Turing patterns on polymerized membranes: a coarse-grained lattice modelling with internal degree of freedom for polymer direction

TL;DR

This study models Turing patterns on polymerized membranes considering internal polymer directions, revealing how membrane anisotropies influence pattern orientation and providing insights into non-equilibrium configurations via hybrid numerical simulations.

Contribution

Introduces a coarse-grained lattice model incorporating internal polymer degrees of freedom to analyze Turing patterns on membranes, highlighting the role of anisotropies in pattern orientation.

Findings

Pattern orientation depends on membrane anisotropies.

Hybrid Monte Carlo and iterative methods effectively simulate membrane-Turing interactions.

Internal degrees of freedom influence membrane relaxation and pattern formation.

Abstract

We numerically study Turing patterns (TPs) on two-dimensional surfaces with a square boundary in ${\bf R}^3$ using a surface model for polymerized membranes. The variables used to describe the membranes correspond to two distinct degrees of freedom: an internal degree of freedom for the polymer directions in addition to the positional degree of freedom. This generalised surface model enables us to identify a non-trivial interference between the TP system and the membranes. To this end, we employ a hybrid numerical technique, utilising Monte Carlo updates for membrane configurations and discrete time iterations for the FitzHugh-Nagumo type Turing equation. The simulation results clearly show that anisotropies in the mechanical deformation properties, particularly the easy axes associated with the stretching and bending of the membranes, determine the direction of the TPs to be perpendicular or parallel to the easy axes. Additionally, by calculating the dependence of the maximum entropy on the internal degree of freedom, we can obtain information on the relaxation with respect to the polymer structure. This crucial information serves to remind us that non-equilibrium configurations can be captured within the canonical Monte Carlo simulations.