From Microscopic to Macroscopic Descriptions of Cell Migration on Growing Domains
Authored by Radek Erban, Christian A Yates, Ruth E Baker
Date Published: 2010
DOI: 10.1007/s11538-009-9467-x
Sponsors:
United Kingdom Engineering and Physical Sciences Research Council (EPSRC)
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Model Documentation:
Other Narrative
Mathematical description
Model Code URLs:
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Abstract
Cell migration and growth are essential components of the development of
multicellular organisms. The role of various cues in directing cell
migration is widespread, in particular, the role of signals in the
environment in the control of cell motility and directional guidance. In
many cases, especially in developmental biology, growth of the domain
also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is
an almost ubiquitous use of partial differential equations (PDEs) for
modelling the time evolution of cellular density and environmental cues.
In the last 20 years, a lot of attention has been devoted to connecting
macroscopic PDEs with more detailed microscopic models of cellular
motility, including models of directional sensing and signal
transduction pathways. However, domain growth is largely omitted in the
literature. In this paper, individual-based models describing cell
movement and domain growth are studied, and correspondence with a
macroscopic-level PDE describing the evolution of cell density is
demonstrated. The individual-based models are formulated in terms of
random walkers on a lattice. Domain growth provides an extra
mathematical challenge by making the lattice size variable over time. A
reaction-diffusion master equation formalism is generalised to the case
of growing lattices and used in the derivation of the macroscopic PDEs.
Tags
behavior
Bacterial chemotaxis
Aggregation
growth
Equations
Pattern-formation
Kinetic-models
Exact stochastic simulation
Diffusion limit
Reinforced random-walks