^{1,a)}and Karsten Kruse

^{1,b)}

### Abstract

Sarcomeres are the basic force generating units of striated muscles and consist of an interdigitating arrangement of actin and myosin filaments. While muscle contraction is usually triggered by neural signals, which eventually set myosin motors into motion, isolated sarcomeres can oscillate spontaneously between a contracted and a relaxed state. We analyze a model for sarcomere dynamics, which is based on a force-dependent detachment rate of myosin from actin. Our numerical bifurcationanalysis of the spontaneous sarcomere dynamics reveals notably Hopf bifurcations, canard explosions, and gluing bifurcations. We discuss possible implications for experiments.

In skeletal muscles, the elementary force producing units are sarcomeres. They consist of interdigitating filaments of molecular motors and actin filaments. Upon activation by a neural signal, the motors move along the actin filaments and thus shorten the sarcomere. Elastic elements maintain the regular arrangement of motor and actin filaments. It has been found experimentally that in the absence of neural signals, sarcomeres can spontaneously oscillate in length. We study a possible mechanism of spontaneous sarcomere oscillations, which is based on a dynamic instability of motors coupled to an elastic element. We find a large variety of nonlinear behavior ranging from simple oscillations to excitable and chaotic dynamics. Our results indicate possible ways to experimentally test the mechanism we propose for spontaneous sarcomere oscillations.

We would like to thank E. Nicola, A. Pikovsky, and M. Wechselberger for discussions.

I. INTRODUCTION

II. DYNAMICS OF A HALF-SARCOMERE

A. Dynamic equations

B. Linear stability

C. Canard explosion of the limit cycle

D. Secondary bifurcations

III. DYNAMICS OF A SARCOMERE

A. Basic oscillation modes

B. Gluing of cycles

C. Chaotic behavior

IV. CONCLUSIONS

### Key Topics

- Bifurcations
- 62.0
- Elasticity
- 11.0
- Explosions
- 11.0
- Muscles
- 11.0
- Relaxation oscillations
- 8.0

## Figures

(a) Illustration of a sarcomere. Bipolar myosin filaments interdigitate with actin filaments, which are attached with their plus-ends to Z-disks. Upon activation of the motors, the actin filaments are pulled toward the M-line resulting in sarcomere contraction. (b) Illustration of the model describing the dynamics of a half-sarcomere. The parallel actin and myosin filaments are, respectively, replaced by single effective filaments. Motors are attached via elastic springs of stiffness with extension to the common backbone, separated from each other by a distance . These effective motors are processive and have a well-defined force-velocity relation, see text. The motors’ detachment rate is force-dependent. A spring of stiffness accounts for the elastic components of the structure.

(a) Illustration of a sarcomere. Bipolar myosin filaments interdigitate with actin filaments, which are attached with their plus-ends to Z-disks. Upon activation of the motors, the actin filaments are pulled toward the M-line resulting in sarcomere contraction. (b) Illustration of the model describing the dynamics of a half-sarcomere. The parallel actin and myosin filaments are, respectively, replaced by single effective filaments. Motors are attached via elastic springs of stiffness with extension to the common backbone, separated from each other by a distance . These effective motors are processive and have a well-defined force-velocity relation, see text. The motors’ detachment rate is force-dependent. A spring of stiffness accounts for the elastic components of the structure.

Hopf bifurcation and canard explosion in a half-sarcomere. The stationary value of the binding probability given by Eq. (6) is unstable for and limit cycle oscillations emerge. At the oscillation amplitude explodes and relaxation oscillations are clearly detectable, see the inset. Parameter values are , , , , , , and (inset, dashed) as well as (inset, solid). Numerical solutions are obtained using AUTO07P (Ref. 16) and the XPPAUT software (Ref. 17).

Hopf bifurcation and canard explosion in a half-sarcomere. The stationary value of the binding probability given by Eq. (6) is unstable for and limit cycle oscillations emerge. At the oscillation amplitude explodes and relaxation oscillations are clearly detectable, see the inset. Parameter values are , , , , , , and (inset, dashed) as well as (inset, solid). Numerical solutions are obtained using AUTO07P (Ref. 16) and the XPPAUT software (Ref. 17).

An excited trajectory in phase space and nullclines for (red) and (blue) in the case of the half-sarcomere with . Inset I: corresponding time course . Inset II: magnification of the phase space around the fixed point. Single arrows correspond to slow dynamics; double arrows correspond to fast dynamics. Other parameters are as in Fig. 2.

An excited trajectory in phase space and nullclines for (red) and (blue) in the case of the half-sarcomere with . Inset I: corresponding time course . Inset II: magnification of the phase space around the fixed point. Single arrows correspond to slow dynamics; double arrows correspond to fast dynamics. Other parameters are as in Fig. 2.

Bifurcation diagram for a half-sarcomere. Stationary states and limit cycles of the half-sarcomere as a function of , respectively, represented by the stationary and extremal values of . The solutions of Eq. (6) and the bifurcating limit cycles are represented by solid lines, the solutions of Eq. (7) are represented by dashed and dot-dashed lines. The arrow indicates the saddle-node bifurcation point. At , we have a subcritical forward Hopf bifurcation. Inset: the oscillation period logarithmically diverges as . Parameters are as in Fig. 2.

Bifurcation diagram for a half-sarcomere. Stationary states and limit cycles of the half-sarcomere as a function of , respectively, represented by the stationary and extremal values of . The solutions of Eq. (6) and the bifurcating limit cycles are represented by solid lines, the solutions of Eq. (7) are represented by dashed and dot-dashed lines. The arrow indicates the saddle-node bifurcation point. At , we have a subcritical forward Hopf bifurcation. Inset: the oscillation period logarithmically diverges as . Parameters are as in Fig. 2.

Limit cycles of sarcomeres. (a) Bifurcation diagram of the sarcomere. (b) Relative phase between the two half-sarcomeres for the limit cycles. Stable states are represented by continuous unstable states by dashed lines. For the system has a stable stationary state . For , loses stability and the oscillatory state with emerges. For , loses stability and the limit cycle with emerges. For an unstable mode with bifurcates from . Inset: detail of the bifurcation diagram around in terms of with . Parameter values are , , , and . Other parameter values are as in Fig. 2.

Limit cycles of sarcomeres. (a) Bifurcation diagram of the sarcomere. (b) Relative phase between the two half-sarcomeres for the limit cycles. Stable states are represented by continuous unstable states by dashed lines. For the system has a stable stationary state . For , loses stability and the oscillatory state with emerges. For , loses stability and the limit cycle with emerges. For an unstable mode with bifurcates from . Inset: detail of the bifurcation diagram around in terms of with . Parameter values are , , , and . Other parameter values are as in Fig. 2.

Gluing bifurcation in a sarcomere. [(a) and (b)] Projection of the system’s trajectories onto the -plane. Arrows indicate where the cycles glue together. (a) The two mutually symmetric limit cycles for , (b) the limit cycle for . (c) Period of the limit cycles before (stars) and after (crosses) gluing. Parameter values are as in Fig. 2 except for .

Gluing bifurcation in a sarcomere. [(a) and (b)] Projection of the system’s trajectories onto the -plane. Arrows indicate where the cycles glue together. (a) The two mutually symmetric limit cycles for , (b) the limit cycle for . (c) Period of the limit cycles before (stars) and after (crosses) gluing. Parameter values are as in Fig. 2 except for .

Chaotic behavior of sarcomeres. (a) Bifurcation scenario according to the Ruelle–Takens route to chaos. Solid lines and dashed lines correspond to stable and unstable states, respectively. Primary and secondary Hopf bifurcations are indicated by arrows. The limit cycle becomes unstable and the oscillatory solution emerges. undergoes another Hopf bifurcation toward the state . For decreasing , this state decays into an apparently strange attractor resulting in chaotic dynamics. Inset: magnification of the transitions to and to chaotic dynamics. (b) State as a function of time and in the plane for . (c) Example of a chaotic solution at . Other parameter values are , , , , , and .

Chaotic behavior of sarcomeres. (a) Bifurcation scenario according to the Ruelle–Takens route to chaos. Solid lines and dashed lines correspond to stable and unstable states, respectively. Primary and secondary Hopf bifurcations are indicated by arrows. The limit cycle becomes unstable and the oscillatory solution emerges. undergoes another Hopf bifurcation toward the state . For decreasing , this state decays into an apparently strange attractor resulting in chaotic dynamics. Inset: magnification of the transitions to and to chaotic dynamics. (b) State as a function of time and in the plane for . (c) Example of a chaotic solution at . Other parameter values are , , , , , and .

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