Nonlinear Analysis of Juxtacrine Patterns

Abstract

We investigate a discrete mathematical model for a type of cell-cell communication in early development which has the potential to generate a wide range of spatial patterns. Our previous work on this model has highlighted surprising differences between the predictions of linear analysis and the results of numerical simulations. In particular, there is no quantitative agreement between the unstable modes derived from linear analysis and the patterns observed numerically. In this paper, we look at the nonlinear model on a domain of two cells with the aim of gaining an insight into behavior in larger systems. We study the existence and stability of spatially heterogeneous steady-state solutions, which correspond to patterns of alternating cell fate on larger domains, as we vary two key parameters. These parameters are measures of the strength of positive feedback in the biological system. By reducing the problem to two coupled nonlinear algebraic equations, we show that a patterned solution exists and is stable on a 2-cell domain for a significant part of parameter space. We compare these results to those obtained from linear analysis and conclude that the behavior of the nonlinear 2-cell system gives a better insight into the results of numerical simulations on large arrays of cells. Furthermore, we conduct a bifurcation analysis of the model on domains of various sizes: we demonstrate that as the domain size increases, the 2-cell pattern becomes unstable for certain parameters, and overall the number of stable patterns increases. This leads us to speculate that on large domains there are many stable patterned solutions to the model of approximately the same periodicity, which is typical of the fine-grained patterns that one sees during early development. Our work predicts that this is a feature of the patterning dynamics rather than a consequence of environmental heterogeneity.