A threshold model for polydactyly

Abstract

We present a cellular automaton-based model for threshold behaviors in vertebrate digit patterning and polydactyly formation. The rules of the model follow classical reactor-diffusion algorithms. Yet it is not physical diffusion that is taken as the required natural agent but the propagation of cellular states, which can be represented by the same differential equations. The bistable cellular states in the model correspond to mesenchymal limb bud cells that can be either "on" or "off" for the cartilage differentiation pathway. Simulation runs demonstrate that reaction rate and cell number have the most decisive influence on the number of digit-like cell activation patterns. Threshold-based effects can generate supernumerary activation stripes via de novo condensation, stabilized bifurcation, and free floaters. All three behaviors are consistent with processes in natural polydactyly formation. It is argued that these effects are rooted in cell-based behaviors, not in gene regulation or globally diffusing morphogens. Our model suggests that the origin of discrete character states, such as individual digits, is a consequence of an additive cell state variable with a normal distribution that is transformed by a growth function with Turing behaviors into discontinuous phenotypic units. We discuss the application of this type of autopod patterning to the mutational, developmental, experimental, and evolutionary occurrences of polydactyly. The model provides a refinement of the previous Hemingway model for digit novelty and supports Turing type pattern formation in the vertebrate limb.