Rheology of crossbridge ensembles
Muscle rheology, or the characterization of a muscle's response to external mechanical perturbations, is crucial to an animal's motor control and locomotive abilities. How the rheology emerges from the ensemble dynamics of microscopic actomyosin crossbridges known to underlie muscle forces is however a longstanding question. Classical descriptions in terms of force-length and force-velocity relationships capture only part of the rheology, namely under steady but not dynamical conditions. Although much is known about the actomyosin machinery, current mathematical models that describe the behavior of a population or an ensemble of crossbridges are plagued by an excess of parameters and computational complexity that limits their usage in large-scale musculoskeletal simulations. In this paper, we examine models of crossbridge dynamics of varying complexity and show that the emergent rheology of an ensemble of crossbridges can be simplified to a few dominant time-constants associated with intrinsic dynamical processes. For Huxley's classical two-state crossbridge model, we derive exact analytical expressions for the emergent ensemble rheology and find that it is characterized by a single time-constant. For more complex models with up to five crossbridge states, we show that at most three time-constants are needed to capture the ensemble rheology. Our results thus yield simplified models comprising of a few time-constants for muscle's bulk rheological response that can be readily used in large-scale simulations without sacrificing the model's interpretability in terms of the underlying actomyosin crossbridge dynamics.