A variational approach to the growth dynamics of pre-stressed actin filament networks

Abstract

In order to model the growth dynamics of elastic bodies with residual stresses a thermodynamically consistent approach is needed such that the cross-coupling between growth and mechanics can be correctly described. In the present work we apply a variational principle to the formulation of the interfacial growth dynamics of dendritic actin filament networks growing from biomimetic beads, an experimentally well studied system, where the buildup of residual stresses governs the network growth. We first introduce the material model for the network via a strain energy density for an isotropic weakly nonlinear elastic material and then derive consistently from this model the dynamic equations for the interfaces, i.e. for a polymerizing internal interface in contact with the bead and a depolymerizing external interface directed towards the solvent. We show that (i) this approach automatically preserves thermodynamic symmetry-properties, which is not the case for the often cited ‘rubber-band-model’ (Sekimoto et al 2004 Eur. Phys. J. E 13 247–59, Plastino et al 2004 Eur. Biophys. J. 33 310–20) and (ii) leads to a robust morphological instability of the treadmilling network interfaces. The nature of the instability depends on the interplay of the two dynamic interfaces. Depending on the biochemical conditions the network envelope evolves into a comet-like shape (i.e. the actin envelope thins out at one side and thickens on the opposite side of the bead) via a varicose instability or it breaks the symmetry via higher order zigzag modes. We conclude that morphological instabilities due to mechano-chemical coupling mechanisms and the presences of mechancial pre-stresses can play a major role in locally organizing the cytoskeleton of living cells.