## Abstract

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.

- muscle contraction
- contractile apparatus
- mechanics
- myosin filament

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 steady-state 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 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

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.

### 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. *N*_{s} such elements are connected in series to form one contractile chain. *N*_{p} such chains are arranged in parallel within the cell. For simplicity, we assume that the muscle is cylindrical with all chains containing *N*_{s} elements in series and that the chains are parallel to the long axis of the cylinder. The product of *N*_{s} and *N*_{p} gives the total number of myosin filaments (or contractile units) in the cell

Adjacent dense bodies in a chain are a distance *l*_{u} 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 *l*_{m} (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., *l*_{m} = *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 *L*_{SEC} 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 *N*_{p} and *l* 3 where α is the proportionality constant between F and *N*_{p}*l. 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 *v*_{u} 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 *L*_{SEC} 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 × *v* and is thus given by *Eq. 5* 5 where β 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 c_{u}, we have the compliance of a complete chain of contractile units given by *N*_{s}c_{u}. Compliances in parallel add as the reciprocals of the individual compliances. Thus the compliance of the contractile element (C_{CE}), shown schematically in Fig. 4, is given by the following equation when we assume that *L* >> *L*_{SEC} 6

The total compliance (C) of the model cell depicted in Fig. 4 is given by *Eq. 7* 7 where C_{PEC} is compliance of the parallel elastic component and C_{SEC} is compliance of the series elastic component.

### Model for Morphometry

We assume that *l*_{m} is small compared with *L*. The probability of any one myosin filament appearing in a randomly chosen cell transverse section is *l*_{m}/*L*. Thus the number of filaments appearing in a transverse section is *Nl*_{m}/*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 (*N*_{m}) 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 (σ) is given by *Eq. 9* 9

*L* and *N*_{s} are related through *l*_{u} (*Eq. 2*) for *L*_{SEC} « *L*. Thus, for *l*_{m} « *L*, σ can be written as follows which can be reorganized for calculation of *N*_{p} as follows

Thus, for very thin slices (*t*/*l*_{m} « 1), *N*_{p} exceeds σ*A* by a factor of *l*_{u}/*l*_{m} 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 σ 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 *N*_{s} and *N*_{p}. 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 *v*_{u} = *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 (*l*_{u}) 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 *l*_{u} 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, *l*_{m} = *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 α 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 C_{PEC} is very much greater than C_{CE} 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, 2, 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 (*N*_{s}) and in parallel (*N*_{p}), 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 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 (*v*_{u}) 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 (*N*_{s}, *Eq. 11*). This increase in *N*_{s} will in turn result in the same extent of increase in P if the term β*v*_{u}*N*_{p}*l* in *Eq. 5* stays constant. The matching increases in power and velocity suggest that the term β*v*_{u}*N*_{p}*l* 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 *N*_{s}, while the number of active cross bridges (*N*_{p}*l*) 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 (σ) and the muscle volume (V), that is 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*), σ 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 *N*_{s}. This is further corroborated by the finding (17) that the rate of ATP consumption also has the same length dependence as *v*, P, and σ (Fig. 6, *middle*).

The fact that σ 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 σ and *v* are changed by the same amounts, it is apparent that α (a measure of cross-bridge activity) was not changed by the adaptation.

Because α′/α = 1, it follows that the coefficient β is also not changed by the process of length adaptation; i.e., β′/β = 1, because, from *Eqs. 3, 4*, and *14*, P′/P = (αβ′F′*v*′)/(α′βF*v*), where α′/α = 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 *N*_{s} 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 *N*_{p} 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 (*N*_{s}) 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 *N*_{s} and *N*_{p}*l,* respectively, then the decrease in muscle power predicted by *Eq. 5* (P = β*v*_{u}*N*_{s}*N*_{p}*l*) 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 (*l*_{m}) and that *N*_{s} 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 *N*_{s} (*Eq. 4*). While the velocity decreases, the force actually increases (26). Thus, not only does *N*_{s} decrease, *l*_{m} 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 *N*_{p} 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 (σ) measurements by taking slices of different thickness from the same tissue sample (assuming *l*_{m}/*L* « 1 in *Eq. 9*)

This equation can be rearranged to make *l*_{m} the subject of the equation

It may not be possible to obtain a good value for *l*_{m}, 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 *l*_{m}.

*Maximal isotonic shortening.* A smooth muscle undergoing isotonic shortening will reach a minimum length (*L*_{min}) when force generated by the muscle (F) equals that of the isotonic load plus the internal load. Under this condition, *l* = F/(α*N*_{p}) (*Eq. 3*) and *L* = *N*_{s}*l*_{u} + *L*_{SEC} (*Eq. 2*). The model (Fig. 3) suggests that, in an isotonic contraction, *l*_{u} will continue to decrease, even when it is shorter than *l*_{m}. Under this condition, *l*_{u} = (*l* + *d*), where *d* is the length of a dense body. Therefore, *L* = *N*_{s}(*l* + *d*) + *L*_{SEC} (*Eq. 2*). The relation between *L*_{min} and F therefore is 18

If the terms [*N*_{s}/(α*N*_{p})] and (*N*_{s}*d* + *L*_{SEC}) 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 *L*_{min} is a linear function of the isotonic load (F). Our preliminary data (10) showed a highly linear relation between *L*_{min} and isotonic loads, especially at the region of high isotonic loads, where *L*_{min} is >40% of the precontracted muscle length.

Length adaptation of smooth muscle can be described as variation in *N*_{s} in proportion to *L* (17). The slope and *y*-intercept of the linear *L*_{min}-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 *L*_{min} at zero isotonic load is proportional to *N*_{s} and, hence, to *L*, provided that *L*_{SEC} is negligible compared with *N*_{s}*d*.

### 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 *L*_{min} (*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

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).

## Footnotes

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- Copyright © 2004 the American Physiological Society