The Ohio State University

During development, cells take on specific fates to properly build tissues and organs. These cell fates are regulated by short and long range signalling mechanisms, as well as feedback on gene expression and protein activity. Despite the high conservation of these signalling pathways, we often see different cell fate outcomes in similar tissues or related species in response to similar perturbations. How these short and long range signals work to control patterning during development, and how the same network can lead to species specific responses to perturbations, is not yet understood. Exploiting the high conservation of developmental pathways, we theoretically and experimentally explore mechanisms of cell fate patterning during development of the egg laying structure (vulva) in nematode worms. We developed differential equation models of the main signalling networks (EGF/Ras, Notch and Wnt) responsible for vulval cell fate specification, and validated them using experimental data. A complex, biologically based model identified key network components for wild type patterning, and relationships that render the network more sensitive to perturbations. Analysis of a simplified model indicated that short and long range signalling play complementary roles in developmental patterning. The rich data sets produced by these models form the basis for further analysis and increase our understanding of cell fate regulation in development.

Strongly nonlinear problems, written abstractly in the form N[u]=0, are typically difficult to analyze unless they possess special properties. However, if we are able to find a quasi-solution u_0 in the sense that the residual N[u_0] := R is small, then it is possible to analyze a strongly nonlinear problem with weakly nonlinear analysis in the following manner: We decompose u=u_0 + E; then E satisfies L E = -N_1 [E] - R, where L is the Fre'chet derivative of the operator N and N_1 [E] := N[u_0+E]-N[u_0]-L E contains all the nonlinearity. If L has a suitable inversion property and the nonlinearity N_1 is sufficiently regular in E, then weakly nonlinear analysis of the error E through contraction mapping theorem gives rise to control of the error E. What is described above is quite routine. The only new element is to determine a quasi-solution u_0, which is typically found through a combination of classic orthogonal polynomial representation and exponential asymptotics.

This method has been used in a number of nonlinear ODEs arising from reduction of PDEs. We also show how it can be extended to integro-differential equations that arise in study of deep water waves of permanent form. The method is quite general and can in principle be applied to nonlinear PDEs as well.

NB. Much of this is joint work with O. Costin and other collaborators.