Actomyosin contraction, aggregation and traveling waves in a treadmilling actin array

• Multiscale perturbation theory leads to a continuum model for the actomyosin array.
• Actin treadmilling is essential for contraction.
• Depending on parameters, periodic or traveling wave patterns evolve in the system.
• Two regimes for contraction with constant rate are identified.
• For maximal contraction, myosin has to concentrate where actin density is minimal.

We use perturbation theory to derive a continuum model for the dynamic actomyosin bundle/ring in the regime of very strong crosslinking. Actin treadmilling is essential for contraction. Linear stability analysis and numerical solutions of the model equations reveal that when the actin treadmilling is very slow, actin and myosin aggregate into equidistantly spaced peaks. When treadmilling is significant, actin filament of one polarity are distributed evenly, while filaments of the opposite polarity develop a shock wave moving with the treadmilling velocity. Myosin aggregates into a sharp peak surfing the crest of the actin wave. Any actomyosin aggregation diminishes contractile stress. The easiest way to maintain higher contraction is to upregulate the actomyosin turnover which destabilizes nontrivial patterns and stabilizes the homogeneous actomyosin distributions. We discuss the model’s implications for the experiment.