Emergence of airway smooth muscle mechanical behavior through dynamic reorganization of contractile units and force transmission pathways

Abstract

Airway hyperresponsiveness (AHR) in asthma remains poorly understood despite significant research effort to elucidate relevant underlying mechanisms. In particular, a significant body of experimental work has focused on the effect of tidal fluctuations on airway smooth muscle (ASM) cells, tissues, lung slices, and whole airways to understand the bronchodilating effect of tidal breathing and deep inspirations. These studies have motivated conceptual models that involve dynamic reorganization of both cytoskeletal components as well as contractile machinery. In this article, a biophysical model of the whole ASM cell is presented that combines 1) crossbridge cycling between actin and myosin; 2) actin-myosin disconnectivity, under imposed length changes, to allow dynamic reconfiguration of “force transmission pathways”; and 3) dynamic parallel-to-serial transitions of contractile units within these pathways that occur through a length fluctuation. Results of this theoretical model suggest that behavior characteristic of experimentally observed force-length loops of maximally activated ASM strips can be explained by interactions among the three mechanisms. Crucially, both sustained disconnectivity and parallel-to-serial transitions are necessary to explain the nature of hysteresis and strain stiffening observed experimentally. The results provide strong evidence that dynamic rearrangement of contractile machinery is a likely mechanism underlying many of the phenomena observed at timescales associated with tidal breathing. This theoretical cell-level model captures many of the salient features of mechanical behavior observed experimentally and should provide a useful starting block for a bottom-up approach to understanding tissue-level mechanical behavior.

mechanisms underlying airway hyperresponsiveness (AHR) in asthma remain poorly understood despite significant research effort and a variety of approaches. The functional problem is bronchoconstriction as a result of airway smooth muscle (ASM) contraction. While contraction of ASM cells occurs rapidly (over the order of seconds), other aspects of ASM mechanics occur over longer timescales, such as isometric force generation (minutes) and length adaptation of the ASM tissue (minutes to hours) (see Ref. 37 and references therein). Much has been made of the fact that ASM cells in vivo reside in a dynamically changing environment due to tidal breathing. Consequently, a large body of experimental and theoretical work has focused on the effect of rapid fluctuations (over seconds) on ASM cells, tissues, lung slices, and whole airways and hence on AHR (2, 11, 23, 24, 27, 29, 32, 34, 45).

Maximally activated ASM tissue-strips subjected to length fluctuations exhibit “banana-shaped” force-length loops (2, 12, 31, 33) characterized by nonlinear strain-stiffening and hysteresis, which are not completely understood. These studies show that length fluctuations reduce the mean contractile force by an amount proportional to the amplitude of the fluctuations. A more recent study shows that the reduction in contractile force may depend on peak transmural pressure rather than amplitude within intact airways (17). Theoretical studies describing acto-myosin crossbridge mechanics using the sliding filament Huxley-Hai-Murphy (HHM) model (9, 12, 31) have been able to account for some of these characteristics. In particular, at the level of a single contractile unit, the HHM model explains the drop in mean contractile force through disruption of crossbridge cycling and it reproduces hysteresis as observed experimentally.

However, the HHM model is unable to account for the banana-shaped force-length loops and strain-stiffening behavior. Part of the problem lies in the fact that extrapolating from contractile-unit level to tissue level (the level of experimental observations) neglects biophysical events that occur beyond crossbridge interactions. These include the possibility of cytoskeletal remodeling, disruption of actin-myosin connectivity, and contractile-unit rearrangement. Furthermore, an underlying assumption in the HHM model, which stems from the original Huxley model, is that filaments are assumed to be infinitely long, resulting in a “flat” force-length curve, i.e., the maximal force generated is independent of filament length. However, when isometric forces are allowed to develop to equilibrium for different lengths over minutes, ASM exhibits characteristic inverted parabolic force-length curves with a peak force occurring at the isometric length (46). Politi et al. (35) represent this by multiplying the contractile force predicted by an HHM model by a length-dependent factor. However, this still does not explain the banana shape of dynamic force-length loops.

The HHM model has recently been modified (9) to account for the possibility that changes in muscle length lead to changes in filament overlap and thus altered ASM force (1, 4, 36, 37). This modified crossbridge model combines the HHM model with quantitative measurements of the distribution of myosin lengths found in ASM in which binding sites become preferentially available within and near the actin overlap region. While this model provides a biophysical basis for the parabolic shape of isometric force-length curves (over intermediate timescales) and length adaptation (by shifting of the force-length curves to an adapted length over much longer timescales), it is still unable to account for the features observed in dynamic force-length loops at the shortest timescales. Aimed at the contractile-unit level, this model may also neglect other, potentially important, events occurring beyond crossbridge cycling.

A picture is emerging, out of experimental data, that supports the view that the actin cytoskeleton of smooth muscle cells is a dynamic structure [evolving through actin (de-)polymerization], which acts in concert with active force-generation mechanisms to play a central role in regulating the development of mechanical tension and the material properties of smooth muscle tissues (13, 14, 30, 48). These and other studies have motivated conceptual models that involve dynamic reorganization of both cytoskeletal components (6, 14, 20) and contractile machinery (18, 25, 36, 38) in response to imposed length changes. Such concepts, however, have not been integrated with previous modeling studies to assess the relative contributions of the different proposed mechanisms. We made a first attempt at this in our recent model of an ASM cell (4) in which we took into account structural components of the whole cell [based on the conceptual model of Kuo and Seow (21)] and used the idea that length changes of the ASM cell could lead to changes in filament overlap length (mentioned above) and hence actin-myosin connectivity. We showed how a single instance of contractile machinery rearrangement (forced by an initial length change) could push the cell into a new state causing more dramatic reduction in contractile force. Strain-stiffening was accounted for through application of a passive nonlinear strain-stiffening of the cytoskeleton. However, other features of the dynamic force-length loops remained elusive; in particular, the cusp in force-length loops appeared at peak strains rather than minimum strains as observed experimentally.

In this article, a model of the whole ASM cell is presented in which actin-myosin disconnectivity allows dynamic rearrangement of myosin filaments (within a dynamically changing lattice of actin filaments) and an investigation of the effect of this on force-length loops. Based on the conceptual models arising from the ultrastructural (21, 22) and experimental studies (25), we explore the concept of parallel force-transmission pathways within the cell, consisting of contractile units that connect up in series in a variety of different configurations (Fig. 1A). Force generated by cycling crossbridges can only be transmitted to the cytoskeleton along those pathways. By assuming that large enough length changes can cause complete actin-myosin disconnectivity for a period of time, a modification of the number and configuration of force-transmission pathways that can exist at that cell length is expected. This captures the notion of parallel-to-serial transitions (and vice versa) that may occur dynamically through a length fluctuation. The above mechanisms are combined in a biophysical model of the ASM cell. The model displays emergent mechanical behavior in force-length loops that capture nearly all the features of experimental observations. The results provide strong evidence that dynamic rearrangement of contractile machinery is a likely mechanism underlying many of the phenomena observed at timescales associated with tidal breathing.

Fig. 1.

A: conceptual model of the airway smooth muscle (ASM) cell in which pathways (shaded regions) consisting of a number of contractile units connected to the cytoskeleton transmit force to the tissue as a whole via extracellular matrix. As depicted here, the pathways are essentially regions that connect one end of the cell to the other via any route that may form out of a random distribution of sparse myosin filaments in a crowded actin filament lattice, but under tension are implicitly assumed to be straight pathways in the model. Force transmission pathways formed in this way are assumed to reconfigure dynamically on imposition of cell length fluctuations. B: simplified representation of the conceptual model illustrated in A. Two types of force transmission pathways are shown: one consisting of contractile units that connect to the cytoskeleton via the nucleus (labeled A) and the other that connects directly to the cytoskeleton (labeled B). Although the pathways contain varying numbers of contractile units with varying contractile-unit lengths this schematic depicts mean numbers of contractile units in series per pathway and parallel pathways of each type. The nucleus is modeled as a linear spring. Viscosity of the cytoplasm is represented as a linear dashpot in parallel with a linear spring representing cytoskeletal stiffness and other contributions to passive stiffness. As an example of internal architectural changes, B shows the cell held at a short length, and C shows the same cell after a length change has been imposed on it, depicting reconfiguration of the force transmission pathways and disconnected contractile units N_C. D: schematic representation of ASM contractile units in series within a force transmission pathway. The pathway depicted here is of type B, and the contractile units connect to the cytoskeleton via membrane adhesion complexes. E: illustrates the reduction in the number of available crossbridges as a result of stretch and the reduction in overlap length between actin and myosin filaments. F: depicts reduction in overlap length via shortening and hence a reduction in the number of available crossbridges. See appendix a for additional definitions.

METHODS AND MODEL DEVELOPMENT

Cell architecture.

The dynamic processes and interactions between cell- and contractile-unit-level events that are expected to occur are outlined before the detailed model description. The ratio of actin to myosin filaments has been found to be as high as 15:1 (21). We thus assume that myosin filaments are spatially distributed across the whole cell in such a way that pathways develop across the cell consisting of a chain of contractile units (each consisting of a myosin filament and 2 actin filaments) and dense bodies that connect up in a variety of different configurations (Fig. 1A). Force generated by cycling crossbridges can therefore only be transmitted to the cytoskeleton along those pathways. Some of the pathways may anchor directly via focal adhesions to the cytoskeleton while others may connect to the cytoskeleton via the nucleus (as depicted in Fig. 1, B and C); ultrastructure data suggest that approximately half of the pathways may connect via the nucleus (21).

Length changes imposed on the cell cause the overlap length between myosin and adjacent actin filaments to decrease (Fig. 1, D–F), thus reducing the number of crossbridges available for binding as included in our previous model (4). In this model, overlap length decreases through increasing strain, until eventually insufficient availability of crossbridges causes the myosin filament to disconnect completely from the actin filament at some threshold strain. These disconnected filaments may remain disconnected while strains are large but grab on to the nearest available pair of actin filaments as the cell returns to original strain. We expect this process to modify the number of force transmission pathways that can exist at that cell length. We assume the number of force transmission pathways drops with an increase in strain and the mean number of contractile units per pathway increases, corresponding to a parallel-to-serial transition of contractile units. In contrast, as the cell is shortened, we would expect serial-to-parallel transitions (consistent with Refs. 36, 38). Furthermore, the fact that the myosin filament lengths are not uniform and therefore distributed about some mean value (9, 26, 36) means that complete disconnectivity for different contractile units could occur at varying values of imposed strains (for instance, occurring more readily for shorter myosin filaments). Passive properties of the cell arise from the nucleus, cytoskeletal actin network, and cross-linking proteins, which we simply represent as an elastic spring in parallel with the force transmission pathways, while the nucleus is represented as an elastic spring in series with a fraction of the contractile units (Fig. 1B).

We assume for simplicity that the contractile units form two types of parallel force transmission pathways (pathways A and B) as depicted in Fig. 1B. The proportion of contractile units (as a fraction of the total number of contractile units in the cell) of each type is denoted N_A and N_B (all variables and parameters are listed in appendix a for reference). Each pathway of type A or B consists of a different number of contractile units in series. At time t, we denote the proportion of contractile units in series per pathway (as a fraction of total contractile unit in the cell) as N_is = (i = A, B). We note that 0 ≤ N_is ≤ 1 for i = A, B. We further assume that there are P_i parallel pathways of i type contractile units (with a mean fraction N_is of contractile units in series in that pathway), so that the total fraction of A- and B-contractile units, N_A and N_B, is given by

$$\begin{array}{cc}{N}_i=P_i{N}_{is},& (i=A,B)\end{array}$$ (1)

where 0 ≤ N_i ≤ 1.

At the contractile unit-level, crossbridge dynamics are governed by the partial differential equations of the HHM model (31) (see Fig. 3, appendix c, and Ref. 4 for details). This model yields the number and stretch of cycling crossbridges and latch bridges at any point in time and hence the magnitude of the contractile force generated by a single contractile unit as a function of time. However, the number of cycling crossbridges and latch bridges can only come from a pool of crossbridges that are available for binding. Sufficient stretch or shortening of the cell is assumed to cause the number of available crossbridges in the overlap region of a contractile unit to drop (as depicted in Fig. 1, E and F). In what follows all contractile unit and cell lengths have been normalized with respect to their lengths at maximal overlap (as in Ref. 4) so that all length measures are effectively stretch ratios. Each contractile unit is assumed to have length Y_i (i = A, B), which is inversely related to the overlap length as shown in Fig. 1E, but the overlap length is not specified explicitly. Above (or below) some threshold value of contractile-unit length Y_i, we assume that the myosin filament detaches completely from the pair of actin filaments it was attached to and reattaches (not necessarily immediately) to other available actin filaments thus changing the mean values of N_A and N_B (Fig. 1C). The threshold value of contractile-unit length is assumed to be proportional to mean filament length since shorter filaments would disassemble at lower threshold values than longer filament lengths (limitations of this assumption are discussed later). Changes in actin-myosin connectivity may be modeled by allowing the number of available crossbridges in a contractile unit to change as a function of deformed contractile-unit length. Denoting ρ_i as the fraction of available crossbridges (relative to the number of crossbridges at maximal overlap) so that 0 ≤ ρ_i ≤ 1, we propose that

$${\rho }_i(Y_i)=\{\begin{array}{cc}0,\hfill & 0\le Y_i<1-\sigma \hfill \\ -\frac{1}{{\sigma }^2}\left[Y_i-(1+\sigma )\right]\left[Y_i-(1-\sigma )\right],& 1-\sigma \le Y_i\le 1+\sigma \\ 0,\hfill & Y_i>1+\sigma \hfill \end{array}$$ (2)

for i = A, B, where 1 ± σ are the threshold values of Y_i at which ρ_i = 0 and at which we assume the contractile unit disassembles (Fig. 2A). The parameter σ (with 0 < σ < 1) is thus a threshold strain since Y_i is normalized with respect to the contractile-unit length at maximal force. By definition, maximal force is generated at Y_i = 1. The value of ρ_i affects the magnitude of the force generated by the contractile unit and is therefore similar to the functional form for a contractile unit-level force-length curve proposed in Ref. 39.

Fig. 3.

Schematic of interactions, within the model, between cell-level and contractile-unit-level events. Crossbridge mechanics in the contractile unit are governed by the Huxley-Hai-Murphy (HHM) model equations (grey boxes) and detailed in appendix c (Eqs. C2a–e). Cell-level response to length change, in the numbers of force transmission pathways and contractile units per pathway, are governed by Eqs. 1, 3, and 5. Solution of the HHM equations to determine n_i(x, t) effectively involves integration over time and space (using a numerical scheme called the Godunov scheme) building history dependence into F_A, F_B, and therefore the total force, T. Full details of this are given in Ref. 4.

Fig. 2.

A: graphical representation of the functional forms used for the variation of available crossbridges ρ_i, i = A, B with changes in contractile-unit length (normalized with respect to the contractile-unit length at maximal force), Y_i, i = A, B. The threshold parameter σ is the strain at which the myosin and actin disconnect completely and thus ρ_i is zero for contractile-unit length Y_i above and below 1 + σ and 1 − σ, respectively. B: parameter value σ feeds into the functional form used to represent the change in total number of contractile units, N_i, i = A, B contained in each type of pathway, A or B as a function of cell length L (dashed curve). Due to the assumed conservation of total number of contractile units, any drop in N_A and N_B causes the number of disconnected contractile units N_C to increase (dot-dash curve). The parameter governing the extent to which contractile units disconnect from force transmission pathways and remain disconnected is governed by the parameter λ as depicted here. μ = 1 produces the function shown here that is symmetric about L = 1. The value of μ = 0.95 used in simulations below shifts the curve slightly to the left. C: graphical representation of the change in the number of parallel pathways P_i as a function of cell length L. ϕ governs the extent to which force transmission pathways can reconfigure from parallel pathways to series. D: corresponding change in the mean number of contractile units in series per pathway N_is as a function of cell length L for increasing ϕ.

Furthermore we assume that there is a third type of contractile unit, N_C, that represent myosin filaments that become detached but do not rebind to actin immediately; again 0 ≤ N_C ≤ 1. This type of contractile unit does not form part of a force transmission pathway and therefore cannot transmit force to the supporting structures. Figure 1C shows schematically the effect of stretching the cell in Fig. 1B; the total number of contractile units is conserved but as the cell is stretched, the mean number of contractile units in series in any force transmission pathway, N_is, increases so that the number of pathways, P_i, decreases concurrently and N_C increases. On the other hand, shortening the cell causes a decrease in N_is and an increase in P_i. Assuming that at time t the total number of contractile units is in equilibrium (see appendix c for details), the following Hill function is proposed for the total number of contractile units as a function of cell length L:

$$\begin{array}{cc}{N}_i(L)={\overline{N}}_i\left[1-\lambda \frac{{(L-\mu )}^m}{{\sigma }^m+{(L-\mu )}^m}\right],& (i=A,B)\end{array}$$ (3)

where N̄_i is the proportion of contractile units of i type at maximal force, λ and μ are positive constants (their effect discussed below), and m is an even integer. This expression accounts for disruption in acto-myosin connectivity by connecting the threshold strain σ for Y_i to cell length L (both variables a measure of stretch) since σ determines the strain at which a contractile unit disassembles and therefore must also be the cell strain at which contractile unit numbers of that type of contractile unit decrease (see Fig. 2A). Setting μ = 1 in (Eq. 3) generates a symmetric functional dependency on L about L = 1 with myosin disconnecting from actin for equal strains regardless of whether it is being lengthened or shortened. Force-length loops from results of model simulations with μ = 1 (see Fig. B1 in appendix b), however, suggest that this symmetry is unlikely to exist in the cell. We therefore assume that actin-myosin pairs disassemble more readily upon stretch than upon shortening; plausible reasons for this are given in the discussion. We account for this asymmetry by using the value μ = 0.95, which shifts the curve to the right compared with the symmetric case shown in Fig. 2B. Results discussed below are therefore for μ = 0.95, with the symmetric case results given in appendix b. Any functional form that allows description of threshold behavior, whereby the response to stretch generates a sharp response at a particular stretch value, could have been used instead of the Hill function; the choice here has been driven by the need to enable parameters within the expression to be readily related to physical mechanisms and also to allow algebraic manipulation. Assuming that the total number of contractile units remains constant, the following holds at any point in time:

$$N_A+N_B+N_C=1.$$ (4)

Thus, upon stretch or shortening, there is a temporary shift in contractile unit population from type A or B to type C; the extent to which this happens is governed by the magnitude of the parameter λ (Fig. 2B).

Consideration of the stochastic redistribution of myosin filaments within the actin filament lattice upon a length change provides a rational basis for determining the resulting number of force transmission pathways. Thus following Ref. 39, we prescribe a plausible functional dependence of P_i on cell length to understand the effect of dynamic rearrangements within the cell, by assuming

$$P_i(L)=\theta e^{-\varphi L},$$ (5)

where θ and ϕ are positive constants (Fig. 2C). Assuming that maximal force is generated at maximum overlap, we assume that there are no unconnected filaments in this state so that N_C = 0. Thus from (Eq. 1), we have

$${\displaystyle \sum _{i=A,B}{\overline{P}}_i{\overline{N}}_{is}=1,}$$ (6)

where all barred quantities henceforth represent the value of that quantity at maximal overlap. For simplicity we set ${\overline{P}}_i$ = $\overline{P}$ for i = A, B for some constant $\overline{P}$ that we prescribe and assume that ${\overline{N}}_i$ = ½ for i = A, B based on ultrastructure measurements (21). This yields the constraint:

$$\begin{array}{cc}\theta {\text{e}}^{-\varphi }=\overline{P},& {\overline{N}}_{is}=\frac{{\overline{N}}_i}{\overline{P}}.\end{array}$$ (7)

with P_i ≥ 1 (see appendix c). N_is follows from Eq. 1. Examples of the functions ρ_i(Y_i), N_i(L), P_i(L), and N_is(L) are shown in Fig. 2 for fixed λ and σ but increasing values of ϕ.

Contractile-unit velocities and force generation.

Dynamic changes in ρ_i and N_i (as a result of length fluctuations of the cell) affect contractile unit shortening velocities and hence force generation; these two contractile unit-level quantities are coupled in the equations governing crossbridge mechanics via the HHM model (4, 31; see Fig. 3 and appendix c). The total dimensional force T* (all starred quantities below are dimensional) exerted by the cell on neighboring cells (which may be tensile or compressive) normalized with respect to E_A^*L̄* to give dimensionless total force T is given by

$$T=\frac{T^{*}}{E_A^{*}{\overline{L}}^{*}}=\alpha P_A{F}_A+\alpha P_B{F}_B+E_C(L-1)+{\mu }_D\dot{L},$$ (8)

where E_A^* is the stiffness of the nucleus, L̄* is the dimensional reference cell length at maximal isometric force, α is a measure of the stiffness of crossbridges relative to the cell nucleus stiffness, F_i(i = A, B), is the dimensionless force generated by a single contractile unit in a force transmission pathway of type i, E_C is the dimensionless cytoskeletal stiffness normalized with respect to E_A^*, μ_D is the dimensionless viscosity, and the dot denotes a time derivative. Note that F_i is a function of both ρ_i, the number of available crossbridges as well as the fraction of attached crossbridges and latch bridges (Fig. 3 and appendix c).

Dimensional contractile-unit lengths are related to cell length via

$$L^{*}=N_{As}^{*}{Y}_A^{*}+Z^{*}=N_{Bs}^{*}{Y}_B^{*},$$ (9)

immediately highlighting the difference between contractile-unit lengths and numbers in the A pathways compared with the B pathways as a result of the presence of the nucleus. As in Ref. 4, the contractile-unit lengths (normalized with respect to their values at maximum isometric force) Y_A and Y_B are related to cell length via

$$Y_A={\eta }_A\frac{L-\beta -\alpha F_A}{1-\beta -\alpha {\overline{F}}_A},$$ (10a)

$$Y_B={\eta }_{B}L,$$ (10b)

where η_i = ${\overline{N}}_{is}/N_{is}$, (i = A, B), β is the resting length of the nucleus normalized with respect to the cell length at maximal force generation, ${\overline{L}}^{\ast }$.

The cell length is typically held constant at L = 1 for a period of time at the start of the experimental protocol, 0 ≤ t ≤ t̄, during isometric force generation, until maximal force has been achieved. After this (t ≥ t̄), sinusoidal length fluctuations are applied to the cell such that

$$L(t)=1+\epsilon \sin (t-\overline{t}),$$ (10c)

where ε is the amplitude of length oscillation, normalized with respect to ${\overline{L}}^{\ast }$.

The velocity of a contractile unit in row B, is given by differentiating Eq. 10b so that

$${\dot{\text{Y}}}_B=\frac{{\overline{N}}_{Bs}}{N_{Bs}}\dot{L}\left[1-\frac{L}{N_{Bs}}\frac{d{N}_{Bs}}{dL}\right],$$ (11)

with N_Bs given by Eqs. 1 and 3. Since N_B is a function of L, which is a prescribed function of t, Ẏ_B is thus a function of t.

Similarly, differentiating (Eq. 10a) with respect to time and combining with an expression for F_A obtained from the HHM equations (see Ref. 4 for algebraic details) yields the following expression for the shortening velocity of row A-contractile units:

$${\dot{\text{Y}}}_A=\frac{{\eta }_A}{N_{As}}\frac{\dot{L}\left[N_{As}-(L-\beta -\alpha F_A)\frac{d{N}_{As}}{dL}\right]-\alpha {\rho }_A{N}_{As}{\mathcal{H}}_1}{(1-\beta -\alpha {\overline{F}}_A)+\alpha {\eta }_A\left({\rho }_A{\gamma }_A{\mathcal{H}}_2+\frac{F_A}{{\rho }_A}\frac{d{\rho }_A}{d{Y}_A}\right)},$$ (12)

where γ is a measure of contractile-unit length at maximal activation relative to the power-stroke length and ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are integrals that depend on the density of the crossbridges in the rapidly cycling and latch bridge pools of A-contractile units (4). If N_is and ρ_i are constant, the expressions Eq. 11 and Eq. 12 reduce to the contractile-unit velocities given in the previously published model. We note that the mean number of contractile units per force transmission pathway N_is affects shortening velocities (Eqs. 11 and 12), while the number of parallel pathways P_i play a role in the total force (Eq. 8), the two quantities coupled via N_i in Eq. 1.

Imposing a length oscillation on the cell, therefore means that the shortening velocities Ẏ_B at any time t may be determined from Eq. 11. On the other hand, Ẏ_A depends on the force F_A generated by the contractile unit via Eq. 12 but the magnitude of F_A depends on the velocity Ẏ_A and so can only be determined through solution of the fully coupled system (full details in Ref. 4). Once F_A and F_B are calculated via the HHM model, the total force of the whole cell T may then be determined as a function of time via Eq. 8. Parameter values used in the simulations are listed in appendix a.

RESULTS

The baseline case is where the cell architecture remains “robust” in the face of length fluctuations (an implicit assumption in current multiscale models of ASM; Refs. 8, 35). This case translates to the model parameter values listed as “robust” in Table A4 in which 1) there is no disconnectivity of myosin filaments from actin (through loss of available crossbridges), i.e., σ >> ε; 2) there are no transitions from parallel to serial arrangements (or vice versa) so that ϕ = 0; and 3) there is no exchange of contractile units among N_A, N_B, and N_C so that λ = 0. This baseline case was fully explored in our previous model (4) with only a single instance of rearrangement. In this section we examine the effect of varying σ, λ, and ϕ to explore the effects of dynamically changing architecture on shortening velocities and force generation and compare them to the robust case. Observed mechanical properties are shown to emerge out of subcellular architectural changes. Finally, the inference of particular subcellular behavior that results in observed tissue-level behavior is demonstrated.

In the fully activated state in which contractile force has reached an isometric steady state value, length fluctuations drive changes in crossbridge populations (4, 31), the pool of crossbridges available for binding, ρ_i, the contractile unit fractions N_i and N_is, and number of pathways, P_i, which in turn modulate the velocity of length fluctuations applied to the cell. The equation governing Ẏ_B is decoupled from the rest of the governing equations and hence is determined first. The resulting velocity waveforms then form an input into the HHM model equations (see Ref. 4), which are solved for the velocity of A pathways, Ẏ_A, the contractile force generated by the A and B pathways, F_A and F_B, respectively, and hence the total force exerted by the cell, T.

Increasing ϕ from 0 while λ = 0 indicates how readily contractile units transition from parallel to serial configurations without spending any time in the “completely” disconnected state N_C. In contrast, increasing λ from 0 while ϕ = 0 indicates how readily contractile units disconnect from A and B pathways completely without reconnecting. Thus ϕ ≠ 0 and λ ≠ 0 combines both effects and the influence of threshold strain σ does not become evident unless λ > 0. We carried out a careful parameter exploration to examine the effect of dynamic changes in subcellular architecture, resulting from length fluctuations applied at the cell level on contractile-unit velocities, Ẏ_i, contractile-unit lengths, Y_i, contractile forces, F_i, and the total force T.

Plotting the total force, T, as a function of cell length, L, for a number of simulations carried out as part of the parameter exploration (Fig. 4), we found that a particular set of architecture parameters (ϕ = 1, λ = 0.6, and σ = 0.18) gave rise to the characteristic banana-shaped tissue-level force-length loops (2, 11) (Fig. 4I). In each of the plots in Fig. 4, we compare the “robust” baseline case (grey loops) with the force-length loops obtained for σ ∼ ε (black and blue loops) and σ >> ε (red loops). Here we note highly interesting emergent mechanical behavior. As previously observed (4, 31), viscoelastic behavior emerges from the crossbridge cycling as evidenced by the hysteresis in the force-length loops for the robust case. The amount and nature of this hysteresis is modified (for ϕ = 0 by increasing λ; Fig. 4, A, D, and G); the maximum force increases slightly, minimum force decreases, and a cusp begins to develop at the low strain end. This cusp is further exaggerated for increasing ϕ (Fig. 4). Differences in behavior between different values of σ only becomes evident for λ > 0 (Fig. 4, D–I) with qualitative differences in elastic behavior for ϕ = 1 (Fig. 4, F and I), significantly exaggerated between σ ∼ ε (black and blue curves) and σ >> ε (red curves). In particular, for σ ∼ ε we see the emergence of strain-stiffening at the high strain end of the force-length loop that is characteristic of the experimentally observed banana-shaped loops. This is entirely a result of acto-myosin disconnectivity and parallel-to-serial reconfigurations of the force-transmission pathways and without resort to a phenomenological stiffening of the cytoskeleton [as we applied in our previous model (4)]. This mechanical behavior may be explained in the following way: as the cell lengthens, since both ϕ > 0 and λ > 0, the number of force transmission pathways, P_i, decreases as both the mean number of contractile units per pathway, N_is, and number of disconnected contractile units, N_C, increases (see later). The decrease in P_i should cause a drop in stiffness, but since contractile units still left connected in the force transmission pathways are highly strained, the large displacement of the crossbridges causes an increase in crossbridge contribution to the total stiffness that dominates over decrease in stiffness through the drop of parallel pathways at this high strain end. We note, however, that this domination of crossbridge stiffness can only occur if a sufficiently small number of contractile units are left in the force-transmission pathways and so is dependent on the specific values of λ and ϕ; indeed this is demonstrated effectively by the parameter exploration in Fig. 4. In the shortening phase of the length fluctuation, the crossbridges contribute relatively little to the stiffness, with cytoskeletal stiffness dominating in this part of the force-length loop. This appears to be a result of the reeling in of slack so that the contractile-unit lengths do not fall much below 1 (see later). We find other, rather contrasting strain-softening behavior for σ >> ε for ϕ = 1 (red loops in Fig. 4, C, F, and I). For this set of parameter values, any contractile units that disconnect from existing pathways transition straight into other pathways. Hence, because none of these contractile units remain disconnected, the decrease in P_i and concurrent increase in N_is causes a reduction in overall stiffness relative to the stiffness at smaller cell lengths.

Fig. 4.

A–I: total dimensionless force T exerted by the cell, resulting from imposed length fluctuations (amplitude ε = 0.04), as a function of dimensionless cell length L for increasing ϕ (left to right) and increasing λ (top to bottom) for different values of the threshold parameter (or mean myosin filament length), σ = 0.8 (red), 0.18 (blue), 0.15 (black) during one length-oscillation cycle. For comparison purposes, the grey curve in each case shows the force-length loop for the robust case ϕ = 0, λ = 0, σ = 0.8. The total force is normalized with respect to the quantity E_A^*L̄^* where E_A^* is the stiffness of the nucleus, and L̄^* is the reference length of the cell at the start of the oscillations (dimensional values and details of the nondimensionalization are given in appendix a).

As summarized in methods and model development, the total force is calculated via Eq. 8 by determining contractile forces F_A and F_B (from the HHM equations) that result from application of length fluctuations at the cell level that translate to modulated contractile-unit velocities Ẏ_A and Ẏ_B. Focusing on the parameter values that generate the banana-shaped loops (ϕ = 1, λ = 0.6, and σ = 0.18), we therefore examine the velocities, contractile unit fractions and contractile forces that contribute to the emergent mechanical behavior of the cell (Fig. 5). In the baseline “robust” case, the cell velocity, which is a result of applying length fluctuations at the cell-level (magenta dot-dashed curve in Fig. 5A) is identical to the velocity of the B-pathway contractile units, Ẏ_B for all values of σ since there are no parallel-to-serial transitions (ϕ = 0) or disconnections (λ = 0) allowed. For increased ϕ (but λ = 0), we found that the amplitude of the velocity oscillations decreased relative to the cell velocity (similar to green curve in Fig. 5A since large σ is effectively identical to λ = 0; see Fig. B2C in appendix b). This is explained through the fact that for large ϕ, contractile units immediately transition from parallel configurations to serial ones for a small change in length so that each B-type contractile unit experiences only small shortening velocities, even at peak cell velocity, before becoming part of another pathway thus effectively adding to the number of serial contractile units (green curve in Fig. 5C). On the other hand for increasing λ (with ϕ = 0), we found the waveform became more nonlinear, steepening and then forming peak contractile-unit velocities that occur at different times relative to the timing of the peaks in the cell velocity waveform (similar to black curve in Fig. 5A; see Fig. B2B in appendix b). The amplitudes of these contractile-unit velocities were significantly increased compared with the robust case (see Fig. B2A in appendix b).

Fig. 5.

Contractile-unit velocities, contractile-unit numbers and contractile forces as a function of (dimensionless) time for ϕ = 1, λ = 0.6 and different values of the threshold parameter, σ, during one length oscillation cycle. A: dimensionless contractile-unit velocity Ẏ_B^*, resulting from length fluctuations (amplitude ε = 0.04) imposed on the cell of velocity L̇ (magenta dashed curves). Contractile-unit velocity is normalized with respect to the product of oscillation frequency ω* = 2 Hz and contractile-unit length at isometric force Ȳ_B^* = 2 μm. B: dimensionless contractile-unit lengths Y_B, resulting from length fluctuations imposed on the cell (magenta dashed curves show L). Contractile-unit length is normalized with respect to its value at isometric force Ȳ_B^* = 2 μm. C: mean fraction of contractile units per pathway of the B type N_B. D: dimensionless contractile-unit velocities Ẏ_A (solid curves) and Ẏ_B (dashed curves). Results for σ = 0.12 are omitted for clarity. E: dimensionless contractile-unit lengths Y_A (solid curves) and Y_B (dashed curves). F: dimensionless contractile force F_A (solid curves) and F_B (dashed curves). The force is normalized with respect to the quantity ρ̄^*K^*h^*2 where ρ̄^* is the number of available crossbridges at maximal overlap (isometric), K^* is the stiffness of an individual crossbridge, and h^*2 is the power stroke length. Parameter values and details of the nondimensionalization are given in appendix a.

The increased amplitude of peak velocities may be explained through the fact that since no parallel-to-serial transitions are possible for ϕ = 0, contractile units that disconnect as a result of increasing cell length do not reconnect for the duration of time the cell stretch L remains above 1 + σ (so that the pool N_C is increased; Fig. 2B). Thus those few contractile units that remain connected (black curve in Fig. 5C) now experience significantly higher strains (black curve in Fig. 5B) and velocities (black curve in Fig. 5A). This results in overlap distances decreasing and more contractile units being thrown off the existing force transmission pathways. The amplification of peak velocity continues until the peak amplitude of the cell fluctuation is reached (black curve in Fig. 5B) after which disconnected contractile units begin to reconnect to the pathways as the cell length returns to L = 1 (black curve in Fig. 5C). Thus the contractile units that are part of existing pathways experience significantly higher shortening velocity than in the robust case until enough contractile units have reconnected to force transmission pathways (black curve in Fig. 5C) bringing Ẏ_B back to L̇ (black curve in Fig. 5A). Therefore, while increased λ causes an increase in velocity amplitude and nonlinear wave steepening, an increase in ϕ serves to attenuate the amplitude, and a combination of both effects results in a shift of peak contractile-unit velocity relative to cell velocity with decreased amplitudes (blue and red curves in Fig. 5A).

We now discuss the effect of increasing threshold strain σ. The influence of σ does not become evident unless λ > 0. Given a value of λ > 0, for values of σ >> ε (the amplitude of cell length fluctuations) Ẏ_B, is almost identical to L̇ (see appendix a), since there is no disconnection between myosin and actin filaments. There is still cycling between the different myosin crossbridge species as per the Hai-Murphy kinetic scheme [giving rise to hysteresis in force-length loops (4, 31); Fig. 4A]. However, smaller values of σ (comparable in magnitude to ε) represent easier disruption of the actin-myosin connection so that changes in the fraction of contractile units in series, N_Bs (black curve in Fig. 5C) in turn modifies the imposed velocity at the contractile unit level as discussed above; contractile-unit velocity waveforms exhibit increasing nonlinearity with increasing σ as indicated by the shift in timing of the peak velocity compared with baseline (see Fig. B2B in appendix b). For ϕ = 0, transitions from parallel-to-serial arrangements are not possible. Thus both N_Bs and N_B drop off for both shortening and lengthening parts of the length fluctuation (black curve in Fig. 2C). For larger ϕ (Fig. 5), however, increased parallel-to-serial transitions occur during lengthening so N_Bs increases on lengthening and decreases during shortening (red curve in Fig. 2C). This variation in N_Bs has the effect on Ẏ_B (Fig. 5A) and Y_B (Fig. 5B) as discussed above. For any pair of values for ϕ and λ peak contractile-unit velocities for σ ∼ ε are significantly greater than σ >> ε (Fig. 5A). As we noted above, the amplitude of cell length fluctuation is attenuated at contractile unit level for increasing ϕ but amplified (for σ ∼ ε), as λ increases. Thus for large ϕ and λ, peak contractile unit strain is increased at peak cell fluctuation (Fig. 5B), but during the cell shortening phase for half of the period shown, the contractile unit does not shorten much beyond the length Y_B = 1, as evidenced by the flat portion of curves in Fig. 5B compared with the cell length curve. This could be thought of as a “reeling in” of slack generated by shortening of the tissue, which has been added empirically to previous models (2) but here is an effect that emerges directly out of consideration of architecture changes of the contractile machinery.

With the use of Ẏ_B as displayed in Fig. 5A, as input into the HHM governing equations, it is then possible to solve for the mean velocity of contractile units in the A-type pathways Ẏ_A and contractile forces F_A, F_B. These are shown in Fig. 5, D–F. Qualitatively, the A-pathway contractile-unit velocities (solid curves) behave in a similar way to those of the B-pathway contractile units (dot-dashed curves) for values of σ ∼ ε (red and blue curves in Fig. 5D) but not for σ >> ε (green curves) where the B-contractile-unit velocities appear to have undergone a period doubling but the A-pathway contractile units have not. Additionally, the B-pathway contractile units have significantly smaller peak velocities than the A-contractile units. It is, however, clear that Ẏ_A has been modified compared with the applied sinusoidal cell velocity (magenta curve in Fig. 5A). There are small quantitative differences between Ẏ_A and Ẏ_B for small ϕ and λ (not shown) which are exaggerated for larger values of ϕ and λ (Fig. 5D). The reason for this rather dramatic difference between Ẏ_A and Ẏ_B for large ϕ and large σ (green curves in Fig. 5D), which is equivalent to λ = 0, is due to the nucleus taking up some of the strain applied to the A pathways during both the lengthening and shortening phases. In contrast the B pathways respond to the whole strain applied to the cell. Thus fewer transitions from parallel-to-serial configurations during lengthening (and serial-to-parallel during shortening) occur in the A pathways since the mean decrease in A-contractile unit overlap lengths will be smaller than in the B-contractile units. Those contractile units that would have detached from B pathways and transitioned straight back into other pathways (for λ = 0), can, for λ > 0 remain disconnected so that B-contractile units begin to experience strains that are closer to those of the A pathways (for σ ∼ ε). Based on these arguments the differences between A- and B-contractile-unit velocities are expected to be further exaggerated for a more compliant nucleus, that is a lower spring stiffness E_A^* (Fig. 1A) but is not explored any further here.

The length fluctuations applied to the cell (or tissue) as given in Eq. 10c and shown in Fig. 5B, translate to contractile-unit length fluctuations shown in Fig. 5E. The length fluctuations of the A-contractile units are qualitatively similar to those of the B-contractile units for small ϕ (not shown) but not for ϕ = 1 (Fig. 5E). The period doubling observed in the velocities for ϕ = 1 is reflected in the length fluctuations and curiously translates to shortening of the B-contractile units during both lengthening and shortening phases of the A-contractile units. Here, although we see the reeling in of slack in the parts of the length fluctuation during which the cell is shortened (L < 1) for both A- and B-contractile units, the effect on the B pathways is much greater than the A pathways (Fig. 5E).

The contractile forces generated by each of the A- and B-contractile units are thus determined. F_A and F_B were found to be qualitatively similar for λ = 0, ϕ = 0, 0.5 (not shown) but are dramatically different for λ = 0.6, ϕ = 1 (Fig. 5F), the difference in period and amplitude between F_A (solid curves) and F_B (dot-dashed curves) reflecting what was seen in the velocities. Large quantitative differences are observed between force generated by the A- and B-contractile units and between different cases of σ. The peak forces occur at increasing times for decreasing σ once again reflecting the steepening of the velocity curves and shifting of the peak velocity to later in the cycle compared with that of the cell velocity. Again, the fact that the nucleus takes up some of the strain imposed on the A pathways results in the B-contractile units generating greater contractile force than the A-contractile units for most of the length fluctuation cycle. The combination all of the cell component forces via (Eq. 8) to calculate the total force exerted by the cell results in the force-length loops discussed above.

A closer look at the contribution of the different types of force transmission pathways (A or B type) to the overall behavior of the force-length loops in Fig. 4I is illustrated in Fig. 6. For both values of σ considered, we see that the dynamics of the A-type force transmission pathways contribute significantly to the hysteresis of the overall force-length loop, while the B-type pathway dynamics appear to contribute significantly to the nonlinear elastic behavior, so that presence of both types of pathways appear to be necessary to generate observed mechanical behavior. The main reason for this difference lies in the A-contractile units not experiencing the full strain applied to the cell (some of it going into stretching the nucleus) and thus actin-myosin disconnectivity plays a smaller role. In contrast, because the whole cell strain is applied to the B-contractile units, there is more actin-myosin disconnection and hence greater parallel-to-serial transitions thus affecting the elastic behavior as discussed above. This is particularly well-demonstrated for σ = 0.8 (Fig. 6B) where the B pathways contribute only to the elasticity with negligible hysteresis.

Fig. 6.

Total dimensionless force T exerted by the cell, resulting from imposed length fluctuations (amplitude ε = 0.04), as a function of cell length L, during one length-oscillation cycle for parameter values ϕ = 1, λ = 0.6 and (A) σ = 0.8 or (B) σ = 0.18. In each case we compare the force that would be generated if both A and B types of force transmission pathways are present (black curves), only A type present (blue curves) and only B type present (red curves) The total force is normalized with respect to the quantity E_A^*L̄^* where E_A^* is the stiffness of the nucleus, and L̄^* is the reference length of the cell at the start of the oscillations.

The force-length loop for ϕ = 1, λ = 0.6, and σ = 0.18 (Fig. 4I) agrees very well with experimentally observed force-length loops of Ref. 2, (Fig. 7), the model having captured significant features of these loops, including hysteresis, strain stiffening at the high strain end and the cusp at the low strain end. We now make more detailed comparisons between the simulation and experimental results. The peak force at the onset of the length oscillations drops rapidly (Fig. 8A) as observed experimentally (Fig. 7), and this has been achieved entirely through increases in overlap length between myosin and actin filaments and the consequent reorganization of force transmission pathways. The recovery of force at the end of the oscillatory period occurs on a similar timescale to that of the initial development of isometric force and in rough agreement with the experimental results. The new steady-state force after the oscillatory period in the model results are, however, higher than that of the experimental results. The mean force in the model drops below isometric force as observed experimentally, but the peak force does not (Fig. 8A) compared with experimental observations. Additionally the transients are much quicker to resolve to the oscillatory steady state in the model results compared with the experiments. Potential explanations for these could lie in additional dynamic changes occurring within the cytoskeletal network, modifying significantly the mechanical properties of the parallel element currently used within the model to represent cytoskeletal stiffness, and could introduce a slower timescale allowing for more plastic changes that are not present in the model so far. Peak force in the model, for these parameter values (as discussed above), is dominated by crossbridge stiffness so differences between model and experiment may point to the value for crossbridge stiffness used in the simulation being an underestimate of the values in reality. Note that asymmetry in the postulated functional forms of disconnectivity (via the parameter μ; see methods and model development) was required for the force-length loops to match experimentally observed data. Force-length loops for symmetric functional forms (with μ = 1) are shown in appendix b (see Fig. B1).

Fig. 7.

Force-time and force-length traces obtained in a strip of rat trachealis muscle. The gray curves show data obtained with 2 Hz length oscillations having amplitudes of ± 1, ± 2, and ± 4% L_ref as indicated, where L_ref is the unstressed, resting length of the tissue strip. Isometric force-time curves are shown in black (at left; solid lines obtained before collecting oscillation data, dashed lines obtained afterwards). The black curves (at right) show the first (solid line) and last (dashed line) force-length loops obtained during length oscillation. [Reproduced from Bates et al. (2).]

Fig. 8.

Total dimensionless force T exerted by the cell, resulting from imposed length fluctuations (amplitude ε = 0.04), as a function of time (A) and cell length L (B) for the whole protocol with 43 ϕ = 1, λ = 0.6, σ = 0.18. Gray curves show transients and solid black the steady state oscillatory solution. The total force is normalized with respect to the quantity E_A^*L̄* where E_A^* is the stiffness of the nucleus, and L̄* is the reference length of the cell at the start of the oscillations.

Based on the set of parameter values that give the best agreement with the force-length loops of Ref. 2 (ϕ = 1, λ = 0.6, and σ = 0.18), we now investigate the effect of different amplitude oscillations and compare them to the force-length loops of Ref. 11 (Fig. 9A). Assuming a constant cytoskeletal stiffness, the force-length loops for increasing amplitude of oscillation (ε = 0.005, 0.01, 0.02, 0.04, and 0.08) exhibit approximately the same average cell stiffness and increased hysteresis (Fig. 9B), but the curves do not appear to follow the trend observed experimentally in which peak force drops with increasing ε; the theoretical model predicts an increase in peak force with a significant portion of the loop for ε = 0.08 exhibiting negative total force (i.e., resisting contraction). If on the other hand, we impose a decrease in average cytoskeletal stiffness with increasing amplitude of oscillation (Fig. 9C), then we find that the theoretical force-length loops agree better with the experimental loops (Fig. 9A). The additional qualitative agreement obtained in this way strongly suggests that both the dynamic assembly and disassembly of force transmission pathways, as well as changes in cytoskeletal stiffness as a result of network reorganization, are needed to explain the mechanical behavior of ASM cells. This is consistent with a number of experimental results in the literature (7, 33, 40) that have demonstrated changes in whole and localized cell stiffness with application of strain, and points to ongoing length adaptation on a slower timescale.

Fig. 9.

A: Experimental force-length loops obtained by applying length oscillations of increasing amplitude (ε = 0.005, 0.01, 0.02, 0.04, and 0.08) to activated ASM tissue strips (from Ref. 11). B: force-length loops obtained from simulations using same parameter values as Figs. 4I and 8, increasing amplitude oscillations and fixed cytoskeletal stiffness, E_C^*. C: force-length loops obtained from simulations as in B but with cytoskeletal stiffness, E_C^* decreasing with increasing amplitude of oscillation. [Reproduced with permission of the American Thoracic Society from Fredberg et al. (11)].

DISCUSSION

Considerable experimental evidence supports the model of an ASM cell in which both the contractile apparatus and the cytoskeleton are malleable. Dynamic changes in cell shape following length changes are thought to drive cytoskeletal and contractile filament reorganization, giving rise to this malleability (13). Furthermore, the reorganization is thought to occur in such a way as to maintain optimal filament overlap at different cell lengths, enabling maintenance of contractile force at different airway calibers (37). In developing the mathematical model presented in this article, a number of the conceptual models arising from ultrastructural (21, 22) and experimental studies (25) have been drawn upon. In particular, the focus is on quantifying the dynamic effects of length fluctuations on the reorganization of force transmission pathways containing force-generating contractile units. In concept, the formation (and disconnection) of force transmission pathways is similar to the formation and dissolution of links (which also consist of chains of contractile units) in the stochastic model of Silveira et al. (39) but here the deformation and dissolution of the force transmission pathway is associated with specific biophysical processes. Doing so has enabled identification of specific mechanical behavior that emerges as a result of cellular level effects that include crossbridge interactions, dynamic disruption of actin-myosin connectivity, and dynamic parallel-to-serial transitions of contractile units.

Previous crossbridge-based mathematical models of force generation by ASM have not been able to reproduce the characteristic force-length loops observed experimentally. By incorporating cell-level mechanisms beyond crossbridge interactions, our model is able to reproduce these loops to a greater extent than has been possible previously (Fig. 8). Moreover the results have highlighted the important role of actin-myosin connectivity in concert with dynamic transitions from parallel-to-serial configurations (occurring on timescales comparable with that of the length fluctuations) on whole cell mechanical behavior. In particular we discover that the nonlinear elastic behavior in the force-length loops (Fig. 4, F and I) emerges from the combination of 1) disconnected contractile units remaining disconnected (Fig. 5C), and 2) the strain experienced by contractile units left in existing force transmission pathways (Fig. 5B). Through this process, crossbridge stiffness dominates at the large strain end while cytoskeletal stiffness dominates at the low strain end, generating the characteristic (instantaneous) strain-stiffening (Fig. 8) observed in the experimental banana-shaped loops (Fig. 7). As the focus has been entirely on active force-generating mechanisms, we have not accounted for possible strain-stiffening of passive elements. For example, passive force-length loops in the study of Ref. 2 point towards this behavior existing in the absence of contractile agonist and such behavior was included phenomenologically in our previous model (4). Since parallel-to-serial transitions are limited to smooth myosin II form, incorporation of such transitions assumes that a significant proportion of the myosin in ASM is in this subgroup.

Although the hysteresis in experimental force-length loops has been demonstrated via previous crossbridge models, we have seen here how markedly this is modulated by dynamic reorganization of contractile structures. The change in hysteresivity is particularly evident as both disconnectivity (characterized by the parameter λ) and parallel-to-serial transitions (via the parameter ϕ) increase (Fig. 4). The relative contributions of the different type of pathways (A or B; see Fig. 1B) to the hysteresis (Fig. 6) suggests that if the kind of force transmission pathway reconfiguration postulated here does occur, then the existence of A-type pathways (i.e., connection of force transmission pathways to the cytoskeleton via the nucleus; Ref 21) appear to play an important role in the mechanical behavior of the cell.

The effect of mean filament length, through the threshold value σ, on the mechanical characteristics of the force-length loops is very clearly demonstrated by the model in cases where significant parallel-to-serial transitions are possible (large ϕ; Fig. 4, F and I). Strain-stiffening type behavior is predicted for values of σ comparable with oscillation amplitude and highly contrasting strain-softening behavior is predicted for σ much larger than the amplitude of oscillation ε. These predictions could be tested by comparing the force-length loops obtained here with force-length loops from experiments on ASM tissue strips treated with drugs such as Latrunculin B, which inhibits actin polymerization (10). Such experiments have previously shown that the effect of Latrunculin B, and hence shorter actin filament lengths, was to significantly augment both tissue strip relengthening and hysteresivity. Indeed, the model predicts this increase in hysteresivity with decreasing σ (Fig. 4, F and I). More detailed comparisons of force-length loops from such experimental studies (in which actin or myosin filament lengths may be manipulated pharmacologically) will be the subject of future work.

Recent, experimental evidence (42) suggests that bundles composed of F-actin and smooth muscle myosin thick filaments spontaneously assemble into contractile units with a well-defined force-velocity relationship. After the self-organization process, Thoresen et al. (43) also found that the maximum tension generated by a contractile unit is determined by the number of motor heads acting in parallel on F-actin, determined by the thick filament length thus lending further support to the model developed here. The self-organization appeared to build contractile units that acted as functional sarcomeres within a disordered actin bundle. Since these sarcomeres are stochastic and dynamic, these functional contractile units are not limited by the same geometrical constraints as those that exist in striated muscle and they demonstrate that contractility can be dynamically regulated by perturbations to thick filament length (43). Here a single threshold parameter σ has been suggested as being related to myosin filament length, but in ASM these lengths have been found to be nonuniform (28) so that a unique value for σ will not exist. Instead, gradual disconnection of contractile units may occur during length oscillations at all amplitudes. Since myosin length distributions are now available from experimental studies (28), in future work we will investigate the role of variability in filament length within the framework of this model. Additional experimental evidence points to the possibility of myosin dimers or short filaments not polymerizing to form long filaments but “line-up” in series behaving as functional filaments (36). This could result in myosin lengths that also vary with stretch or shortening and could provide a plausible physical reason for the net asymmetry we have assumed in the functional form assumed for N_i; upon shortening it may be possible for dimers to reconnect to actin filaments with the correct polarity (3) onto the other side of the dense body.

The focus here was on dynamic transitions of the contractile machinery, which has enabled identification of the possible origins of certain mechanical characteristics. While the model has been able to capture some fluidization observed experimentally (through the drop in mean force), the peak force in the model, however, still increases with amplitude of oscillation (Fig. 9B). One possible explanation is that we have not accounted for nonlinear behavior that could result from the cytoskeleton remodeling dynamically with length fluctuations on the same timescale as that of the contractile machinery, thus contributing further to the extent of fluidization. Additionally, decreased mean cell stiffness with increased amplitude of oscillation was only possible in the model through a prescribed decrease in cytoskeletal stiffness (Fig. 9C). This suggests that loading history must play a role in the overall mechanical behavior, which is currently not accounted for in this model. It is possible for contractile machinery disassembly, in concert with cytoskeletal reorganization, to generate a greater resting length of the cell. In the absence of length fluctuations, there is a steady-state maintenance of force and here there is indeed an increase in the population density of the latch bridges or slowly cycling crossbridges (A_M; not shown), and this has been previously demonstrated (31). However, the model does not account for a transfer of generated force to passive elements since this would likely require adaptation of the cytoskeleton. Longer term changes that could emerge out of potential feedback from the reorganized subcellular structures to the cytoskeleton have also not been accounted for. Taken together these modeling results, however, strongly implicate cytoskeletal involvement in length adaptation and mechanical plasticity, experimental evidence of which is well documented (13).

It would be desirable to understand the physical significance of the parameters we have investigated. We have already established that σ (the mean threshold strain at which actin-myosin disconnectivity occurs) is related to actin and myosin filament lengths. The extent to which parallel-to-serial transitions could occur is characterized by ϕ. We speculate that ϕ is related to cytoskeletal stiffness. If the cytoskeleton is very stiff, then length changes imposed on the cell would not be transmitted to the subcellular structures, preventing reconfiguration of the force transmission pathways so that ϕ is small. In contrast if the cytoskeleton is malleable and compliant, force transmission pathways could readily reconfigure so that ϕ would be larger in these cases. To test the hypothesis that ϕ is related to cytoskeletal stiffness, and thereby test the model predictions presented here, a possible experiment could involve rapid transient stretch (known to cause cytoskeletal fluidization) to a tissue strip followed by length oscillations in the presence of agonist. The aim would be to compare force-length loops obtained in this way with force-length loops obtained upon application of length oscillations without the initial transient stretch. λ is a measure of the extent to which disconnected filaments remain disconnected. It is not clear what physical factors might govern this parameter. Again, we speculate that one factor that might impair the ability of a disconnected myosin filament to “grab on” to the nearest actin filament could be the effective diffusivity of myosin filaments, within a potentially dynamically changing actin filament lattice, which may be determined by the crowdedness in the cell, not accounted for here. Another factor may involve coupling between focal adhesions and the cytoskeleton that appears to require actin polymerization near the focal adhesions (30). If this is impaired (possibly via drugs such as Latrunculin B), then the effect would be like that of disconnected units unable to transmit contractile force to the cytoskeleton (giving a larger λ). The notion of ASM fragility has been discussed in the literature consisting of forced and spontaneous rearrangements of subcellular and cytoskeletal structures on different timescales (19). The model results suggest that σ may be a measure of forced rearrangements while ϕ is a measure of spontaneous rearrangement.

Significant challenges still lie ahead. Accounting for length and force adaptation on different timescales in this framework requires some history-dependent effects that have not been included here. The functional forms for disconnectivity and parallel-to-serial transitions are assumed to be purely length dependent and therefore instantaneous responses to contractile-unit events. However, timescales other than those associated with crossbridge dynamics must operate in reality; in particular rates of actin and myosin polymerization upon ASM activation have not been incorporated and the assumption of rapid disconnectivity rates leading to quasi-steady equilibrium of the different types of contractile units may not be realistic. Most importantly the dynamics of cytoskeletal mechanics have not been accounted for here, the focus having been on contractile apparatus, but this is an important aspect of whole ASM cell mechanics that will need to be considered in future work. To understand how these cell-level effects transmit to the tissue level and affect airway hyperresponsiveness, there is also a need to consider force transmission pathways between the cell and the extracellular matrix (ECM). This would require further model development to account for other elements representing mechanical properties of focal adhesions and the ECM. Additional challenges lie in simplifying complex models of the kind presented here that still captures the physics but can be incorporated into a continuum solid mechanics framework required for multiscale models (5, 35).

The biophysical cell-level model developed here captures many of the salient features of mechanical behavior observed experimentally. In particular the specific nature of observed hysteresis and nonlinear elastic behavior emerges directly out of acto-myosin disconnectivity and dynamic reorganization of contractile machinery as a result of imposed length fluctuations. These results also confirm experimental observations that cytoskeletal malleability plays an additional important role in regulating force generation but that its influence on mechanical behavior is likely to occur on a slower timescale to that of contractile apparatus rearrangement.

GRANTS

B. S. Brook is supported by the New Investigator Research Grant from the Medical Research Council, UK (G0901174).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: B.S.B. conception and design of research; B.S.B. performed experiments; B.S.B. analyzed data; B.S.B. interpreted results of experiments; B.S.B. prepared figures; B.S.B. drafted manuscript; B.S.B. edited and revised manuscript; B.S.B. approved final version of manuscript.

Appendix A: LIST OF VARIABLES AND PARAMETERS

Tables A1, A2, A3, and A4 provide a list of variables and parameters.

Table A1.

List of dimensional variables

Dimensional/Original Variable	Variable Name
Mean number of contractile units of i type	Ni*, i = A, B, C
Mean number of contractile units in series per pathway of i type	Nis*, i = A, B
Mean number of parallel pathways of i type	Pi*, i = A, B
Contractile-unit length	Yi*, i = A, B
Contractile-unit velocity	Ẏi*, i = A, B
Cell length	L*
Cell velocity	L̇*
Length of the nucleus	Z*
Number of available crossbridges per unit length of myosin filament	ρi*, i = A, B
Total force generated by cell	T*
Contractile force generated by a single contractile unit within a pathway	Fi*, i = A, B
Time	t*

Table A2.

List of dimensional parameters

Dimensional Parameter	Parameter Name	Parameter Value
Total number of contractile units in the cell	NS*	(Only fractions of total needed in model)
Reference number of parallel path-ways in the cell	Piref*	=1
Average length of individual contractile units at maximum overlap	¯YA* ∼ ${\overline{Y}}_B^{\ast }$	≈2 μm (21)
Average length of an ASM cell at maximum activation	¯L*	≈100 μm (41)
Undeformed length of nucleus	Z0*	≈10 μm (21)
Stiffness of the cell nucleus	EA*	≈0.7 Nm−1 (21)
Extensional viscosity of the cell	μD*	≈3 Ns/m (4)
Stiffness of the cytoskeleton (varies with frequency)	EC*	≈0.1–10 Nm−1 (44)
Number of available crossbridges at max overlap in each contractile unit	ρ̄i*	≈6 × 107m−1 (4)
Stiffness of each crossbridge	K*	=1.8 × 10−3 Nm−1 (47)
Power-stroke length	h*	=15.6 nm (31)
Frequency of oscillation	ω*	=2 Hz (2)
Amplitude of oscillation	Lε*/Ni* × 100	0.5, 1, 2, 4, 8%

Numbers in parentheses indicate reference numbers.

Table A3.

List of normalized variables

Normalized Variable	Variable Name	Normalization
Proportion of contractile units of i type	Ni(i = A, B, C)	= Ni*/NS*
Proportion of contractile units in series per pathway of i type	Nis* (i = A, B)	= Nis*/NS*
Proportion of parallel pathways of i type	Pi(i = A, B)	= Pi*/Piref*
Contractile-unit length (stretch ratio)	Yi(i = A, B)	= Yi*/Ȳi*
Cell length (stretch ratio)	L	= L*/L̄*
Length of nucleus	Z	= Z*/Z0*
Proportion of available crossbridges	ρi(i = A, B)	= ρi*/ρ̄i*
Total force generated by the cell	T	= T*/(EA*L̄*)
Contractile force generated by a contractile unit within a pathway	Fi(i = A, B)	= Fi*/(ρ̄*K*h*2)
Time	t	= t*ω*

Table A4.

List of dimensionless parameters

Dimensionless Parameter	Parameter Name	Parameter Value
Dimensionless threshold strain at which actin-myosin pairs disconnect	σ	0.12, 0.15, 0.18, 0.8 (robust)
Maximum fraction of disconnected contractile units that remain disconnected	λ	0.0 (robust), 0.3, 0.6
Extent of parallel-to-serial transitions and vice versa	ϕ	0.0 (robust), 0.5, 1.0
Dimensionless amplitude of length oscillation	ε = Lε*/L̄*	0.005, 0.01, 0.02, 0.04, 0.08
Dimensionless cytoskeletal stiffness	EC = EC*/EA*	0.001, 0.1, 0.2, 0.3, 0.4, 0.5
Ratio of crossbridge properties to cell level properties	$\alpha ={\overline{\rho }}_i^{*}{K}^{*}{h}^{*2}/E_A^{*}{\overline{L}}^{*}$	0.004
Loss tangent or hysteresivity parameter	${\mu }_D={\mu }_D^{*}{\omega }^{*}/E_A^{*}$	0.001
Ratio of undeformed nucleus length to cell length	$\beta =Z_i^{*}/{\overline{L}}^{*}$	0.1
Fraction of contractile units in each set of force transmission pathways at maximal activation	N̄i (i = A, B) = N̄i*/NS*	0.5
Fraction of force transmission pathways of each type at maximal activation	P̄ = Pi*/P̄iref*	10

Appendix B: ADDITIONAL FIGURES

Figures B1 and B2 provide additional supporting data.

Fig. B1.

Total dimensionless force T exerted by the cell (for μ = 1; symmetric N_i), resulting from imposed length fluctuations (amplitude ε = 0.04), as a function of dimensionless cell length L for increasing ϕ (left to right) and increasing λ (top to bottom) for different values of the threshold parameter (or mean myosin filament length), σ = 0.8 (red), 0.075 (blue) during one length oscillation cycle. For comparison purposes, the black curve in each case shows the force-length loop for the robust case ϕ = 0, λ = 0, σ >> ε. The total force is normalized with respect to the quantity E_A^*L̄^* where E_A^* is the stiffness of the nucleus, and L̄^* is the reference length of the cell at maximal isometric force.

Fig. B2.

Maximum amplitude of contractile-unit velocity Ẏ_B (A) and difference in timing of peak Ẏ_B and peak L̇ (B), for ϕ = 0, as a function of λ for different values of threshold parameter σ. C: maximum amplitude of Ẏ_B, for λ = 0, as a function of ϕ; there are no differences between the different values of σ since its effect only becomes evident for λ > 0.

Appendix C: MODEL EQUATIONS

The underlying model development, nondimensionalization of the governing equations and algebraic details are given in Ref 4. Here we highlight modifications and extensions required for the present model development and detail the consequent differences in the governing equations.

Dimensional governing equations.

At time t* (starred quantities are dimensional and bars denote quantities at maximally activated steady-state isometric conditions) the length of an individual cell L^*(t) and the lengths of the individual contractile units Y_A^*(t^*) and Y_B^*(t^*) satisfy

$$L^{*}=N_{As}^{*}{Y}_A^{*}+Z^{*}=N_{Bs}^{*}{Y}_B^{*},$$ (C1a)

where N_As^* and N_Bs^* are the mean number of contractile units constituting each of the A and B force transmission pathways respectively. We assume that the contractile units in each pathway can be represented by the same average characteristics, effectively allowing us to treat them as identical. The total number of contractile units in the cell at any point in time is N_S^* = N_A^* + N_B^* + N_C^* where the total number of contractile units in all force transmission pathways of i type is given by N_i^* = N_is^*P_i^* for i = A, B, with P_i referring to the number of force transmission pathways that exist of each type. We assume that contractile unit types transition from A, B, and C ensuring that N_S^* is conserved. Typical parameter values for these quantities and others that follow below are taken from Ref. 4.

The total force, T^*, that the cell exerts (which may be tensile or compressive) on its neighbors is given by the sum of the forces in each of the parallel contributions. Balancing forces, it follows that

$$T^{*}=P_A^{*}{F}_A^{*}+P_B^{*}{F}_B^{*}+E_C^{*}(L^{*}-{\overline{L}}^{*})+{\mu }_D^{*}{\dot{L}}^{*},$$ (C1b)

$$P_A^{*}{F}_A^{*}=E_A^{*}(Z^{*}-Z_0^{*}),$$ (C1c)

where the dot denotes a time derivative. L̄^* is the reference cell length at maximal isometric force generation and Z₀^* is the unstressed nucleus length, respectively.

The contractile unit forces F_A^* and F_B^* and the shortening velocities of each contractile unit type are calculated using the HHM (12, 31). The Hai-Murphy four-state model (15, 16) is based on observations that myosin exists in four distinct states: 1) free unattached myosin (M^*), 2) phosphorylated unattached myosin (M_p^*), 3) phosphorylated myosin attached to actin (AM_p^*) (so-called rapidly cycling crossbridges), and 4) dephosphorylated myosin attached to actin (AM^*) giving rise to very slowly cycling or noncycling crossbridges (also called latch bridges). The equations that relate force generation by the contractile units to their shortening velocity and method of solution are detailed in Ref. 4; the dimensionless versions are given below for completeness.

Nondimensionalization.

As in Ref. 4, we express cell, contractile unit and nucleus lengths as stretch ratios relative to their maximally activated steady state or reference configuration. We normalize cytoskeletal stiffness with respect to the stiffness of the nucleus E_A^*, total force with respect to E_A^*L̄^* and contractile force with respect to total number of crossbridges per unit length of myosin filament and crossbridge stiffness (see Ref. 4 for details). Normalizing the number of contractile units N_i^* and N_is^* with respect to N_S^* gives contractile unit fractions N_i = N_i^*/N_S^* for i = A, B, C so that N_A + N_B + N_C = 1. Additionally, we normalize the number of parallel pathways with respect to a reference number of pathways (P_i^ref* = 1) so that P_i = P_i^*/P_i^ref* and hence N_i = N_isP_i. Applying these normalizations yields the dimensionless equations given in the main text. Algebraic manipulations and solution of governing equations follows that detailed in Ref. 4.

The dimensionless equations governing the HHM model of force generation are

$$\frac{\partial n_i}{\partial t}+{\gamma }_i{\dot{Y}}_i\frac{\partial n_i}{\partial x}=S_i{n}_i,$$ (C2a)

$${\displaystyle \sum n_i=1,}$$ (C2b)

for i = A, B, where x is an internal variable representing the displacement of the crossbridges from their unstressed position, n_i = (M_i, M_pi, AM_pi, AM_i) is the fraction (in x space) of crossbridges in each pool (i.e., M_i = M_i^*/ρ_i^* etc.). S_i is the matrix of dimensionless rate constants and describes the probabilities of transitions among the four states and how those probabilities depend on x. The dimensional version of S_i is given in Fredberg et al. (12); as in Ref. 4, the dimensionless version is

$$S_i(x,t)=\left(\begin{array}{cccc}-k_1& k_2& 0& g(x)\\ k_1& -k_2-f_p(x)& g_p(x)& 0\\ 0& f_p(x)& -k_5-g_p(x)& k_6\\ 0& 0& k_5& -k_6-g(x)\end{array}\right).$$ (C2c)

The dimensionless x-dependent rate functions are given by

$$\left[f_p(x),g_p(x),g(x)\right]=\{\begin{array}{cc}(0,g_{p2},g_2),\hfill & x<0\hfill \\ (f_{p1}x,g_{p1}x,p_{1}x),\hfill & 0\le x\le 1\\ \left[0,(g_{p1}+g_{p3})x,(g_1+g_3)x\right],& x>1\hfill \end{array},$$ (C2d)

where f_p1 = f_p1^*/ω^* and so on, and all the parameter values for the rate constants are taken from Ref. 4. The dimensionless force generated by each contractile unit is then given by

$$F_i(t)={\rho }_i(Y_i){\displaystyle {\int }_{-\infty }^{\infty }xq\cdot n_i(x,t)dx,}$$ (C2e)

for i = A, B, where ρ_i = ρ_i^*/ρ̄_i^* and q ≡ (0 0 1 1).

Quasi-steady equilibrium limit for transitions among N_A, N_B, and N_C.

The transitions between these three types of contractile units may be viewed in terms of “binding” and “unbinding” of the different species and written as

$$N_A\underset{k_{CA}(L)}{\overset{k_{AC}(L)}{\rightleftharpoons }}{N}_C\underset{k_{BC}(L)}{\overset{k_{CB}(L)}{\rightleftharpoons }}{N}_B,$$ (C3a)

where k_iC, k_Ci, i = A, B are cell-length dependent rate functions. Differential equations governing this reaction are thus given by:

$$\frac{d{N}_A}{dt}=-k_{AC}(L)N_A+k_{CA}(L)N_C,$$ (C3b)

$$\frac{d{N}_B}{dt}=-k_{BC}(L)N_B+k_{CB}(L)N_C,$$ (C3c)

subject to

$$\begin{array}{cc}{\overline{N}}_A+\overline{N_B}=1,& {\overline{N}}_C=0.\end{array}$$ (C3d)

Therefore, assuming that at any time during the length fluctuation the contractile units are in quasi-steady equilibrium, i.e., $\frac{d{N}_i}{dt}$ ≈ 0, for i = A, B leads to

$$N_i=\frac{k_{Ci}}{k_{iC}}{N}_C,$$ (C3e)

for i = A, B. Substituting for N_A and N_B from (Eq. 3) and applying (Eq. C3d) yields

$$\begin{array}{cc}\frac{k_{Ci}}{k_{iC}}={\overline{N}}_i\frac{f(L)}{1-f(L)},& \text{where}\hspace{0.17em}f(L)=1-\frac{\lambda {(L-\mu )}^m}{{\sigma }^m+{(L-\mu )}^m},\end{array}$$ (C3f)

for i = A, B. This is therefore the equivalent of assuming that the ratio of the “on” rate to the “off” rate is given by the cell-length-dependent expression above and that a quasi-steady assumption allows us to use (Eq. 3) as a description of the instantaneous response to a length change which is itself varying with time. Implicit in this is that the reactions in Eq. C3a are faster than the crossbridge kinetics described by S_i.