A Strain-Dependency of Myosin Off-Rate Must Be Sensitive to Frequency to Predict the B-Process of Sinusoidal Analysis

Abstract

Muscle force arises as the result of many myosin molecules, each producing a force discrete in magnitude and in time duration. In previous work we have developed a computer model and a mathematical model of many myosin molecules acting as an ensemble and demonstrated that the time duration over which myosin produces force at the molecular level (referred to here as “time-on”) gives rise to specific visco-elastic properties at the whole muscle level. That model of the mechanical consequences of myosin-actin interaction predicted well the C-process of small length perturbation analysis and demonstrated that the characteristic frequency 2πc provided a measure of the myosin off-rate, which is equal to the reciprocal of the mean time-on. In this study, we develop a mathematical hypothesis that a strain-dependence of the myosin off-rate at the single molecule level can result in a negative viscous modulus like that observed at low frequencies, i.e., the B-process. We demonstrate here that a simple monotonic strain-dependency of the myosin off-rate cannot account for the observed B-process. However, a frequency-dependent strain-dependency, as may occur when visco-elastic properties of the myosin head are introduced, can explain the observed negative viscous modulus. These findings suggest that visco-elastic properties of myosin constitute the specific molecular mechanisms that underlie the frequency-dependent performance of many oscillatory muscles such as insect flight muscle and mammalian cardiac muscle.

Dynamics of cross-bridge cycling, ATP hydrolysis, force generation, and deformation in cardiac muscle

“…For a complete description of the model simplification approach please see Palmer [13]. Simplification is performed by expanding the exponentials as quadratic polynomials, multiplying Equations (4)–(6) by s i , and integrating from −∞ to +∞ which yields: …”

“…Often the computational expense of these distributed models can be circumvented using a moment distribution approach to reduce a system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) [9, 10]. Previous studies have applied this approach [9–13], however none of these have accounted for the effect of metabolites on the kinetics. Similarly, a previously developed kinetic model [14] that does account for metabolite concentrations (energetic state) on XB cycling does not account for the coupling of deformation/strain and kinetics.…”

“…To account for the effect of deformation on XB cycling, transition rates between different conformations are affected by stretch/strain as experienced by the myosin-actin complex during shortening or lengthening. The model is constructed around a four-state XB scheme [15, 16] which is less parsimonious than previous two-state or three-state XB models [3, 13, 17–19]. However, this formulation is the most economical that could fit all the sinusoidal perturbation experiments described below.…”

Resistance training alters skeletal muscle structure and function in human heart failure: effects at the tissue, cellular and molecular levels

“…The frequency portion of the B‐process (2π b ) has been historically interpreted as the apparent rate of myosin force production or, in other words, the rate of myosin transition between the weakly and strongly bound states (Kawai et al 1993; Zhao & Kawai 1993). We have recently put forth the interpretation that 2π b may represent the mechanical rate constant of the viscoelastic stiffness of the myosin head (Palmer 2010). Both interpretations point to a P i ‐dependent mechanical characteristic of the myosin lever arm between the pre‐ and post‐power stroke states.…”

An analysis of deformation‐dependent electromechanical coupling in the mouse heart

“…The dependence of tension on velocity involves at least two different processes, known as the B process and the C process, which work on different time scales to affect the crossbridge cycle. However, the response of tension to strain rate is complex (Kawai & Brandt, 1980), and the mechanistic effects controversial, with both an increase and decrease in the rate of strongly to weakly bound crossbridges with respect to strain rate having been proposed to explain this process (Yadid & Landesberg, 2010; Palmer, 2010).…”