Growing domains

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.

Linking scales

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. We have developed a reaction-diffusion master equation framework, in which the lattice evolves with time, to study the effects of growth on the individual level.

  • R. E. Baker, C. A. Yates and R. Erban (2010). From microscopic to macroscopic descriptions of cell migration on growing domains. Bull. Math. Biol. 72(3):719-762. DOI
  • C. A. Yates, R. E. Baker, R. Erban and P. K. Maini (2012). Going from microscopic to macroscopic on nonuniform growing domains. Phys. Rev. E. 86(2):021921 DOI
  • C. A. Yates and R. E. Baker (2013). Isotropic model for cluster growth on a regular lattice. Phys. Rev. E 88(2):023304. DOI

Pattern formation

We have extended this approach by taking a higher-level description  that involves a Lagrangian-based division of the domain, and used it to study the effects of noise upon pattern formation on growing domains.

fig8b

  • T. E. Woolley, R. E. Baker, E. A. Gaffney and P. K. Maini (2011). Power spectra methods for a stochastic description of diffusion on deterministically growing domains. Phys. Rev. E 84(2):021915. DOI
  • T. E. Woolley, R. E. Baker, E. A. Gaffney and P. K. Maini (2011). The influence of stochastic domain growth on pattern nucleation for diffusive systems with internal noise. Phys. Rev. E 84(4):041095. DOI
  • T. E. Woolley, R. E. Baker, E. A. Gaffney and P. K. Maini (2011). Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation. Phys. Rev. E. 84(4):046216. DOI

Domain growth in volume exclusion models

More recently, we have focussed on incorporating domain growth into our moment dynamics framework. This work has led to some interesting observations regarding the sensitivity of the models to the specific details of the growth algorithm.

  • R. J. H. Ross, C. A. Yates and R. E. Baker (2016). The effect of domain growth on spatial correlations. bioRxiv
  • R. J. H. Ross, C. A. Yates and R. E. Baker (2016). How domain growth is implemented determines the long term behaviour of a cell population through its effect on spatial correlations. bioRxiv

Adventures in the world of mathematical biology