Developmental Incongruity as a Dynamical Representation of Heterochrony

The theory of heterochrony provides us with a generalized quantitative perspective on the dynamics of developmental trajectories. While useful, these linear developmental trajectories merely characterize changes in the speed and extent of growth in developmental time. One open problem in the literature involves how to characterize developmental trajectories for rare and incongruous modes of development. By combining nonlinear mathematical representations of development with models of gene expression networks (GRNs), the dynamics of growth given the plasticity and complexity of developmental timing are revealed. The approach presented here characterizes heterochrony as a dynamical system, while also proposing a computational motif in GRNs called triangular state machines (TSMs). TSMs enable local computation of phenotypic enhancement by producing nonlinear and potentially unexpected outputs. With a focus on developmental timing and a focus on sequential patterns of growth, formal techniques are developed to characterize delays and bifurcations in the developmental trajectory. More generally, growth is demonstrated using two conceptual models: a Galton board representing axial symmetry and a radial tree depicting differential growth. These techniques take into consideration the existence of multiple developmental genotypes operating in parallel, which ultimately characterize the exquisite phenotypic diversity observed in animal development.