Partial differential equations for self-organization in cellular and developmental biology

Abstract

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field.Mathematics Subject Classification: 35−02; 35−R99; 92−02; 92B05; 92C15; 92C17; 92C37

A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell–cell contacts

“…However, these PDE models are not capable of simulating processes of a cellular scale. We finally remark that some reviews on PDE modeling of biological processes were written by Geris et al (2010), Geris et al (2009), Baker et al (2008), Murray (2004), where both mechanical balances and transport through random walk, tenso-taxis and chemo-taxis were taken into account.…”

Laser Assisted Bio-printing (LAB) of Cells and Bio-materials Based on Laser Induced Forward Transfer (LIFT)

“…Moreover, the engineering of realistic tissue constructs will help further understanding of tissue physiology and function; this, in turn, will refine tissue engineering strategies and optimize blueprints. To this end, coupling mathematical modeling [57] to the design of micropatterns of cells and morphogens by the abovementioned technologies could be fruitful. Cell-patterning techniques could be promising tools for the fabrication of organotypic tissues, which could elucidate basic cell biology mechanisms, further drug evaluation and toxicity testing in vitro, and become useful platforms within the clinic.…”

The Fokker-Planck Equation

“…This can be seen from the exercise below. The Keller-Segel system [1,8,10,13] for chemotaxis is the most famous model in this area and assumes that cells move with a combination of a random (Brownian) motion and an oriented drift, which is a chemical gradient of a molecule emitted by the cells themselves and diffused in the medium. Therefore, the Keller-Segel system corresponds to a singular kernel K. / (the fundamental solution of the Laplace equation).…”