Mathematical modeling of myosin, muscle contraction, and movement

This special issue features 7 original research articles that delve into the application of mathematical modeling to understand cardiac contractile dynamics across a range of spatial scales.

At the sarcomeric level, McCabe et al. [1] developed a multiscale model to explore the mechanistic effects of 2’-deoxy-ATP (dATP), an inotrope with therapeutic potential, on the dynamics of crossbridge cycling. The authors start by constraining crossbridge association rates using Brownian dynamics simulations before embedding these rates within a mechanistic Monte Carlo Markov chain model. Simulations reveal that the effect of dATP on force production can be captured if both the powerstroke and crossbridge detachment rates were increased. Interestingly, simulations of twitch dynamics show that the effects of dATP on crossbridge kinetics alone were not sufficient to capture relaxation kinetics, suggesting a possible effect of dATP on calcium dynamics.

A metabolic by-product of dATP following hydrolysis by myosin is 2’-deoxy-ADP (dADP). Given the therapeutic promise of dATP to treat hypo-contractility, assessment of the effects of dADP are warranted. Using molecular dynamics simulations of myosin protein structure, Childers et al. [2] were able to delineate atomic-level insights on the allosteric transmission of information in the myosin structure. They found that dADP increased myosin interactions with actin and could dissociate from the binding pocket at a faster rate than ADP.

While myosin is typically considered to be the protagonist of the crossbridge duo, the actin molecules that comprise the thin filament are an important regulator of crossbridge dynamics. To investigate how variations in thin filament length and regulatory unit (RU) density affect ensemble cross-bridge behaviour and force production, Fenwick et al. [3] used a spatially explicit model of the half-sarcomere to simulate the effects of uniform and random RU knockout schemes on force generation. They found that the dynamics of force production is not simply influenced by the raw number of RUs available for myosin binding but is dependent on the spatial pattern in which RUs are distributed.

While still at the sarcomere scale, the work by Malingen et al. [4] investigates an entirely different perspective of the sarcomere, focussing not on the kinetics of the actin and myosin filaments, but instead on the fluid that surrounds them. They developed a finite element model of fluid flow within the sarcomere to characterise flow fields and drag forces as a function of filament overlap. Their calculations reveal that the magnitude of drag forces is surprisingly small relative to the force generated by a myosin motor, and that energetic cost arising from viscous shearing against lattice proteins is minimal.

Moving up in spatial scale, Jarvis et al. [5] propose a number of properties at the molecular level that are required to explain mechanical phenomena at the muscle fiber level. Starting with the requirement of a weakly-bound crossbridge state, the authors found that they could fit the force-velocity or the force transient data, but not both concurrently. Addition of force-dependent detachment to the weakly-bound state, and an elastic element in series with the crossbridges led to a satisfactory fit of the data. These results suggest a key role involving the interaction of crossbridges and a series elastic element in explaining the force response after stretch. This study, along with that of McCabe et al. [1], highlights the utility of mathematical modeling as a tool to systematically inter-rogate underlying physiological mechanisms.

At the trabecula level, myocardial relaxation has been reported to be modified by the rate at which the muscle is lengthened. Building on these studies, Schick et al. [6] used a combination of in vitro trabecula experiments and computational modeling to show that reduction of preload can further increase the rate of muscle relaxation. These results suggest a potential avenue for clinical treatment of diastolic dysfunction by reducing ventricular volume to augment relaxation.

At the level of the myocardium, the mechanical characteristics of individual cardiomyocytes are not homogeneous. In particular, Clark et al. [7] report substantial variations in the rate of relaxation among cardiomyocytes. They simulated the effect of cellular relaxation heterogeneity on tissue-level relaxation kinetics by varying the proportion of slow and fast relaxation contractile units and found a disproportionate impact of fast-relaxing cells on overall tissue relaxation. The model predictions were validated using a novel engineered heart tissue platform derived from induced pluripotent stem cells.

Together the collection of studies in this special issue provide a snapshot of the latest advancements in mathematical modelling of cardiac contractile dynamics from molecular interactions to tissue level mechanics.