J Appl Physiol 96: 469–476, 2004.
First published September 23, 2003; 10.1152/japplphysiol.00736.2003.
Mathematical description of geometric and kinematic aspects
of smooth muscle plasticity and some related morphometrics
R. K. Lambert,1 P. D. Paré,2 and C. Y. Seow2
1
Institute of Fundamental Sciences-Physics, Massey University, Palmerston North, New Zealand;
and 2The McDonald Research Laboratories/The iCAPTURE Centre, St. Paul’s Hospital/Providence
Health Care, University of British Columbia, Vancouver, British Columbia, Canada V6Z 1Y6
Submitted 16 July 2003; accepted in final form 15 September 2003
MECHANICAL ACTIVITY OF MUSCLE cells stems from subcellular
structural changes. In striated muscle, the structure-function
relation has been accurately delineated (7, 13–15). In smooth
muscle, the structural basis for contraction is poorly understood. Because a sarcomere-like structure or any regularly
repeating filament array is lacking, the structural change associated with smooth muscle contraction has not been precisely
described. Another impediment to construction of a coherent
structure-function model in smooth muscle is the plastic behavior of its contractile machinery, which includes the contractile filaments and the scaffolding cytoskeleton that supports
the actomyosin interaction and transmits force intracellularly
and throughout the tissue. The reported plastic behavior of
airway smooth muscle (ASM) indicates that the muscle structure must be malleable, so that it can accommodate large
changes in cell dimensions (3, 8, 17–18, 22–23, 26, 30–32). It
has been suggested that the large-scale structural change of
which smooth muscle is capable is beyond the range that could
be accomplished by the mechanism of filament sliding alone
(4). Disassembly and reassembly of contractile units, as well as
restructuring of the cytoskeleton, may be a strategy whereby
smooth muscle cells adapt to large changes in cell length (8,
11, 17, 23, 31, 32). Thus the conventional approach employed
by researchers in studying the structure-function relation of
striated muscle may not be appropriate for smooth muscle. The
dynamic nature of smooth muscle structure implies that static
structure-function correlation can only be found under steadystate conditions whereas, during the process of adaptation, the
structure-function relation in smooth muscle can only be characterized in a time-dependent manner. A mathematical model
that describes not only the static, but also the dynamic, behavior of the contractile apparatus and cytoskeleton of smooth
muscle will therefore be helpful in our quest to understand how
smooth muscle contracts.
We present a model of ASM that provides a starting point
for correlating geometric arrangement of contractile units
within a muscle cell and the kinematics associated with the
arrangement. The model does not address the mechanisms
underlying restructuring of the cytoskeleton and the contractile
filaments associated with the plastic adaptation of the muscle.
Assumptions regarding these structural and functional changes
are based on evidence gathered from other studies (3, 5, 6, 8,
11, 17, 18, 20, 23, 26, 32–34). This model is not designed to
test cross-bridge kinetics, transient changes in muscle properties due to variation in the state of activation or cross-bridge
phosphorylation, the cross-bridge mechanism of the forcevelocity relation, or the contribution by the various components within the muscle’s contractile units to the overall compliance of the muscle. The model is not designed to assess
functional consequences of structural changes that are nonplastic or irreversible, nor is it designed to elucidate the mechanisms preventing normal plastic adaptation in the muscle.
ASM was chosen as the basis for our model because 1) the
plastic behavior was first reported in this tissue (3, 8, 23) and
2) for this tissue there is ultrastructural evidence for plasticity
(11, 17, 18, 24). Tracheal smooth muscle contains relatively
little connective tissue, and the muscle cells are aligned parallel
to each other along the longitudinal axis of the muscle bundle;
these features make the preparation ideal for mechanical studies that involve large changes in muscle length. How well
tracheal smooth muscle serves as a model for the muscle in
Address for reprint requests and other correspondence: C. Y. Seow, Dept. of
Pathology and Laboratory Medicine, St. Paul’s Hospital, Univ. of British
Columbia, Vancouver, BC, Canada V6Z 1Y6 (E-mail: [email protected]).
The costs of publication of this article were defrayed in part by the payment
of page charges. The article must therefore be hereby marked “advertisement”
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
muscle contraction; contractile apparatus; mechanics; myosin filament
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Lambert, R. K., P. D. Paré, and C. Y. Seow. Mathematical
description of geometric and kinematic aspects of smooth muscle
plasticity and some related morphometrics. J Appl Physiol 96:
469–476, 2004. First published September 23, 2003; 10.1152/
japplphysiol.00736.2003.—Despite considerable investigation, the
mechanisms underlying the functional properties of smooth muscle
are poorly understood. This can be attributed, at least in part, to a lack
of knowledge about the structure and organization of the contractile
apparatus inside the muscle cell. Recent observations of the plasticity
of smooth muscle and of morphometry of the cell have provided
enough information for us to propose a quantitative, although highly
simplified, model for the geometric arrangement of contractile units
and their collective kinematic functions in smooth muscle, particularly
airway smooth muscle. We propose that, to a considerable extent,
contractile machinery restructures upon activation of the muscle and
adapts to cell geometry at the time of activation. We assume that,
under steady-state conditions, the geometric arrangement of contractile units and the filaments within these units determines the kinematic
characteristics of the muscle. The model successfully predicts the
results of experiments on airway smooth muscle plasticity relating to
maximal force generation, maximal velocity of shortening, and the
variation of compliance with adapted length. The model is also
concordant with morphometric observations that show an increase in
myosin filament density when muscle is adapted to a longer length.
The model provides a framework for design of experiments to quantitatively test various aspects of smooth muscle plasticity in terms of
geometric arrangement of contractile units and the muscle’s mechanical properties.
470
MATHEMATICAL MODEL OF SMOOTH MUSCLE
more peripheral airways is unclear, although we have some
preliminary evidence that muscle from intralobular airways
adapts similarly to tracheal muscle (19, 29). From a clinical
point of view, dysfunction of ASM has been implicated in the
pathophysiology of asthma and other obstructive airway diseases; a better understanding of the contraction mechanism and
plastic behavior of this tissue will shed light on the mechanism
of airway hyperresponsiveness in asthma and other obstructive
airway diseases. The existing data are not sufficiently complete
to formulate a model that explains all aspects of smooth muscle
properties. However, there are sufficient data to generate a
model to test how much of what we already know can be
explained and to make predictions of what might be observed
in yet-to-be-performed structural and functional experiments.
THE MODEL
Structural Model
A myosin filament in smooth muscle is believed to be a
side-polar structure (1, 12), as opposed to the bipolar structure
of its counterpart in striated muscle. As illustrated in Fig. 1,
cross bridges on one side of the myosin filament interact with
an actin filament possessing the “right” polarity, while the
bridges on the other side interact with a different actin filament
possessing an opposite polarity. The simplistic model suggests
that one myosin filament can only interact with two actin
filaments. In reality, this may not be the case. As illustrated in
Fig. 2, multiple actin filaments may originate from the same
dense body and possess the same polarity; therefore, they are
able to interact with one myosin filament simultaneously. This
is possible only if the “neck” region of a myosin cross bridge
is flexible in all directions; this will enable the bridges from a
myosin filament to attach to several of the surrounding actin
filaments.
Mechanically (in terms of force and shortening produced by
a muscle), there is no difference whether a myosin filament
interacts with multiple actin filaments (Fig. 2) or with one actin
filament (Fig. 1). For simplicity, our mathematical description
of a contractile unit is based on the model shown in Fig. 1.
Our conceptual model for one unit of the contractile apparatus of smooth muscle consists of an actin filament attached at
Fig. 2. Schematic drawing of a partial view of a contractile unit of smooth
muscle. For clarity, only cross bridges on the “top” side of the thick filament
are showing. Inset (bottom right): whole cross section of a side-polar thick
filament; dotted line separates bridges that interact with actin filaments of
different polarity. “Neck” region of a cross bridge is assumed to be flexible in
all directions, and the angle (␪, deviation from the midposition, which is
perpendicular to the dotted line) is determined by positions of thin filaments
that interact with cross bridges. It is assumed that there is a limit for ␪ and that
a cross bridge cannot “swing” beyond the dotted line and interact with thin
filaments of “wrong” polarity.
one end to a dense body (or dense plaque). The other end is
free. A myosin filament overlaps some of this actin filament,
and cross bridges attach one to the other (Fig. 3). The myosin
filament is not necessarily completely overlapped by the actin
filament. A second actin filament is attached to the myosin and
oppositely oriented to the first. The overlap distance is the
same for both actin filaments. Cross-bridge activity causes both
actin filaments to slide past the myosin filament in such a way
as to bring the adjacent dense bodies in series closer together
and shorten the muscle. Ns such elements are connected in
series to form one contractile chain. Np such chains are arranged in parallel within the cell. For simplicity, we assume
that the muscle is cylindrical with all chains containing Ns
elements in series and that the chains are parallel to the long
axis of the cylinder. The product of Ns and Np gives the total
number of myosin filaments (or contractile units) in the cell
N ⫽ N pNs
Fig. 1. Schematic drawing of a contractile unit of smooth muscle with a
side-polar myosin filament. Double arrows indicate directions of movement of
actin filaments and dense bodies due to cyclic action of myosin cross bridges.
J Appl Physiol • VOL
Adjacent dense bodies in a chain are a distance lu apart. We
denote the distance between adjacent ends of two adjacent
myosin filaments in a chain as i and the length of a myosin
filament as lm (Fig. 3). Equation 1 gives the relation between
these lengths
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l u ⫽ lm ⫹ i
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We have developed a model of the contractile apparatus of
smooth muscle that correlates available static and quasi-static
functional data with structural data. Although we have primarily used data generated from ASM of several species, the
elements of the model could be applied to any smooth muscle
type that exhibits plastic behavior. The data that are available
are from functional and morphometric studies.
471
MATHEMATICAL MODEL OF SMOOTH MUSCLE
Fig. 3. Schematic diagram of adjacent contractile units in series. lu, Length of 1 contractile unit; lm, length of a myosin filament,
l, distance over which myosin and actin filaments are overlapped and, thus, permit formation of cross bridges; i, separation of ends
of adjacent myosin filaments; d, length of a dense body.
L ⫽ N slu ⫹ LSEC
(2)
Functional Model
We assume that the steady-state isometric force generated in
one contractile unit is proportional to the number of active
cross bridges in the unit, which in turn is proportional to the
length of overlap of a myosin filament with one actin filament
(l). We also assume that the cross bridges are uniformly
activated on all myosin filaments. (Full activation of the cross
bridges is not required by the model.) It is a requirement of
mechanical equilibrium (steady state) that this force be the
same everywhere along the chain. Thus the total force generated in one cell of smooth muscle (F) is proportional to Np and l
F ⫽ ␣Npl
(3)
where ␣ is the proportionality constant between F and Npl.
Equation 3 indicates that F is proportional to the total number
of active cross bridges acting in parallel in one muscle cell. The
velocity of shortening of a cell (v) can be calculated by
differentiating the expression for length (Eq. 2) with respect to
time. Thus we obtain Eq. 4, in which vu is the velocity of
shortening of one contractile unit. If it is assumed that the
cross-bridge cycling rate is constant and, therefore, that all
contractile units have the same shortening velocity, we have
v ⫽ N svu
(4)
We assume that LSEC does not change during the steadystate contractile process (although it may decrease during the
transient shortening that occurs when a muscle is released from
isometric to isotonic conditions to measure v).
The power (P) developed by a cell is proportional to F ⫻ v
and is thus given by Eq. 5
P ⬀ Fv ⫽ ␤vuNsNpl
(5)
where ␤ is a constant of proportionality. Equation 5 indicates
that in the steady state where the cross-bridge cycling rate is
constant, the power is proportional to the total number of active
cross bridges in a cell.
The compliance of a chain of elements in series is equal to
the algebraic sum of the compliances of each element. Thus,
denoting the compliance of a single contractile unit by cu, we
have the compliance of a complete chain of contractile units
given by Nscu. Compliances in parallel add as the reciprocals of
the individual compliances. Thus the compliance of the contractile element (CCE), shown schematically in Fig. 4, is given
by the following equation when we assume that L ⬎⬎ LSEC
C CE ⫽ 共Ns/Np兲cu
⬇ 共c u/Nplu兲L
(6)
The total compliance (C) of the model cell depicted in Fig.
4 is given by Eq. 7
Fig. 4. Schematic diagram of components of a muscle cell. CE, contractile
element responsible for generating active force; PEC, parallel elastic component responsible for carrying resting tension when CE is not active; SEC, series
elastic component, which transmits force but does not itself generate active
force. L, length of muscle cell.
J Appl Physiol • VOL
C ⫽ CCECPEC/共CCE ⫹ CPEC兲 ⫹ CSEC
(7)
where CPEC is compliance of the parallel elastic component
and CSEC is compliance of the series elastic component.
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The hallmark of fully adapted muscle is that the overlap
between myosin and actin filaments is maximal (to provide
maximal force generation by the muscle); i.e., lm ⫽ l. Adaptation of ASM occurs when the muscle is stimulated repeatedly
at a fixed length (23, 30, 31). Once ASM is fully adapted,
overlap of actin and myosin is optimal.
The length of the muscle cell (L) is given by Eq. 2 (Fig. 4),
in which LSEC accounts for any length that is not part of a
contractile unit (e.g., some passive tethering at each end of a
chain or between 2 contractile units in series)
472
MATHEMATICAL MODEL OF SMOOTH MUSCLE
Model for Morphometry
We assume that lm is small compared with L. The probability
of any one myosin filament appearing in a randomly chosen
cell transverse section is lm/L. Thus the number of filaments
appearing in a transverse section is Nlm/L. Morphometry is
never performed on a cross section but, rather, on a slice. (A
cross section is the surface of a cut. A slice has two cross
sections: one on each face.) The number of points contained in
a slice of thickness t through a random collection of N points
in a cylinder of transverse cross-sectional area A and volume V
is NtA/V. Thus the number of myosin filaments appearing in a
histological slice at right angles to the longitudinal axis of the
cell (Nm) is given by the following equation
N m ⫽ Nlm/L ⫹ N共1 ⫺ lm/L兲tA/V
(8)
␴ ⫽ N/V关lm ⫹ 共1 ⫺ lm/L兲t兴
(9)
L and Ns are related through lu (Eq. 2)
lu ⬇ L/Ns
for LSEC ⬍⬍ L. Thus, for lm ⬍⬍ L, ␴ can be written as follows
␴ ⫽ 共N plm/Alu兲共1 ⫹ t/lm兲
which can be reorganized for calculation of Np as follows
N p ⫽ ␴Alu/关lm共1 ⫹ t/lm兲兴
Thus, for very thin slices (t/lm ⬍⬍ 1), Np exceeds ␴A by a
factor of lu/lm
N p ⫽ ␴A共lu/lm兲
(10)
Fig. 5. Schematic diagram of a muscle cell with diameter D and length L.
Myosin thick filaments are shown as thick lines within the cell lying parallel
to L. t, Thickness of a transverse section defined by the 2 parallel dotted lines
within the cell. An infinitely thin section (represented by 1 dotted line) will cut
across a certain number (in this example, 4) of thick filaments as predicted by
the first term of Eq. 8 (Nlm/L). With a finite section thickness of t, an additional
number of thick filaments (in this example, 1) will be intersected by the
section, as predicted by the second term of Eq. 8 [N(1 ⫺ lm/L)tA/V].
We assume that the speed of contraction of the individual
contractile units is not affected by adaptation, so that vu ⫽ v⬘u.
[This assumption is supported by the finding that force-velocity
curves obtained from a muscle adapted to different lengths
have an identical shape (23).] Thus we predict that the ratio of
shortening speeds is the same as the ratio of the number of
contractile units in series
v⬘/v ⫽ N⬘s/Ns
(11)
The ratio of muscle lengths before and after adaptation (Eq.
2) is
L⬘/L ⫽ 共N⬘sl⬘u ⫹ LSEC兲/共Nslu ⫹ LSEC兲
Plastic Adaptation of Muscle
A unique feature of smooth muscle compared with skeletal
muscle is its ability to adapt to an alteration in muscle length.
That is, with an increase in muscle length, there is a time- and
activation-dependent recovery of force generation and an increase in shortening velocity (17, 23). We assume that the
volume (V) of an intact muscle cell is constant, regardless of
the contractile state of the muscle or of any adaptation that has
happened or is taking place. Myosin filaments appear to form
and grow during activation of ASM (11, 17, 26). Our model is
concerned with maximally activated ASM, and we assume that
polymerization of myosin is a significant aspect of adaptation.
The adaptation affects the commonly measured physiological
variables: F, C, and v. A and ␴ are measured morphometrically.
When the muscle is allowed or caused to adapt to a new length,
any of these variables can change as a result of changes in Ns
and Np. We denote the value of a variable after adaptation by
attaching a prime to it. Thus, from Eq. 4, we obtain the
following equation
v⬘/v ⫽ 共v⬘uN⬘s兲/共vuNs兲
J Appl Physiol • VOL
We assume that the noncontractile lengths contribute negligibly to this ratio. Hence
L⬘/L ⬇ 共N⬘sl⬘u兲/共Nslu兲
(12)
Equation 12 indicates that length adaptation is associated
with a variation in the number of contractile units in series, if
the length of the contractile units (lu) stays the same. On the
other hand, by combining the results for L and v (Eqs. 11 and
12), we can estimate the change in lu
l⬘u/lu ⫽ 共v/v⬘兲共L⬘/L兲
(13)
Adaptation of smooth muscle to a new length could also
result in alteration of power output of the muscle. This can be
described as follows
P⬘/P ⫽ 共␤⬘v⬘uN⬘pN⬘sl⬘兲/共␤vuNpNsl兲
(14)
Plasticity of L
A hallmark of plasticity is that maximal isometric force can
be achieved at any muscle length within the adaptable length
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As illustrated in Fig. 5, the first term on the right-hand side
of Eq. 8 represents the number of myosin filaments intersected
by an infinitely thin slice (i.e., the number in a cross section).
The second term indicates the additional number of myosin
filaments appearing in a slice of finite thickness. Hence, the
number of filaments per unit area (␴) is given by Eq. 9
MATHEMATICAL MODEL OF SMOOTH MUSCLE
range; restructuring of the contractile apparatus ensures that
optimal overlap of contractile filaments is preserved. We therefore assume that in fully adapted muscles the myosin filaments
are optimally overlapped with the actin filaments. That is, lm ⫽
l. When we also assume that l/L ⬍⬍ 1, t/l ⬍⬍ 1, and cell volume
is conserved at different cell lengths, we obtain the following
(from Eqs. 3 and 9)
F⬘/F ⫽ 共␣⬘N⬘pl⬘兲/共␣Npl兲
(15a)
␴⬘/␴ ⬇ 共N⬘pN⬘sl⬘兲/共NpNsl兲
(15b)
Equation 15, a and b, can be combined to yield Eq. 16
F⬘/F ⫽ 共␴⬘/␴兲共v/v⬘兲共␣⬘/␣兲
(16)
C ⬇ 共cu/Nplu兲L ⫹ CSEC
(17)
If the coefficient of L is unchanged by adaptation, the model
predicts a linear relation between C and L in experiments in
which the muscle is adapted to each new L. The line does not
pass through the origin. Thus a direct proportionality between
compliance and length should not be observed.
DISCUSSION
We have developed a relatively simple model based on the
assumption that a contractile unit is made of a side-polar
myosin filament (28) and the associated actin filaments and
dense bodies (12), as shown schematically in Figs. 1–3. The
model predicts many functional and morphological features of
smooth muscle. In contrast to striated muscle, where the
number of contractile units (half-sarcomeres) and their structural arrangement relative to one another (in series or in
parallel) are invariable within a mature cell, we postulate that
ASM has a more malleable contractile apparatus and, when
sufficient time is allowed for adaptation to occur, is able to
adjust the number of contractile units in series (Ns) and in
parallel (Np), and even the total number of units (N), to
accommodate large changes in cell dimensions to preserve
optimal overlap of contractile filaments. This model, however,
does not address the mechanism by which plastic remodeling
of the contractile apparatus is achieved. Besides the freedom
for varying the number and arrangement of contractile units in
a cell, another degree of freedom allowed in our model is the
variable length of myosin filaments. In this regard, our model
is focused on the polymerization of myosin on activation.
(Again, the model does not address the molecular mechanism
of thick filament formation.) It is known that actin also polymerizes during contractile activation (20); the extent of polymerization is less than that for myosin filaments (11). Actin
polymerization is likely important in the process of thin filament attachment to the dense bodies and plaques and, therefore, is important for plasticity of the cytoskeleton. It must be
pointed out that plastic changes in contractile filaments have to
be accompanied by plastic changes in cytoskeleton, because
J Appl Physiol • VOL
the contractile filaments are supported by the cytoskeletal
scaffold. A contractile apparatus is defined here as the contractile filaments plus the scaffolding cytoskeleton. The in-series
and in-parallel rearrangement of the contractile units described
above therefore requires plasticity of the cytoskeleton and the
contractile filaments. The mathematical description of plasticity of the contractile apparatus presented in this study is based
on an implicit assumption that the plastic restructuring of the
cytoskeleton occurs concurrently with the plastic restructuring
of the contractile filaments. In this study, myosin receives more
attention than actin (or other cytoskeletal elements), because
myosin filament density in ASM appears to have a linear
correlation with force generation (18), whereas there are no
quantitative data regarding correlation between actin filament
density and force or any other mechanical parameters. The fact
that there are many more thin filaments than thick filaments in
smooth muscle suggests that after the cytoskeletal scaffold is
firmly in place, variation in the number and length of the thick
filaments may be the controlling factor for optimizing the
mechanical performance of the muscle. The linear correlation
between thick filament density and mechanical and energetic
aspects of muscle behavior is very amenable to modeling, and
it is therefore a focus of the present mathematical description.
We assumed that the overlap regions between one myosin
filament and its two attached actin filaments were the same.
The underlying assumption here is that all cross bridges are
identical and contribute equally to the development of tension.
Because the force at any point in one chain of contractile units
must be the same as the force at any other point, the number of
cross bridges pulling one way on the myosin filament must be
the same as the number of cross bridges pulling the other way.
Thus the regions of overlap must be equal in length. If the
myosin attaches to more than one actin at one end (Fig. 2) but
not at the other, the sum of the overlap lengths of the multiple
actins at one end must equal the overlap length of the single
actin at the other end to maintain mechanical equilibrium.
Application of the Model to Observations
Changes in force-velocity parameters, muscle compliance,
muscle energetics, and myosin filament density in ASM
adapted over a large length range. Functional study of ASM
adapted to different lengths (17, 23) revealed that isometric
force (F) was length independent within the adaptable range
(Fig. 6, top); i.e., F⬘/F ⫽ 1. This is consistent with a model that
assumes a constant number of contractile units in parallel (and
variable numbers of contractile units in series, as discussed
below) in a muscle cell adapted to different lengths. Because
the change in power output is given by P⬘/P ⫽ (F⬘/F)(v⬘/v), and
with F⬘/F ⫽ 1, any change in power due to length adaptation
has to be matched to the same extent by the change in velocity.
This was indeed observed (Fig. 6, middle). If shortening
velocity of individual contractile units (vu) is assumed to be
unaffected by the process of length adaptation, then the increase in v has to be matched by the same extent of increase in
the number of contractile units in series (Ns, Eq. 11). This
increase in Ns will in turn result in the same extent of increase
in P if the term ␤vuNpl in Eq. 5 stays constant. The matching
increases in power and velocity suggest that the term ␤vuNpl
was not changed by length adaptation.
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In Eq. 16, the force ratio, the area filament density ratio, and
the velocity ratio are experimentally measurable. Thus this
relation enables us to determine the change in ␣ as a result of
adaptation.
The change in compliance with adaptation to a new length
can be predicted from Eqs. 6 and 7 with the assumption that
CPEC is very much greater than CCE
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MATHEMATICAL MODEL OF SMOOTH MUSCLE
and because V is constant in intact smooth muscle adapted to
different lengths (17), myosin content is proportional to myosin density. As shown in Fig. 6 (middle), ␴ was increased with
adapted muscle length, suggesting that the myosin content was
indeed increased with cell length. Furthermore, the increase in
M was in exact proportion to the increase in muscle shortening
velocity and power, suggesting that the augmented mechanical
performance of the muscle at longer lengths is a direct consequence of the increase in Ns. This is further corroborated by the
finding (17) that the rate of ATP consumption also has the
same length dependence as v, P, and ␴ (Fig. 6, middle).
The fact that ␴ increased by the same extent as shortening
velocity (Fig. 6) also indicates that the cross-bridge activity
was unchanged by the adaptation process. This follows from
Eq. 16, which is reproduced in a reorganized form
␣⬘/␣ ⫽ 共␴⬘/␴兲共v/v⬘兲共F⬘/F兲
If the increase in power output of a muscle adapted at a
longer length is due to an increase in Ns, while the number of
active cross bridges (Npl) in parallel remains the same, there
has to be an increase in the total content of myosin filaments to
allow for the increase in N. Because the myosin filament
content or mass (M) is proportional to the myosin filament
density (␴) and the muscle volume (V), that is
M ⬀ ␴V
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Fig. 6. Length-dependent muscle behaviors observed experimentally and explained by the model. Lengths are normalized by a reference length (Lref),
which is close to the in situ length of the muscle and corresponds to a resting
passive tension of ⬃1–2% of maximal isometric force. F And E represent
findings by Pratusevich et al. (23) in 2 separate experiments; other symbols
represent findings by Kuo et al. (17). All data groups have their own reference
value. Top: force-length relation from 3 experiments. Dotted line, length
independence of force as described (see DISCUSSION) by a model where length
adaptation is achieved by variation of Ns. Middle: velocity values (F, E, and ‚).
䊐, Myosin filament density in cross sections (␴); }, power; , rates of ATP
consumption. Power and ATP rate at 1.5 Lref are shown slightly off the length
mark for visibility of the data. All values (velocity, power, number of filaments
per unit area, and ATP rate) have the same linear dependence on adapted
muscle length, consistent with a model of length adaptation where Ns is
variable (see DISCUSSION). Best linear fit (solid line) has a y-intercept of
0.484 ⫾ 0.078 (SE) and a slope of 0.571 ⫾ 0.044. Bottom: muscle compliancelength relation. Best fit (solid line) has a y-intercept of 0.680 ⫾ 0.105 and a
slope of 0.321 ⫾ 0.057.
Because F is unchanged by the adaptation and ␴ and v are
changed by the same amounts, it is apparent that ␣ (a measure
of cross-bridge activity) was not changed by the adaptation.
Because ␣⬘/␣ ⫽ 1, it follows that the coefficient ␤ is also not
changed by the process of length adaptation; i.e., ␤⬘/␤ ⫽ 1,
because, from Eqs. 3, 4, and 14, P⬘/P ⫽ (␣␤⬘F⬘v⬘)/(␣⬘␤Fv),
where ␣⬘/␣ ⫽ 1, F⬘/F ⫽ 1, and P⬘/P ⫽ v⬘/v.
Pratusevich et al. (23) found a linear relation between the
muscle compliance and length. This indicates that the coefficient of muscle length (L) in Eq. 17 was also not affected by
the process of length adaptation. The linear relation is reproduced in Fig. 6 (bottom).
On the basis of the models described above, length adaptation affects only Ns in a cell.
Changes in force-velocity parameters after disruption of
contractile apparatus. We recently observed that oscillatory
strains applied to relaxed ASM decreased isometric force in the
subsequently induced contraction (30). The decrease in isometric force was accompanied by a similar extent of decrease in
the density of myosin thick filaments observed in transverse
sections of the muscle (18), suggesting that the force decrease
could be due to depolymerization of the thick filaments. Forcevelocity measurements revealed that, after a length oscillation,
there was an increase in shortening velocity, despite a decrease
in isometric force (32). We can use our model to explain these
results. We suggest that oscillation caused depolymerization
and fragmentation of the thick filaments, with the breaking of
some chains of contractile units. Thus Np was decreased, which
partly explains the decrease in isometric force. Because the
shortening velocity increased, we conclude that the number of
thick filaments (and, therefore, contractile units) in series (Ns)
within a cell increased. Thus, because the cell length was the
same after and before oscillation, we must conclude that l
decreased. The decrease in l also contributed to the decrease in
isometric force. The increase in velocity observed by Wang et
al. (32) was 8.2 ⫾ 3.3% (mean ⫾ SE), and the decrease in
isometric force was 17.8 ⫾ 0.8%; if these values reflected
changes in Ns and Npl, respectively, then the decrease in
muscle power predicted by Eq. 5 (P ⫽ ␤vuNsNpl) would be
11 ⫾ 2.9%, which is not different from the actually measured
decrease in maximal power output of the muscle: 13.6 ⫾
2.3% (32).
MATHEMATICAL MODEL OF SMOOTH MUSCLE
Model Predictions
Indirect measurement of myosin filament length. The model
indicates that it may be possible to obtain the absolute value of
the average length of the myosin filaments from density (␴)
measurements by taking slices of different thickness from the
same tissue sample (assuming lm/L ⬍⬍ 1 in Eq. 9)
␴ 2 /␴ 1 ⫽ 共l m ⫹ t2兲/共lm ⫹ t1兲
This equation can be rearranged to make lm the subject of the
equation
l m ⫽ 共␴1t2 ⫺ ␴2t1兲/共␴2 ⫺ ␴1兲
It may not be possible to obtain a good value for lm, because
the calculation involves twice taking differences of numbers
that will be similar. Thus the calculation will be very prone to
noise and may require a large sample size to obtain an accurate
estimate of lm.
Maximal isotonic shortening. A smooth muscle undergoing
isotonic shortening will reach a minimum length (Lmin) when
force generated by the muscle (F) equals that of the isotonic
load plus the internal load. Under this condition, l ⫽ F/(␣Np)
(Eq. 3) and L ⫽ Nslu ⫹ LSEC (Eq. 2). The model (Fig. 3)
suggests that, in an isotonic contraction, lu will continue to
decrease, even when it is shorter than lm. Under this condition,
lu ⫽ (l ⫹ d), where d is the length of a dense body. Therefore,
L ⫽ Ns(l ⫹ d) ⫹ LSEC (Eq. 2). The relation between Lmin and
F therefore is
L min ⫽ Ns关F/共␣Np兲 ⫹ d兴 ⫹ LSEC
⫽ 关N s/共␣Np兲兴F ⫹ 共Nsd ⫹ LSEC兲
If the terms [Ns/(␣Np)] and (Nsd ⫹ LSEC) remain constant
during an isotonic contraction (a reasonable assumption if no
adaptation occurs during a brief contraction) and if the elastic
internal load is negligible (a reasonable assumption if the
muscle has not shortened a lot), Eq. 18 predicts that Lmin is a
linear function of the isotonic load (F). Our preliminary data
(10) showed a highly linear relation between Lmin and isotonic
loads, especially at the region of high isotonic loads, where
Lmin is ⬎40% of the precontracted muscle length.
Length adaptation of smooth muscle can be described as
variation in Ns in proportion to L (17). The slope and yintercept of the linear Lmin-F plot (Eq. 18) therefore will vary
according to the adapted muscle length; the linear relation,
however, will be maintained. This means that the minimum
length of an adapted muscle undergoing isotonic contraction
will be longer at longer initial (adapted) lengths and vice versa.
Equation 18 shows that Lmin at zero isotonic load is proportional to Ns and, hence, to L, provided that LSEC is negligible
compared with Nsd.
Physiological and Clinical Applications of the Model
The model that we have developed, although overly simplistic, provides guidance for experiment design seeking correlation between changes in mechanical function and possible
alterations in the configuration of the arrays of contractile
units, a very useful function considering the scanty ultrastructural clues we obtain from smooth muscle in our effort to
unravel the secret of the contraction mechanism. In skeletal
muscle, structural data are often used to explain functional
properties. In smooth muscle, we may have to adopt a reverse
strategy, i.e., use clues from functional studies to seek supporting structural evidence. To make sense of the seemingly
random and uninterpretable structural data from smooth muscle, we need a model that straddles function and structure; a
precise functional state needs to be clearly defined before
meaningful interpretation of structural data in that functional
state can be obtained. For example, if the model is correct, the
density of myosin filaments is a function of adapted muscle
length (Eq. 15). Thus comparison of filament densities among
different cells will be meaningful only when the relative cell
length is taken into account.
The prediction that ASM will shorten to a shorter minimal
length if it is adapted at a shorter initial (resting) length (Eq.
18) has significant clinical implications. One of the leading
hypotheses in asthma research regarding the mechanism of
airway hyperresponsiveness (16, 25) is that in asthmatic airways the smooth muscle has been adapted to pathologically
short lengths, perhaps due to chronic stimulation by inflammatory mediators and (or) inflammation-driven remodeling of the
airway wall structure resulting in decoupling of the ASM layer
from the tethering provided by the lung parenchyma. Adaptation of ASM to short lengths therefore leads to shorter Lmin
(Eq. 18) and may cause complete airway closure when the
narrowing airways cross a critical caliber where little wall
tension is required to overcome the transmural pressure and to
collapse the airways, as predicted by the Laplace law.
GRANTS
(18)
J Appl Physiol • VOL
This study was supported by Canadian Institutes of Health Research
Operating Grants MT-13271 (to C. Y. Seow) and MT-4725 (to P. D. Paré).
96 • FEBRUARY 2004 •
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Lengthening of thick filaments during activation and development of isometric force. During the initial phase of force
development in an isometrically stimulated ASM, the shortening velocity (measured by abruptly releasing the muscle to
various isotonic loads) decreases (26). The experimenters’
interpretation of the data was that the myosin thick filaments
increased in length (lm) and that Ns decreased. Ultrastructural
evidence of increasing thick filament length during contraction
(5, 6, 11, 33, 34) supports this interpretation. An alternative
explanation is based on the “latch-bridge” hypothesis (2, 27).
Our present model of ASM supports the explanation that
invokes restructuring of the contractile units. We assume that
the shortening velocity of a contractile unit remains the same
after the muscle has been fully activated. Full activation
indicated by power output and myosin light chain phosphorylation is achieved in ASM before isometric force reaches a
plateau (21). Thus a decrease in shortening velocity must be
caused by a decrease of Ns (Eq. 4). While the velocity decreases, the force actually increases (26). Thus, not only does
Ns decrease, lm and also l, the length of overlap of actin and
myosin, must also increase to generate the increasing force. An
alternative method for increasing the force would be for Np to
increase (9). Hence, it is not necessary to postulate the existence of latch bridges to explain the observed change in
contractile characteristics during ASM contraction. This conclusion only applies to ASM and to contractions that last for
only a few seconds, and not to prolonged contractions encountered in vascular and some other smooth muscles.
475
476
MATHEMATICAL MODEL OF SMOOTH MUSCLE
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