Incorporating Stochastic Gene Expression, Signaling-Mediated Cell-Cell Interactions, and Regulated Cell Proliferation in Models of Coordinated Tissue Development

Formulating quantitative and predictive models for tissue development requires consideration of complex, stochastic gene expression dynamics, their regulation via cell-cell interactions, and cell proliferation. Including all of these processes in a practical mathematical framework requires complex expressions that are difficult to apply. We construct a theory that incorporates intracellular stochastic gene expression dynamics, signaling chemicals that influence these dynamics and mediate cell-cell interactions, and cell proliferation and its accompanying differentiation. The dynamics of cellular states are described by a Waddington vector field, where no assumption is made. We provide a procedure to systematically simplify the theory by applying explicit assumptions—this can reduce the theory to a master equation or even to a system of ordinary differential equations. We define an epigenetic fitness landscape that describes the proliferation of different cell types and elucidate how this fitness landscape is related to Waddington's vector field. We illustrate the applicability of our framework by analyzing two model systems: an interacting two-gene differentiation process and a spatiotemporal organism model inspired by planaria.