Diffusion patterns on domains representing developing Xenopus retina

The retinotectal map of mature Xenopus frogs has linear metrics. The regions of the Xenopus retina were modeled during embryonic and early larval development with a set of standard geometrical shapes. A disk was used to model the entire early retina, rings of various thickness were used to model the ciliary margin (the growth zone of the larval retina), and portions of a sphere were used to model the entire late embryonic and larval retina. Gradients produced by several reaction-diffusion pattern-generating configurations were considered: Gierer-Meinhardt and Kauffman-type activator-inhibitor systems; an ideal Turing-like two morphogen system; a point source with diffuse sinks; and variously placed point source-point sink combinations. A few special arrangements produced linear metrics: a point source-point sink system produced linear angular metrics on a thin ring; and ideal Turing system produced linear cartesian metrics on a thin ring; and carefully configured multiple sources and sinks produced linear metrics on a disk. These cases could lead to linear map metrics, but are biologically implausible. Thus linear map metrics are probably not a direct consequence of linear positional values, and more generally, most embryonic tissues probably use non-linear gradients of positional information. Formation of linear map metrics may be functionally important for Xenopus frogs, accounting in part for the evolutionary preservation of mechanisms for interactions among neighboring retinal fibers.