The Compliance-induced Slowdown of Stress Relaxation in Active Sarcomeres

Abstract

Muscle contracts when stimulated but also generates transient stress in response to externally imposed length changes. The stress-relaxation timescale of the perturbation-induced stress is vital for the everyday use of muscle. If stresses are long-lived, muscle is stiff like an elastic solid and useful for behaviors such as maintaining posture. But if stresses are rapidly dissipated, muscle yields like a liquid and enables rapid postural changes. Although common experience suggests that muscle routinely transitions between a solid-like and liquid-like response, current sarcomere models cannot account for large changes in the stress-relaxation time. We show by modeling muscles as half-sarcomeres with cycling crossbridges on a compliant filament backbone that filament compliance leads to a many-fold increase in the stress-relaxation time compared with a rigid backbone. The slowdown depends on a single dimensionless ratio l/p of the filament overlap l to an emergent length scale p that depends on how strain is shared between the filament backbone and crossbridges. This parameter depends on the mean number of attached crossbridges such that greater activation implies a greater value of l/p. We find that the stress-relaxation time is nearly constant for l/p ≪ 1, but increases as (l/p)2 when l/p ≫ 1. We estimate l/p > 5 for mammalian sarcomeres, which leads to a 10x slowdown of stress relaxation relative to having a rigid backbone. The physical basis is that in a compliant filament model, the attachment of one crossbridge distorts neighboring crossbridges and induces changes in its kinetics. Our predicted effect of filament compliance hypothesizes a new mechanism for muscle’s ability to maintain an elastic behavior on timescales much longer than crossbridge cycling. These effects may scale up to the whole fiber based on titin that mediates inter-sarcomeric interactions.