  The reaction-diffusion waves of proteins are known to be involved in
fundamental cellular functions, such as cell migration, cell division, and
vesicular transportation. In some of these phenomena, pattern formation on the
membranes is induced by the coupling between membrane deformation and the
reaction-diffusion system through curvature-inducing proteins that bend the
biological membranes. Although the membrane shape and the dynamics of the
curvature-inducing proteins affect each other in these systems, the effect of
such mechanochemical feedback loops on the waves has not been studied in
detail. In this study, reaction-diffusion waves coupled with membrane
deformation are investigated using simulations combining a dynamically
triangulated membrane model with the Brusselator model extended to include the
effect of membrane curvature. It is found that the propagating wave patterns
change into nonpropageting patterns and spiral wave patterns due to the
mechanochemical effects. Moreover, the wave speed is positively or negatively
correlated with the local membrane curvature depending on the spontaneous
curvature and bending rigidity. In addition, self-oscillation of the vesicle
shape occurs, associated with the reaction-diffusion waves of
curvature-inducing proteins. This agrees with the experimental observation of
liposomes with a reconstituted Min system, which plays a key role in the cell
division of Escherichia coli. The findings of this study demonstrate the
importance of mechanochemical coupling in biological phenomena.
