| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
1Institute 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
 |
ABSTRACT |
|---|
 TOP ABSTRACT THE MODELDISCUSSIONGRANTSREFERENCES |
|---|
muscle contraction; contractile apparatus; mechanics; myosin filament
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 force-velocity 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 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 |
|---|
|
TOPABSTRACTTHE MODELDISCUSSIONGRANTSREFERENCES |
|---|
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 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
 |
|
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
 | (1) |
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)
 | (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
 | (3) |
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
 | (4) |
We assume that LSEC does not change during the steady-state 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 x v and is thus given by Eq. 5
 | (5) |
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
 | (6) |
The total compliance (C) of the model cell depicted in Fig. 4 is given by Eq. 7
 | (7) |
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
 | (8) |
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
 | (9) |
|
L and Ns are related through lu (Eq. 2)
 |
can be written as follows
 |
 |
Thus, for very thin slices (t/lm « 1), Np exceeds A by a factor of lu/lm
 | (10) |
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
 |
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
 | (11) |
The ratio of muscle lengths before and after adaptation (Eq. 2) is
 |
We assume that the noncontractile lengths contribute negligibly to this ratio. Hence
 | (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
 | (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
 | (14) |
Plasticity of L
A hallmark of plasticity is that maximal isometric force can be achieved at any muscle length within the adaptable length 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)
 | (15a) |
 | (15b) |
Equation 15, a and b, can be combined to yield Eq. 16
 | (16) |
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
 | (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 |
|---|
|
TOPABSTRACTTHE MODELDISCUSSIONGRANTSREFERENCES |
|---|
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.
|
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
 |
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
 |
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. Force-velocity 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).
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.
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)
 |
This equation can be rearranged to make lm the subject of the equation
 |
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
 | (18) |
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 y-intercept 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 |
|---|
|
TOPABSTRACTTHE MODELDISCUSSIONGRANTSREFERENCES |
|---|
|
FOOTNOTES |
|---|
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.
|
REFERENCES |
|---|
|
TOPABSTRACTTHE MODELDISCUSSIONGRANTSREFERENCES |
|---|
This article has been cited by other articles:
|
|
 |
|
  J. E. Speich, A. M. Almasri, H. Bhatia, A. P. Klausner, and P. H. Ratz Adaptation of the length-active tension relationship in rabbit detrusor Am J Physiol Renal Physiol, October 1, 2009; 297(4): F1119 - F1128. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 F. Ali, L. Chin, P. D. Pare, and C. Y. Seow Mechanism of partial adaptation in airway smooth muscle after a step change in length J Appl Physiol, August 1, 2007; 103(2): 569 - 577. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
J. E. Speich, C. Dosier, L. Borgsmiller, K. Quintero, H. P. Koo, and P. H. Ratz Adjustable passive length-tension curve in rabbit detrusor smooth muscle J Appl Physiol, May 1, 2007; 102(5): 1746 - 1755. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
P. S. P. Silveira, J. P. Butler, and J. J. Fredberg Length adaptation of airway smooth muscle: a stochastic model of cytoskeletal dynamics J Appl Physiol, December 1, 2005; 99(6): 2087 - 2098. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 C. Y. Seow Myosin filament assembly in an ever-changing myofilament lattice of smooth muscle Am J Physiol Cell Physiol, December 1, 2005; 289(6): C1363 - C1368. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 A. M. Herrera, B. E. McParland, A. Bienkowska, R. Tait, P. D. Pare, and C. Y. Seow `Sarcomeres' of smooth muscle: functional characteristics and ultrastructural evidence J. Cell Sci., June 1, 2005; 118(11): 2381 - 2392. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
P. J. Sterk Heterogeneity of airway hyperresponsiveness: time for unconventional, but traditional, studies J Appl Physiol, June 1, 2004; 96(6): 2017 - 2018. [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
, 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 
1-2% of maximal isometric force. 
And 
represent findings by Pratusevich et al. (
). 
, 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 compliance-length relation. Best fit (solid line) has a y-intercept of 0.680 ± 0.105 and a slope of 0.321 ± 0.057.