Fractional models for the migration of biological cells in complex spatial domains

Abstract

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models are successfully used to model anomalous diffusion and spatial heterogeneity in different physical contexts. We develop a fractional model based on the Fisher--Kolmogoroff equation, analyse it for its wavespeed properties, and relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments. References

• R. J. Adler, R. E. Feldman and M. S. Taqqu. A practical guide to heavy tails: Statistical techniques and applications. Birkauser, 1998.
• I. J. Allan and D. F. Newgreen. The origin and differentiation of enteric neurons of the intestine of the fowl embryo. The American Journal of Anatomy, 157, 137--154, 1980. doi:10.1002/aja.1001570203
• B. J. Binder, K. A. Landman, M. J. Simpson, M. Mariani and D. F. Newgreen. Modeling proliferative tissue growth: A general approach and an avian case study. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 78(3), 1--13, 2008. doi:10.1103/PhysRevE.78.031912
• M. A. Breau, A. Dahmani, F. Broders--Bondon, J. P. Thiery and S. Dufour. $\beta 1$ integrins are required for the invasion of the caecum and proximal hindgut by enteric neural crest cells. Development, 136, 2791--2801, 2009. doi:10.1242/dev.031419
• K. Burrage, N. Hale and D. Kay. An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM Journal on Scientific Computing, 34(4), A2145--A2172. doi:10.1137/110847007
• N. R. Druckenbrod and M. L. Epstein. The patterns of neural crest advance in the cecum and colon. Developmental Biology, 287, 125--133, 2005. doi:10.1016/j.ydbio.2005.08.040
• H. Engler. On the speed of spread for fractional reaction-diffusion equations. International Journal of Differential Equations, 2010, Article ID 315421, 2010. doi:10.1155/2010/315421
• M. Ilic, F. Liu, I. Turner and V. Anh. Numerical approximation of a fractional-in-space diffusion equation (II)--with nonhomogeneous boundary conditions. Fractional Calculus and Applied Analysis, 9, 333--349, 2006. http://eprints.qut.edu.au/23835/
• P. K. Maini, D. L. S. McElwain and D. Leavesley. Travelling waves in a wound healing assay. Applied Mathematics Letters, 17, 575--580, 2004. doi:10.1016/S0893-9659(04)90128-0
• R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R. E. Baker, P. K. Maini and P. M. Kulesa. Multiscale mechanisms of cell migration during development: theory and experiment. Development, 139, 2935--2944, 2012. doi:10.1242/dev.081471
• J. D. Murray. Mathematical Biology I and II. Springer Verlag, 2003.
• M. J. Simpson, D. C. Zhang, M. Mariani, K. A. Landman and D. F. Newgreen. Cell proliferation drives neural crest cell invasion of the intestine. Developmental Biology, 302, 553--568, 2007. doi:10.1016/j.ydbio.2006.10.017
• H. M. Young, A. J. Bergner, R. B. Anderson, H. Enomoto, J. Milbrandt, D. F. Newgreen and P. M. Whitington. Dynamics of neural crest-derived cell migration in the embryonic mouse gut. Developmental Biology, 270, 455--473, 2004. doi:10.1016/j.ydbio.2004.03.015
• P. Zhuang, F. Liu, V. Anh and I. Turner. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis, 47(3), 1760--1781, 2009. doi:10.1137/080730597