## Abstract

Although the structure of the contractile unit in smooth muscle is poorly understood, some of the mechanical properties of the muscle suggest that a sliding-filament mechanism, similar to that in striated muscle, is also operative in smooth muscle. To test the applicability of this mechanism to smooth muscle function, we have constructed a mathematical model based on a hypothetical structure of the smooth muscle contractile unit: a side-polar myosin filament sandwiched by actin filaments, each attached to the equivalent of a Z disk. Model prediction of isotonic shortening as a function of time was compared with data from experiments using ovine tracheal smooth muscle. After equilibration and establishment of in situ length, the muscle was stimulated with ACh (100 μM) until force reached a plateau. The muscle was then allowed to shorten isotonically against various loads. From the experimental records, length-force and force-velocity relationships were obtained. Integration of the hyperbolic force-velocity relationship and the linear length-force relationship yielded an exponential function that approximated the time course of isotonic shortening generated by the modeled sliding-filament mechanism. However, to obtain an accurate fit, it was necessary to incorporate a viscoelastic element in series with the sliding-filament mechanism. The results suggest that a large portion of the shortening is due to filament sliding associated with muscle activation and that a small portion is due to continued deformation associated with an element that shows viscoelastic or power-law creep after a step change in force.

- mathematical model
- contractile unit
- sliding-filament mechanism

the structure of the contractile unit in smooth muscle is unknown, despite the common belief that the cycling-cross-bridge/sliding-filament mechanism of contraction, which operates in striated muscle, is also responsible for smooth muscle contraction. The predominant model for the sarcomere-equivalent contractile unit in smooth muscle is a side-polar myosin (thick) filament sandwiched by two oppositely oriented actin (thin) filaments, each attached to a dense body that is believed to function like a Z disk in striated muscle (18, 21, 27), as shown in Fig. 1. An important assumption associated with the model is that the sliding of the thick and thin filaments relative to each other is caused by repetitive interaction of the myosin heads (cross bridges) of the thick filament with the thin filaments, in the same manner described for striated muscle (22, 23). Such a cross-bridge mechanism for smooth muscle myosin has been observed in experiments where the interaction of isolated individual myosin heads with thin filaments was visualized and quantified directly (14, 15, 28). As predicted by Huxley's 1957 model (22), the actomyosin cross-bridge interaction should result in a load-dependent shortening velocity that can be described by a hyperbolic function. Such a hyperbolic relationship between load and shortening velocity has been demonstrated in isolated single cells (43) and cell bundles (17, 39) of smooth muscle. It appears therefore that there is strong evidence for actomyosin interaction as the molecular mechanism in smooth muscle contraction, as in striated muscle. However, how the contractile filaments are organized to produce force and sliding in smooth muscle is not clear, and the model shown in Fig. 1 remains a hypothetical concept.

One study designed to test the sliding-filament model (Fig. 1) in smooth muscle has provided some additional supporting evidence (18). In that study, it was assumed that the amount of overlap between the thick and thin filaments would decrease linearly as the muscle shortened; the data revealed a linear dependence of active force on muscle length (18), consistent with the model prediction. With the establishment of relationships between force and velocity and between length and force, we are able to integrate the information and predict the time course of muscle shortening under a constant load. If the sliding-filament mechanism is indeed responsible for smooth muscle shortening, a model based on that mechanism should account for the shortening produced by the muscle. In the present study, we have developed a mathematical description for a model that consists of contractile units such as that shown in Fig. 1 in series with components that exhibit elastic, viscoelastic, and power-law creep behavior. The model prediction was then compared with data from experiments.

## MATERIALS AND METHODS

To test the models described below, data were obtained from airway smooth muscle (ASM) that included length records of isotonic shortening after quick releases from the isometric force plateau and the force-velocity properties.

### Muscle Preparation

Ovine tracheas obtained from a local abattoir were used in these experiments. The use of tissue was approved by the Committee on Animal Care of the University of British Columbia. Soon after the animals were killed, tracheas were removed and put in physiological saline solution (PSS; pH 7.4, 118 mM NaCl, 4 mM KCl, 1.2 mM NaH_{2}PO_{4}, 22.5 mM NaHCO_{3}, 2 mM MgSO_{4}, 2 mM CaCl_{2}, and 2 g/l dextrose). After transportation to the laboratory, tracheas were cleaned and kept in PSS at 4°C for ≤4 days before use. Muscle strips for experiments were prepared from a ∼2-cm-long tracheal segment taken from the middle portion of the trachea. Before dissection, the tracheal rings were examined under the dissecting microscope for signs of muscle contracture. A wrinkled epithelial layer above the muscle layer was an indication that the ASM was in a contracted state. Only the tracheas with relaxed ASM were used for the experiments. The in situ length of the relaxed tracheal smooth muscle bundle connecting the C-shaped cartilage ring was measured. The tracheal rings were then cut open, and adventitial connective tissue and the epithelial layer were removed from the muscle layer. Muscle strips (∼6 mm long, 1.5–2 mm wide, ∼0.2–0.3 mm thick) were dissected, and aluminum foil clips were affixed on both ends for attachment to the force-length transducer.

The muscle strip was mounted vertically in an organ bath between two hooks. The bottom hook was fixed, and surgical thread (size 6) was used to connect the upper hook to the lever arm of a force-length transducer. The muscle strip was then immersed in PSS aerated with 95% O_{2}-5% CO_{2} at 37°C. Warm water was circulated through the jacket around the muscle bath to maintain the 37°C bath temperature. The muscle strip was stretched to its previously measured in situ length, which was used as a reference length (*L*_{ref}) for normalization of all length measurements. Muscle strips were equilibrated before initiation of the experiments to allow the muscle to recover from mechanical and metabolic perturbations caused by dissection, lack of perfusion, and low storage temperature. During the equilibration period, the muscle strip was activated every 5 min with an 8-s electrical field stimulation (EFS). Every 5 min, PSS in the muscle bath was replaced with fresh PSS prewarmed to 37°C. Equilibration was considered complete when stimulations produced a stable maximal EFS-induced force with low (2.09 ± 0.028 and 1.43 ± 0.020 mN for *muscles A* and *B*, respectively) resting tension. The resting tension was averaged over a 5-s time period before the muscle was stimulated. The EFS-induced force referred to active force only, that is, total force minus resting force. The equilibration process took ∼1.5 h.

### Experimental Procedure

After equilibration, the muscle was stimulated with 100 μM ACh for ∼6 min as a control observation to determine how quickly the force reached the plateau value (F_{max}) and how stable the plateau remained over time (Fig. 2, *top*). On the basis of the control observation, an experimental protocol was adopted. The muscle was stimulated (100 μM ACh) for 2 min and then allowed to shorten against a preset isotonic load (Fig. 2, *bottom*). This process was repeated five more times for isotonic loads that ranged from 5% to 75% F_{max}. The isotonic contraction lasted 2.5–3.5 min. The initial shortening velocity associated with each isotonic shortening (shortening rate measured 150 ms after the quick release) was determined and used later for construction of force-velocity curves. The 150-ms time delay was implemented to minimize contamination of velocity measurements by the viscoelastic recoil associated with the sudden change in load. Data were collected at a rate of 20 Hz. After each shortening, the muscle was equilibrated using an 8-s EFS every 5 min for 30 min before the next isotonic contraction. To test whether the shortening velocity changes between 2 and 6 min after ACh stimulation, we applied isotonic quick releases at 2, 3, 4, 5, and 6 min after ACh stimulation. Because we are only interested in the relative change in velocity, a single isotonic load of 20% F_{max} was chosen for each release at different times resting. The results are shown in Fig. 3.

### Models

#### Contractile unit model.

As shown in Fig. 1, the model of a contractile unit of smooth muscle consists of a side-polar thick filament (6) flanked by thin filaments of opposite polarity. The thin filaments are anchored to dense bodies that are believed to be the Z-disk equivalents in smooth muscle (4, 11, 26, 42). Our recently reported findings suggest that dense bodies may not be the Z-disk equivalents in smooth muscle (46); we therefore use the term “Z-disk equivalent body” or “Z body” to describe the cellular element that functions as the anchorage site for thin filament attachment. Shortening of the contractile unit is brought about by the sliding of the thin filaments in opposite directions. Three assumptions are associated with the contractile unit model: *1*) the thick filament spans the length of the thin filament lattice, from Z body to Z body; *2*) the ability of the contractile unit to generate force is directly proportional to the amount of overlap between the thick and thin filaments; and *3*) the ends of the thick filament sliding past the Z bodies can no longer contribute to force generation. Bound by these assumptions, the model predicts that the ability of the contractile unit to generate force is directly proportional to its own length, as shown in Fig. 4. A linear relationship between length and force has been found to exist in ASM (18); this finding is consistent with the model proposed above.

The length-force relationship shown in Fig. 4 can be represented by a mathematical expression, F(*L*) = F_{i} + [(F_{max} − F_{i})*/*(*L*_{max} − *L*_{i})]·(*L* − *L*_{i}), or, by expressing length and force values as fractions of *L*_{max} and F_{max}, respectively, we have
_{max}, *L*_{max}, isotonic load (F_{i}), and muscle length at F_{i} (*L*_{i}) are defined in the legend of Fig. 4, and (1 − F_{i})/(1 − *L*_{i}) = *S*, the slope of the linear curve in Fig. 4.

#### Mechanical model for muscle preparation.

The muscle preparation used in this study is a bundle of trachealis dissected from ovine trachea that contained tens of cells in series and hundreds in parallel embedded in extracellular matrix containing collagen and elastin fibers, with overall dimensions as follows: ∼5 mm long, 1 mm wide, and 0.3 mm thick. Figure 5 is a mechanical representation of the muscle preparation. In our experiment, the resting tension of the muscle at the reference length was negligible (<3% F_{max}); therefore, the parallel viscoelastic element was assumed to have no effect on the time course of the muscle's shortening and, for simplicity, was not included in our mathematical analysis. Also in our experiment, the muscle was first activated at *L*_{max} isometrically and then by an isotonic quick release from F_{max}. The following mathematical analysis was performed only on the isotonic shortening after the quick release.

#### Mathematical description of the output of the model in terms of length change due to the sliding-filament mechanism.

We start with the Hill equation (19) describing the relationship between shortening velocity and load
*a* and *b* are the Hill constants, *V* is shortening velocity, F_{i} is isotonic load, and F(*L*) is the force generated by the muscle at the moment the velocity is measured. It should be pointed out that the hyperbolic force-velocity relationship discovered by Hill (19) was later shown by Huxley (22) to originate from cross-bridge interactions within the sarcomeres. The hyperbolic relationship is therefore one of the cornerstones of the sliding-filament cross-bridge theory of muscle contraction. Since in smooth muscle the force produced by a shortening muscle decreases linearly with muscle length, as described by *Eq. 1*, we substitute F(*L*) in *Eq. 2* with *Eq. 1* and set *V* = −d*L*/d*t* (shortening is taken as a negative change in length, thus the negative sign)
*Eq. 3* yields
*L*_{contraction} in Fig. 5). More details of the mathematical derivation are presented in the appendix.

An implicit assumption associated with *Eq. 4* is that, during the isotonic contraction, the rate of shortening is independent of time and is only dependent on the load seen by the cross bridges (i.e., load per bridge). However, shortening velocity of smooth muscle has been shown to decrease over time, especially during the early phase of contraction, and this decrease is independent of external load (8, 16, 36). As shown in Fig. 3, even in the sustained phase of contraction (2–6 min after activation), where there was a flat force plateau, a slight, but significant, decrease in velocity continued. The velocity decline could be mathematically described as an exponential decay (solid line in Fig. 3); it could also be described as a linear decrease (dashed line in Fig. 3) with almost equal goodness of fit (at least for the time period considered).

*Equation 5*, which includes the time-dependent velocity decline, is a modified form of *Eq. 3*
*ct*) describes the velocity decline over the time interval where isotonic shortening occurs, with *time 0* defined as the start time of the isotonic shortening. The constant *c* is the slope of the linear velocity decline (dashed line in Fig. 3). The reason for the use of a linear, instead of an exponential, function to describe the velocity decline is based purely on the simplicity associated with the mathematical description. It should be pointed out that the term (1 − *ct*) describing the velocity decline with time is purely empirical; no underlying mechanism for the change in velocity has been assumed.

Integration of *Eq. 5* yields
*L*_{contraction} in Fig. 5). When *c* = 0, *Eq. 6* becomes *Eq. 4* (see appendix for more details of derivation of *Eq. 6*).

#### Mathematical description of the output of the model in terms of length change due to deformation of the in-series viscoelastic element after a step change in load.

##### case 1: linear spring in parallel with newtonian viscous element.

After a step change from an isometric load to a lighter isotonic load, the viscoelastic element in series with the contractile element will undergo creep (in this case, shortening). As described in the appendix, the creep displacement can be modeled as
*K*_{1} is the spring constant and *L* denotes *L*_{creep} (as illustrated in Fig. 5).

##### case 2: power-law creep.

As elaborated in more detail in the discussion, deformation of smooth muscle cells due to a change in force can be described by power-law creep (or, in this case, shortening). Therefore, instead of the viscoelastic deformation described in *case 1*, power-law creep is used in this case to represent the displacement produced by the in-series viscoelastic element (shown in Fig. 5, *L*_{creep}) and can be described mathematically as
*A* and γ are constants and *L*_{m} is the initial length of *L*_{creep} (Fig. 5) under isometric force before the isotonic quick release.

Curve fitting with the two possible creeps [viscoelastic (*case 1*) and power law (*case 2*)] is presented in results.

#### Mathematical description of combined shortenings produced by filament sliding, viscoelastic or power-law deformation, and recoil of the series elastic component.

Because the contractile element, the viscoelastic element, and the series elastic component are in series (Fig. 5), displacements produced by these elements after an isotonic quick release are additive. If we assume that there is no time-dependent intrinsic slowing of shortening velocity, the time course of isotonic shortening can be described mathematically in the following equation for contraction modeled by a contractile element operating with the sliding-filament mechanism in series with elements displaying Newtonian viscoelastic creep and elastic recoil (i.e., summation of *Eqs. 4* and *7*)
*C*_{1} is a constant (see appendix for detailed derivation).

For the contraction modeled by a contractile element operating with the sliding-filament mechanism (again, with the assumption of no slowing of time-dependent intrinsic shortening velocity) in series with elements displaying power-law creep and elastic recoil (i.e., summation of *Eqs. 4* and *8*)
*C*_{2} is a constant.

If we assume that there is an intrinsic time-dependent slowing of velocity during the isotonic shortening, *Eq. 9* becomes
*Eq. 10* becomes

#### Curve fitting.

In the curve-fitting experiments, the maximal rate of shortening occurred near the beginning of an isotonic contraction. For each isotonic load, we measured the rate of shortening 150 ms after the muscle was released from isometric contraction, and the pair of data (isotonic load and shortening velocity) constituted a force-velocity point. Six such points were obtained at 5, 10, 15, 30, 50, and 75% of F_{max}. The data were then fit with Hill's hyperbolic equation (*Eq. 2*) (19). A nonlinear method (SigmaPlot version 10.0) was used to perform the curve fitting.

The experimental data constituting the time course of isotonic shortening to each load were also fit using SigmaPlot version 10.0, according to the mathematical models developed above (see *Mathematical description of combined shortenings produced by filament sliding, viscoelastic or power-law deformation, and recoil of the series elastic component*). In fitting the data with *Eq. 9*, none of the fitting parameters (constants in the equation) were constrained. That is, the partition of the two exponential components in terms of their amplitudes and rate constants was left entirely to the fitting algorithm. However, in fitting the data to *Eq. 10*, we kept the same amplitude and rate constant for the first exponential term obtained from curve fitting with *Eq. 9* and let other constants in the equation vary. The rationale for this stemmed from our assumption that the two models represented by *Eqs. 9* and *10* shared the same contractile element (thus, the same exponential component) but differed in the “creep” component: one was due to Newtonian viscoelastic creep (*Eq. 9*), and the other was due to power-law creep (*Eq. 10*).

## RESULTS

As shown in Fig. 2, stimulation of the muscle with ACh (100 μM) resulted in a sustained contractile force, which we termed F_{max}. At the plateau of contraction, the muscle was allowed to shorten isotonically against loads of varying percentages of F_{max}, so that a force-velocity relationship could be obtained. Two ASM preparations (*muscles A* and *B*) from different tracheas were used in this study; two sets of data were obtained from these two preparations: six isotonic shortening length-force records and six force-velocity points. Experimental data presented in this section are primarily from *muscle A*; only part of the data from *muscle B* are presented graphically, because the models fit both sets of data (from *muscles A* and *B*) equally well. The Hill constants were as follows: *a* = 0.148 ± 0.032 (F_{max}) and *b* = 0.048 ± 0.016 (*L*_{ref}/s) for *muscle A* and *a* = 0.199 ± 0.021 (F_{max}) and *b* = 0.050 ± 0.007 (*L*_{ref}/s) for *muscle B*. The force-velocity curves for *muscles A* and *B* are presented in Fig. 6, *A* and *B*. Linear regression on the length-force relationships (taken from the maximally shortened lengths and their corresponding isotonic loads) for *muscles A* and *B* was carried out to determine the slope *S* (Fig. 6, *C* and *D*). For *muscle A*, *S* = 1.204 ± 0.046 (F_{max}/*L*_{ref}); for *muscle B*, *S* = 1.234 ± 0.055 (F_{max}/*L*_{ref}).

The shortening traces could not be modeled satisfactorily with just one single-exponential or power-law process; this is demonstrated in one example shown in Fig. 7, *A* and *B*. Interestingly, fitting the data with *Eq. 4* (which assumes no decrease in the cross-bridge cycling rate during the isotonic shortening) or *Eq. 6* (which assumes a decrease in the cycling rate) produced virtually the same results, with the latter improving the goodness of fit marginally (but significantly from 0.9057 to 0.9152; Table 1) and little overall improvement in the quality of data description (Fig. 7, *A* and *C*). Figure 8 shows paired comparisons between data fit with *Eqs. 4* and *6*. Although there is a statistically significant improvement in the goodness of fit in all traces using *Eq. 6* (Table 1), there is no visually discernable improvement in the quality of fit. In fitting the data with *Eq. 6* (Fig. 8*B*), we allowed the constant *c* (negative slope of the linear fit from Fig. 3) to float freely. (Note that the sets of data for Figs. 3 and 8 were not obtained from the same muscle preparations.) Nevertheless, the *c* value from Fig. 3 (0.0007 *L*_{ref}/s) is comparable with the *c* values obtained from Fig. 8 (Table 1). Taken together, the intrinsic, time-dependent slowing of velocity alone cannot account for the characteristic features of isotonic shortening; on the other hand, a single-exponential function can account for most of the isotonic shortening, albeit not perfectly.

In our next step of curve fitting, we expanded *Eq. 4* to include another exponential term (*Eq. 9*) or a power-law term (*Eq. 10*). From the perspective of the model shown in Fig. 5, during an isometric contraction, the contractile element shortens while the viscous and series elastic elements lengthen. After the isotonic quick release, the viscous and series elastic elements shorten. The series elastic element shortening is instantaneous, whereas the process of viscous element shortening is described in our model as either viscoelastic recoil following an exponential time course or a creep following a power-law time course. The results are shown in Fig. 9. Data from *muscle A* fit by *Eq. 9* are shown in Fig. 9*A*; the same data fit by *Eq. 10* are shown in Fig. 9*B*. The fitting parameters are listed in Tables 2 and 3. Judging from the goodness of fit (*R*^{2}) for all fitted curves (*muscle A*; Tables 2 and 3), a significantly better fit (*P* = 0.0001, 2-way ANOVA) was produced by *Eq. 9* than *Eq. 10*. For *muscle B*, again, *Eq. 9* gave a better fit than *Eq. 10* (*P* = 0.0001).

We then proceeded to fit the data with *Eqs. 11* and *12*. A comparison of goodness of fit among data fit with *Eqs. 9–12* for one trace (isotonic shortening at 15% F_{max} from *muscle A*) is shown in Fig. 10. Although *Eqs. 11* and *12* produced slightly better fits (judging from the *R*^{2} values) than *Eqs. 9* and *10*, respectively, the improvement cannot be appreciated by eye. The *R*^{2} values for all six traces fit with *Eqs. 11* and *12* are listed in Tables 4 and 5, respectively. Two-way ANOVA on the *R*^{2} values indicate a better fit to the data by *Eq. 12* than *Eq. 11* for *muscles A* (*P* = 0.0383) and *B* (*P* = 0.0097). *F*-tests were also carried out for the comparisons between *Eqs. 9* and *11* and between *Eqs. 10* and *12* in terms of the goodness of fit; the results are listed in Table 6.

We also examined the goodness of fit in data fit with three exponential terms (by adding another exponential term to *Eq. 9*) and with two exponential terms plus a power-law term (by adding another exponential term to *Eq. 10*). The *R*^{2} values are reported in Tables 7 and 8. With an additional exponential term, as expected, the fits are slightly better from a statistical point of view. However, again, the improvement is marginal in terms of the overall accuracy and precision of data description.

Figure 11*A* shows the correlation between the experimentally measured shortening velocities (from *muscle A*) and the rate constants (obtained from curve fitting) of the first exponential term in *Eq. 9* (which is the same as the exponential term in *Eq. 10*). This indicates that the rate constants varied in the same load-dependent manner as did the shortening velocities. Figure 11*B* shows the correlation between the same rate constants in Fig. 11*A* and the constants calculated from the exponent [*bS*/(F_{i} + *a*)]; the individual constants within the exponent were obtained independently from force-velocity and length-force relationships of the corresponding muscles.

## DISCUSSION

The major finding of this study is that the time course of isotonic shortening can be described accurately by a simple model with three components in series: *1*) the contractile element operating with a sliding-filament mechanism, *2*) an element that undergoes Newtonian viscoelastic or power-law creep when subjected to a step change in force, and *3*) a series elastic component (Fig. 5). The results (Figs. 7 and 8) demonstrate that the data can be described mathematically, although not perfectly, with one exponential or power-law process. The goodness of fit can be improved drastically by inclusion of another exponential process in the curve fitting, as demonstrated in the fits produced by *Eqs. 9* and *10* (Fig. 9). An interesting and somewhat unexpected result was the trivial improvement in the goodness of fit (Figs. 7 and 8, Table 1) when we took into account the load-independent decrease in velocity (possibly due to slowing of cross-bridge cycling rate over time) during the time course of isotonic shortening (Fig. 3). The results suggest that the change in slope as a function of time in isotonic shortening traces has little to do with the slowing in the intrinsic (load-independent) shortening velocity of the muscle (Fig. 3). This conclusion is also supported by the marginal improvement in the goodness of fit using *Eq. 11* over *Eq. 9* or *Eq. 12* over *Eq. 10* (Fig. 10, Tables 2⇑⇑⇑–6). Adding another exponential term to *Eqs. 9* and *10* again produced marginal improvement in the goodness of fit (Tables 7 and 8), and the improvement was not discernable by eye (graphs not shown).

In fitting the data with the various equations that we have developed from the models, we have observed, in many cases, that a statistically significant improvement in the goodness of fit (*R*^{2}) did not accompany a real improvement in the “goodness” of the model in accommodating or explaining the data. Figure 10 illustrates a good example. In the absence of a convincing reason, we have little justification to use equations more complicated than *Eqs. 9* and *10*. Therefore, we concentrate on the explanation of data based on the models represented by *Eqs. 9* and *10*.

### Assumptions Associated With the Models

Many assumptions were made with regard to the model of the contractile unit (Fig. 1) and the model for the muscle tissue preparation (Fig. 5). Comparison of the model output with experimental data has provided opportunities to assess the validity of these assumptions. One important determinant for the proposed structure of the contractile unit of smooth muscle is the side-polar thick filament; evidence for such filaments in smooth muscle was provided by several key studies (6, 37, 40, 45). The row-polar thick filament suggested by Hinssen et al. (20) could also work for the proposed contractile unit. Evidence for the dense bodies acting as anchorage sites for the thin filaments was provided by some early studies (4, 11, 26, 42). A recent study (46) showed that dense bodies formed long cable-like structures within smooth muscle cells and were capable of bearing passive tension. This finding questions the commonly assumed role for dense bodies, although it does not rule out the possibility that some of the dense bodies function as the Z-disk equivalents. The studies cited above have provided evidence for the structural elements necessary for constructing the smooth muscle contractile unit akin to the sarcomere in striated muscle, but how exactly these elements are put together to form a contractile unit remains unclear, and any proposed structural models can only be regarded as hypothetical without new evidence. Because an unambiguous confirmation of the proposed structure in this study (Fig. 1) is not likely to come from structural studies soon, we have opted to seek functional evidence to verify the model.

Studies from our laboratory have shown that, in contrast to striated muscle, the ascending limb of the length-force relationship in ASM is linear (18), as predicted by the model (Fig. 1). The linear relationship has also been observed in the present study (Fig. 6). The present study has provided further functional evidence to suggest that the sliding-filament mechanism as illustrated in Fig. 1 is at the heart of the contractile apparatus of smooth muscle. This conclusion, of course, depends on the assumptions we made regarding the models shown in Figs. 1 and 5. One of the important assumptions with regard to the model shown in Fig. 1 is that there are no gaps in the overlap between the thick and thin filaments. That is, the thick filament spans the whole distance between the Z bodies. This is to ensure that as the muscle (or contractile unit) shortens, the model would predict a linear decrease in active force with muscle length, as demonstrated in experiments (Fig. 6). This is important for *Eq. 1* to be valid. Several assumptions were made for the tissue model (Fig. 5), primarily, because of the lack of necessary information and, secondarily, for the purpose of simplifying the mathematical analysis. One of the assumptions is that the parallel viscoelastic element in the tissue model (depicted in gray in Fig. 5) does not affect the time course of isotonic shortening. This assumption would be clearly invalid if there was a substantial amount of resting tension in the muscle preparation at the chosen reference length because of the inevitable power-law creep (or viscoelastic recoil) that would occur after the isotonic quick release (24, 31). In our experiments, the resting tension of the muscle preparations was kept at <3% of F_{max} (Fig. 2); this has allowed us to ignore the effect of the parallel element. Resisting force stemming from compression of this parallel element does not seem to be a factor because of the linear relationship we have observed in the ascending portion of the length-force curve for the muscle (Fig. 6) (18).

### Model Predictions and Experimental Results

Curve fitting with *Eqs. 9* and *10* has produced a reasonably accurate mathematical description for the time course of isotonic shortening in ASM (Fig. 9). In fitting the data with *Eq. 9*, it was clear that the shortening at lower loads was dominated by the first exponential term with a rate constant more than an order of magnitude higher than that of the other exponential term (Table 2). In addition, the rate constants of this faster exponential process decreased with increasing load in the same fashion as the decrease in shortening velocity (Fig. 11*A*). Note that the curve fitting was not constrained in any way, so the emergence of this load-dependent rate constant associated with the fast exponential process was not a guaranteed result from curve fitting. We took this as evidence suggesting that, embedded in an isotonic shortening trace, there was a component that stemmed from displacements due to the sliding-filament mechanism. We have preserved this fast exponential component in the data fitting with *Eq. 10*, with an implicit assumption that the difference between the two models (represented by *Eqs. 9* and *10*) is in the slow exponential component (second exponential term in *Eq. 9*) and the power-law component (in *Eq. 10*).

Although the data are better fit, statistically, by *Eq. 9* than *Eq. 10*, the goodness of fit associated with *Eq. 10* is so close to unity that we cannot dismiss the model represented by *Eq. 10*. We can only conclude at this point that both models are adequate in explaining the time course of isotonic shortening.

In *Eqs. 9* and *10*, the rate constant for the fast exponential process can be calculated from measurements of experimental records, because the rate constant is defined by *bS*/(F_{i} + *a*). Instead of letting the curve-fitting algorithm determine the rate constant, we could have used the calculated value. Figure 11*B* shows a correlation plot between the rate constants determined by curve fitting and those calculated from the term *bS*/(F_{i} + *a*). A good correlation was found (*R*^{2} = 0.989); however, the values obtained from curve fitting are higher than those calculated from *bS*/(F_{i} + *a*). This suggests that the calculation based on the term *bS*/(F_{i} + *a*) underestimates the true rate constant for the fast exponential process. If we recall that the Hill constants *a* and *b* were obtained from the force-velocity relationship (Fig. 6) and that the velocities were measured 150 ms after the quick release to avoid contamination by the viscoelastic recoil, this may have led to an underestimation of the true rate of muscle shortening, which would manifest as a smaller value for the constant *b*. Other unidentified factors could also contribute to the underestimation of the rate constant. However, the linear relationships shown in Fig. 11 indicate that the fast exponential term in *Eqs. 9* and *10* is related to the characteristic load-dependent shortening velocity of the muscle.

There are some limitations of the models presented in this study. Besides the limitations that stem from the simplifying assumptions described above, our models are based primarily on two steady-state relationships: the force-velocity relationship and the length-force relationship. The models are therefore incapable of describing tension or velocity transients or any non-steady-state muscle behavior. The validity of our model depends on the validity of the force-velocity and length-force relationships. Although these two relationships are macroscopic manifestations of cross-bridge interactions within the confines of the contractile units that possess filaments with finite tensile stiffness, it is beyond the scope of our model to describe the kinetics of cross-bridge interaction (32) and how filament extensibility may affect the interaction (33).

### Comparison With Previous Models

Numerous models have been published to describe and explain certain features of smooth muscle function (1–3, 5, 9, 13, 16, 24, 25, 27, 34, 38, 44). Some of these models are focused on the cross-bridge kinetics or manifestation of mechanical properties stemming from the kinetics and are not specifically designed to test the sliding-filament mechanism within the confines of a contractile unit. None of the models uses quantitative comparison of model output with isotonic shortening data as a strategy to verify the model. However, many agreements can be found between our model and previous models. Bates and Lauzon (3) showed that the force response to length oscillation in smooth muscle could be modeled by a simple serial arrangement of a contractile element (governed by a hyperbolic force-velocity relationship) and a series elastic component representing noncontractile tissue property. Our model is similar to the model of Bates and Lauzon, except for an additional viscoelastic element added in series (Fig. 5). To match the isotonic shortening traces, we found that the added viscoelastic element is crucial in producing a more precise fit (cf. Figs. 8 and 9).

For cultured or isolated smooth muscle cells, the stress relaxation or creep compliance after a step change in strain or stress varies with time according to a weak power law (7, 10, 34, 41). In muscle strips with cells embedded in extracellular matrix, the viscoelastic properties of the preparation are modified to some extent compared with the cultured or single-cell preparations (25). In our model, we have partitioned the total amount of isotonic shortening and attributed a portion of the displacement to an element displaying power-law creep (*Eq. 10*). In a parallel model represented by *Eq. 9*, we have attributed the same portion of displacement to viscoelastic recoil. Both models explain the isotonic data adequately.

It is important to point out that the viscoelastic or power-law component identified in the isotonic shortening does not necessarily represent passive tissue mechanical properties. It is more likely that the bulk of the component comes from cytoskeletal reorganization after the sudden change of load in the muscle (7, 10, 24, 31, 41). Also, if we take into account the time-dependent decrease in velocity (Fig. 3), either as a linear or an exponential decrease (over the defined time period in Fig. 3), the mathematical description remains a single-exponential function (i.e., *Eq. 4* is transformed to *Eq. 6*). This suggests that the slowing in velocity (due to slowing of cross-bridge cycling rate or other reasons) cannot be modeled by addition of an exponential or a power-law term. In other words, the second exponential term in *Eq. 9* or the power-law term in *Eq. 10* has nothing to do with the time-dependent decrease in shortening velocity in our model. Also, because *Eq. 6* remains a single-exponential function, it has limited capability to accommodate the data that clearly require at least a double-exponential function for accurate description (Figs. 7⇑–9).

One of the leading explanations for the time-dependent decrease in velocity in smooth muscle is the “latch bridge” theory (8, 16). By combining the model of Hai and Murphy (16) with the model of Huxley (22), Mijailovich et al. (32) were able to produce a model that explains well many aspects of the dynamic behavior of smooth muscle subjected to force perturbation, except the “plasticity” exhibited by the muscle. A crude comparison can be made here. *Equation 4* in the present study is an adaptation of the Huxley model (22) (which gives rise to the force-velocity relationship) to smooth muscle (which possesses a linear ascending limb in the length-force relationship). *Equation 6* is analogous to the model of Mijailovich et al., which incorporates the time-dependent decrease in velocity. *Equation 6* and the model of Mijailovich et al. do poorly in accounting for factors stemming from structural remodeling during a prolonged contraction. Our answer to an improved description of a prolonged smooth muscle contraction is addition in series to the sliding-filament mechanism a component that exhibits power-law or viscoelastic behavior.

### Conclusion

The simple models we have developed in this study based on known length-force and force-velocity relationships in ASM are able to describe the time course of isotonic shortening accurately, suggesting that a large portion of the shortening is due to cross-bridge interaction, which gives rise to the force-velocity properties, and the sliding of filaments, which produces a linear decrease in force with length change. This can be taken as evidence supporting the existence of a sliding-filament mechanism in smooth muscle, but with a contractile unit configuration that is different from that of striated muscle.

## GRANTS

This work was supported by Canadian Institutes of Health Research Operating Grants MOP-13271 and MOP-37924.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

## ACKNOWLEDGMENTS

We thank Pitt Meadows Meats (Pitt Meadows, BC, Canada) for the supply of fresh sheep tracheas.

## APPENDIX

#### Derivation of Eq. 4 from Eq. 3.

To obtain *L* as a function of time (*t*), we start by rearranging *Eq. 3*
*L* represents only the length of the contractile apparatus (i.e., *L*_{contraction} in Fig. 5), and the displacement is solely produced by the sliding-filament mechanism.

#### Derivation of Eq. 6 from Eq. 5.

By rearranging *Eq. 5*
*c* = 0, *Eq. 6* becomes *Eq. 4*.

#### Newtonian creep.

For a linear spring (with a spring constant *K*_{1}) in parallel with a dashpot (with a Newtonian coefficient of viscosity *v*), the relationship between the forces in the spring and dashpot after a step release in load can be expressed as
*L* denotes displacement of the spring-dashpot assembly (i.e., *L*_{creep} in Fig. 5). Under isometric conditions (just before the isotonic release), the spring is maximally stretched and bears the isometric force (F_{max}). Designation of the length of the spring under isometric condition as *L*_{m} results in *K*_{1}*L*_{m} = F_{max}. Rearranging and integrating *Eq. A1*
*K*_{1}*L*_{m} = F_{max} = 1
*Equation A2* thus describes the Newtonian creep after an isotonic quick release.

#### Power-law creep.

The deformation (or creep) of a viscoelastic element after a step change in load can also be modeled as a power-law creep
*A* is a constant and the exponent γ has a value between 0 and 1. When γ = 1, the deformation is Newtonian in nature; when γ = 0, the deformation is that of an elastic solid. Power-law creep occurs when 0 < γ < 1. Note that *L* represents *L*_{creep}, as shown in Fig. 5.

#### Combined displacement produced by sliding-filament mechanism, creep, and series elastic recoil.

As depicted in Fig. 5, total shortening of the muscle (*L _{total}*) can be represented by a linear summation of displacements produced by the sliding-filament mechanism (

*L*;

_{contraction}*Eq. 4*or

*6*), the creep (

*L*;

_{creep}*Eq. A2*or

*A3*), and the displacement of the series elastic component (

*L*)

_{SEC}*time 0*) for muscle shortening with Newtonian creep can be expressed as

*Eq. A4*becomes

*Eq. A4*becomes

*Eq. A5*becomes

*Eq. A8*becomes

*Eq. A9*becomes

- Copyright © 2011 the American Physiological Society