A continuous-binding cross-linker model for passive airway smooth muscle

Abstract

Although the active properties of airway smooth muscle (ASM) have garnered much modeling attention, the passive mechanical properties are not as well studied. In particular, there are important dynamic effects observed in passive ASM, particularly strain-induced fluidization, which have been observed both experimentally and in models; however, to date these models have left an incomplete picture of the biophysical, mechanistic basis for these behaviors. The well-known Huxley cross-bridge model has for many years successfully described many of the active behaviors of smooth muscle using sliding filament theory; here, we propose to extend this theory to passive biological soft tissue, particularly ASM, using as a basis the attachment and detachment of cross-linker proteins at a continuum of cross-linker binding sites. The resulting mathematical model exhibits strain-induced fluidization, as well as several types of force recovery, at the same time suggesting a new mechanistic basis for the behavior. The model is validated by comparison to new data from experimental preparations of rat tracheal airway smooth muscle. Furthermore, experiments in noncontractile tissue show qualitatively similar behavior, suggesting support for the protein-filament theory as a biomechanical basis for the behavior.

Figure 1

Illustration of cross-linker function. Here the distribution of bound cross-linkers is given for a single half-sinusoidal stretch of 17% of reference length for 0.25 s, beginning at 0.5 s. The steady-state distribution persists for the first 0.5 s, followed by the single transient stretch displacing the bound cross-linkers. After the stretch, recovery toward steady state begins.

Figure 2

Time courses of strain-induced fluidization. Tissue is stretched by 17% from reference length and held for 10 s; then, 2-Hz oscillations of 0% (baseline), 10%, 20%, and 40% of the stretch amplitude occur for 60 s. Recording continues for a further 30 s after the stretch. Experimental data (left) and model predictions (right) for the full oscillatory time course (upper) and the mean data averaged with a 0.5-s moving average to show the mean effect of the oscillations (lower).

Figure 3

Time courses of strain-induced fluidization (layout as in Fig. 2) for skin and tendon, with initial stretch amplitudes of 12.5% and 2%, respectively.

Figure 4

Final values for strain-induced fluidization. The ratio of force after oscillations (F_osc) to force during stress relaxation without oscillation (F_con) for the data in Fig. 2, (left), and equivalent data for skin and tendon (right). Errors are given as ± 1 SD.

Figure 5

Force recovery between transient stretches. Tissue at equilibrium is subject to a transient stretch of 17% of reference length lasting 0.25 s, followed by a rest period and a second, identical stretch. The ratio between peak forces above baseline is measured as a function of the time between stretches. Data are given for both the model prediction and ASM experiment, with errors shown as ± 1 SD.

Figure 6

Recovery of force after a transient stretch. Tissue is stretched by 17% of reference length for 0.25 s, half-sinusoidally, generating a drop in basal force upon return to L_ref and subsequent recovery. Results are given both for the ASM experiment (gray line) and model predictions (black line).

Figure 7

Stress relaxation. The cross-linker is stretched by 10% of the reference length and held for 1000 s. Although the relaxation is multiexponential, across these five decades of time the behavior is a reasonable approximation to a power law.