Generic mechanisms of sequential pattern formation in an expanding cell population

In contrast to Turing's original proposal, developmental patterns are often formed sequentially in the biological world. In somitogenesis, for example, somites emerge one by one following an oscillating process localized at the posterior end of the elongating body axis. An adaptation of the morphogen gradient theory to such a situation was made by Cooke and Zeeman in the ``clock and wavefront’’ model, where the presomitic cells undergo rhythmic oscillations before committing to a specific differentiation path triggered by a moving determination front. Pattern formation generated by an oscillating front has also been demonstrated recently in a bacterial system where the motility of individual cells is controlled by a synthetic circuit (Liu et al., Science 334, 238 (2011)). In tumor growth, the expansion velocity of a solid tumor may show oscillatory behavior under suitable conditions.

The mathematics to analyze sequential pattern formation is under-developed as the procedure to reduce the full nonlinear dynamics of a spatially extended system onto one or a few dominant unstable modes is cumbersome and usually not analytically tractable. In this respect, a lot can be gained by developing phenomenological theories of genotype (cellular-level behavior) to phenotype (population dynamics) mapping that focus on physical mechanisms of instability in the frontal region. In this project, we will examine in detail two extreme scenarios for a pulsating front: oscillations arising from a reaction cycle localized at the front, and (quasi) spatial separation of fast and slowly diffusing groups. Reduction of the full dynamics to a set of ODEs for the localized unstable mode will be carried out both numerically and analytically under physically motivated approximations. The resulting ``amplitude equations’’ will be investigated. Different types of transitions afforded by the amplitude equations, i.e., first order or continuous, will be examined. With a more complete understanding of the mathematical structure behind the dynamical instability, systematic comparisons between the model behavior and experimental observations can be carried out, and the role of various molecular components and their interactions (with relevant rate constants) in the patterning process quantified. We aim at generic descriptions of pulsating fronts in these systems. Lessons learned may very well be used to guide future design of synthetic molecular circuits to achieve patterns with desired features.