Polarization of concave domains by traveling wave pinning

<div><p>Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3-dimensional domains can serve as a generic gradient-maintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reaction-diffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation.</p></div

Fronts pinned in the local widening of the 3D cylindrical domain.

<p>The width of the simulation domain is 2. (A) An example stable stationary solution for <i>D</i> = 0.02 and <i>f</i>(<i>u</i>) = (1 − <i>u</i>)(<i>u</i> + 1)(<i>u</i> + <i>ϵ</i>), <i>ϵ</i> = <i>ϵ</i>_<i>max</i> = 0.264. Cross-section plane containing axis of symmetry is shown. The surface <i>f</i>(<i>u</i>) = 0 defines the position of the stationary front. (B-C) the curvature <i>κ</i> = 1/<i>R</i> of the stationary fronts calculated from numerical simulations for <i>f</i>(<i>u</i>) = (1 − <i>u</i>)(<i>u</i> + 1)(<i>u</i> + <i>ϵ</i>), in the domain with the diameter ratio 0.2 (B) and 0.4 (C) for five values of <i>D</i>: 0.00125, 0.005, 0.02, 0.08, 0.32 (colors from dark blue to red) versus the analytical result given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190372#pone.0190372.e018" target="_blank">Eq (8)</a> (black line overlapping with dark blue line) in the <i>D</i> → 0 limit. The front surface position is determined by its radius of curvature via <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190372#pone.0190372.e012" target="_blank">Eq (3)</a>. Inserts in (B) and (C) show cross-section of the stationary front surfaces with the cylinder symmetry plane for <i>D</i> = 0.005 and six values of <i>ϵ</i>/<i>ϵ</i>_max: 0.1, 0.2, 0.4, 0.6, 0.8, 1; <i>ϵ</i>_max = 0.134 for diameter ratio 0.2 (B) and <i>ϵ</i>_max = 0.0832 for diameter ratio 0.4 (C).</p