A simple reaction–diffusion system as a possible model for the origin of chemotaxis

Abstract

Chemotaxis is a directed cell movement in response to external chemical stimuli. In this paper, we propose a simple model for the origin of chemotaxis – namely how a directed movement in response to an external chemical signal may occur based on purely reaction–diffusion equations reflecting inner working of the cells. The model is inspired by the well-studied role of the rho-GTPase Cdc42 regulator of cell polarity, in particular in yeast cells. We analyse several versions of the model to better understand its analytic properties and prove global regularity in one and two dimensions. Using computer simulations, we demonstrate that in the framework of this model, at least in certain parameter regimes, the speed of the directed movement appears to be proportional to the size of the gradient of signalling chemical. This coincides with the form of the chemical drift in the most studied mean field model of chemotaxis, the Keller–Segel equation.

Keywords:

• Chemotaxis
• Reaction–diffusion system
• Cdc42
• Gradient tracking
• Regularity

Acknowledgments

AK has been partially supported by the NSF-DMS award 2006372. The authors thank Daniel Lew for patiently teaching them some of the relevant biology (all inadequacies are our fault) and helpful discussions. We are grateful to anonymous referees for useful comments that helped improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Science Foundation  Division of Mathematical Sciences [2006372].