CALCIUM MOVEMENT IN THE SARCOMERE AND ITS CONNECTION TO
MUSCLE CONTRACTION: A PILOT STUDY
Neil Goldsmith
A Thesis
Submitted to the Graduate College of Bowling Green
State University in partial fulfillment of
the requirements for the degree of
MASTER OF SCIENCE
November 2008
Committee:
Dr. Lewis Fulcher, Advisor
Dr. Ronald Scherer
Dr. Donald Cooper
Dr. John Laird
ii
ABSTRACT
Dr. Lewis Fulcher, Advisor
The human body uses calcium as an activator for muscle contraction. A muscle contraction
begins with the release of calcium from the terminal cisternae into the sarcomere. The
interaction of calcium with myoplasmic proteins then causes the muscle to contract. A
biophysical model of the sarcomere will be developed in order to use the model to connect
chemical concentrations with force production by the muscle. Since the sarcomere is the
base contractile unit of muscle, it should therefore be the appropriate starting point for such
a model. The model includes calcium release, diffusion, binding, and uptake. Magnesium
concentrations are also modeled as they compete for calcium binding sites on parvalbumin,
ATP, and troponin. The binding of calcium to troponin is of special importance because
it results in unblocking of the actin sites. The actin will then interact with myosin in a
multi-step process that is well understood but poorly quantified. This interaction leads to
contraction of the sarcomere and thus the production of force.
iii
Dedication
To my grandmother. Someone once told her I could do anything... and she never let me
forget it.
iv
Acknowledgement
I gratefully and thankfully acknowledge the Physics department at Bowling Green State
University. I would like to personally thank Dr. Fulcher whose unending curiosity serves as
an example of what scientist should be. I would also like to thank Dr. Laird. He always
had time to listen and a wise word of advice to offer.
v
TABLE OF CONTENTS
Page
CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
CHAPTER 2. MUSCLE DATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1
Overview of muscle tension data . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1.1
Twitch measurements in the canine vocalis . . . . . . . . . . . . . . .
18
2.1.2
Tetanic contraction in the canine vocalis . . . . . . . . . . . . . . . .
21
2.1.3
Twitch contraction in canine posterior cricoarytenoid muscle . . . . .
22
2.1.4
Twitch contraction in thyroarytenoid muscle . . . . . . . . . . . . . .
24
Overview of biochemical data . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.1
ATP utilization and its temperature dependence . . . . . . . . . . . .
26
2.2.2
Effects of strain on actomyosin kinetics . . . . . . . . . . . . . . . . .
27
CHAPTER 3. SUMMARY OF EARLIER MODELS . . . . . . . . . . . . . . . . . .
29
2.2
3.1
Cannell and Allen model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2
Baylor and Hollingworth’s 1998 model . . . . . . . . . . . . . . . . . . . . .
35
3.3
Baylor and Hollingworth’s 2007 model . . . . . . . . . . . . . . . . . . . . .
38
3.4
Shorten et al’s 2007 model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
CHAPTER 4. THE PRESENT MODEL . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.1
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.2
SERCA pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.3
Calcium release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.4
Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.5
Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
vi
CHAPTER 5. RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . .
52
5.1
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
5.2
Towards a better model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
vii
LIST OF FIGURES
Figure
Page
1.1
Muscle organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Example twitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Dual twitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
Steady frequency stimulus contraction . . . . . . . . . . . . . . . . . . . . .
5
1.5
External view of sarcomeres . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.6
Myosin power stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.7
Cannell and Allen’s model geometry
. . . . . . . . . . . . . . . . . . . . . .
10
1.8
Comparison of BH07 calcium release to Shorten et al.’s calcium release . . .
12
1.9
Force production kinetic scheme . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.10 Shorten et al’s model vs. experimental data . . . . . . . . . . . . . . . . . .
13
1.11 Titze’s predicted muscle tension curve vs. experimental data . . . . . . . . .
14
2.1
Example twitch with metric overlay . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
50% relaxation time versus strain in the canine vocalis . . . . . . . . . . . .
19
2.3
Force versus elongation in canine vocalis . . . . . . . . . . . . . . . . . . . .
20
2.4
Mean contraction time for various species . . . . . . . . . . . . . . . . . . . .
22
2.5
Temperature effect on twitch . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.6
Latency time of a twitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.7
Twitch of a canine thyroarytenoid muscle . . . . . . . . . . . . . . . . . . . .
25
2.8
Temperature dependence of resting force and ATP utilization . . . . . . . . .
27
3.1
Sarcoplasmic calcium depletion . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2
SERCA kinetic scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.1
SERCA models theoretical comparison . . . . . . . . . . . . . . . . . . . . .
47
viii
5.1
Surface graphs of calcium concentration gradient . . . . . . . . . . . . . . . .
55
5.2
Spatially averaged calcium concentration during a twitch . . . . . . . . . . .
57
5.3
Force producing crossbridge concentration for a twitch . . . . . . . . . . . .
58
5.4
Force producing crossbridge concentration for multiple twitches . . . . . . .
59
ix
LIST OF TABLES
Table
Page
3.1
Half-sarcomere dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.2
Crank’s finite difference approximation equations. . . . . . . . . . . . . . . .
34
3.3
Reaction constants and concentrations of chemicals and proteins from Ref. [11]. 35
3.4
Diffusion constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.5
Constants and total concentrations of chemicals and proteins from Ref. [13] .
37
3.6
Concentration of myoplasmic constituents . . . . . . . . . . . . . . . . . . .
40
3.7
Rate constants for reactions shown in Fig. 3.2 and the troponin-calcium binding reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.8
Rate constants for force production . . . . . . . . . . . . . . . . . . . . . . .
42
4.1
Half-sarcomere dimensions in present model . . . . . . . . . . . . . . . . . .
45
4.2
Rate constants for reactions in the present model . . . . . . . . . . . . . . .
50
1
CHAPTER 1
INTRODUCTION
Figure 1.1 shows a top-down organizational view of skeletal muscle. The present study
will focus on the myofibril, an element of the muscle fiber, but some features at levels above
the myofibril are important to note. The tendons and to a lesser degree the epimysium,
perimysium, and blood vessels all add an elastic nature to the muscle. This elastic nature is
an important feature of the muscle because it allows the muscle to contract without moving
the limb to which it is attached. In the following pages data for isometric and isotonic muscle
contractions will be presented. Some care is needed to understand the isometric contraction
since the muscle length does not change. In this situation the muscle shortens while the
elastic elements are lengthened until the force of the contraction is balanced by the pull of
the elastic elements. This allows the contracting parts of the muscle to shorten without
moving the attached load. Since parts of the muscle shorten, it is not truly an isometric
contraction.
A second type of muscle contraction is carried out under isotonic conditions. In an
isotonic contraction the muscle contracts against a finite, constant load. In this situation
the muscle will develop tension equal to the load, the elastic elements will transfer that
force to the load, and the load will move. This would be the type of contraction where one
picks up a weight with a slow uniform motion. A sphincter muscle serves as an example of
an intermediate type of contraction. A sphincter is a ringlike muscle that serves to close
bodily passages through contraction. Sphincter muscles are not anchored to a load so they
do not perform an isotonic contraction, and they are required to reduce their size to close the
passage that passes through them, thus their contraction cannot be considered isometric.
Our understanding of a muscle that undergoes a contraction that is neither isotonic or
isometric is important because the muscles of interest for this thesis are all skeletal muscles of
the larynx and in voiced sounds they are required to perform in a manner that has elements
2
Figure 1.1: Organizational levels of skeletal muscle, taken from Raul654, Deglr6328, and
Rama [1].
of both isometric and isotonic contractions. Some examples are the lateral cricoarytenoid
and the posterior cricoarytenoid. The lateral cricoarytenoid muscle rotates the arytenoid
cartilage medially when it contracts, which relaxes and adducts the vocal folds [2]. The
posterior cricoarytenoid is an antagonistic muscle to the lateral cricoarytenoid. The posterior
cricoarytenoid rotates the vocal process of the arytenoid cartilage laterally and abducts the
vocal folds [2]. In both of these cases the muscle creates a tension in a pseudo-isometric
manner, until it has created enough force to move the load to which the muscle is attached,
and then it contracts in a pseudo-isotonic fashion. Finally, when the muscle has moved the
3
attached load to the desired position it returns to a pseudo-isometric contraction. In the
first instance, the isometric contraction is not a true isometric contraction because the load
it is acting against is not infinite, and the muscle is shortening even though the load is not
moving. Once the muscle has generated sufficient force, then it will start to move the load.
As the load is moved it will become greater either by the stretching of an antagonist muscle
or the elongation of an attached elastic element such as a ligament or a tendon. This makes
the load variable, which means that the contraction is not isotonic. Once the force of the
load reaches the force of the muscle, the muscle returns to a state similar to an isometric
contraction. Again, it is not a true isometric contraction because the muscle will constantly
be tensing and relaxing instead of holding a steady length.
The force these laryngeal muscles exert in response to a stimulus has been measured.
The force has even been measured in vivo [3,4], which gives the observer the opportunity to
witness a contraction that is neither truly isometric or isotonic. In the body the stimulus is
an action potential, but the same response can be observed by the application of electricity
to the muscle. This was discovered in 1756 by Leopoldo Marco Antonio Caldani [5,6]. An
alternate source credits Luigi Galvani with the discovery in 1780 when he noticed that the
limb of a dead frog would move in response to electricity [7]. Galvani’s find led to the art
of corpse reanimation in Europe, and was quite the show for several years afterwards [7]. A
single action potential or electrical pulse will elicit a physical response that looks similar to
the force versus time curve in Fig. 1.2. The time from the onset of mechanical contraction to
the maximum developed force is the contraction time, which is different for different muscles.
Muscles with a very short contraction time are called fast-twitch muscles, and those with
considerably lomger contraction times are called slow-twitch muscles.
If a second action potential reaches the muscle before it has adequate time to relax, the
muscle will exert a greater force than it had previously exerted. In Fig. 1.3 a small piece
of muscle tissue was stimulated with two pulses 5 ms apart and then with two pulses 30
4
Figure 1.2: Representative graph of muscle in response to a single stimulus.
Figure 1.3: Summation of two twitches: (dotted trace) two pulses separated by 5 ms, (solid
trace) two pulses separated by 30 ms, taken from Titze [8].
5
ms apart. The trace of the 5 ms delay shows that greater force is produced than a simple
summation of the two individual twitches, suggesting the importance of nonlinear effects.
Peaks associated with the individual twitches can be seen in the 30 ms delay case. The height
of the first peak is approximately 0.18 N, while the summation of two twitches separated by
the 5 ms stimulus delay is greater than 0.5 N, more than double the force of a single twitch.
Figure 1.4: Muscular response to stimulation by a periodic sequence of pulses at frequencies
between 50 - 90 Hz, taken from Titze [8].
If a muscle is stimulated at a frequency that is high enough, it will eventually achieve
a limit cycle between the relaxation process and the contraction process. At this point the
relaxation of muscle cells will be balanced by new contractions in response to the repeated
stimulus. This can be seen in Fig. 1.4. It is clearest on the 50 Hz trace at any time beyond
0.15 s. This allows the muscle force to oscillate with a small amplitude about a steady force.
6
Also of note in Fig. 1.4 is a saturation effect with increasing frequency, that is, the 10 Hz
increases in stimulation frequency are much less effective in increasing the force at higher
frequencies than at lower frequencies.
A muscle fiber is made up of many myofibrils, and each myofibril is made up of many
sarcomeres. The sarcomere is the smallest complete unit of muscle contraction and relaxation. Within the sarcomere are proteins called actin and myosin. The interaction of these
proteins is ultimately reponsible for the force produced by the muscle [9]. In Fig. 1.1 one
notices that the actin and myosin are parts of long protein chains. The actin chain is called
the thin filament, and the myosin chain is called the thick filament. The inhibition of the
interaction between the thick and the thin filament is the process that relaxes the muscle.
This inhibition is accomplished by the formation of the troponin-tropomyosin complex [9], a
combination of proteins that acts as a barrier between actin and myosin and will not allow
them to interact [9].
The tropomyosin inhibition is removed when troponin binds to calcium [9]. At resting
calcium levels within the sarcomere, most of the troponin is not bound to calcium and
this leaves the tropomyosin in a position to inhibit the thin filament from binding to the
thick filament. The calcium required to overcome this inhibition and thus cause a muscle
contraction is stored within the sarcoplasmic reticulum and the terminal cisterna [9]. These
are the membranes that surround the sarcomeres in Fig. 1.5. While they are considered
different from each other, they are continuous and unless otherwise stated will be referred to
as the sarcoplasmic reticulum. Two terminal cisterna are separated by a transverse tubule.
The terminal cisternae and the transverse tubule form a structure called the triad of the
reticulum or the triadic junction. When an action potential travels down the transverse
tubule it signals the terminal cisternae to release calcium into the extramyofibrillar space
[9]. The extramyofibrillar space is a small gap between the outer wall of the sarcomere and
the region of the sarcomere containing troponin, and filaments of actin, and myosin, which is
7
Figure 1.5: Sarcomeres, myofibrils, and their surrounding structures, taken from Guyton
and Hall [9].
called the myofibrillar space. Unless the distinction is necessary, the extramyofibrillar space
and the myofibrillar space will be referred to as the myoplasmic space. The extramyofibrillar
space promotes the diffusion of calcium by allowing it to diffuse laterally without the binding
of troponin to slow its progress. The calcium will diffuse throughout the sarcomere and bind
to troponin. Troponin then releases the barrier to interaction of the thin filament with the
thick filament and the actin and myosin react with each other [9].
When a myosin head (Fig. 1.1) attaches to an actin molecule they form a crossbridge [9].
In Fig. 1.6 one sees the myosin head of the crossbridge rolling forward towards its hinge. This
is the power stroke [9]. This motion drags the actin towards the center of the sarcomere
and contracts the sarcomere. The myosin head can then bind to adenosine triphosphate
8
Figure 1.6: The conformational change in the myosin head that creates force in a muscle by
a rotation, taken from Guyton and Hall [9].
(ATP) and release itself from the actin filament. The myosin head will then hydrolyze the
ATP into adenosine diphosphate (ADP) and a free phosphate group . The myosin uses the
energy released in hydrolyzing ATP to return it to its pre-power stroke position so that it
is ready to attach to the actin again. If the actin is available then it will repeat the process
until the supply of ATP runs out, or the actin is sufficiently inhibited. This process of
crossbridge cycling is known as the walk-along mechanism of muscle contraction [9]. While
the basic steps are generally agreed upon, a specific kinetic scheme to accurately describe all
the available data has proven elusive.
As the crossbridge cycling occurs and the muscle contracts, calcium buffers in the sarcomere are binding calcium and the Sarcolemmal Endoplasmic Reticulum Calcium ATPase
pump, called the SERCA pump, removes calcium from the sarcomere to the sarcoplasmic
reticulum [9]. Its operation is believed to remove two calcium ions with the hydrolysis of
one ATP molecule. The SERCA pump and the buffers act quickly to lower the calcium concentration back to near equilibrium levels. At equilibrium the action of the SERCA pump
is balanced by a leak of calcium into the extramyofibrillar space. The source of this leak is
uncertain but it is believed to be either calcium diffusion through the membrane separating
the sarcoplasmic reticulum and the extramyofibrillar space or a backflow of calcium through
9
the SERCA pump [10]. In either case, the SERCA pump maintains a gradient across this
membrane of approximately five orders of magnitude.
Once the calcium is returned to the sarcoplasmic reticulum it diffuses to achieve equilibrium, and a buffer in the terminal cisterna binds to it, in order to store it until the next
action potential initiates the process again. The buffer in the terminal cisterna is called
calsequestrin [11]. The buffers in the sarcomere are phosphocreatine, ATP, parvalbumin,
and two non-regulatory binding sites on troponin that are similar to parvalbumin. Some
models use the transport of calcium into the sarcoplasmic reticulum by the SERCA pump as
a buffer, since the calcium spends some time bound to the pump and during that time is in
neither the myoplasm nor the sarcoplasm [12]. Not every model uses all of these buffers, and
they certainly are not the only proteins that bind calcium within the sarcomere. The buffers
listed above are those that play the most important roles in calcium regulation within the
sarcomere.
In a 1984 paper, Cannell and Allen [11] created a computer model of the process previously described. It was based on a half sarcomere of frog skeletal muscle. It assumed the full
sarcomere had mirror symmetry around the M-line, the middle of the sarcomere. They also
used a cylinder as a reasonable approximation to the shape of their half-sarcomere. This
geometry is often used and is frequently a part of artists’ depictions of the sarcomere, as in
Figs. 1.1 and 1.5. Cannell and Allen further assume axial symmetry, or invariance under rotations about the central axis, which allows them to create a model that is a two-dimensional
plane, with no reference to the angle about the central axis. This plane is divided into cells.
It has cells to represent the extramyofibrillar space, the myofibrillar space, the terminal cisternae, and the sarcoplasmic reticulum. All cells contain some concentration of calcium in
the ionized state, Ca2+ . Each cell of the myoplasmic space contains parvalbumin, and troponin is present in only the myofibrillar space. Calsequestrin is present within the terminal
cisternae and nowhere else. Crank’s finite difference approximation was used to handle the
10
diffusion of calcium [11]. This diffusion occured within the entire myoplasmic space and also
within the sarcoplasmic reticulum.
Figure 1.7: Geometry and divisions in the Cannell and Allen model [11].
The permeability of the membrane between the extramyofibrillar space from the terminal
cisternae, denoted by P, in response to a stimulus was approximated by the function
P = Pmax [1 − e−t/τon ]L [e−t/τoff ],
(1.1)
where P has units of distance per time. In the Cannell and Allen version, L is equal to 1 and
Pmax was 0.062 mm/s−1 . Equation 1.1 is the general form that has been used by others. The
uptake of the calcium by the SERCA pump, denoted by U in Fig. 1.7, was approximated
by the first-order Hill equation, and it removed the calcium from the extramyofibrillar space
into the adjacent sarcoplasmic reticulum cell.
Baylor and Hollingworth [13] recorded some improvements on Cannell and Allen’s
model. In their first paper, published in 1998, they showed the importance of ATP as a
11
buffer and transporter of calcium within the sarcomere. The diffusion of parvalbumin was
also considered, and the calcium release equation was altered to match the shorter duration
of the pulse in more recent data [13]. In a second paper, Baylor and Hollingworth [12] altered
the model to more accurately describe the contraction of fast-twitch mouse fibers. A time
delay, T, was added to the release function. The time variable t was replaced by t − T , and
at any time t < T the release was equal to zero. This parameter was added to account for
a delay between the action potential reaching the muscle and the measured release of the
calcium. Another difference was the location of the calcium release. In all previous works,
the triad of the reticulum was located at the Z-line (see Figs. 1.5 and 1.7). In mammals
this structure is offset from the Z-line by 0.5µm, and the new model incorporates this more
realistic release point. The final important difference was the addition of a buffering SERCA
pump. In all previous models the SERCA pump acted instantaneously to move a given
number of moles of calcium from the extramyofibrillar space to the sarcoplasmic reticulum,
and calcium never spent any time bound to the pump proteins. It is more realistic to believe
that the SERCA pump would bind the calcium and for a certain time, and the calcium would
not be present in any of the cells shown in Fig. 1.7 during that time. A kinetic scheme was
offered to account for this buffering ability.
The models of calcium movement and regulation described above are important in any
attempt to model a muscle, but thus far we have not considered the production of force.
This task was undertaken by Shorten et al. [14] to model fatigue in muscle contraction. In
this model additional parameters are used to describe the calcium release, but they produce
calcium transients similar to those produced by Baylor and Hollingworth [12], as shown in
Fig. 1.8. While differences are noted between the two functions used to release the calcium in
the two models, it is unlikely that these differences are important for the kind of approximate
treatment we develop.
Figure 1.9 is the kinetic scheme Shorten et al. [14] used to describe the force production
12
Figure 1.8: Calcium transient in the sarcomere, left graph is from the measurements of
Shorten et al. [14], right graph is from the calculation of Baylor and Hollingworth [12]. The
frequency of the stimulus is 67 Hz. The concentrations are micromolar.
Figure 1.9: A kinetic scheme to model force production in the sarcomere, taken from Shorten
et al. [14].
13
Figure 1.10: Production of tension in a muscle, taken from Shorten et al. [14].
within a sarcomere. In Fig. 1.10 one can see reasonable agreement between the model’s
predicted tension and the experimental tension for one type of muscle. There are differences
between the graphs, but they are close enough to support the use of the scheme of Fig. 1.9 as
a foundation for the production of tension. An addition to the scheme which may achieve a
better fit is the Fenn effect [15]. The Fenn effect describes the ability of a muscle to adjust its
energy cost to the prevailing mechanical constraints. The energy required is supplied by the
hydrolysis of ATP. The muscle seems to adjust the turnover rate of the crossbridges to match
the function it is called to perform. This would have an effect on the transition in Fig. 1.9
from post power stroke to the detached state. If a muscle is shortening and exerting more
force than its load, the crossbridges turn over at a greater rate because fewer crossbridges
are required to produce force but more detached crossbridges are needed to perform the
walk-along mechanism. If the muscle is exerting less force than is needed to move a load, the
crossbridges will stay attached and as more crossbridges form, more force is produced. Once
the muscle has formed enough crossbridges to overcome the opposing force, it will start to
cycle the crossbridges and begin the walk-along mechanism. This effect is not accounted for
14
in the model of Shorten et al. [14].
Figure 1.11: Comparison of measured muscle tension with Titze’s parameterization, taken
from Titze [8].
Our interest in the physiology of generating force by muscles began with the study of
Titze’s paper [8] concerned with neurophysiological sources of jitter in human speech. His
approach was based on mathematical representations of individual muscle twitches, and it
seemed reasonable that a better understanding of those twitches would furnish a more solid
foundation for possible causes of jitter and shimmer. However, this undertaking proved
too ambitious for us, and we decided to focus on constructing a reliable model of calcium
movement in the sarcomere and its role in generating force. We believe that our model has
great potential for applications, including the original goal of an explanation for jitter and
shimmer.
In Fig. 1.11 one can see Titze’s [8] mathematical representation of muscle twitch. The
curve appears to rise too quickly and decrease too slowly. Titze himself discussed this
in his paper and offered ideas on how to improve the fit. Using his model he concluded
that neurophysiological sources could be a major factor in the production of jitter and
15
shimmer and provided estimates for the percentage of jitter that could be attributed to
neurophysiological sources [8]. Improvement on the basic force curve for a muscle would be
able to offer a better estimate to the magnitude of neurologic sources role in jitter. One could
undertake this by simply altering the mathematical representation with added parameters,
as was suggested by Titze [8]. While this certainly could be successful, it would be limited in
its applications. Limitations such as this are the impetus for us to develop a more in-depth
and fundamental model based on the physiological factors that produce force.
In order to test the validity of our model, we will focus on the calcium transient and will
attempt to make our calcium transient fit those measured by Baylor and Hollingworth [12].
Their experiments were done using a fluorescent dye and appear to be the most recent data
on the subject. Attempts will be made to create an accurate model for a buffering SERCA
pump. The SERCA pump is a saturable pump and is believed to be second order. This
allows us to attempt a fit to the second-order Hill equation. Finally, the tension produced
in response to a stimulus will be matched to the experimentally measured time courses of
various other muscles. These muscles will all be laryngeal muscles but we expect our model
to have a generality that makes it appropriate for a wide variety of muscles.
16
CHAPTER 2
MUSCLE DATA
2.1
Overview of muscle tension data
Theory and experimental data drive each other. If data are found that have no previous
explanation, the theorist goes to work and comes up with a new idea to explain the scientific
relationships underlying the data. The new theory is then able to make predictions that
can be tested experimentally. Such progress in the interplay between experiment and theory
drives our understanding of the physical world. A concrete example of this method is Einstein’s general theory of relativity. Einstein intuitively knew that a light ray was bent in a
gravitational field, and yet the universal law of gravitation required that gravity act between
two massive objects. Since light has no rest mass, the theory of gravity and the data on light
were not consistent in some sense. To reconcile these two disparate ideas, Einstein created
his general theory of relativity. With this theory he was able to make predictions about the
angular separation of two stars that could be measured during a solar eclipse. During the
next eclipse, Arthur Eddington made careful measurements of the locations of these stars
and measurements supported Einstein’s theory. The experimental data that drive our theoretical work are muscle data. Most of these data are in the form of tension versus time
graphs. Many of the graphs have the same geometric shape, which is an indication that
the same mechanism produces the force, the actomyosin cycle. The graphs differ in several
details, and careful experimental work offers good metrics to quantify these differences.
The five measures that will be presented here are latency period, peak tension, time to
peak tension, time between peak tension and half maximum tension, and tetanic contraction
period. The latency period is the time between the onset of electrical activity in the muscle
and the beginning of muscle contraction. Peak tension is simply the maximum force produced
by a single twitch, and time to peak tension is the time it takes from the onset of mechanical
17
contraction to reach that maximum force. Time to half-maximum tension is the time it
takes for the muscle to relax to half of its peak tension value. The tetanic contraction time
is the amount of time it takes under a periodic stimulus to reach tetanus. In Fig. 1.4 one
can see the muscle reaching tetanus when the tension levels off. The muscle stimulated at
a frequency of 50 Hz reaches tetanus a short time after 150 ms and it took 9 stimulii, and
thus 9 twitches, to get to that level.
Most skeletal muscles of the body are composed of a mixture of fast-twitch and slowtwitch muscle fibers, with still other fibers graduated between these two extremes [9]. The
fibers derive their names from the contraction time for each type of fiber. Fast-twitch
fibers are larger than slow-twitch fibers and are used for rapid development of a very strong
contraction [9]. Slow-twitch muscle fibers have a smaller diameter and since muscle strength
is primarily determined by muscle cross-sectional area, more of this type of fibers are required
to match the strength of contraction of a muscle with predominantly fast-twitch fibers [9].
This decreased strength of each contraction is made up for by a longer relaxation time [9].
The relative amounts of the different muscle fibers in a given muscle work together to give
a muscle its characteristic contraction. Furthermore, one can train a muscle to react more
like either a fast- or slow-twitch muscle. A muscle will adjust as the demands placed on
it change. Experiments have shown that the muscle’s contractile proteins in some smaller,
more active muscles can be totally replaced in as little as two weeks [9]. These changes at
the protein level may help explain how muscle types differ. One type of muscle may have a
greater concentration of a certain protein or a different isoform of a protein that performs
the same task but does it faster or slower than another isoform. An example of this comes
from a paper written by Andruchov et al. [16], where they studied the effects of different
isoforms of myosin on cross-bridge kinetics.
There are two other differences between muscle data that are not intrinsic to the muscle.
The temperature at which the data are collected has an effect on the resulting tension data.
18
Cooper et al. [4] took data that show a lower force produced and shorter contraction time
at higher temperatures for the canine posterior cricoarytenoid muscle. The temperature also
has an effect on the rate of ATP utilization, which plays a direct role in the dissociation of
the actomyosin complex into actin and myosin [17]. Another difference is the strain placed
on the muscle while contracting. The actomyosin kinetics are also altered as the strain on the
muscle is changed [18] and [19]. ATP utilization measurements [17], which were made using
fluorescent dyes, establish an additional feature of muscle contraction, that is, a shortening
of the muscle leads to a greater rate of ATP utilization. This is related to the Fenn Effect
[15] that was discussed in Chapter 1.
2.1.1
Twitch measurements in the canine vocalis
Alipour-Haghighi, Titze, and Durham [20] performed in vitro measurements on the
twitch characteristics of the canine vocalis muscle. This muscle is part of the thyroarytenoid.
The characteristics they used to describe the twitch were the peak tension, the contraction
time, and the 50% relaxation time, denoted by F, C, and R in Fig. 2.1.
Figure 2.1: Schematic representation of twitch response of a canine vocalis muscle, taken
from Alipour-Haghighi, Titze, and Durham [20].
A small piece of the muscle was held between a sample holder and an ergometer arm,
19
which allowed the adjustment of the strain on the muscle prior to stimulating the sample with
parallel plate electrodes. After each pulse, the muscle was allowed to relax for 5 minutes. A
time of this length is required since the muscle tension will relax on the order of 100 ms, but
it may take several seconds to minutes for the calcium distribution to return to equilibrium.
The contraction times varied between 22 and 32 ms, with an average of 25.7 ms. The
50% relaxation time varied between 17 to 37 ms, with an average of 25.2 ms. Interestingly,
they found no correlation between the strain placed on the muscle prior to stimulation and
the resulting contraction time for the twitch, which led them to conclude that contraction
time was independent of strain. An increase in contraction time was found to be associated
with tissue from larger animals. The 50% relaxation time showed a slight increasing trend
with strain as can be seen in Fig. 2.2. The active force was determined by how much of the
Figure 2.2: The 50% relaxation times versus strain for different canine vocalis muscles. Each
letter represents a different canine, taken from Alipour-Haghighi, Titze, and Durham [20].
20
total force after stimulation was due to passive force. The passive force was the tension in
the muscle before any stimulation. This would be due to elastic elements within the muscle
and had a direct correlation with the elongation of the muscle, as one might expect from an
elastic tissue. Upon stimulation, the muscle would develop greater tension, that is, the total
tension. In order to determine active force, the passive force was subtracted from the total
tension.
The total force minus the passive force gives the active force, or the force produced by
the muscle in response to stimulation. Total tension, passive force, and active force curves
are represented in Fig. 2.3. Active force decreases as the elongation of the muscle increases.
Figure 2.3: Force versus elongation in canine vocalis showing the relationship of the active
and passive forces, from Alipour-Haghighi, Titze, and Durham [20].
This decrease supports the sliding filament model of muscle contraction. As the elongation
increases, the overlap between the actin and the myosin filaments decreases, and thus the
21
potential number of force-producing cross-bridges is reduced.
2.1.2
Tetanic contraction in the canine vocalis
Alipour-Haghighi, Titze, and Perlman [21] studied the characteristics of the canine
vocalis muscle in vitro under stimulation frequencies high enough so that the muscle was
unable to relax before the next stimulation. Since a muscle relaxation time is about 100 ms,
any stimulation frequency over 10 Hz should show an augmentation of the force. At these
frequencies one will see a rise in the peak force up to a certain point where the force rounds
off and creates a sigma-shaped curve. Such a curve can clearly be seen at 50 Hz in Fig. 1.4.
A sample of canine vocalis muscle was placed into an egometer that allowed AlipourHaghighi, Titze, and Perlman to control the strain on the muscle and observe how that strain
affected the tetanic contraction time. The samples were then stimulated to elicit a tetanic
response and the time for this response to develop was recorded. No mention was made
of the actual rate of stimulation. This data was probably omitted because the result could
be evaluated without it. They found that the time it takes for the muscle to reach tetanus
increased with increasing strain. At resting length the muscle would have zero strain and
the tetanic contraction times ranged from 100 to 200 ms. The strain of the muscle in vivo at
any given time is unknown. It would depend on many factors, including the state of tension
of any antagonist muscles. This fact taken together with the wide range of measured tetanic
contraction times makes these data lack predictive power. What the data do accomplish is
to point to a correlation between strain and tetanic contraction time. As strain increases,
so does the tetanic contraction time. This serves as more of a qualitative measure than a
quantitative measure.
These measurements were done under isometric conditions, essentially an infinite load.
Under isotonic conditions, a similar response is observed up to the point where the force is
equal to the load. At that time the load will start to move, the muscle contracts, and the
22
tension stays constant. There is no explanation for why the muscle does not accelerate, but it
is an important feature of muscle contraction that should be addressed by model calculations.
A possible explanation is that the Fenn effect [15] alters the number of attached crossbridges
and thus the force produced. This could possibly be even more biophysically relevant than
the tetanic contraction time and therefore worth noting.
2.1.3
Twitch contraction in canine posterior cricoarytenoid muscle
Cooper et al. [4] measured the dynamic properties of the canine posterior cricoaryteniod
muscle in vivo. This differs significantly from the previous studies and allows one to see the
dynamics of the muscle as it functions in the body. An interesting point noted by AlipourHaghighi, Titze, and Perlman’s paper on tetanic contraction was an increase in contraction
time as the mass of the animal increased [21]. Cooper et al. [4] plotted the contraction time
of the cricothryoid muscle as a function of the breath duration of a given species as shown
in Fig. 2.4.
Figure 2.4: Mean contraction time for various species plotted versus the breath duration,
taken from Cooper et al. [4].
23
Figure 2.5: The effect of temperature on contraction times and twitch strength, taken from
Cooper et al. [4]
The duration of the respiratory cycle is an exponential function of the animal mass. The
plot shows that as the duration of the respiratory cycle increases, so does the contraction
time, which connects the contraction time of the cricothyroid muscle to the mass of the
animal.
Measurements of the twitch were made at different temperatures, and they showed that
the muscle contraction time and peak tension varied with temperature. As the temperature
increases, the contraction time and the peak tension decrease. The data are presented in
Fig. 2.5. These measurements were done on the same dog to eliminate any differences in
contraction time due to the size of the dog. The mean value for the contraction time was
determined to be 33.3 ms. Cooper et al. [4] also measured latency time, which was the delay
between the onset of electrical activity in the muscle and the onset of a contraction. The
average latency time was 3.3 ms. This latency period is partly responsible for the horizontal
parts of the curves near t = 0 in Fig. 2.6. Since the horizontal section of the graph includes
a conduction delay along the nerve as well as latency time, the value of the time derived
24
from this horizontal section will be larger than the latency period due to chemical processes
within the sarcomere.
Figure 2.6: The recurrent laryngeal nerve was stimulated at 0 ms and no contraction occurs
until a short time later, marked by a vertical dash, taken from Cooper et al. [4].
2.1.4
Twitch contraction in thyroarytenoid muscle
Measurements were done by two separate groups using different techniques on the canine
thyroarytenoid muscle. Cooper, Pinczower, and Rice [3] measured the muscle in vivo and
got a contraction time of around 12 ms 1 , as we estimated from Fig. 2.7. Alipour-Haghighi,
Titze, and Durham [20] measured the contraction time of this muscle in a different dog,
using in-vitro techniques, and found a contraction time greater than 20 ms.
It is possible that the differences come from the in vitro versus in vivo techniques. It is
also possible the canines varied in mass and this can affect the contraction time, as discussed
1
D. S. Cooper, personal communication, 18 Aug 2008
25
Figure 2.7: Twitch of a canine thyroarytenoid muscle, taken from Cooper, Pinczower, and
Rice [3].
in connection with Fig. 2.4. This may not be able to explain the difference, because the
mass connection to the contraction time was established for a different muscle and has not
been established for the thyroarytenoid. Although the origin of this observed difference is
unclear, our model has the potential of giving insight into such differences.
2.2
Overview of biochemical data
It is necessary for a muscle to have ATP in order to relax, but not necessary to have it
for the production of force. If a muscle contracts in the absence of ATP it will reach a given
tension and stay at that tension. This happens when a human dies and the muscle uses up
its last ATP sources, with no way to replenish them. This state is known as rigor mortis. In
a live healthy muscle the ATP store is constantly being replenished. Of particular interest
is the binding of ATP and its subsequent hydrolysis by myosin. In hydrolyzing the ATP, a
myosin head is able to store some of the energy from the breaking of the phosphate bond
and then release it during the power stroke. This creates a paradox in that ATP is necessary
for relaxation but supplies the energy for contraction. That is why the ATP hydrolysis is
coupled with the removal of calcium from the muscle and the subsequent inhibition of actin
26
by troponin to produce the relaxation of the muscle cell.
The ATP hydrolysis reaction is interesting and requires special comment due to its
dependence on the mechanical state of the system [15]. When ATP is hydrolyzed, it is not
100% efficient in transferring energy. Thus, a temperature change in the muscle is a window
into the amount of ATP being hydrolyzed at a given time. It was shown in 1923 by Wallace
O. Fenn that a muscle will increase its energy liberation in proportion to the amount of
work done [15]. This requires the muscle to contract, lest no work will be done. Work is
defined as a force applied over a distance and during an isometric contraction the load on the
muscle does not move so no work can be done. This does not mean that during an isometric
contraction, ATP is not hydrolyzed. In fact, ATP hydrolysis in an isometric contraction
acted as a baseline for the measurements done by Fenn [15]. It was later shown that the
temperature and strain played a role in establishing this baseline [15]. Thus, it would seem
that an explanation of the muscular data and its dependence on temperature and strain are
intimately linked to the cycling of the crossbridges.
2.2.1
ATP utilization and its temperature dependence
In Fig. 2.8 A, one can see that as temperature increases the resting force exerted by the
muscle increases in an apparently exponential manner. The resting force is the same force
that was referred to as passive force in Alipour-Haghighi, Titze, and Durham [20]. This same
relationship is also noted in Fig. 2.8 B where the resting rate of ATP utilization increases
with increasing temperature. Although the vertical axes of these two graphs differ in scale
and units, they appear to be behaving similarly. This may mean that as ATP utilization goes
up the force produced goes up. This appears to be at odds with the idea that ATP hydrolysis
is linked to the relaxation process in muscles. No explanation of these characteristics was
offered as the goal of the Hilber, Sun, and Irving’s [17] research was experimental. A well
developed model will potentially be able to explain these results.
27
Figure 2.8: Temperature dependence of resting force and ATP utilization, taken from Hilber,
Sun, and Irving [17].
2.2.2
Effects of strain on actomyosin kinetics
Siththanandan, Donnelly, and Ferenczi [18] measured the effects of strain on actomyosin
kinetics. ATP was removed from a muscle sample and saturating levels of calcium were
introduced to produce rigor within the muscle. This created an environment within the
muscle in which the actomyosin complex could form and go through the power stroke, but
no ATP was present to detach the myosin from the actin. A caged ATP compound was
allowed to diffuse into the muscle so that upon excitation by laser, ATP would be liberated
and cause a rapid relaxation that would not be diffusion-limited. In this way the kinetics of
28
the reaction could more clearly be measured. Two such runs were done at different tensions.
At greater initial tension, the rate of decrease in tension due to ATP detachment of cross
bridges was greater, presumably because each crossbridge supported more force. It was also
noted that crossbridge detachment rate increased with increasing tension. Siththanandan,
Donnelly, and Ferenczi [18] used their measurements to create a kinetic scheme for actomyosin
cycling. These kinetics are not used in our model, but the relationship between strain and
crossbridge kinetics is important in developing our model.
29
CHAPTER 3
SUMMARY OF EARLIER MODELS
Previous models of the sarcomere have focused primarily either on calcium diffusion and
buffering or on tension production. There is surprisingly little published research linking the
two in skeletal muscle. Considering the importance of calcium in the production of tension,
it would seem important to develop a model that allows progress on both problems simultaneously. Several factors operate to make the development of such a model less likely. Much
of the muscle research being done focuses on cardiac muscles. The tension development
research generally looks into the actomyosin system and uses phenomenological approximations for any calcium considerations. The calcium regulation models avoid steps that do not
directly impact the calcium. For instance, after troponin binds calcium no more attention is
paid to the troponin except to track when it releases the calcium. Such simplifications are
necessary as the computers used in the modeling have a finite ability to process information,
and the more information one includes the greater the calculation time becomes. The influence of such limitations is dwindling as computers become more powerful, and models have
thus become more and more complex.
Cannell and Allen [11] created a model of calcium diffusion and regulation in frog skeletal
muscle, hereafter referred to as CA84 model. One of the motivations for the model was to
estimate the error in the measurement of myoplasmic calcium when an aequorin dye was
used. Aequorin is a fluorescent dye that has been widely used as an intracellular calcium
indicator but the relation between the light response and calcium concentration is nonlinear.
After the release of calcium into the sarcomere, a large calcium gradient in the myoplasm
develops. Thus, when the light signal from the whole preparation is collected, the resulting
light signal is not proportional to the mean concentration of calcium.
Baylor and Hollingworth [13] attempted to improve on this model by the use of data
from a fluorescent calcium indicator called furaptra. This model will be referred to as BH98.
30
They found the calcium release time to be much briefer than Cannell and Allen thought.
Both models have a peak calcium release at 1.7 ms, but the BH98 release controlled τon and
τoff (see Eqn. 1.1) in such a manner as to give a shorter pulse while retaining the same peak
release time. A larger amount of calcium was also released in BH98 because of more recent
data suggesting a larger calcium transient. This larger transient was somewhat offset by
the addition of ATP as a buffer within the sarcomere. Diffusion of ATP and parvalbumin
was considered, which helped to decrease the magnitude of the calcium gradient that would
develop in the sarcomere in response to a stimulus. Baylor and Hollingworth also introduced
a competition between calcium and magnesium for binding sites on parvalbumin, ATP, and
the non-regulatory binding sites of troponin.
Baylor and Hollingworth [12] updated BH98 to adjust for differences between amphibian
and mammalian skeletal muscle. This updated model will be called BH07. The main differences between BH98 and BH07 are the location of the triadic junction and the waveform of
the calcium release. In mammals the triadic junction is located approximately 0.5 µm from
the end of the sarcomere, but it is located at the end of the sarcomere in the amphibian
skeletal muscle. The release equation was altered in order to match the signal from furaptra, similar to the adjustments made in their previous paper. The most dramatic change in
BH07 was the addition of a buffering SERCA pump. This was a logical addition because
the SERCA pump has to bind calcium in order to transport it across the sarcoplasmic reticulum membrane. A SERCA pump binds two calcium ions before pumping them into the
sarcoplasmic reticulum. This requires that time must pass between the binding of the first
calcium and the binding of the second calcium and also when the calcium is transported
across the membrane. The time necessary for the binding and transporting is time when
the calcium is not present in either the myoplasm or the sarcoplasmic reticulum, it is bound
to the SERCA pump in the membrane between them. A buffering SERCA pump readily
accounts for the time when these calcium ions are not present.
31
The models discussed above all stop short of creating a physical force from the crossbridge formation within the sarcomere. A model developed by Shorten et al. [14], hereafter
referred to as SODS07, takes this step. Using a more complex model of calcium release, they
were able to achieve calcium transients similar to those of Baylor and Hollingworth, depicted
in Fig. 1.8. To produce force they used the kinetic scheme detailed in Fig. 1.9, where the
post-power stroke concentration of attached crossbridges is proportional to force. This is one
of the few papers that address the entire process of force production within the sarcomere,
beginning with the calcium release. SODS07 force production will serve as a basis for force
production in our model.
Before summarizing the previous models it is worthwhile to discuss some conventions
used in describing them. The first symbol used is a closed bracket ([ ]). This will appear
with a chemical symbol inside and should be read as the concentration of that chemical.
For instance [Ca2+ ] is the concentration of ionized calcium. All concentrations will be given
in units of moles per liter, or molar (M). This should not be confused with lower case m
which is reserved for units of meters. When the symbol for a chemical shows up without the
brackets, such as Ca2+ , its units will be moles.
Another convention that is used often in biochemical models is the Hill equation. This
equation takes the basic form,
d[S]
[S]nH
.
= Vmax nH
nH
dt
[S] + Km
(3.1)
The Hill equation is useful in describing the action of saturable pumps, such as the SERCA
pump. The fractional part on the right-hand side of the Eqn. 3.1 approximates the saturation
of the substrates at the binding site, and Vmax is a combination of the forward rate for that
particular pump and maximum available concentration of binding sites on the pump. Each
concentration is raised to the nH power, where nH is the Hill coefficient and is a measure
of cooperativity. For instance, if a pump moves one atom at a time, then nH would equal
one. If the pump moves two atoms at a time, then nH would equal two. Sometimes there
32
are other factors that can alter this value beyond how many atoms it pumps at a time, but
for now this approximation is adequate. The final constant is Km and is called the Michaelis
constant. It is the concentration where the pump works at half of its maximum value.
3.1
Cannell and Allen model
Cannell and Allen’s 1984 model was designed to explore the intracellular processes that
produce myoplasmic calcium concentrations during a single twitch and subsequently a rapid
series of stimuli creating a tetanus [11]. They also wanted to measure the error that arises
when using aequorin as a calcium indicator. These goals were achieved in CA84, but since it
was based on an amphibian muscle, and we are interested in mammalian muscle, some additional considerations will be required. The methods described in CA84 have been adapted
to each model that will be presented, attesting to the general nature of their framework.
Figure 1.7 shows the geometry used in CA84 to approximate the half-sarcomere. The
half-sarcomere was divided into 12 radial and 10 longitudinal elements for a total of 120 cells.
The sarcoplasmic reticulum made up 10 of these cells along the top row and the terminal
cistern was allocated a single cell. The extramyofibrillar space made up the 10 cells in the
second row and the remaining 100 cells were used to define the myofibrillar space. The
geometric parameters of the half-sarcomere are listed in Table 3.1.
Table 3.1: Half-sarcomere dimensions.
Half-sarcomere dimensions
Radius = 0.5µm
Length
= 1.1µm
Volume
= 0.86µm3
Percent volume of compartments
Myofibrillar space
=
85.0%
Extramyofibrillar space
=
6.0%
Terminal cisternae
=
3.5%
Longitudinal SR =
5.5%
The release of calcium was calculated by multiplying Eqn. 1.1 by the area of the terminal
33
cistern and the calcium concentration difference between the terminal cisternae and the
adjacent extramyofibrillar space. This yields the equation,
d[Ca2+ ]
= P A([Ca2+ ]tc − [Ca2+ ]x ),
dt
(3.2)
with L = 1 in Eqn. 1.1 to determine P. Values chosen for A, Pmax , τon , and τoff were 0.54
x 109
mm2
,
liter of muscle
0.062
mm
,
s
1 ms, and 5 ms, respectively [11]. The released calcium atoms
went from the terminal cisternae cell into the adjacent extramyofibrillar cell. In Fig. 1.7
the terminal cisternae could be the the cell in the upper left-hand corner and the recipient
cell would be directly below it. The calcium was then allowed to diffuse radially through
the myofibrillar space and laterally through the sarcoplasmic reticulum and the myofibrillar
space. Thus the calcium concentration satisfies a diffusion equation of the form,
∂
[Ca2+ ] = D∇2 [Ca2+ ] + source terms,
∂t
where D = 7 x 10−4
mm2
,
s
(3.3)
the diffusion constant, and the source terms represent local
concentration changes in the due to interaction with other proteins in the medium.
Crank [22] developed a numerical method for the solution of the diffusion equation where
he approximated the derivatives in Eqn. 3.3 by finite differences. The spatial derivatives in
the cylindrical geometry involved concentration differences in neighboring cells. For the geometry of Fig. 1.7, these differences take the form of the six equations listed below, since
it is important to treat the boundary cells differently from those in between. The lateral
solid boundary cell (Eqn. 3.4), lateral central cell (Eqn. 3.5), lateral mirror boundary cell
(Eqn. 3.6), radial solid boundary cell (Eqn. 3.7), radial central cell (Eqn. 3.8), and radial
mirror boundary cell (Eqn. 3.9) are the six types of cells that require individual equations.
A mirror boundary cell is one that is adjacent to the M-line for lateral diffusion and adjacent
to the central axis of the cylinder for radial diffusion. A solid boundary cell is an extramyofibrillar cell for radial diffusion, and any cell adjacent to the Z-line for lateral diffusion. Let
[Ca2+ ]v be the calcium conentration in the v th cell. Then the derivatives of Eqn. 3.3 form
the parts of the equations listed in Table 3.2.
34
Table 3.2: Crank’s finite difference approximation equations.
d[Ca2+ ]v
dt
d[Ca2+ ]v
dt
d[Ca2+ ]v
dt
d[Ca2+ ]v
dt
d[Ca2+ ]v
dt
d[Ca2+ ]v
dt
=
=
=
=
=
=
D
([Ca2+ ]v+1 − [Ca2+ ]v )
2
dx
D
([Ca2+ ]v−1 − 2[Ca2+ ]v + [Ca2+ ]v+1 )
dx2
D
([Ca2+ ]v−1 − [Ca2+ ]v )
dx2
4D
([Ca2+ ]v+1 − [Ca2+ ]v )
dr 2
D
((2v + 1)[Ca2+ ]v+1 − 4v[Ca2+ ]v + (2v + 1)[Ca2+ ]v−1 )
2vdr 2
D
((2v − 1)[Ca2+ ]v−1 − (2v + 1)[Ca2+ ]v )
2
2vdr
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
In addition to its release and diffusion, it is also important to consider the binding of
calcium to proteins. The three proteins considered in CA84 are troponin, parvalbumin, and
calsequestrin. Troponin has four binding sites for calcium. Two sites are regulatory and
have a low affinity for calcium. The other two sites are high affinity sites that are similar
to parvalbumin. Parvalbumin has two binding sites that will bind calcium or magnesium.
Calsequestrin only binds calcium and is confined to the terminal cistern. Calcium, and in
some cases magnesium, was assumed to bind to each protein through the simple kinetic
scheme,
k
on
Ca + S ←→
SCa,
koff
(3.10)
where S (short for substrate) denotes one of the proteins. The equation describing the change
in concentration of the protein-substrate complex is
d[SCa]
= kon [Ca][S] − kof f [SCA]
dt
(3.11)
Table 3.3 lists the relevant constants controlling the chemical reactions and initial values for
the concentrations.
The final ingredient in CA84 was the inclusion of a SERCA pump to return the concentration of calcium in the myoplasmic space to equilibrium. This pump is modeled by a
35
Table 3.3: Reaction constants and concentrations of chemicals and proteins from Ref. [11].
Concentration (µM)
On rate (M−1 s−1 )
Off rate (s−1 )
Troponin regulatory (total sites)
70 (140)
1.2 x 108
120
Parvalbumin and
940
Protein
Calcium
2.5 x 108
parvalbumin like sites
1
Magnesium
Calsequestrin
6.6 x 104
6
240
0.2
31000
Free metal concentrations
Resting myoplasmic [Ca2+ ]
0.06
Resting SR [Ca2+ ]
1500
Myoplasmic [Mg2+ ]
3300
first order Hill equation, meaning nH is equal to one, Vmax is 1.0
mM
s
and Km is 1.0 µM.
Given these three constants and the calcium concentration in the extramyofibrillar space,
the SERCA pump will work continuously. Thus, a counterflow of calcium is required to
achieve equilibrium. This is modeled using a leak that takes the form of equation 3.2 with a
constant permeability. At equilibrium, the amount of leak will match the amount of calcium
pumped and the net flux across the membrane will be zero.
3.2
Baylor and Hollingworth’s 1998 model
BH98 added several features to CA84 in order to consider the role of ATP in the regulation of myoplasmic calcium [13]. In BH98 the terminal cisternae and the sarcoplasmic
reticulum were not considered. Equation 1.1 controlled the release of calcium into the extramyofibrillar space at the same point as in CA84 but the calcium was allowed to come from
a source that was not part of the model. Accordingly, the calcium removed by the SERCA
36
pump was only recorded as a loss in the myofibrillar space, but no gain was recorded elsewhere.
Using data gathered from furaptra, Baylor and Hollingworth noted a distinctly briefer
time course for the release of calcium and a much greater peak concentration of myoplasmic
calcium. The constants in Eqn. 1.1, τon , τoff , and L were 1.50 ms, 0.63 ms, and 5. The
value of Pmax was chosen to give a peak release rate of 141
µM
,
ms
although an exact value for
this constant was not given. The spatially averaged peak concentration for one twitch was
approximately 18 µM, more than a factor of 2 larger than that of CA84.
The cylinder of the half-sarcomere in BH98 was longer than that for CA84. It had
a length of 2µm. Baylor and Hollingworth also added a space near the M-line where no
troponin was present. This was due to the length of the half-sarcomere being double that
of the thin filament. They did not treat this as an extra space, but simply a space lacking
troponin. The half-sarcomere of BH98 was divided into 6 horizontal elements and 3 radial
elements. Diffusion was handled using an approximation from Fick’s law,
F lux = −A D
∆[Ca2+ ]
,
∆x
(3.12)
where the flux measures the change in concentration per unit time, A is the area of the
boundary between two cells, D is the appropriate diffusion constant, ∆x denotes the center
to center distance between the two compartments, and ∆[Ca2+ ] is the difference between
the calcium concentrations in the adjacent compartments. A similar equation describes any
of the other molecules that diffuse within the sarcomere. Equation 3.12 performs the same
role in BH98 that Eqn. 3.3 does in CA84. More species were allowed to diffuse in BH98.
The diffusive species and their diffusion constants are given in Table 3.4. All calcium and
magnesium bound species are assumed to have the same diffusion constants as unbound
species.
For the sake of comparison, Table 3.5 is included to compare concentrations and constants with CA84. Some of the differences can be explained by different volumes containing
37
Table 3.4: Diffusion constants
Species
Diffusion constant
10−4 mm
s
Free Ca2+
3.0
Parvalbumin
0.15
ATP
1.4
2
Table 3.5: Constants and total concentrations of chemicals and proteins from Ref. [13]
Concentration (µM) On rate (M−1 s−1 )
Protein
Troponin (regulatory)
240
Parvalbumin
1500
Off rate (s−1 )
0.885 x 108
115
Calcium
0.417 x 108
0.5
Magnesium
0.00033 x 108
ATP
8000
3
Calcium
0.1364 x 108
30000
Magnesium
0.015 x 108
Free metal concentrations
Resting myoplasmic [Ca2+ ]
Myoplasmic [Mg2+ ]
0.1
1000
45
38
the species and some differences reflect recent data. A different temperature was also used.
The final difference between BH98 and CA84 was the SERCA pump. Baylor and
Hollingworth used a second-order Hill equation with a time-dependent function to turn
it on, that is,
−t
d[Ca2+ ]
Vmax [Ca2+ ]nH
ems
,
= [1 − e τ ]N
nH
dt
[Ca2+ ]nH
+
K
ems
m
(3.13)
where N, τ , nH, and Km are equal to 10, 1 ms, 2, and 1 µM respectively. No reason was
given for the function that turns the pump on other than to allow time for the calcium to
diffuse. The squaring of the concentration terms is reasonable because the SERCA pump is
believed to transport two calcium ions across the sarcoplamic membrane making it a second
order pump. This assumption is consistent with data from Lytton et al. [23], where the
Hill coefficient was reported as 2.1 ± .1. Baylor and Hollingworth also use a Vmax of 1.5
mM
s
which is larger than that used by Cannell and Allen, but not too different. Baylor and
Hollingworth also used the same Michaelis constant as Cannell and Allen.
3.3
Baylor and Hollingworth’s 2007 model
Baylor and Hollingworth’s 2007 model (BH07) was actually two models with different
geometry [12]. One model, the one of interest, had a length of 1.2 µm and a radius of 0.484
µm. The second model is longer with a smaller radius. We will only concern ourselves
with the shorter model because it has the geometry closest to CA84. The release point
was also adjusted to more closely match that of mammals. The triadic junction is located
about 0.5 µm from the Z-line in mammals. This releases the calcium into a region of
the extramyofibrillar space that allows it to diffuse in three directions as opposed to the
two directions previously allowed in the CA84 and BH98. A consequence of this choice
is a quicker diffusion of the calcium. Collectively, this increased freedom for diffusion of
the unbound calcium, the diffusion of ATP and pavalbumin bound calcium, and the extra
binding of calcium, one can see how the sarcomere is able to relieve the calcium gradient
39
that develops when a large amount of calcium is released into the myoplasmic space at a
point.
The release of calcium was handled as before in BH98 but when trying to stimulate
the muscle a second time, it was noted that the release of calcium was far less. Since
BH98 and BH07 do not track the calcium in the sarcoplasmic reticulum, they were forced
to introduce an independent parameter to describe this second release. The shortcoming of
this ad hoc approach is that it only works for stimulation at 67 Hz and only for the fivefold
stimulus that they considered. Using any other frequency or number of stimuli will require
additional inputs from measurements of a muscle under those conditions. This is a major
limitation of this model. The decrease in amount of calcium released as more stimulii come is
a consequence of the fact that there must be a limited source of calcium in the sarcoplasmic
reticulum, and it will decrease with each stimulus if adequate time is not available to restore
equilibrium. This effect was noted in CA84, as we can see from Fig. 3.1.
Figure 3.1: The depletion of calcium in the sarcoplasmic reticulum in response to a single
stimulus and a train of stimulii at 30 Hz, taken from Cannell and Allen [11].
Baylor and Hollingworth modelled a buffering SERCA pump to account for the interval
of time during which the calcium is bound to the SERCA pump. This choice supersedes the
limitations of the Hill approximations used in BH98 and CA84. BH07’s kinetic scheme for
the SERCA pump also includes competition for binding sites on the pump from magnesium
and hydrogen. Magnesium and hydrogen were assumed to stay at constant concentrations
40
Figure 3.2: A kinetic scheme for a buffering SERCA pump, taken from Baylor and Hollingworth [12].
throughout a muscle contraction. The kinetic scheme for BH07’s SERCA pump is presented
in Fig. 3.2, where E is the SERCA pump. The concentration of each myoplasmic constituent
is listed in Table 3.6.
Table 3.6: Concentration of myoplasmic constituents
Constituent
Concentration (µM)
resting [Ca2+ ]
0.050
resting [Mg 2+ ]
1000
resting [H + ]
0.1
troponin
120
SERCA pump
120
parvalbumin
750
ATP
8000
Table 3.7 lists the forward and backward rate constants for all reactions pertaining to
the BH07 SERCA pump. The calcium and magnesium reactions with the pump all had time
dependence determined by their individual rate constants. Hydrogen was assumed to react
41
several orders of magnitude quicker than calcium and magnesium. Accordingly, hydrogen
would return to equilibrium well before the next timestep in the simulation and so its binding
was considered to always be in equilibrium with the available binding sites on the SERCA
pump. This made hydrogen strictly concentration-dependent.
Table 3.7: Rate constants for reactions shown in Fig. 3.2 and the troponin-calcium binding
reaction
Reaction
Forward
Reverse
Ratio
M−1 s−1
s−1
µM
Ca2+ + E ⇀
↽ CaE
1.74 x 108
6.97
0.04
Ca2+ + CaE ⇀
↽ Ca2 E
1.74 x 108
8.71
0.05
Mg 2+ + E ⇀
↽ MgE
8.71 x 104
4.36
50
Mg 2+ + MgE ⇀
↽ Mg2 E
8.71 x 104
87.1
1000
H+ + E ⇀
↽ HE
instantaneous
instantaneous
(pK=8)
H + + HE ⇀
↽ H2 E
instantaneous
instantaneous
(pK=8)
H + + H2 E ⇀
↽ H3 E
instantaneous
instantaneous
(pK=6)
H + + H3 E ⇀
↽ H4 E
instantaneous
instantaneous
(pK=5)
3.48 M
0
0
Ca2+ + T rop ⇀
↽ CaT rop
1.77 x 108
1544
8.72
Ca2+ + CaT rop ⇀
↽ Ca2 T rop
0.885 x 108
17.1
0.194
Ca2 E ⇀
↽ E + (2 Ca2+ )
Special note should be made on the third reaction from the bottom; Ca2 E ⇀
↽ E+
(2 Ca2+ ) is the transport step where the SERCA pump releases the bound calcium into the
sarcoplasmic reticulum. The units on this rate constant are different from other forward
rate constants. Since it proceeds in the absence of any interaction with another chemical or
protein, its units are s−1 . The reactions in the two bottom rows describe the rate of reaction
of calcium with the regulatory binding sites on troponin. One notices that the rate constants
change when the troponin has bound one calcium. This is a refinement over previous models
42
that used the same rate constants for both regulatory sites
3.4
Shorten et al’s 2007 model
We only wish to focus on the force production part of Shorten et al.’s work [14], as
much of their work involved complications unnecessary for our purposes. Figure 1.9 presents
the kinetic scheme proposed by Shorten et al. The constants controlling this scheme are
presented in Table 3.8.
Table 3.8: Rate constants for force production
Parameter
Definition
Value (ms−1 )
k0off
Troponin without calcium moving from unblocked to blocked
0.15
on
kCa
Troponin with 2 calcium bound moving from blocked to unblocked
0.15
off
kCa
Troponin with 2 calcium bound moving from unblocked to blocked
0.05
f0
rate of crossbridge attachment
1.5
fp
rate of pre-power stroke crossbridge detachment
15
h0
forward rate of power stroke
0.24
hp
reverse rate of power stroke
0.18
g0
rate of post power stroke crossbridge detachment
0.12
A point that we may be able to improve in this scheme is an alteration of the rate of
post-power stroke crossbridge detachment. In a more complex scheme of crossbridge cycling
presented by Gordon, Regnier, and Homsher [24], there are three intermediates that occur
between the power stroke and crossbridge detachment. The first is a strain-sensitive forward
rate that is greater than 500 s−1 at low strain and between 3 and 10 s−1 at high strain.
The next is a release of ADP and is greater than 100 s−1 . After this release the crossbridge
binds ATP at a rate termed “very rapid,” but no value was given and the final step is the
actual detachment which happens at greater than 1000 s−1 . It is possible that the 0.12 ms−1
43
(120 s−1 ) detachment rate from Shorten et al. [14] is primarily a result of the rate constant
controlling the ADP release. As strain increases, the strain sensitive constant decreases far
enough that it becomes the limiting step. This could be incorporated using a variable rate
of post-power stroke crossbridge detachment.
44
CHAPTER 4
THE PRESENT MODEL
Each model presented is made up of five basic components, and each component is a
little model in and of itself. The five parts are the geometry, the calcium release, the diffusion
of chemicals, the chemical interactions, and the SERCA pump which returns the model to
equilibrium. These five mini-models work together to create a larger, more comprehensive
model. Each of these five basic parts of the model will be presented in a section below. This
compact organization should allow the reader an easy way to look up individual parts of the
model. In our model, we try to stay as close to Baylor and Hollingworth’s 2007 model as
possible. It is the most recent paper we have found on the subject and should offer the best
numerical values for the parameters and the most reliable data. Additions were made in an
attempt to allow the model to be able to reproduce tension vesus time data for a muscle.
Our model was created using the C++ programming language, and a half-second simulation
of calcium movement and force production in the sarcomere takes approximately one-half
hour to run on a 3.8 GHz PC.
4.1
Geometry
The geometry of our model follows that of Cannell and Allen’s 1984 model [11]. Cannell
and Allen’s model contains cells to represent the sarcoplasmic reticulum, which is not present
in the other models. The release point was altered so that the triadic junction was located
approximately 0.5 µm from the Z-line, in accord with BH07 [12]. To allow for more exact
geometry, the extramyofibrillar space was considered separate from, but adjacent to, the
myofibrillar space. This allowed us to set its radius, and that radius would not change
regardless of how many radial subdivisions we chose for the myofibrillar space. In the final
model we allowed for 6 radial subdivisions of the myofibrillar space and one radial subdivision
45
of both the extramyofibrillar space and the sarcoplasmic reticulum.
The model contained 11 horizontal subdivisions. These were the same across all portions
of the sarcomere. This number was chosen to allow for a better approximation for the location
of the troponin. Our model length is 1.1 µm, and the actin filament, which is surrounded
by the troponin complex, extends 1.025 µm from the Z-line [25]. Therefore, all cells in the
myofribrillar space that are within 1 µm of the Z-line contain troponin, but the last row of
cells adjacent to the M-line do not.
Table 4.1: Half-sarcomere dimensions in present model
Half-sarcomere dimensions
Divisions
Radius
=
0.5 µm
Radial Subdivisions
Length
=
1.1 µm Myofibrillar space
Volume
=
0.86 µm3
Extramyofibrillar space
1
Terminal cisternae cell volume
=
0.0301 µm3
Sarcoplasmic reticulum
1
Sarcplasmic reticulum cell volume
=
0.00473 µm3
6
Horizontal Subdivisions
All spaces
11
Cell to adjacent cell distance
Radial
Myofibrillar space
Extramyofibrillar space
△rmy
=
0.0768 µm
△r
=
0.0160 µm
△x
=
0.10 µm
Horizontal
All spaces
The terminal cisternae were modeled as one cell, embedded in the sarcoplasmic reticulum, but much larger in volume than a sarcoplasmic reticulum cell. This size difference is
a consequence of the division of the sarcoplasmic reticulum into ten horizontal cells, eleven
divisions minus the cell that is assigned to the terminal cisternae. The effects of this volume
difference will be discussed in the section on diffusion. Since the triadic junction is moved
46
inward from the Z-line in our model, we are treating the two terminal cisternae as one cell.
In reality they are separated by the T-tubule system, but to model this would require unneccessary geometrical complexity in our model, which is not likely to contribute anything
essential. The parameters used in describing the geometry are listed in Table 4.1.
4.2
SERCA pump
We used the buffering SERCA pump modeled in BH07. Adjustments to the rate constants for the calcium binding and turnover rate have been altered to more closely approximate the pump data from Lytton et al. [23]. The kinetics of our pump will have a common
feature with other chemical reactions in that every chemical and protein presented in one
of the following sections on kinetics will bind calcium or magnesium. In the general case
the bound species then may, or may not, diffuse before releasing the bound chemical. The
only way for a bound species to transport calcium to a different cell is through diffusion.
The SERCA pump will bind calcium and move that calcium to another space without the
process of diffusion. This makes it fundamentally different from any other kinetic scheme
included in our model.
Both rate constants for the binding of calcium by the proteins of the SERCA pump in
the present model are 7 times greater than those in BH07. This brings the pump velocity
versus negative logarithm of the calcium concentration (pCa) curve closer to the expected
value for SERCA1a, which is the SERCA pump found in fast-twitch skeletal muscle [23].
A comparison of the Baylor and Hollingworth’s pump, the present model’s pump, and the
expected SERCA pump in Fig. 4.1. The factor of seven was determined by setting the model’s
equilibrium myoplasmic calcium concentration equal to the Michaelis constant, where the
pump should be half saturated. The on rate for both steps of calcium binding to the pump
were increased until a half-saturation was accomplished, at a factor of 7 increase to the on
rates. The turnover rate of the SERCA pump was raised from the BH07 value of 3.48 s−1 to
47
Figure 4.1: Comparison of SERCA pump models with an ideal second-order saturable pump
4.2 s−1 . This adjustment was needed to increase the pump speed so that it reached a correct
maximum velocity of 1 mM/s [23].
In our model we use a leak through the sarcoplasmic membrane to balance the pumping
action of the SERCA pump at equilibrium. This leak has been explained by some as a flux
of calcium back through the pump instead of a leak through the membrane. Peinelt and
Apell [26] presented the SERCA kinetics quoted by Baylor and Hollingworth, but Baylor and
Hollingworth only use the portion of their kinetics that pertains to the myoplasmic side of the
sarcoplasmic membrane. Peinelt and Apell went on to offer two schemes to model calcium
binding on the sarcoplasmic side, but these are not utilized by Baylor and Hollingworth.
Binding on the sarcoplasmic side by the SERCA pump was also included in the scheme
presented by Bartolommei et al. [27]. If one considers that the calcium concentration in the
sarcoplasm is five orders of magnitude greater than in the myoplasm, it is reasonable that
this binding on the sarcoplasmic side could be considerable. This would reduce the number
of calcium binding sites available to cells in the extramyofibrillar space. In an attempt to
stay close to the model of Baylor and Hollingworth, while still improving it, we increased the
48
on rates for binding calcium by the pump so that it would start out with more sites already
bound and thus fewer sites available to bind more calcium. The increase in the on rates is
an approximation to model the binding of calcium on the sarcoplasmic side without having
to include sarcoplasmic intermediates into our kinetic scheme.
These changes in the SERCA pump kinetics improve the fit of the SERCA pump to
experimental values while still keeping the original BH07 scheme intact. We feel it is important not to stray too far from BH07 since most of our assumptions about the regulation of
calcium concentrations come from their work. Furthermore, any changes in one component
of the model will lead to changes in another component of the model. In this instance, our
SERCA pump velocity increases at a faster rate with lower calcium concentrations. On the
whole, these facts lead to a considerably faster SERCA pump than BH07 modeled. Thus, we
will need to adjust the calcium release function in order to achieve the same peak calcium
transient at the same time as BH07.
4.3
Calcium release
With the updated SERCA pump, the BH07 calcium release made the spatially-averaged
calcium peak 0.8 ms too soon. To achieve the same peak spatially averaged calcium transient
at the same time as BH07, that is, 18 µM at 3.7 ms, the parameters in Eqn. 1.1 were adjusted.
Once a satisfactory release function was achieved, the difference between the terminal
cisternae calcium concentration and the adjacent extramyofibrillar space calcium concentration was recorded at 0.2 ms intervals. The value of the release function at every 0.2 ms
interval was then divided by the difference to create a new set of data points, which when
multiplied by this difference would give back the same function. Another curve fit was then
done to create a gradient-dependent calcium release function. This step was done so that
as the terminal cisternae calcium concentration depleted due to release, the release would
decrease in proportion to it. This is in spirit to the approach of CA84, but the calcium
49
concentration in the terminal cisternae was treated as an independent variable in BH07, a
limitation that we did not want to include. Equation 4.1 shows the final form of the release
function used in our model,
R = C([Ca2+ ]TC − [Ca2+ ]MY )[1 − e−(t+T )/τon ]L [e−(t+T )/τoff ],
(4.1)
where [Ca2+ ]TC is the terminal cistern calcium concentration, and [Ca2+ ]MY is the adjacent extramyofibrillar space calcium concentration. The quantities L, C, τon , τoff , and T in
Eqn. 4.1 are 4.8, 1.88 x 10−6 , 30 ms, 0.4 ms, and 1.4 ms, respectively. For any time t < T the
calcium release was taken to be zero. The release, R, is in moles and has to be divided by the
volume of the extramyofibrillar cell into which it is released in order to get the appropriate
increment in concentration. This allowed the model to keep track of concentrations instead
of moles. The same process was used to determine the loss of calcium concentration by the
terminal cisternae. The focus on concentration is required because the diffusion equation
(Eqn. 3.3) deals with concentrations.
4.4
Diffusion
Diffusion was considered by using Crank’s finite difference approximation to the derivatives of the diffusion equation [22]. This was chosen over Fick’s approximation because
Crank’s method is a numerical solution to the diffusion equation (Eqn. 3.3) using finite differences, which gives a detailed treatment of the space and time dependence of the diffusing
ions, whereas Fick’s law only gives averages over certain volumes. The numerical solution
will be more accurate, but will also have some inherent error. Fick’s law would bring the
advantage of handling diffusion across incremental spaces of different size. Crank’s approximation requires that radial subdivisions be evenly spaced, and this presents a challenge when
one tries to include spaces of uneven radial displacements.
In our model this problem arises between the extramyofibrillar space and the myofibrillar
50
space. In Table 4.1, one can see that the myofibrillar radial increment is almost 5 times
greater than the extramyofibrillar radial displacement. To handle this situation, Eqn. 3.7 was
solved using an average of the two radial increments to calculate the change in concentration
between the two cells. The myofibrillar cell was then treated as a boundary cell and solved
accordingly.
4.5
Kinetics
The kinetics used in the present model are mostly the same as in BH07. In Baylor and
Hollingworth’s 1998 model, they accounted for magnesium binding to ATP, which was not
present in their 2008 model. We have kept magnesium binding in our model. Since our model
includes the sarcoplasmic reticulum, we also have included calsequestrin kinetics. Table 3.6
contains the concentrations of chemicals and proteins used in our model. The concentration
of calsequestrin was 31 mM and the sarcoplasmic reticulum had a calcium concentration of
1.5 mM. Table 4.2 lists the reactions used in our model and rates that control those reactions.
Table 4.2: Rate constants for reactions in the present model
Reaction
Forward
Reverse
M−1 s−1
s−1
Ca2+ + Calseqeustrin ⇀
↽ CaCalseqeustrin
240
0.2
Ca2+ + Parvalbumin ⇀
↽ CaParvalbumin
0.417 x 108
0.5
Mg2+ + Parvalbumin ⇀
↽ MgParvalbumin
0.00033 x 108
3
Ca2+ + ATP ⇀
↽ CaATP
0.417 x 108
0.5
Mg2+ + ATP ⇀
↽ MgATP
0.13 x 108
390
Ca2+ + T rop ⇀
↽ CaT rop
1.77 x 108
1544
Ca2+ + CaT rop ⇀
↽ Ca2 T rop
0.885 x 108
17.1
51
The method of force production was handled using the kinetic scheme presented in
Fig. 1.9. The constants from Table 3.8 were used in implementing this scheme. One will
notice that the Fenn effect is not included in our model. This was attempted, but since it
did not improve the results (in fact it made them worse) this idea was left out of our model.
More discussion on this point will be offered in the results and discussion section.
52
CHAPTER 5
RESULTS AND DISCUSSION
5.1
Results
A pilot study is designed to determine the feasibility of a full-scale treatment. In the
present study we are trying to determine the possibility of creating an accurate model of
muscle twitches using previous computer models of the sarcomere. If an accurate model could
be developed then it would provide a good foundation for a study of jitter and shimmer.
The utility of a successful model would not be limited to the study of jitter and shimmer,
although these topics were the primary motivation for our interest in this area. In fact,
the broad scope of research where this model could be used suggests we should not try to
model a specific muscle. Instead, we are attempting to develop a model that fits the general
characteristics described in Chapter 2, without a focus on any one specific data set. If we
are successful, then subsequent studies could use our framework and kinetics specific to their
muscle of interest to specialize the model to their research. Conversely, a successful model
may be able to determine kinetics if twitch data of a specific muscle is given.
That being said, at present the model cannot be used as a reliable model of muscle
twitch. While the present model offers several important improvements over previous models,
these improvements were not able to come together into a model capable of producing
quantitative results that matched experimental data well. This conclusion is mainly based
on the final force versus time curve in response to a single stimulus. Before we are able to
discuss these results we need to discuss, more thoroughly, the small pieces of the model that
combine to create the composite model. These smaller parts are the calcium release, SERCA
pump, calcium diffusion and geometry, and the kinetics. A discussion of these is necessary
to fully understand the force results from the model.
The geometry and the method of handling the calcium diffusion are dependent upon each
53
other. If one chooses all the cells of the model to have the same radial and axial displacements,
Crank’s finite difference approximation offers certain advantages. This is due to the fact that
fewer calculations are needed to solve for the diffusion at a given time when compared with
Fick’s law. Using Crank’s method, a single cell requires two calculations, one for diffusion
in the radial direction and one for diffusion in the axial direction. Fick’s law requires four
such calculations, one for radial diffusion towards the outside of the cylinder, one for radial
diffusion towards the center of the cylinder, and one for diffusion in each axial direction.
The downside of Crank’s method, as it pertains to our model, is the limitation presented
by requirements on the number of radial divisions in such a model. The extramyofibrillar
radial increment is so small that the total radius of the myoplasmic space would need to be
divided into 30 elements to get a reasonable approximation to the size of the extramyofibrillar
radial increment. While this is possible in the present model, it increases calculation time
dramatically. Thirty radial divisions represent 24 more than the number used in our model,
and each additional radial subdivision adds 11 cells in the axial direction that need to be
calculated, for a total of 264 more cells that have to be considered. Each new cell has two
equations to control its diffusion and 7 equations, at the minimum, to control the reactions
that change the cell concentrations of their chemical contents. Multiplying the additional
264 cells by the 7 chemical equations creates an additional 1848 calculations. Multiply this
by 2 for diffusion of each chemical in the radial and axial directions and the this brings the
minimal increase in calculations to 3696. This alone would not be overwhelming, but these
extra calculations are done once for every iteration through the program. The program will
iterate 2000 times to create a half-millisecond of simulated time. Clearly, the addition of
cells to the model has to be done judiciously, or the model would have a run time that would
limit its usefulness.
The other option is the use of Fick’s law. This allows variable sizes for the cells, allowing
for more customizable cell sizes. The trade-off for this advantage is a requirement of more
54
calculations for each cell. This increase in calculations would be more than compensated
for by the decrease in calculations because of the smaller number of cells, leading to an
overall decrease in the number of calculations. This would make Fick’s law a better choice
if it were not for another consideration that arises with the size of the timestep. Since
the diffusion equations for each cell are solved separately, a smaller timestep is required to
keep the approximation from reducing the calcium concentration in the cell too much in a
given timestep. This smaller timestep is also necessary because Fick’s law only gives the
instantaneous flux. Taking this flux as the average flux over a given timestep is reasonable,
assuming the timestep is small. The bigger the timestep, the more calcium diffused in a
given timestep, and it is possible to exceed the amount of calcium available. Since flux
is dependent on calcium concentrations, the greater the change in calcium concentrations
during a timestep, the more Fick’s law becomes an unreliable way to solve for diffusion over
time.
Two methods to deal with the challenge presented by the small extramyofibrillar space
and calcium diffusion were examined. The size of the extramyofibrillar space was maintained
in both. In the first, Fick’s law was used to handle the diffusion of calcium between the
extramyofibrillar cell and the adjacent myofibrillar cell. This adjacent myofibrillar cell was
then treated as a boundary and the myofibrillar space was divided into six radial divisions
and diffusion within the myofibrillar space was handled by Crank’s method. In order to
keep this method from diverging, the timestep had to be more than 100 times smaller than
the value that was used in the final model. A 100-fold decrease in timestep would increase
the number of calculations by 100-fold over the same simulated time period. The second
method, and ultimately the method we selected, was described in the previous chapter. To
examine the validity of this method, mass conservation was checked, and a contour map of
the calcium concentration was produced. The present model showed a gain in calcium equal
to less than 1% using a 100 Hz stimulation for 0.5 seconds. This extra mass was lost upon
55
return to equilibrium. The system did not retain this extra mass upon return to equilibrium.
The contour maps were examined in sequential time steps to see if the concentration gradient
Figure 5.1: Change in concentration gradient of calcium from 3.8 ms on the left to 4.8 ms
on the right
developed and diminished in a logical manner. A representative of these contour maps is
given in Fig. 5.1. Each line is an iso-concentration line and the concentration drops with
increasing distance from the calcium release point. The extramyofibrillar space is represented
by the cells across the top of the figures and one sees a greater range of concentrations moving
laterally outward from the calcium release. This is due to the lack of troponin binding in
this region. This can also be seen in the radial diffusion at the M-line.
An issue arose when handling the calcium release using the small extramyofibrillar space.
The calcium release was calculated as the number of moles moving out of the terminal
cisternae and into the extramyofibrillar cell. The number of moles had to be divided by the
volume to make it into a concentration change. The small volume of the extramyofibrillar
cell made it important to keep the timestep small so as not to release too much calcium
into the extramyofibrillar space. Too large a release would raise the concentration in the
56
extramyofibrillar space above the terminal cisternae. Additionally, if the difference between
the extramyofibrillar cell that receives the calcium and any adjacent myoplasmic cell is too
great, the concentration will not be numerically stable.
The calcium release was handled using the double exponential form from Baylor and
Hollingworth. This calcium release serves as a starting point and thus is an approximation.
It was set in a manner that allowed for the peak spatially averaged myoplasmic calcium
concentration to be 18 µM at 3.7 ms. We assume the same restrictions when determining the
parameters in Eq. 1.1. Different parameters were needed in order to get the same release with
the incorporation of more chemical reactions and a different method of handling diffusion.
An additional consideration when adjusting the release was the depletion of sarcoplasmic
reticulum calcium. After the first calcium release, an additional stimulus will release much
less calcium. One possible reason for this is the aforementioned depletion. The release in
our model does not deplete the sarcoplasmic reticulum enough, but does release considerbly
more calcium than the original release of Baylor and Hollingworth.
Another departure from the methods of Baylor and Hollingworth was an adjustment
of the SERCA pump kinetics. The reason and justification for this change was previously
described but more needs to be said about the features of the pump velocity curve (Fig. 4.1).
At a pCa of 7, the expected pump velocity is considerably slower than the velocity of the
pump in the present model. Conversely, at a pCa of 5, the theoretical pump works much
faster than the pump used in the present model. This would affect the spatially averaged
calcium concentration by increasing the the approach to equilibrium at any concentration
less than 1 µM and slowing it at any concentrations above this. Figure 5.2 shows the
spatially-averaged calcium concentration in the present model. The peak is at the correct
magnitude and time but the concentration at larger times is considerably less than reported
by Baylor and Hollingworth [12], who had a concentration at 25 ms of between 1 and 3 µM.
The present model gives a value at this time around 0.5 µM.
57
Figure 5.2: Single twitch spatially averaged calcium concentration in the myofibrillar space
The difference in concentrations at later times is important in bringing to light an issue
with the kinetic scheme used to determine the force producing crossbridges. Figure 5.3 shows
the concentrations of force-producing crossbridges in response to the calcium released during
a single twitch. It peaks at 22.5 ms, has a latency period of 3.3 ms, and a half-relaxation
time of 57.5 ms. To determine if the magnitude is reasonable, the calcium concentration was
set to 1 mM. This high concentration saturates the troponin and thus allows for the greatest
number of force producing crossbridges. The peak concentration in response to a twitch
was divided by the maximal concentration at saturating calcium concentrations; this gave a
value of 16%. If one examines Titze’s data (Fig. 1.3 and Fig. 1.4), the force of the muscle
approaches a 1 N asymptote in Fig. 1.4. In Fig. 1.3, the force of the first twitch peaks at
approximately 0.18 N. This would give a single-twitch force divided by the maximum force
of 18%.
On the whole, the data on a single twitch matches favorably with several checkpoints.
The one major difference is in the half-relaxation time. The present model’s half-relaxation
time is 20.5 ms greater than the largest published value. This is especially problematic when
58
Figure 5.3: Force-producing crossbridge (FPXB) concentration in response to calcium release
during a twitch
considered in the light of the greater rate with which the present model moves calcium out
of the myoplasmic space. An excellent model of the SERCA pump would pump calcium out
slower at tenths of a micromolar concentrations. This would leave more calcium in the sarcomere for a longer time and accordingly, more calcium bound to troponin. The more calcium
is bound to troponin, the greater is the concentration of force-producing crossbridges. This
points to an issue with the kinetic scheme used to calculate the force-producing crossbridges,
it does not relax fast enough.
The slower rate of relaxation in the model becomes a big issue when the calcium release is repeated at a given frequency. Figure 5.4 shows the response to several stimulation
rates, and its clear that our model is problematic. The concentration of force-producing
crossbridges does not reproduce the features seen in Fig. 1.4 of Alipour-Haghighi, Titze, and
Perlman’s data [21]. Another issue is that the saturation level is the same for all frequencies of stimulation. The frequency only determines the time at which the force reaches the
maximum level. We suspect that this discrepancy is a result of the kinetic scheme used to
model the force production.
59
Figure 5.4: Force producing crossbridge concentration in response to a stimulus frequency
5.2
Towards a better model
An important consideration that needs to be addressed in future models is the temperature dependence of the model. This dependence can be seen in two places, the diffusion
constant and the rate constants controlling the reactions. The diffusion constant is proportional to the increase or decrease in absolute temperature, as can be seen in Eqn. 5.1, the
Stokes-Einstein relation,
D=
kb T
,
γ
(5.1)
where γ is a constant for a given species, kb is the Boltzmann constant, and T is the absolute
temperature. Diffusion in this model is dealt with statistically, as is necessary since the
number of molecules is simply too large to be followed on a computer. If one were to seek
a more microsopic approach with atoms or molecules, the higher temperature would be due
to incresed kinetic energy of each molecule. This would cause more collisions between atoms
and molecules and thus more reactions per unit time, which would lead to an increase in
the forward rate constants controlling those reactions. The increase in temperature also
60
increases the reverse rate as well as the forward rate constants. Without studying the
temperature dependence of an individual reaction, it would be difficult to even approximate
these effects. Temperature effects on a single set of reactions are probably small, but as
more reactions are considered the importance of temperature dependence would probably
grow. The temperature dependence of diffusion and kinetics is the likely cause of changing
characteristic twitches with change in temperature (See Fig. 2.5).
Of course, before temperature dependence may be addressed, an effective model needs
to be created. As previously stated, the present model is not free from inconsistencies and
would not be a reliable framework to build on. Our model needs to be improved before
additions can be made. One of the assets of the present model has proved to be a liability as
well. It includes the greatest number of reactions and most accurate geometric descriptions
of any model we have found in the literature. This is a positive in that we are able to see how
a change in one area affects another area of the model. The downside to this approach is the
difficulty of deciding what parts of our model are most problematic. It is most likely that
a solution to this overall problem will reside in multiple smaller studies aimed at isolating
individual problems.
The first of such studies should include a realistic geometry, correct calcium concentrations, a calcium release function, and a method for handling diffusion. For this phase,
getting the exact form of the release may not be important, since it is only a means to study
diffusion. If one evenly divides the myoplasmic space and tracks the calcium concentration
using Crank’s method to calculate it, you can get the concentrations at a given point at a
given time. I suggest tracking the concentration at the center near the Z and M line and a
centrally located cell. Next alter the geometry to accurately describe what is seen in the cell
and try different methods of handling the diffusion. This will allow one to check concentrations at specific points at specific times and see if it matches Crank’s widely-used method.
Alternately, one could use Fick’s law as the foundation for diffusion calculations. In either
61
case, one should be able to verify the mass conservation and whether the method is capable
of reproducing realistic patterns of diffusion.
Once an acceptable method for handling diffusion with a more intricate geometry is
achieved, the focus should turn to the SERCA pump. At this point, the concentrations
of magnesium and protons need to be added to the model. Then the model should be
able to return to equilibrium after a calcium release. In order to test different models
for the SERCA pump, a Gauss-Jordan elimination subroutine should be used to determine
equilibrium conditions. This was done in the present model due to the large number of linear
differential equations created in trying to use the buffering model of the SERCA pump. A
possible, and simple, method for altering the SERCA pump would be an intermediate form
of the pump that is open to binding on the saroplasmic side of the sarcoplasmic membrane.
This would allow it to bind calcium on the sarcoplasmic side and possibly release this calcium
back into the myoplasm. Such a pump would be closer to the actual conditions between the
sarcoplasm and the myoplasm than using a leak to balance pump activity. A protein-free
membrane is highly impermeable to ions, thus the permeability is due to the proteins in
the membrane and not a leak. These proteins are highly selective to the ions they interact
with [28]. This explains why the SERCA pump would bind magnesium and protons, but
not pump them across the membrane. That is why the movement of calcium across the
membrane, in either direction, should be handled with a SERCA pump.
The issue that one would encounter when adding intermediates into a kinetic scheme
would be the determination of the rate constants that control the formation and loss of the
intermediates. Thus, it might be wise to simply use different published kinetic schemes until
one that fits the theoretical pump adequately is observed. This possibility is the reason the
Gauss-Jordan elimination subroutine is so important. It will make the procedure for finding
equilibrium concentrations described by the rate equations much more efficient.
Once suitable models for the SERCA pump and for the diffusion are completed, then
62
the buffers and troponin need to be programmed in. It is necessary not to try to solve the
issue of force production in the absence of the buffers because they serve to help decrease
the amount of free calcium, which in turn should help relax the muscle. This will also have
an effect on the release functions. Unfortunately, the force production is so connected with
the other processes of the program, it is impossible to remove it from them. This may again
be a situation where determining the best force production kinetic scheme is a matter of
testing all schemes until a satisfactory one is found.
63
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