A continuum-mechanical skeletal muscle model including actin-titin interaction predicts stable contractions on the descending limb of the force-length relation.

Thomas Heidlauf,Tobias Siebert,Thomas Klotz,Oliver Röhrle,Christian Rode,Silvia Blemker

doi:10.1371/journal.pcbi.1005773

Abstract

Contractions on the descending limb of the total (active + passive) muscle force—length relationship (i. e. when muscle stiffness is negative) are expected to lead to vast half-sarcomere—length inhomogeneities. This is however not observed in experiments—vast half-sarcomere—length inhomogeneities can be absent in myofibrils contracting in this range, and initial inhomogeneities can even decrease. Here we show that the absence of half-sarcomere—length inhomogeneities can be predicted when considering interactions of the semi-active protein titin with the actin filaments. Including a model of actin—titin interactions within a multi-scale continuum-mechanical model, we demonstrate that stability, accurate forces and nearly homogeneous half-sarcomere lengths can be obtained on the descending limb of the static total force—length relation. This could be a key to durable functioning of the muscle because large local stretches, that might harm, for example, the transverse-tubule system, are avoided.

Highlights

The isometric active force—length relation of muscle fibres [1] and many muscles [2,3,4] is composed of strikingly linear segments
Passive forces in single muscle fibres [11,12,13] and some entire skeletal muscles, e. g. the rabbit extensor digitorum longus, extensor digitorum II, and soleus [3, 4] or the frog semitendinosus [14], appear only at lengths that correspond to the descending limb of the active force—length relation [13, 15] leading to a descending limb in the total force—length relation
For different initial half-sarcomere lengths, lhs[0], i. e., the half-sarcomere lengths at which the activation started, quasi-static (i. e. the force—velocity relation was omitted) active stretch experiments have been performed for different active stretch increments, Δlhs = lhs − lhs[0]

Summary

Introduction

The isometric active force—length relation of muscle fibres [1] and many muscles [2,3,4] is composed of strikingly linear segments. From a mechanical point of view, the negative slope of the force—length relationship at long muscle lengths (descending limb) was and is of special interest for muscle physiologists and modellers due to its inherent instability It has been suggested early [7] that contractions on the descending limb of the force—length relation will result in unstable behaviour leading to the formation of short and long half-sarcomeres in series during fixed-length muscle contractions (cf Fig 1 top). G. the rabbit extensor digitorum longus, extensor digitorum II, and soleus [3, 4] or the frog semitendinosus [14], appear only at lengths that correspond to the descending limb of the active force—length relation [13, 15] leading to a descending limb in the total force—length relation Simulation of these muscles is challenging using continuum-mechanical finite element models, which typically superimpose the passive stress tensor with an active stress that includes the active force—length relation [16, 17]. The descending limb of the resulting total stress—stretch relation (Fig 1 top) causes instability of these models, which explains why previous continuum-mechanical models focused on muscles with significant passive forces occurring at short muscle length, i. e., that have no descending limb in the total force—length relation [17,18,19,20,21]

Methods

Results

Discussion

Conclusion