Two-state model of acto-myosin attachment-detachment predicts C-process of sinusoidal analysis

Abstract

The force response of activated striated muscle to length perturbations includes the so-called C-process, which has been considered the frequency domain representation of the fast single-exponential force decay after a length step (phases 1 and 2). The underlying molecular mechanisms of this phenomenon, however, are still the subject of various hypotheses. In this study, we derived analytical expressions and created a corresponding computer model to describe the consequences of independent acto-myosin cross-bridges characterized solely by 1), intermittent periods of attachment (t(att)) and detachment (t(det)), whose values are stochastically governed by independent probability density functions; and 2), a finite Hookian stiffness (k(stiff)) effective only during periods of attachment. The computer-simulated force response of 20,000 (N) cross-bridges making up a half-sarcomere (F(hs)(t)) to sinusoidal length perturbations (L(hs)(t)) was predicted by the analytical expression in the frequency domain, (F(hs)(omega)/L(hs)(omega))=(t(att)/t(cycle))Nk(stiff)(iomega/(t(att)(-1)+iomega)), where t(att) = mean value of t(att), t(cycle) = mean value of t(att) + t(det), k(stiff) = mean stiffness, and omega = 2pi x frequency of perturbation. The simulated force response due to a length step (L(hs)) was furthermore predicted by the analytical expression in the time domain, F(hs)(t)=(t(att)/t(cycle))Nk(stiff)L(hs)e(-t/t(att)). The forms of these analytically derived expressions are consistent with expressions historically used to describe these specific characteristics of a force response and suggest that the cycling of acto-myosin cross-bridges and their associated stiffnesses are responsible for the C-process and for phases 1 and 2. The rate constant 2pic, i.e., the frequency parameter of the historically defined C-process, is shown here to be equal to t(att)(-1). Experimental results from activated cardiac muscle examined at different temperatures and containing predominately alpha- or beta-myosin heavy chain isoforms were found to be consistent with the above interpretation.

FIGURE 1

Frequency and time domain representations of the force response to length perturbation analysis. (A) The complex modulus recorded at frequencies ranging 0.125–250 Hz (+) for human epicardial myofilaments at pCa 4.5 and 37°C is presented here as a Nyquist plot (viscous versus elastic moduli) and is characterized well by Eq. 1A (inset) represented by the solid line. (B) The complex modulus like that shown in panel A has been attributed to three underlying processes. The graphical representation of the A-process (······) is displayed as a linear relationship between the viscous and elastic moduli. The B-process (–––) and C-process (··-··-··) are represented as semicircles, which are indicative of exponential responses in the frequency domain. (C) A recorded tension change after a step length change of 0.1% muscle length can be fit closely by Eq. 1B (inset), which represents the step response of the tension per unit length change. (D) The time courses of the individual A-, B-, and C-processes in Eq. 1B are illustrated here in the time domain.

FIGURE 2

Strains on the acto-myosin cross-bridge. (A) Myosin binding sites, indicated by short vertical lines, occur at 5.5 nm spacings along the actin filament (49). No force, f = 0, results when myosin is detached (det) from actin. (B) An extension or compression of the elastic element may be required for the myosin head to bind to actin. The resultant force, f, is proportional to the stiffness of the elastic element, k_stiff, and the displacement undergone by the myosin head to find to an actin binding site, d_site. (C) Force is generated when the myosin power stroke imposes unitary displacement, d_uni, on the elastic element and is additive to that force resulting from d_site. (D) An additional cross-bridge-dependent force is transmitted through the half-sarcomere when the elastic element undergoes an externally applied length change, L_hs.

FIGURE 3

Examples of the number of occurrences generated for the random variables t_att and t_det. (A) The probability density function for t_att with a mean value of 21 ms was constructed by summing two random numbers generated from two independent single-exponential probability density functions having mean values of 20 ms and 1 ms. (B) The probability density function for t_det with a mean value of 210 ms was generated in a similar fashion from two independent single-exponential probability density functions having mean values of 200 ms and 10 ms. (C) Alternating attached and detached states were constructed over a 1-s time period for each of 20,000 acto-myosin cross-bridges. One such second for one such cross-bridge is illustrated here.

FIGURE 4

The simulation of individual cross-bridge force responses to length perturbations. Sinusoidal length perturbations of the half-sarcomere at 1 Hz (A), 10 Hz (B), and 100 Hz (C) were simulated with amplitude 1 nm and no phase relative to time zero. The force response of a single cross-bridge occurred only during the time of attachment (att) as indicated by the horizontal bars in panels D–F. At 1 Hz (D), the time of attachment is relatively short such that the deflection in force effectively reflects the instantaneous velocity of the perturbation during the time of attachment. At 10 Hz (E) the time attached is long enough to display some curvature in the force deflection, and at 100 Hz (F) the time attached is so long that the several cycles of the length perturbation are displayed in the force deflection. The sum of 20,000 individual force responses give an ensemble response at 1 Hz (G), 10 Hz (H), and 100 Hz (I), which can be fit with a sine wave (solid line) to provide amplitude and phase of the force response of the ensemble for the virtual half-sarcomere. The fitted sine waves at 1 Hz (J), 10 Hz (K), and 100 Hz (L) can be partitioned into two components that are in-phase (solid line) and out-of-phase (dotted line) with respect to the length perturbation, and the respective magnitudes represent the elastic and viscous moduli.

FIGURE 5

The Nyquist plot, i.e., viscous versus elastic moduli, of simulated data (+) at several frequencies of perturbation between 1 and 250 Hz conform to the expression given by Eq. 16.

FIGURE 6

Examples of mouse cardiac muscle undergoing sinusoidal length perturbation analysis. (A–C) Plots of elastic modulus versus frequency (A), viscous modulus versus frequency (B) and viscous versus elastic moduli (C) of data recorded from myofilaments containing α-MHC and examined at 37°C. (D–F) Plots of data from myofilaments containing α-MHC and examined at 17°C. The characteristic dips in elastic and viscous moduli occurred at lower frequencies at 17°C compared to 37°C, which reflects slower cross-bridge kinetics at the lower temperature. (G–I) Plots of data from myofilaments containing β-MHC and examined at 17°C. The characteristic dips in elastic and viscous moduli occurred at lower frequencies with β-MHC compared to α-MHC, which reflects the slower cross-bridge kinetics associated with the β-MHC.

FIGURE 7

A visco-elastic equivalent of two-state model of acto-myosin cross-bridge kinetics with a single-exponential distribution for t_att. The effective elastic stiffness, k_e, of the half-sarcomere is proportional to the fraction of myosin heads attached, the total number of heads, and the mean stiffness of the cross-bridge elastic elements. The effective viscosity, k_v, due to the intermittent binding of myosin to actin is additionally proportional to the mean time of cross-bridge attachment.