 Same as Movie S1, but N = 35.
Movie S3 Video representing the mechanochemistry of an actin
filament with N = 8 binding sites at a time resolution of dt = 1 ms. dt
is the time resolution Δt used in the analysis of Vf 2f . To cover more time
in a video, however, between each two frames of the video, the system’s
state is advanced by Δt (at the video frame rate of 30 frames per second,
this would mean that a video would run at real time if a dt of 1 s/30 would
be used). The top most, blue line represents the actin sliding progress.
The red, second line from the top represents the frame to frame actin
sliding velocities (Vf 2f ). The white, third from the top display represents
the current displacement of the the N myosin binding sites away from
the unstrained position, x0m , while displacement above the horizontal line
indicates x0m − a > 0, Δstep
is not displayed. pre power stroke binding sites
m
appear as solid circles, post power stroke binding sites as hollow circles,
unbound binding sites are not drawn. The white, bottom most display
shows a time course of these binding site displacement. Units are indicated
109
by the scale bars next to the displays, scale bar ticks indicate the 0 point.
The model parameters are given in the top left of the video.
Movie S4 Same as Movie S3, but N = 15 and dt = 1 ms.
Movie S5 Same as Movie S3, but N = 35 and dt = 1 ms.
Movie S6 Same as Movie S3, but N = 8 and dt = 10 ms.
Movie S7 Same as Movie S3, but N = 15 and dt = 10 ms.
Movie S8 Same as Movie S3, but N = 35 and dt = 10 ms.
Movie S9 Same as Movie S3, but N = 8 and dt = 100 ms (Δt used in
the analysis of our experiments).
Movie S10 Same as Movie S3, but N = 15 and dt = 100 ms (Δt used
in the analysis of our experiments).
Movie S11 Same as Movie S3, but N = 35 and dt = 100 ms (Δt used
in the analysis of our experiments).
3.4
Thesis Revision Comments
These comments were added in response to the thesis reviewers’
comments. As this thesis is based on existent publications, answers to
questions concerning this published article were added here in response to
comments instead of making changes directly within the article.
The density of motors on the motility surface was not explicitly
considered in this study. A deliberate decision was made to construct the
model from the perspective of myosin binding sites on actin, to which it
does not matter which myosin binds, only with what initial strain. The
concentration of myosin on the motility surface is included in the effective
attachment rate of a myosin to a free myosin binding site, which was
lowered to reflect reductions in myosin concentration in the assay within the
mathematical model.
110
Within this article, the step lengths of 4 nm and 2 nm were used for
the myosin main power stroke and the minor mechanical step preceding
myosin detachment, respectively [174]. We also assumed that these were
the reaction coordinate distances leading up to the chemical reactions
underlying the mechanical transitions. In works by Veigel et al. and Kad et
al. [94], however, the effective distance to the transition state for the minor
detachment could be quantified as ≈ 2.6 nm, differing somewhat from the
measurable myosin step length for that specific transition. It is important
to point out that a difference between the distance to the transition state,
which enters calculations of strain-dependent transition rates, and the
measurable step length can exist.
It also needs to pointed out that it is not necessarily the work associated with a mechanical transition that determines its strain dependence.
Instead, it is the work needed to reach the transition state of a given reaction. This can, for example, mean that the transition state to initiate a
specific reaction can be reached after a minimal mechanical distance, after
which the rest of the mechanical step will be executed independently of
strain.
It should be clarified that, within our mathematical model, the straindependence of the main power stroke as well as the minor mechanical step
preceding detachment is based on the mechanical work that would be done
if the transition was completed. It is not the current mechanical force
on a myosin that leads to arrest of the mechanical transitions, meaning
that there is no high and continuous force loaded onto myosins that are
arrested in their kinetics. If a high force was loaded to arrest the kinetics,
simultaneously a forced detachment (ripping off) of the myosins would
result from this force, as observed by Capitanio et al. [24]. We did not
111
include such a detachment in our mathematical model development. Lastly,
as described in several works [174, 94, 24], the changes in the rate of
mechanical transitions depends crucially on the direction of the strain.
Assisting (pulling) loads accelerate stepping, retarding loads slow down
stepping. However, in our model formulation, these loads enter as negative
or positive differences in the mechanical work in the system, which might
have obscured this point.
Differently from the claims made in this article, the model of Walcott
et al. could in fact explain an increase in actin sliding velocity even when a
plateau in motile fraction is reached [178]. It was therefore incorrect to state
that other models cannot explain this observation. Further, Walcott et al.
also modelled intermediate groups of ≈ 50 myosin motors, which was stated
differently in our article.
Another explanation of the differences of observations in tribead
assays and our motility assays could be the relative geometry of the myosin
carrying object and the actin filament. In the tribead assay, an initial
myosin has to bind to actin which exhibits large vertical fluctuations,
but then brings actin in a close binding range. In a motility assay, actin
filaments are either bound by myosins continuously or float away from the
motility surface and out of microscope focus. This implies physical and
geometrical differences, which might also contribute to the differences in
observations made in these two different experimental setups.
112
Chapter 4
Published Article Two
This article addresses the specific goal 3 as outlined in 2.3 (p. 70): the
impact of actin and tropomyosin isoforms on actin sliding and underlying
chemical kinetics and mechanical coupling within groups of myosin motor
proteins is assessed.
4.1
Article Information
Title: Molecular Mechanical Differences between Isoforms of Contractile
Actin in the Presence of Isoforms of Smooth Muscle Tropomyosin Journal
Information: PLoS Computational Biology, Volume 9, Number 10, e1003273
(2013).
Authors: Lennart Hilbert1,2,3 , Genevieve Bates3 , Horia N. Roman4,3 ,
Jenna L. Blumenthal3 , Nedjma B. Zitouni3 , Apolinary Sobieszek5 , Michael
C. Mackey1,6,2 , Anne-Marie Lauzon7,4,3,∗ ; (1) Dept. Physiology, McGill
University, Montréal, Québec, Canada; (2) Centre for Applied Mathematics
in Bioscience and Medicine, Montréal, Québec, Canada; (3) MeakinsChristie Laboratories, McGill University, Montréal, Québec, Canada;
(4) Dept. Biomedical Engineering, McGill University, Montréal, Québec,
Canada; (5) Institute for Biomedical Aging Research, Austrian Academy of
Sciences, Innsbruck, Austria; (6) Dept. Physics and Dept. Mathematics,
McGill University, Montréal, Québec, Canada; (7) Dept. Medicine, McGill
University, Montréal, Québec, Canada;
∗
Corresponding author, [email protected];
113
4.2
Main Article
Abstract
The proteins involved in smooth muscle’s molecular contractile mechanism – the anti-parallel motion of actin and myosin filaments driven
by myosin heads interacting with actin – are found as different isoforms.
While their expression levels are altered in disease states, their relevance
to the mechanical interaction of myosin with actin is not sufficiently understood. Here, we analyzed in vitro actin filament propulsion by smooth
muscle myosin for α-actin (αA), α-actin-tropomyosin-αβ (αA-Tmαβ),
α-actin-tropomyosin-β (αA-Tmβ), γ-actin (γA), γ-actin-tropomyosin-αβ
(γA-Tmαβ), and γ-actin-tropomoysin-β (γA-Tmβ). Actin sliding analysis
with our specifically developed video analysis software followed by statistical assessment (Bootstrapped Principal Component Analysis) indicated
that the in vitro motility of αA, γA, and γA-Tmαβ is not distinguishable. Compared to these three ’baseline conditions’, statistically significant
differences (p < 0.05) were: αA-Tmαβ – actin sliding velocity increased
1.12-fold, γA-Tmβ – motile fraction decreased to 0.96-fold, stop time elevated 1.6-fold, αA-Tmβ – run time elevated 1.7-fold. We constructed a
mathematical model, simulated actin sliding data, and adjusted the kinetic
parameters so as to mimic the experimentally observed differences: αATmαβ – myosin binding to actin, the main, and the secondary myosin power
stroke are accelerated, γA-Tmβ – mechanical coupling between myosins is
stronger, αA-Tmβ – the secondary power stroke is decelerated and mechanical coupling between myosins is weaker. In summary, our results explain
the different regulatory effects that specific combinations of actin and
smooth muscle tropomyosin have on smooth muscle actin-myosin interaction
kinetics.
114
Author Summary
Dependent on the required physiological function, smooth muscle
executes relatively fast contraction-relaxation cycles or maintains longterm contraction. The proteins driving contraction – amongst them actin,
tropomyosin, and the contraction-driving myosin motor – can show small
changes in the way they are constructed, they can be expressed as different
“isoforms”. The isoforms are supposedly tailored to support the specific
contraction patterns, but for tropomyosin and actin it is unclear exactly
how the isoforms’ differences affect the interaction of actin and myosin
that generates the muscle contraction. We measured actin movement
outside the cellular environment, focusing on the effects of different isoform
combinations of only actin, myosin, and tropomyosin. We found that
the actin isoforms cause differences in the mechanical interaction only
when tropomyosin is present, not without it. Also, all different actintropomyosin combinations affected the mechanical interactions in a different
way. In our experiments we could not directly observe the mechanical
interactions of actin, tropomyosin, and myosin, so we reconstructed them in
a mathematical model. With this model, we could determine in detail how
the different actin-tropomyosin combinations caused the differences that we
observed in our experiments.
Introduction
Smooth muscle contractile protein expression
Differential expression of smooth muscle contractile proteins has
been associated with organismal development [85], contractile phenotypes
[47, 106, 48], and pathologies, e.g. preterm labour, hypertrophic bladder,
or airway hyper-responsiveness [133, 108, 43]. While the role of the smooth
muscle myosin isoforms has been extensively investigated [105, 55, 43], the
115
Figure 4–1: In vitro motility assay and video analysis. A) Purified
smooth muscle myosin motors are immobilized on a microscope cover slip
and propel fluorescent actin filaments in the presence of ATP. For conditions
whose protein combinations contained tropomyosin (Tab. 4–1), tropomyosin
was added into the assay buffer in excess of actin. B) Filament images are
extracted from and tracked across consecutive video frames. The filament
trace velocity (V ) is determined from the trace resulting from the whole
tracking of a filament (blue line). The frame-to-frame velocities (Vf 2f ) are
determined from the centroid displacements between every two consecutive
frames (centroids – red crosses, displacements – red lines). C) The motile
fraction (fmot ), stop times (tstop – beige regions), and run times (trun – light
blue regions) are determined from Vf 2f time courses.
functional implications of the differential expression of specific actin and
actin-regulatory protein isoforms remain elusive [48].
Smooth muscle actin
In smooth muscle, actin isoforms are expressed from four different
genes, yielding “vascular muscle” α- and “enteric muscle” γ-actin, as well as
nonmuscle (cytoplasmic) α- and γ-actin. The muscle isoforms are associated
with the contractile apparatus, the nonmuscle isoforms with cytoskeletal
structures [133]. Muscle α-actin is generally associated with tonic, γ-actin
with phasic smooth muscles [133, 124, 166]. An anti-proportional relationship between the absolute levels of α- and γ-actin has been established [47].
Disease-related expression differences in α- vs. γ-actin have been found
[108]. Functional differences between α- and γ-isoforms were searched for
in molecular mechanics experiments, but, to our knowledge, no differences
were detected [100, 172, 136, 71]. Insight from tissue level mechanics seems
lacking, too [48].
116
Smooth muscle tropomyosin
Smooth muscle tropomyosin affects the weak to strong binding of ATPactivated myosin to actin: tropomyosin can be in an ON state supporting
myosin strong binding, or an OFF state hindering myosin strong binding
[124, 127]. When regulated by caldesmon-calmodulin, dependent on the
caldesmon-calmodulin activation state, smooth muscle tropomyosin is
stabilized in the open or the closed state, increasing or decreasing the rate
of myosin cycling compared to the rate without any tropomyosin being
present [124, 62]. Tropomyosin forms chains along actin filaments by a head
to tail overlap of consecutive tropomyosin molecules. This overlap leads
to an increased cooperativity in the switching between the ON and the
OFF state. Compared to striated muscle tropomyosin isoforms, a stronger
cooperativity between tropomyosin displacement due to stronger end to end
binding between tropomyosin molecules is observed, as well as a greater
bias for the ON conformation [183, 127, 111]. Similar to striated muscle
tropomyosin, smooth muscle tropomyosin facilitates cooperative binding of
myosin to actin: above a critical ratio of myosin heads per actin monomer,
myosin heads cooperatively displace tropomyosin into the ON state so that
further myosin binding is facilitated; below a critical density or activation by
phosphorylation, tropomyosin remains mostly in the OFF state [60, 61] and
inhibits myosin cycling [111, 124].
Tropomyosin is expressed from the same two genes in nonmuscle,
striated muscle, and smooth muscle cells. In smooth muscle, alternative
splicing yields two smooth muscle specific isoforms (tropomyosin-α and
tropomyosin-β), one from each gene [155]. In vivo, tropomyosin-α and
tropomyosin-β mostly occur as αβ heterodimers, making functional differentiation between the isoforms difficult [124, 155]. In disease states, however,
117
expression differences between both isoforms can be observed [108], raising
the question of functional differences between these two isoforms, especially
in interaction with other differentially expressed contractile protein isoforms.
Crystallized N-terminal fragments of tropomyosin-α and tropomyosin-β
displayed differences in the heterodimerization properties of tropomyosin-α
vs. tropomyosin-β and a greater head to tail overlap of tropomyosin-α
than that of tropomyosin-β [142]. These structural results were interpreted
as indication of negligible differences in tropomyosin’s interface for actin
binding and more important differences in the surfaces available for mediation of actin-myosin interactions as well as the binding of other proteins
[142]. However, actin affinity (in terms of K1/2 binding constants) of smooth
muscle tropomyosin-α was found to be ≈ 10 times greater than that of
tropomyosin-β [27, 26].
In this study, we use an in vitro motility assay to investigate differences in the propulsion of “vascular” α-actin vs. “enteric” γ-actin by
smooth muscle myosin in the presence of smooth muscle tropomyosin-αβ,
tropomyosin-β, or in the absence of tropomyosin, see Fig. 4–1 A and Tab.
4–1. We develop and simulate a mathematical model to establish the differences in actin-myosin interaction kinetics that underlie the experimentally
observed differences.
Results
Actin length resolved features of in vitro motility
Using our specifically developed analysis software, we extracted the
following features of actin sliding: mean sliding velocity (V ), the motile
fraction (fmot ), the average run time (trun ), and the average stop time (tstop )
(Fig. 4–1 B, C). These features were extracted for the different experimental
118
Protein combination
Short name Flow through chambers Videos
α-actin
αA
23
69
γ-actin
γA
25
75
α-actin and tropomyosin-αβ αA-Tmαβ
20
60
γ-actin and tropomyosin-αβ γA-Tmαβ
22
65
α-actin and tropomyosin-β
αA-Tmβ
17
51
γ-actin and tropomyosin-β
γA-Tmβ
21
62
Table 4–1: Experimental conditions. Actin and smooth muscle
tropomyosin isoform combinations used in each condition, with abbreviated short name and number of experiments and videos.
Figure 4–2: In vitro actin sliding features resolved by filament
length. Panels A-D show the actin sliding features average sliding velocity
(V ), motile fraction (fmot ), stop time (tstop ), and run time (trun ), respectively. Sliding window range 0.3-3.25 μm, window width 0.59 μm, 50 equally
spaced windows, 500 bootstrap data sets per condition, gray areas are 95%
confidence intervals. Inset in panel D: number of filaments within length
windows, counted separately for each protein combination. Note that L does
not start at 0 μm, but at 0.6 μm.
conditions (Tab. 4–1) and resolved by actin filament length (L) (Fig. 4–
2). For αA-Tmαβ a consistent V increase is apparent (Fig. 4–2 A). fmot ,
trun , and tstop do not immediately suggest consistent differences, (Fig. 4–2
B-D). In spite of high filament counts (Fig. 4–2 D, inset), the width of
the confidence intervals compared to potential differences makes a direct,
conclusive inference difficult, especially for trun and tstop at L > 2μm.
119
Baseline conditions and regulated conditions
Figure 4–3: Regulation occurs in three actin and tropomyosin isoform dependent modes. A, B) For each condition (Tab. 4–1), the main
data set (solid black symbols) and the bootstrap data sets (hollow colored
symbols) demonstrate the location and variation in the first three Principal Components (PCs). A convex hull is drawn around all bootstrap data
sets belonging to each condition (thin solid lines). C, D) Solid lines connect
conditions that show no statistically significant differences, the absence of a
connecting line indicates significant separation. E) Linkage in a tree describing agglomerative hierarchical clusters of all bootstrap data, suggesting the
use of four clusters for further analysis. F, G) Unsupervised classification of
bootstrap data into four clusters (represented by color and symbol shape),
enclosed in convex hulls (solid lines). H) Contribution of each experimental
condition to the four clusters. I) Summary scheme.
The L resolved features represent a simultaneous measurement of 200
values, whose interdependence cannot be judged a priori. We applied a
Principal Component Analysis (PCA) to reduce the dimensionality of our
data and remove correlations between values, which would otherwise inflate
statistical significance. Transformation into the three Principal Components
120
(PCs) explaining most of the variance indicates that consistent differences
between the experimental conditions exist (Fig. 4–3 A, B). Our statistical
analysis detected no differences between αA, γA, and γA-Tmαβ, which
will therefore be referred to as baseline conditions that show no effect; αATmαβ, γA-Tmβ, and αA-Tmβ are all different from the baseline conditions,
as well as from each other (Fig. 4–3 C, D). To support the conclusions from
our statistical analysis, we executed a hierarchical cluster analysis. Based
on the relatively large reduction of linkage when going from four to five
clusters, a number of four clusters was chosen (Fig. 4–3 E). In the PC space,
the four clusters appear similar to the above separation into one baseline
and three regulated conditions (Fig. 4–3 F, G). Indeed, the four clusters
form a clear representation of the αA, γA, γA-Tmβ baseline conditions, and
the three distinctly regulated conditions αA-Tmαβ, γA-Tmβ, and αA-Tmβ,
(Fig. 4–3 H). Thus, two independent methods of statistical assessment
indicate that only αA-Tmαβ, γA-Tmβ, and αA-Tmβ are significantly
regulated, while for each of them the regulation affects actin sliding in the in
vitro motility assay in a distinctly different manner (Fig. 4–3 I).
Molecular mechanical effects of regulation: Next, we wanted to
attribute the differences that had been detected using PCA to molecular
mechanical features. Thus, we evaluated the motility features’ fold changes
relative to αA, averaged over L. For αA-Tmαβ, V is statistically significantly increased to 1.12 times the baseline value (Fig. 4–4 A). For γA-Tmβ,
fmot is decreased to 0.96-fold, tstop is increased by a factor of 1.6 relative to
the baseline value, though both changes show up only as strong trends (Fig.
4–4 B, C). For αA-Tmβ, tstop is elevated 1.3-fold, which also shows up as a
strong trend only (Fig. 4–4 D). When fmot and trun are analyzed together,
the joint fold changes for γA-Tmβ become statistically significant (Fig. 4–4
121
Figure 4–4: Fold changes in in vitro motility features. All fold changes
are relative to αA, averaged over L, error bars are 95% confidence intervals.
A-D) Motility features averaged over whole L range. V is statistically significantly elevated for αA-Tmαβ. E) Statistically significant differences for γATmβ become apparent by using 95% confidence bands in a twodimensional
space spanned by fmot and tstop (red and blue area, projection of bootstrap
data points onto vector connecting both conditions). F) trun is statistically
significantly elevated for αA-Tmβ in the short L range. Windowing parameters as in Fig. 4–2, except for panel F: 0.325 − 1.0 μm, window width
0.1 μm, 25 windows.
E). When only short actin is considered, tstop is statistically significantly
elevated to 1.7 times the baseline value (Fig. 4–4 F). Note that each condition’s differences are found in different features, which is coherent with the
PCA finding that the regulated conditions are each affected by tropomyosin
in a distinct manner.
Kinetics underlying regulation
To theoretically understand the regulatory effect that tropomyosin
has on actin-myosin interactions, we constructed a mathematical model
of the kinetics of a myosin coated surface interacting with actin filaments
of different length L. Stochastic simulations of our model produce Vf 2f
time courses (Fig. 4–1 C). Averaging these time courses gives V , all other
features of actin sliding can be extracted in exactly the same way as from
experimental data. Our model is an extension of our earlier model of the
122
Figure 4–5: Simulated in vitro actin sliding features. A-D) Actin sliding features plotted vs. filament length. Motility features were extracted
from simulated actin sliding in the same way as from the experimental data
(Fig. 4–2).
group action of myosins propelling actin filaments in the in vitro motility
assay [78]. Briefly, the model assumes that myosin moves actin by two
mechanical steps, the main power stroke and a secondary mechanical
step preceding myosin detachment [23, 174]. When several myosins are
simultaneously bound to the same actin filament, they are mechanically
coupled via the filament. Thus, the individual myosins’ steps cause a change
in the mechanical configuration of the overall system of bound myosins and
the actin filament. Consequently, mechanical work might have to be exerted
on or might be released from the actin-myosin system during the execution
of an individual myosin’s mechanical step. This mechanical work affects
the strain dependent rates of both mechanical transitions, the main power
stroke and the secondary predetachment step. The overall number of myosin
binding sites that are accessible on a given actin filament (N ) is assumed
to be proportional to L. Using the helix repeat of actin (0.0355μm) as an
approximate binding site distance [163, 23], the N ranges were adjusted to
correspond to the L ranges used in the different analysis steps. For details
regarding our mathematical model, see Supplementary Material.
A set of model parameters was determined to mimic the baseline
condition (Fig. 4–5). These baseline parameters were altered so as to mimic
123
Figure 4–6: Fold changes in in vitro actin sliding features in model
simulations. A-D) Simulated motility features averaged over whole N
range. The fold changes were calculated in the same way as for the experimental data (Fig. 4–4). The altered conditions αA-Tmαβ, γA-Tmβ, and
αA-Tmβ are normalized by the baseline condition (αA, γA, γA-Tmαβ). E)
tstop fold change for low L range (N = 10, 16, . . . , 40).
the changes in L resolved features that were observed experimentally for the
αA-Tmαβ, γA-Tmβ, and αA-Tmβ conditions (Fig. 4–5). The scalar fold
changes in motility features were determined in the same way as from the
experimental data (Fig. 4–6). The L resolved motility features as well as the
fold changes capture the experimentally observed differences between the
baseline conditions and the conditions that exhibited statistically significant
effects.
The changes in model parameters that were necessary to mimic the
experimentally observed differences point towards the aspects of actinmyosin interaction kinetics that are changed in the different conditions (Fig.
4–7). For αA-Tmαβ, all kinetic rates (ka , kp , kd ) are increased 1.15-fold. For
γA-Tmβ, the impact of mechanical coupling between myosins on the rate of
the mechanical transitions (cf ) is increased by a factor of 1.2. For αA-Tmβ,
124
cf is reduced to 0.8 of the baseline value, and kd is reduced to 0.75 of the
baseline value.
Figure 4–7: Fold changes in model parameters. The model parameters
of the regulated conditions (αA-Tmαβ, γA-Tmβ, and αA-Tmβ) are shown,
normalized by the parameters determined for the baseline condition, whose
values are displayed to the right of the bars.
Discussion
Regulation of molecular mechanics by actin and tropomyosin
isoform combinations
We investigated in vitro the relevance of actin and smooth muscle
tropomyosin isoforms to the mechanical action of smooth muscle myosins
on actin. In accordance with prior studies [100, 172, 136, 71, 48], no
differences between actin isoforms could be detected. However, the sequence
differences between actin isoforms are confined to regions of interaction with
regulatory proteins [75], suggesting potential mechanochemical differences
in the presence of such regulatory proteins. In vitro studies in solution
(i.e. not on a motility surface) showed a different binding affinity between
actin and smooth muscle tropomyosin [27, 26]. Here, we establish that, in
the presence of both tropomyosin-αβ and tropomyosin-β, the molecular
mechanics differ between α- vs. γ-actin. Thus, the sequence differences
between actin isoforms not only affect actin-tropomyosin interactions, but
125
also actin-myosin mechanochemistry. Importantly, we found that γ-actin is
significantly regulated only by tropomyosin-β, while α-actin is regulated by
both tropomyosin-αβ and tropomyosin-β.
More specifically, the regulation by tropomyosin has distinct effects
on in vitro molecular mechanics in three regulated actin-tropomyosin
combinations (experimentally determined), suggesting three different modes
by which tropomyosin regulation affects actin-myosin mechanochemistry
(determined by model parameter adjustment): (1) αA-Tmαβ – V is
increased 1.2-fold. This is caused by a 1.15-fold increase in the myosin
attachment rate to actin, the unstrained myosin main power stroke rate,
and the unstrained rate of detachment of unloaded myosin from actin. (2)
γA-Tmβ – fmot is reduced to 0.96-fold and tstop is increased 1.6-fold. This
is caused by an increase in the impact that myosin-to-myosin mechanical
coupling has on rates of mechanical steps of myosin by a factor of 1.2.
(3) αA-Tmβ – trun is increased 1.7-fold for short actin. This is caused by
a decrease in the unstrained rate of detachment of myosin from actin to
0.75 times the baseline value and a decrease to 0.8-fold in the impact that
myosin-to-myosin mechanical coupling has on rates of mechanical steps of
myosin.
Note that no quantitative adjustment, e.g. minimization of sum of
squared errors, was used to determine the model parameter changes stated
above. In consequence, the numeric parameter changes stated above should
be understood as qualitative indicators of the general nature of changes in
actin-myosin interaction kinetics.
Comparison to existent knowledge on regulation
The changes in kinetic parameters determined for αA-Tmαβ using our
model-based assessment are in line with what is known for this condition
126
from ATPase assays with skeletal muscle myosin and actin. Sobieszek
determined that gizzard smooth muscle tropomyosin increases the ATPase
Vmax , while the affinity of myosin for the actin-tropomyosin complex was not
affected at myosin:actin ratios of less than one myosin head per 4 to 6 actin
monomers – which is the relevant regime for our experiment [160]. These
observations were attributed to increases in the rates of the kinetic steps
after myosin binding to the actin-tropomyosin complex, which is concurrent
with the general increase in the unstrained kinetic rates we observed for
αA-Tmαβ. Williams et al. found results that are similar to Sobieszek’s
and were measured at low myosin concentrations and low ionic strengths
corresponding to those used in our motility assays [183].
Sufficient evidence exists to state that smooth muscle tropomyosin
does regulate smooth muscle myosin interactions with actin, and thus, the
resulting molecular mechanics [124, 16, 61, 60]. Regarding the functional
relevance of the smooth muscle tropomyosin isoforms, however, several not
mutually exclusive mechanisms by which the isoforms could affect molecular
mechanics have been put forward [155]:
1. Differences in the molecular structure of tropomyosin, in the commonly observed dimerization of tropomyosin, or in end-to-end binding
of the dimers, i.e. differences attributed to tropomyosin only, not
to other binding partners (high sequence and structure variation in
end-to-end binding domains [142], impaired long chain formation in
head-to-tail overlap region mutants [26]);
2. Differences in the location and configuration of tropomyosin dimers attaching to the actin filament surface, leading to increased or decreased
blocking of other actin binding partners, i.e. differences attributed
to the interaction of tropomyosin and actin (α- vs. β-isoform lead
127
to 10-fold differences in actin-tropomyosin dissociation constant [27],
Tropomyosin-αβ dimers exhibit specificity in their orientation when
bound to actin [6]);
3. Differences that are directly attributed to the interaction between
actin binding proteins and tropomyosin. (α- vs. β-tropomyosin lead
to almost two-fold difference in myosin(S1)-actin dissociation constant
[27], Troponin specific binding site that occurs in skeletal, but not
smooth muscle, β-isoform of tropomyosin [26], binding between
smooth muscle myosin and smooth muscle tropomyosin without actin
present[129]).
With regards to smooth muscle contraction, smooth muscle myosin is
the most central interaction partner of actin. We investigated its mechanical
action on actin in the background of different actin and tropomyosin isoforms’ interaction. Because we found that tropomyosin isoforms are indeed
relevant to the regulation of actin-myosin interactions, all three mechanisms
are possible for actin-tropomyosin-myosin interactions. However, the observed difference between the tropomyosin isoforms depends on the actin
isoform. This suggests direct interactions between the actin filament and
tropomyosin, highlighting the second mechanism.
Concluding remarks
Our mathematical model does not include tropomyosin-mediated
myosin binding cooperativity. Binding cooperativity is often assessed by
changing the myosin-actin ratio or the myosin activation level [124, 61,
60, 111]. Within the scope of this study, one detectable effect of binding
cooperativity differences would be a shift in the actin length at which
bifurcations between nonmotile and motile behavior occur [78]. These
bifurcation lengths depend on the number of myosins effectively bound to
128
actin and would be affected by cooperativity-mediated changes in the the
effective rate of myosin binding to actin. We found no significant shifts
in these lengths between the conditions, and therefore no indication of
differences in binding cooperativity.
Like any automation of a manual analysis procedure, our video analysis
software makes the analysis of large data sets feasible and prevents differences occurring between different days or operators. A specific advancement
is the automated machine learning-based approach to quality control of the
filament traces. Further, a result management framework was devised, which
allows keyword-based queries into annotated data sets and the application of
custom analysis functions. Utility functions allow the creation of customized
MatLab scripts to interact with results. This supports customized analyses
of existent data sets also by computational scientists without their own
motility assays, as well as the “high throughput” necessary for determining
statistical distributions and L resolved curves of motility features. The MatLab scripts with instructions are released as open source (In Vitro Motility
Assay Automated Analysis – ivma3 , http://code.google.com/p/ivma3/).
FIESTA is another openly accessible analysis software that can be used for
in vitro motility assays [146]. It reaches nanometer precision and allows
interactive assessment of filament motility in a graphical user interface.
Differently, our software provides less precise image analysis and tracking
at the benefit of fast processing of a high number of experiments and the
possibility to execute specific analyses on large data sets in an automated
fashion.
The statistical assessment uses bootstrapping to maintain the high
filament count that is necessary for a high L resolution while still giving
account of the variation present in the experiment. To explore the results
129
and counteract inflation of statistical significance resulting from L resolved
analysis, PCA was used on the bootstrapped data sets. We could not find
existent examples of this combination of PCA and bootstrapping – other
studies estimate the variation of PCA itself [170, 185], or assess the variation
of bootstrap scores (loadings) [25, 138].
More detailed assessment of in vitro motility and the observed specificity of regulation require more specific theoretical explanations of the
molecular mechanochemistry underlying these observations. Our relatively
simple stochastic model generated data sets that were analyzed in the same
way as actual experimental data, indicating how the different actin and
tropomyosin isoform combinations affect actin-myosin interaction kinetics.
While providing a perspective beyond mere presentation of our experimental
findings, the simplicity of our model as well as the procedure by which
model parameters were adjusted to mimic the experimental observations call
for future work. From an experimental perspective, molecular mechanical assays using expression and site-directed mutagenesis of actin and tropomyosin
seem promising.
Materials and Methods
Experimental Methods
Protein purification and preparation. Contractile proteins were
purified from tissues donated from the slaughterhouse as specified below.
Myosin was purified from pig stomach antrum as described previously
by Sobieszek [161]. α-actin was purified from chicken pectoralis acetone
powder as described by Pardee and Spudich [137]. γ-actin was purified
from turkey gizzard following a previously reported protocol by Ebashi [42].
Actin was fluorescently labelled by incubation with tetramethylrhodamine
isothiocyanate (TRITC P1951, Sigma)-phalloidin [181]. Tropomyosins were
130
purified by ammonium sulfate precipitation and then collected by isoelectric
precipitation at pH 4.616. Tropomyosin-αβ was purified from chicken
gizzard, tropomyosin-β from the phasic region of pig stomach.
Myosin phosphorylation. Myosin (5 mg/ml) was thiophosphorylated
with CaCl2 (6.75mM), calmodulin (3.75 μM, P2277, Sigma-Aldrich Canada),
myosin light chain kinase (0.08 μM), MgCl2 (10 mM) and ATP γ-S (5 mM)
by incubation with all reagents for 20 minutes at room temperature, kept at
4◦ C overnight, and then stored in glycerol at −20◦ C.
In vitro motility assay. Flow-through chambers and buffers were
prepared and used as previously described by Léguilette et al. [109]. The
oxygen scavenger contained 0.16 mg/ml glucose Oxidase, 0.045 mg/ml
Catalase, 5.75 mg/ml glucose. Nonfunctional myosin heads were removed by
ultra-centrifugation of purified myosin (42,000 rpm, 4◦ C, 32 min, TLA-42.2
rotor in Optima L-90K ultracentrifuge, Beckman Coulter, Fullerton, CA).
In parallel with ultra-centrifugation of myosin, the buffers used to perfuse
labelled actin in the regular motility assay protocol were prepared on ice to
contain 0.6 μM α- or γ-actin and, where applicable, 6 μM tropomyosin-αβ
or tropomyosin-β (Tab. 4–1). Before incubation in the flow-through chambers, myosin was diluted three-fold to 0.17 mg/ml by addition of myosin
buffer. In each execution of the motility assay, 12 or 16 flow-through chambers were recorded. Batches of four flow-through chambers were incubated
according to randomized conditions up until methylcellulose buffer perfusion
and stored in a light-protected and humidified container. These flow-through
chambers were then separately perfused with methylcellulose immediately
before insertion into the microscope stage, while being preheated to 30◦ C
during this last perfusion step (XH-2002 Small Slide Warmer, Premiere).
131
Video recording. As soon as microscope focus could be achieved,
actin motility was visualized with an inverted microscope (IX70, Olympus),
recorded with an image-intensified CCD camera (KP-E500, Hitachi, 30 fps),
and digitized with a custom-built recording computer (Norbec Communication, Montreal, QC, Canada; Pinnacle Studio DV/AV V.9 PCI Capture
Card).
Video Analysis
We developed an automated video analysis software which executes the
following steps. Raw video data are preprocessed (image enhancement and
frame merging to a time resolution of Δt = 0.333 s) and turned into binary
images. Filament objects and their properties are extracted from individual
frames using connected components methods. Filaments are tracked
throughout consecutive frames based on their centroid position and area.
Frame-to-frame velocities (Vf 2f ) are calculated from centroid displacements
between two consecutive frames. Filament length (L) and travelled path
lengths are determined based on a transformation of image objects into
rectangles of same area and perimeter, the longer edge representing lengths.
A filament’s mean trace velocity (V ) is determined by dividing the total
distance that the filament’s tip has travelled by the time the filament was
present for (T ). Filament traces with filament crossing events or signs of
irregular motion were removed by a machine-learning algorithm, which was
trained on subset of our data that we scored by hand. The automated video
analysis was assessed using computer-generated mock motility videos, the
automated quality control was evaluated against hand-scored data sets. For
details see Supplementary Material.
132
Statistical Analysis
Statistical significance was assumed for p < 0.05. Statistical comparisons were executed by bootstrapping of the compared statistic; statistical
significance was assumed where no overlap exists between the 95% confidence intervals of the compared conditions. For details see Text Supplementary Material.
Acknowledgements
Frederic Simard gave advice on the machine learning and parallel code
design. Lea Popovic and Daniel Zysman discussed the statistical analysis.
We thank Del R. Jackson for helpful discussions of models of in vitro
motility.
133
4.3
Supplementary Methods
Video Analysis
Preprocessing of raw motility videos. Starting from raw motility
videos, n consecutive frames are merged (brightness average of grey scale
images). This lowers the number of frames to analyze, improves the signal to
noise ratio by reducing brightness fluctuations, and reduces the influence of
Brownian motion type noise (Fig. 4–8 A-C). In this study, every 10 frames
were averaged into one, giving an effective time resolution of Δt = 0.333 s.
In each averaged frame, the brightness values are shifted by a scalar value
so as to give the same brightness median across all frames to counteract
photobleaching and other lighting fluctuations. The brightness range of each
video is rescaled to [0, 1], so that the brightest pixel has the value 1, the
darkest pixel 0.
Extraction of image objects. Each video frame is thresholded into
a binary image, whose connected components are extracted (four neighbour
connectivity). Objects of the size of one pixel are removed from further
analysis right away.
Filament tracking. Filaments are tracked from one frame to the next
based on a dissimilarity matrix M̂. For two consecutive frames (n − 1 and
n), the dissimilarity matrix is an Nn−1 × Nn matrix, where Nn is the number
of filaments in frame n. The matrix elements are
Mk,l = (Akn−1 − Aln )2 + (ĉkn−1 − ĉln )2 ,
(4.1)
that is, the dissimilarity matrix M̂ = (Mk,l ) is determined from the area
Akn−1 (number of pixels times area per pixel) and weighted centroid ĉkn−1
(center of mass of grey scale pixel values) of the k-th filament in the old
(n − 1) frame, and Aln and ĉkn of the l-th filament in the new (n) frame.
134
Figure 4–8: Brownian motion type displacement at different time
resolutions Δt. A) Mean velocities expected from sliding at a constant
velocity v = 0.7μm/s (solid line) and a velocity vb = 0.2 μm
100 ms/Δt
s
resulting from Brownian motion type displacements (dashed line). B) Means
of two Gaussians (νmax and νlow ) fitted to the Vf 2f velocity distribution extracted from four videos of α-actin sliding, [My] = 0.167 mg/ml. C) Vf 2f
histograms for different time resolutions. The two populations that should
be observable in in vitro motility [123] are visible only for sufficiently high
Δt, while too low Δt the two populations are not visually separable. Inset:
computation time on a single and two processors (“cores”).
All ĉ are twodimensional vectors, and potentiation by 2 indicates the scalar
product formed by the dot product of a vector with itself (i.e. the sum of
squared elements).
The properties of a filament in a specific single frame n are stored
under a Metaindex m. Given that in each frame n a Metaindex m uniquely
references a single filament, the Metaindices can be used for filament
tracking – a filament holding the same Metaindex m in two frames is
understood as the same filament tracked over several frames.
The task of tracking, formulated in the above way, is to use M̂ to assign
Metaindices in a new frame n based on those from an old frame n − 1. First,
the row minimum is found for all rows of M̂. All values except these row
135
minima are set to zero, e.g.
⎞
⎛
⎛
⎜ 1 0 0
⎜ 1 3 4 ⎟
⎟
⎜
⎜
⎜ 2 5 6 ⎟→⎜ 2 0 0
⎟
⎜
⎜
⎠
⎝
⎝
0 2 0
3 2 4
⎞
⎟
⎟
⎟.
⎟
⎠
(4.2)
In this matrix, the three different possibilities by which a Metaindex can
be assigned to a filament in the next frame are shown: (1) In the first and
the second rows, the minimum is at the first position. This indicates that
two Metaindices from the current frame point to the same filament in the
next frame. In this case the filament in the next frame (in our example with
index 1) is assigned a new Metaindex. (2) In the third row, the minimum is
at the second position, so the Metaindex of the filament with index 3 in the
current frame is forwarded to the filament with index 2 in the next frame.
(3) The filament with index 3 in the next frame is not assigned a Metaindex
from Metaindex forwarding, therefore it is assigned a new Metaindex.
Whenever a filament in the next frame is not assigned a Metaindex, a
new Metaindex is introduced. It is simply +1 greater than the greatest
Metaindex existent at that point, which ensures uniqueness of Metaindices.
Two more checks are executed during Metaindex forwarding: A length
change check and a separation check. For the length change we compare
the size between two filaments that are about to be assigned the same
Metaindex in two consecutive frames. We check if 1) the ratio of the lengths
of the n and n + 1 filament is greater than a defined parameter Crel or
smaller than 1/Crel , or 2) the lengths of the n − 1 and the n filament
differ by more than a defined parameter Cabs . If so, we conclude that an
unreasonably big length change occurred and assign a new Metaindex
for the n filament. A separation check ensures that two filaments which
were chosen to be assigned the same Metaindex in consecutive frames
136
will actually form a connected trace. If they share at least one common
pixel, the trace is connected, and the check is passed. If the filaments in
consecutive frames do not share any pixels, they are assumed to be different
filaments and a new Metaindex is introduced for the filament in the second
frame. Similar approaches to following filaments were used in earlier works
[84, 122, 146, 125]
Frame to frame velocity. The frame to frame velocity (Vf 2f ) between
two consecutive frames is the centroid displacement divided by the time
resolution Δt. Vf 2f is used where time courses of filament motion are desired
(Fig. 4–1 B). However, for longer filaments an increasing underestimation of
sliding velocity occurs due to the centroid cutting corners when the filament
turns [184].
Filament length and trace based sliding velocity. Filaments and
traces, both of elongated shape, are transformed into rectangles of same area
(A) and perimeter (P ). The object length then is the longer x+ of the two
rectangle edges [145]
P
x± = ±
4
$
P2
− A.
16
(4.3)
The trace based velocity (V ) is calculated from the distance that a filament
travelled along its trace during the time it was observed and tracked for (T ).
The trace is constructed by merging the filament’s binary images from all
frames (Fig. 4–1 B). The filament length (L) is determined by averaging
over the filament length in all frames that the filament was observed in. L is
subtracted from the trace length, resulting in the distance travelled by the
filaments tip along its sliding path (Fig. 4–1 B). This trace length is divided
by T to give V . Unlike the centroid method, V is not affected by systematic
errors resulting from curved motion.
137
Features of actin motility. The mean sliding velocity is calculated
as the mean of trace velocities V . Trace velocities are used instead of the
centroid based frame to frame sliding velocities (Vf 2f ) to prevent velocity
underestimation for long filaments (Fig. 4–1 B) [184]. The motile fraction
(fmot ) is the fraction of Vf 2f greater than a threshold velocity Vt (Fig. 4–1
C). Based on another threshold velocity Vtt , the Vf 2f are scored as stopped
(Vf 2f ≤ Vtt ) or motile (Vf 2f > Vtt ). Periods spent in the nonmotile state
without transition into the motile state are called stop times tstop , conversely
periods spent in the motile state without transition into the nonmotile state
are called run times trun (Fig. 4–1 C). In this study, Vt = 0.125 μm/s and
Vtt = 0.225 μm/s were used.
Assessment of video analysis
Influence of random displacement. Filament centroids get displaced due to directed filament motion and random fluctuations, which
were approximated as a Brownian motion type influence on Vf 2f . The mean
velocity of directed sliding (ν) is independent of the time resolution Δt.
The velocity from Brownian motion is proportional to (Δt)−1/2 (Fig. 4–8 A,
B). This implies that excessively lowering Δt will obscure Vf 2f distribution
features which are visible at sufficiently high Δt (Fig. 4–8 C). For sufficient
Δt, a motile and a nonmotile population in the Vf 2f can be distinguished
[123], indicating that a sufficiently large Δt was chosen to discern these
features of in vitro motility.
Accuracy of L and V values. Computer generated mock motility
videos were used to assess the analysis’s accuracy. V and L were faithfully
detected above the diffraction limit. Below the diffraction limit, indicated
by complex L or V (Eq. 4.3), V and L become quantitatively unreliable,
but filaments are still qualitatively ordered by L (Fig. 4–9 A). At a too
138
Figure 4–9: Validation of automated video analysis. Mock videos with
filaments of known L and V were computer generated and subsequently
analyzed. Black crosses represent input (L, V ) combinations, black circles
detected filaments, and blue boxes indicate detected filaments for which
length and velocity values were real numbers (without imaginary parts). A)
Analysis at 6 frames per second, no optical noise, Brownian displacement
or change of direction assumed. Only small deviations from input (L, V )
values can be observed, complex solutions occur at low L. While complex
solutions do not produce accurate L values, data points are still successfully
ordered along the L axis. B) As in A), but analysis at 3 frames per second.
For high sliding velocities V , the filament length is overestimated due to motion of the filament in frames that are merged (motion blur). C) Filaments
created with fluorophore brightness fluctuations, Brownian displacement of
fluorophores, and curved filament motion.
coarse time resolution, L is overestimated as filament motion over several
consecutive frames “stretches” filament images similar to a motion blur (Fig.
4–9 B). L and V values are mostly insensitive to curved sliding of filaments,
see Fig. 4–9 C and Hamelink et al. [70]. However, less of the filaments
successfully pass the rejection criteria (Fig. 4–9 C).
L below the diffraction limit. In the rectangular transformation
to determine filament and trace lengths (Eq. 4.3) complex solutions with
√
an imaginary component = 0 arise for P/4 > A. This is the case where
filaments that are shorter than the diffraction limit of ≈ 0.2μm appear
in the videos as approximately circular objects (Fig. 4–9 A-C). Below
the diffraction limit filament and trace lengths are not quantitatively
precise. Still, longer filaments can be expected to hold a greater number of
fluorophores, increasing the brightness and in turn the optical area of the
139
Figure 4–10: Robustness of velocity estimates to filament width.
Four α-actin videos were analyzed with different Black-White thresholds (BW ). Increasing BW decreases filament width (W ) and filament length (L). The mean sliding velocity (V ) is mostly unaffected for
(0.2 ≤ BW ≤ 0.35)). Outside this range, V is affected as well. For
BW < 0.15 an increasing number of valid filaments is detected, while computation time increases sharply (inset). Data shown are arithmetic means
with standard errors.
filament, resulting in a greater image object. In consequence, the real parts
P/4 could be used to qualitatively sort affected filaments according to L,
even though quantitatively accurate L could not be inferred (Fig. 4–9 A-C).
Relevance of filament width and digitization threshold. Based
on analysis results obtained with different Black-White thresholds (BW ), V
and L averages depend on filament width (W ) (Fig. 4–10). However, in a
Black-White threshold range between 0.2 and 0.35, V , W , and the number
of analyzed filaments seem mostly unaffected. For a low BW < 0.15,
the computation time increases sharply, indicating that optical noise and
brightness fluctuations are analyzed as objects (Fig. 4–10, inset). L depends
140
Figure 4–11: Assessment of automated filament rejection. A) The
predicted error rate reduces towards a plateau above 100 trees in the decision tree ensemble, indicating that 150 trees will ensure maximally achievable performance. B) Receiver Operating Characteristic (ROC) curve for
crossvalidation between motility data recorded on two different days. C)
Cost optimization to determine acceptance threshold above which filaments
are kept in the data set. Line styles: solid – overall cost of misclassification,
dashed – false positive rate, dash-dotted – false negative rate. Colors: Gray
– training on manual scoring from Day 1, assessment on manual scoring
from Day 2; black – training on Day 2, assessment on Day 1. Cost: false
positive – 5, false negative – 1. 150 trees were used in B and C. Filter for
corner detection: Gaussian, parameters [21, 1], 2.5, maximal number of corners: 200, parameters for corner detection: sensitivity factor 0.2, quality
level 0.15.
on BW across all BW values. This can be understood when taking into
consideration that a lower BW leads to an increase in the visual area
covered by objects, giving a “blobbier” appearance. In consequence, also the
head and tail tips of filaments reach out further, effectively increasing L. For
BW > 0.55 the number of valid filaments drops sharply, indicating that an
excessively high BW effectively removes most filaments.
Machine Learning Quality Control
Filament traces can be rendered unfit for analysis by filament crossings,
filaments “swirling” around attachment points on the motility surface,
lateral displacement [173], or other features that lead to inaccurate V
and L estimation. Initially, such traces were removed manually, leading
to acceptance (1) or rejection (0) of specific traces. Decision tree models
were then trained on such manually generated data sets, to replace time
141
consuming manual inspection of filament traces. To make trace images more
uniform and thereby easier to categorize, trace images were rotated so that
all images’ main axes pointed in vertical direction. The binary image data;
length, width, solidity, perimeter, and area of the trace; and the number and
position of corners found using MatLab’s corner function were extracted
as features. These features were used to train the decision tree models on
manually scored data. Using > 100 decision trees did not lead to a further
decrease in the prediction error rate, indicating ≈ 100 as the number of
decision trees necessary to achieve the maximally possible classification
performance (Fig. 4–11 A). Models used in this work were based on 150
decision trees, ensuring a sufficient number of decision trees. Receiver
Operating Curves for the classification of data from one day based on
training on data from another day show that a good margin for successful
classification exists (Fig. 4–11 B). Based on cost optimization for different
data set combinations, we chose an acceptance quorum of 0.7 trees accepting
a filament trace (Fig. 4–11 C). False positives (traces that should have been
rejected) were visually inspected and showed little or no features deviating
from an undisturbed, elongated shape (data not shown).
Statistical Analysis
The raw data are motility assay videos recorded for the different
conditions (Tab. 1), in the following the conditions are indexed by C. In
each condition, a number (V C ) of different videos were recorded, in the
following the videos are indexed by v = 1, . . . , V C . Let v be a sequence of
length V C , whose elements each refer to one video of C. In a general case, a
video can occur in v once, more than once, or also not at all. The meaning
of v is to describe a data set made up of the different videos referenced in v,
e.g. v = (1, 2, . . . , V C ) represents the original data set in which each video
142
of condition C is referenced exactly once. Resolution by L is executed using
a sliding window approach: an overall L range (L ∈ [Lmin , Lmax ]) is set,
then the center of a window of constant width is moved along this range at
equally sized steps. At each step, all filaments from the videos contained
in v (multiply referenced videos are included multiply) within the windows
current L range are gathered, the specific motility feature is calculated for
all filaments, and the arithmetic mean of these feature values is stored as the
according element of a result vector (RC
v ). In this study, 50 length windows
were applied for each of the four features (Fig. 4–2), resulting in an RC
v with
a length of 200 elements. Within this formalism, RC
v=(1,2,...,V C ) refers to the
original data set with all videos contained exactly once; this can be likened
to a “mean” in a scalar measurement. However, when the videos referenced
by v are assigned by random draws with replacement, effectively bootstrap
resampled RC
v are created, which serve as empirical distributions around the
“mean”.
Statistical assessment in Principal Component space. PCA
was applied to reduce feature dimension and remove correlations between
different elements in the RC (v omitted from notation) values to prevent
inflation of statistical significance. First, the bootstrap RC from all C were
pooled to execute the PCA on all bootstrap data at once. The elements
of the bootstrapped RC were transformed into z-scores (centered on 0 by
subtraction of the mean and normalized by standard deviation in each
element) and the first 3 PCs were considered in the statistical assessment
(Fig. 4–3 A, B).
After transformation into PC space, the overlap of 95% intervals of the
empirical distributions of the bootstrapped RC was assessed to detect if
statistically significant differences with p < 0.05 are present. To compare
143
two conditions C = I and C = II, the original (not bootstrapped) RI and
RII PC coordinates were determined using the PCA transformation matrix
determined based upon the bootstrapped RC , giving rI and rII , respectively.
A vector d = rII − rI connecting both C = I, II was determined and the
bootstrapped RI and RII were projected onto that vector using a simple
scalar (dot or inner) product. Then, empirical 2.5% and 97.5% percentiles
for each C were constructed from these onedimensional values (Ward’s
bootstrap confidence intervals). Where no overlap between these percentiles
existed, statistical significance was inferred (p < 0.05) (Fig. 4–3 C, D).
While PCA and bootstrapping are standard methods, their combined
application is seen more rarely. In other studies, for example, bootstrapping
is used to estimate variance in the PCA procedure itself [170, 185], or a data
set of several measurements is bootstrapped into mock data sets for each of
which an independent PCA is executed to determine statistics of the PCA
scores [25, 138].
Hierarchical clustering. The bootstrapped RC in PC space were
pooled across all C, and then subjected to agglomerative hierarchical
clustering to rediscover the conditions (Fig. 4–3 E-H). The Ward method
and an Euclidean metric were used.
Length averaged fold change analysis. αA was chosen as a
reference condition (C = 0), by which the other conditions were normalized.
All bootstrapped RC are divided elementwise by the values of the reference
condition’s R0 original (not bootstrapped) feature vector sC values. From
the normalized bootstrapped RC , the V , fmot , tstop , and trun are then
averaged over L (arithmetic mean), resulting in scalar fold change values.
To estimate the rate of type I errors (false positive rate), the rate
of rejection of the null hypothesis in absence of an actual difference was
144
Figure 4–12: Assessment of Type I Error rate in the length averaged fold change analysis. Shown are empirical cumulative probabilities
of the difference in confidence interval limits (ΔCI ) for comparing two random resamples of the baseline condition (αA). A Type I Error (detection of
a statistically significant difference in the absence of an actual difference) is
indicated by ΔCI > 0. For the four features of in vitro motility (V , fmot ,
tstop , trun ), no ΔCI > 0 could be detected. The ΔCI distributions are several distribution widths below 0, which further indicates that ΔCI
>
0
should occur very rarely. Distributions were created from 300 comparisons of
resamples.
145
determined. We executed length averaged fold change analyses as described
above, with the difference of comparing the baseline condition (αA) to itself.
Due to the bootstrapping employed in this procedure, two executions of the
procedure correspond to a random reshuffling of the same data set. This
preserves the same original data set, thus creating two data sets that have
no difference between them but do account for the fluctuations being present
in the recorded data. The rejection of the null hypothesis, i.e. the detection
of statistically significant differences, applied when the confidence intervals
did not overlap. This would be the case if
CI1,1 − CI2,2 > 0 or CI2,1 − CI1,2 > 0,
(4.4)
where CIi,j are the limits of the confidence intervals. i ∈ {1, 2} refers to
the first and second resample, respectively. j ∈ {1, 2} refers to the smaller
and greater limit of the confidence interval, respectively. In consequence, a
significant difference would be detected if
ΔCI = max (CI1,1 − CI2,2 , CI2,1 − CI1,2 ) > 0.
(4.5)
Empirical distributions of ΔCI indicate that, given our specific assessment
method, number of measurements, and fluctuations in our data, the detection of statistically significant differences in the absence of actual differences
is highly unlikely (Fig. 4–12).
Mathematical Model and Simulation
We use a mathematical model developed in one of our earlier studies
[78]. Here, we give only the essential assumptions, expressions, and descriptions. We simulate N myosin binding sites (N = L/0.0355 μm [163, 23])
which are mechanically coupled via the actin filament as a rigid backbone.
The myosin binding sites are referred to by an index m = 0, . . . , N , and have
146
a chemical state cm ∈ {0, 1, 2} (pre power stroke state, post power stroke
state, unbound state) and a mechanical displacement from their unstrained
position xm ∈ R (in units of μm). Based on this, σm describes if a binding
site m currently has a myosin bound to it:
⎧
⎪
⎨ 0 if cm = 0,
σm =
⎪
⎩ 1 if cm ∈ {1, 2}.
(4.6)
Further, the unstrained position of a myosin bound to binding site m is
shifted by mechanical steps executed by the myosin. The current step length
Δstep
is
m
Δstep
m
⎧
⎪
⎨ 0 nm if
=
⎪
⎩ 4 nm if
cm ∈ {0, 1},
(4.7)
cm = 2.
This means that only the main power stroke of 4 nm length affects the
equilibrium position, while the secondary power stroke of 2 nm is followed by
an immediate detachment (ATP saturated buffer) [174].
The kinetics of the individual binding sites consist of a unidirectional
reaction cycle (0 → 1 → 2 → 0), with transition rates ka (attachment,
0 → 1), kp (main power stroke, 1 → 2), and kd (detachment, 2 → 0) [174].
Due to actin acting as a rigid backbone, the xm result from an interaction between all myosins currently bound to actin (σm = 1). The length
travelled by the actin filament (a, in units of μm) also results from this
mechanical interaction. For a binding site going from cm = 0 to cm = 1, an
unstrained position
x0m ∼ N (a, w)
147
(4.8)
is assigned, which is a random variable drawn from a normal distribution
around the mean a (attachment without any strain) with a standard
deviation of w.
The force exerted by myosin on an individual binding site m is
fm = −σm xm = −σm K(a − x0m − Δstep
m ).
(4.9)
K is a spring constant, we assume linear elasticity. After each kinetic
transition, we assume that an instantaneous force equilibrium between all
attached myosins is attained,
N
fm = 0 ⇒ a =
σm (x0m
+
Δstep
m )
%
natt , natt =
σm .
(4.10)
m=1
a is left unchanged if natt = 0.
Transition rate calculation. The current chemical state cm as
well as the work that has to be exerted for the next kinetic transition
(ΔWm ) determine the rate at which a binding site m will undergo the next
transition, rm :
⎧
⎪
⎪
if
ka
⎪
⎪
⎨
2
1→2
rm =
if
kp0 e− K cf ΔWm
⎪
⎪
⎪
⎪ 0 − K2 cf ΔWm2→0
⎩
if
kd e
cm = 0,
cm = 1,
(4.11)
cm = 2.
2cf /K is a coefficient quantifying how strongly mechanical work exerted
during a mechanical transition affects the rate of this mechanical transition.
When a, x0m and cm are known, the overall mechanical work stored in the
actin-myosin system can be calculated:
W =
N
m=1
σm
2
K
a − x0m − Δstep
.
m
2
148
(4.12)
m denotes the general summation over all myosin binding sites, irrespective
of the current value of m.
We now assume that a is continuously changed to the system’s mechanical equilibrium position throughout the execution of a mechanical step. It
follows that
ΔWm = Wmaf ter − Wmbef ore
(4.13)
can be calculated from the work before (Wmbef ore ) and after (Wmaf ter ) a
mechanical step of size d occurring at binding site m,
d 2
step 2d
K
0
ΔWm = 2
+ (a − xm − Δm ) natt
m
natt
d
− Δstep
2d + x0m 2d .
+d2 − a + natt
m
(4.14)
The simulation is initialized with a = 0, xm = 0, and randomly assigned
cm . Filament motion is simulated by iteration of the following steps.
1. Instantaneous force equilibration by adjustment of a.
2. Current transition rates rm are calculated based on the system’s
current state.
3. The next transition time and the binding site undergoing a transition
is stochastically drawn based on a Gillespie scheme [56]. The according
binding site is advanced by one step in the reaction cycle.
This sequence is repeated until either a maximal number of kinetic transitions per binding site (nsteps ) or a maximal simulation time (tmax ) is
reached. The iteration results in a sequence of travelled lengths a, and
elapsed times t. In the actual video analysis from the motility assay, these
exact time courses are not accessible, instead only the difference in travelled
length at time intervals Δt (time between consecutive frames in our videos)
149
can be assessed. Accordingly, Vf 2f were extracted from the stochastic
simulation using an imposed time resolution of Δt.
Actin sliding was simulated for N = 5, 6, . . . , 90. At each N , actin
sliding was simulated till 100 s of simulation time or 100, 000 · N chemical
transitions were reached. Features of actin motility were extracted for each
N , followed by a running window analysis (Start: N = 10, end: N = 90,
window width of 20). Considering a binding site distance of ≈ 0.035 μm
[23, 163], this approximates the L range we analyzed in our experiments.
Correction for nonspecific binding to motility surface. When
this model was simulated, above N ≈ 25, trun and tstop could not be
measured due to Vf 2f not falling below the threshold for determining trun
and tstop . Occasional nonspecific stopping events occur for longer filaments
[78]. We introduce these stopping events by inserting Poisson distributed
Vf 2f = 0 μm/s values into the Vf 2f time trace:
ΔT = −T log r,
(4.15)
where ΔT is the time between two consecutive stop events, r a random
number drawn from a uniform distribution r ∈ [0, 1]. T is the mean time
between two consecutive stop events,
⎧
⎪
⎨
∞
if
T =
⎪
⎩ 20 s + 2 s if
N −30
N ≤ 30,
(4.16)
N > 30,
which has been adjusted to limit trun as observed in our experiments (Fig.
4–2 D).
Parameter adjustment. The baseline model parameters (Fig. 4–7)
were taken from the literature and then adjusted to fit the L resolved V and
fmot (ka [19, 57], kp [174], kd [174], an additional factor of 5.5 was used to
150
Figure 4–13: Changes in L resolved features for changing the model
parameters. Each model parameter (one per row) was changed from 0.25
to 1.75 times (ka , kp , kd ) or 0.75 to 1.25 times (cf , w) its baseline value (15
equally sized steps, indicated by solid lines shaded from black representing
the lowest value to light grey representing the highest value; dashed line
represents baseline). The resulting changes in the motility features V , fmot ,
tstop , and trun (one per column) were used to determine model parameters
that mimic the experimentally observed differences.
151
account for higher temperature in our experiment [78]), or estimated from L
resolved V and fmot , starting from values determined in our earlier study of
skeletal muscle myosin [78] (cf , w). The altered parameters mimicking the
αA-Tmαβ, γA-Tmβ, and αA-Tmβ conditions were estimated based on Fig.
4–13 and comparison with L resolved features and scalar fold changes in the
conditions with statistically significant differences in the experiment (Fig.
4–2 and 4–3, respectively).
152
Chapter 5
Discussion
5.1
Conclusions
In the first article, a formerly unknown complex dependence of un-
loaded actin sliding motion on the number of myosin protein motors that are
capable of simultaneously interacting with an individual actin filament was
detected experimentally and explained theoretically. The findings describe
two qualitative transitions in the interaction between a group of myosins and
an actin filament that they simultaneously work on. Small groups of myosin
( 10) hinder each others mechanochemical cycle progression, large enough
groups ( 30) exhibit sustained activity. Intermediate size groups alternate
between these two states, so that the entire group is mostly either inactive
or active. It is not clear how this translates into the actin-myosin interfaces
driving cellular contraction, and therefore the contractile properties of
smooth muscle cells. However, these findings indicate that alterations in the
number of closely associated myosin motors in the cellular contractile ultrastructure might affect cell mechanical properties due to these qualitative
transitions.
In the second article,differences between contractile actin isoforms in
the presence of smooth muscle tropomyosin were identified, thus answering
the long standing question of molecular mechanical differences between actin
isoforms positively. Further, it was confirmed that a complex interaction
between actin and tropomyosin isoform exists. Finally, a complete toolbox
and detailed approach for the mathematical model based inference of
individual myosin chemistry and intermyosin mechanical coupling from the
153
actin sliding motions of actin filaments propelled by mechanically coupled
groups of myosin was provided.
5.2
Future Directions
The major implications of this thesis are in two directions. This thesis’s
findings were made at the level of molecular mechanical interactions, where
the new methods give new tools for investigation, and the findings raise
questions with respect to the mechanical coupling in intermediate size
groups of motors. The physiological relevance, however, arises due to the
fact that these findings further the knowledge of the underlying mechanisms
of smooth muscle cellular mechanics. Thus, the second direction concerns
the implications that these findings might have at the cellular level.
5.2.1
Active State of Myosin Group Action
The experimental and theoretical findings indicate that a group of
myosin working on an actin filament can be in two distinct states. One
is the actively cycling state, the other one the fully arrested state. For
intermediate groups sizes, switching between the two states is observed,
which occurs on a time scale that is clearly slower than the underlying actinmyosin interactions. Even though the mechanically realistic mathematical
model is capable of reproducing and predicting all measurements, a simple
and intuitive description of the actively cycling and the arrested state and
the dynamic alternation between these states has not yet been found. The
heuristic reduction to lowdimensional dynamical systems, likely of the slow
fast variable type, presents itself as a promising avenue towards a simplified
formalism to describe the existence of and alternation between the actively
cycling and the arrested state of myosin group kinetics.
A different perspective on the actively cycling and the arrested state
is that of a dynamically formed dissipative structure. Such structures
154
dynamically establish themselves in the face of a build up of potential
energy gradients, creating complex dynamic phenomena that contribute to
the accelerated dissipation of the built up potential energy. Viewing the
active state as an example of such a dissipative structure, one can address
the conditions for and the mechanism of the emergence of the dissipative
structure. Specifically, one should address the (1) minimum number of
simultaneously bound myosins, (2) the minimum chemical free energy from
ATP hydrolysis (which is the source of the potential energy gradient in this
system), and (3) potential scenarios that break down the clear separation
between existence and nonexistence of the dissipative structure. One recent
laser trap study addresses the influence of ATP, ADP, and phosphate on
myosin group kinetics – however not from the perspective of an emergent
dissipative structure that manifests itself as a distinct motile state [33].
The history of muscle molecular mechanics is marked by the desire to
explain tissue level phenomena based on elementary, microscopic mechanisms. This is similar to the program of statistical mechanics, which deduces
macroscopic thermodynamic properties and relationships from atomic level
particles. Just as in statistical mechanics, the transition from the microscopic to the macroscopic largely rests upon the assumption that basic
particles are independent from each other. The findings made in this thesis
indicate that the behavior observed at an intermediate level – the motility
assay with 100 motors that are coupled while working on the same actin
filament – is rooted in coupled kinetics of the myosin motors. It would be
interesting to assess the impact of this coupling on models of tissue level
muscle mechanics, but the transition is hindered because the myosins as
the basic units of the macroscopic behavior are no longer independent. As
long as no local effects are present, meaning that a rigid coupling via actin
155
has to be assumed, a mean field approach can be utilized. When localized
instabilities in myosin kinetics, such as the stop and go behavior we found
for intermediate size groups, occur, such a mean field approach would
no longer be possible. Other hindrances include long-range regulation by
tropomyosin, which makes the straight-forward application of mean field
theory impossible; the unknown impact of biochemical factors (phosphate,
ADP) on myosin group kinetics; or the unknown kinetic and mechanical
influence of other proteins in the cellular ultrastructure.
In the model formulation proposed in this thesis, several simplifying
assumptions were made – which in the light of recent experimental findings
should be considered more closely, or treated in a more realistic way. It was
found that upon deviation from its unstrained position, myosin develops
a restoring force in a highly nonlinear manner [95], while in thie theses
the response was assumed to be purely linear. Also, the dependence of the
rate of the mechanical steps on mechanical strain might be much more
complicated then assumed in our model, as seen in recent findings from
ultrafast loading of skeletal myosin mechanical transitions [24].
Lastly, it could be of interest to assess how mixtures of different myosin
types and isoforms perform with respect to the actively cycling and the
fully arrested state of group kinetics [176]. Structural studies have shown
differences in the conformations taken by skeletal and smooth muscle
when bound to actin, which might affect their impact in the context of
transmitting and receiving mechanical strains in a group of myosins coupled
to the same actin filament. The combination of mixture in vitro motility
assays with the video analysis methods and mathematical model developed
in this thesis might give new insights into the relevance of these structural
differences.
156
5.2.2
Classification of Regulatory Proteins
Given that all methodical steps for the molecular mechanical assessment
of actin regulatory proteins are now described and available, the detailed
and streamlined investigation of different proteins and their isoforms has
become more easily feasible. The assessment of newly recorded as well
as already existent in vitro motility data could provide rapid insight into
underlying molecular mechanical mechanisms.
As with all molecular level, in vitro investigations, extrapolation of the
findings to the cellular and systemic level poses considerable challenges.
In the particular case of this thesis, one can ask what impact specific
regulatory proteins’ molecular level function might have on the assembly
and performance of molecular contractile units, the molecular properties
of individual smooth muscle cells, or in shaping the tissue mechanics in
disease. Here, multiscale models connecting actin-myosin interaction, a
detailed structural description of the smooth muscle contractile units, and
the ultrastructure of the whole muscle cell provide a link between molecular
level regulation and tissue mechanics [21, 20].
5.2.3
Challenges in Plasticity Theory
The most established explanation of cell mechanical property changes
at the level of the contractile ultrastructure is a series to parallel transition
of the myosin thick filaments upon muscle activation and contraction (Fig.
2–9). Recently, two major challenges to this theory have emerged.
An in vitro study by Thoresen et al. that reconstituted contractile
bundles between microspheres instead of dense bodies, found a surprising
dependence of unloaded contraction velocity on myosin thick filament
length [169]. The length of skeletal as well as smooth muscle myosin thick
filaments was preset by the polymerization protocol for the preparation of
157
thick filaments. For both myosin types, an increase in myosin thick filament
length led to an increase in unloaded shortening velocity. According to the
series to parallel transition theory, longer thick filaments should lead to a
slower unloaded contraction. In this study, it was suggested that the myosin
thick filament should probably not be seen as the basic contractile unit, but
rather complexes of thick filaments held together by actin filaments, which
could potentially lead to larger contractile units at shorter thick filament
length.
Another challenging finding comes Liu et al. who recently determined
length distributions of smooth muscle cell myosin thick filaments [115].
First, the measured exponential distributions had a surprisingly low average
length of 200 nm or even less. This puts the length of actin-myosin interfaces
within the actin length ranges at which qualitative transitions occurred
in our experiments, even if a higher effective concentration in vivo due
to the ultrastructural organization of actin and myosin filaments would
shift the transitions in dynamic behavior to lower actin-myosin interface
lengths than see in vitro. Second, even more surprisingly, the average myosin
filament length was lowered at the isometric contraction plateau, which is
the opposite of what is expected from the series to parallel transition theory.
This discrepancy was not addressed in that study – even though it comes
from the same research group as the theory it calls into question. This
observation is a fundamental challenge to the series to parallel transition
theory, and again suggests that the myosin thick filament can not be
interpreted as the basic contractile unit in smooth muscle.
A suggestion brought forward by Thoresen et al. is that the contractile
unit length is indeed not proportional to the myosin thick filament length,
158
but depends on it in a complex fashion [169]. This would explain the observations made by Liu et al. but can hardly count as a specific hypothesis –
some dependence of contractile unit length on thick filament that matches
the experimental observations can always be found. In this situation, it
seems appropriate to search hypotheses that are based on known molecular
mechanisms and give a consistent explanation what a contractile unit consists of, and how its length depends on the length of myosin thick filaments.
Such a description of the contractile unit will likely include its dynamic
plasticity (polymerization and depolymerization of constituent actin and
myosin filaments, ultrastructural plasticity) and the resulting mechanical
properties. Given the central role that myosin motor groups play in shaping
these processes and properties, the findings made in this thesis likely hold
fundamental relevance. It could, for example, be important that small,
kinetically arrested groups of myosin could establish permanent linking
bridges between actin filaments.
Recent mathematical models provide a starting point for this integration and interpretation of the findings made in this thesis in the cellular
context. Donovan introduced the myosin thick filament length distributions
and the ability of filaments to bind and unbind dependent on actin-myosin
relative sliding into classical models of smooth muscle contraction [36]. This
can account for the observation of immediately reduced force development
moving the muscle away from its resting length, and an adjustment to a
new resting length over longer time courses. However, the model extension
essentially corresponds on the concept of actin-myosin filament overlap as
known from skeletal muscle. Further, the model does not account for force
hysteresis during rapid length oscillations. It is therefore questionable if it
mechanistically describes the molecular mechanical mechanisms underlying
159
mechanical plasticity in response to mechanical protocols. Differently, Brook
has combined a classical crossbridge theory model with changing length
of basic contractile units made up of actin-myosin interfaces, unbinding
of myosin thick filaments from actin upon stretch and shortening, and the
establishment of ultrastructures of force transmission on the cellular scale
in a detailed, multicompartment model [20]. This latter model accounts for
rapid oscillations force hysteresis.
In both models, it is crucial that myosin thick filaments detach from
actin thin filaments upon a change in muscle cell length. This means that,
when the muscle is stretched or shortened, a mechanism must be proposed
that causes this detachment. In both cases, it is assumed that the unbinding
of myosin thick filaments from actin thin filaments is driven by changes
in their overlap: similar to a skeletal and cardiac muscle sarcomere, it is
thought that displacing the myosin filament relative to the actin filament
reduces the effective length at which the filaments are in contact. At too low
contact length, myosin filaments detach.
The assumption of detachment as a result of reduced overlap is questionable. The amorphous structure of smooth muscle contractile units, the
overabundance of actin filaments (relative content of 15:1 actin filaments
per myosin thick filament), and the lack of longitudinal register that defines
overlap in skeletal and cardiac muscle sarcomeres all suggest that alternative
actin filaments should be readily available for binding of myosin filament
ends that “come loose” from the actin filament they are mainly attached
to. The results presented in this thesis show that below a certain number
of myosin proteins working together, the myosin mechanochemistry is shut
down. The experiments and simulations described situations without external mechanical loading of this myosin motor group. Experiments with
160
externally loaded myosin groups indicate that group kinetics stall at a
force plateau, and that high external forces lead to abrupt detachment of
whole myosin groups [95, 139]. Stalling under load suggests that the critical
myosin group size that is necessary for active group kinetics increases under
load. This would mean that, in loaded cells and tissues, myosin thick filament kinetics would shut down, effectively binding myosin thick filaments to
actin permanently. A further increase in load, as would result from an external stretch, would lead to forcible detachment of myosin thick filaments from
actin. A decrease in load, as would result from externally imposed shortening, would unblock the myosin group kinetics, allowing a synchronized
detachment of all myosins and a detachment of the myosin thick filament
from actin. This constitutes an alternative mechanism – independent of
overlap between actin and myosin filaments – for the prolonged binding of
myosin thick filaments upon contraction at rest length, and the reduction of
force upon increase or decrease of the externally applied force.
161
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