Energetics of cytoskeletal gel contraction

Cytoskeletal gels are prototyped to reproduce the mechanical contraction of the cytoskeleton in vitro. They are composed of a polymer network (backbone), swollen by the presence of a liquid solvent, and active molecules (molecular motors, MMs) that transduce chemical energy into the mechanical work of contraction. These motors attach to the polymer chains to shorten them and/or act as dynamic crosslinks, thereby constraining the thermal fluctuations of the chains. We describe both mechanisms thermodynamically as a microstructural reconfiguration, where the backbone stiffens to motivate solvent (out)flow and accommodate contraction. Via simple steady-state energetic analysis, under the simplest case of isotropic deformation, we quantify the mechanical energy required to achieve contraction as a function of polymer chain density and molecular motor density. We identify two limit regimes, namely, fast MM activation (FM), and slow MM activation (SM). FM assumes that MMs provide all the available mechanical energy 'instantaneously' and leave the polymer in a stiffened state, i.e. the MM activity occurs at a time scale that is much smaller than that of solvent diffusion. SM assumes that the timescale for MM activation is much longer than that of solvent diffusion. To achieve the same final contracted state, FM requires the largest amount of work per unit reference volume, while SM requires the least. For all intermediate cases where the timescale of MM activation is comparable with that of solvent diffusion, the required work ranges between these two limits. We provide all these quantities as a function of chain density and MM density. Finally, we compare our results on contraction energetics with experiments and observe good agreement.