Correspondences Between Parameters in a Reaction-Diffusion Model and Connexin Functions During Zebrafish Stripe Formation

Abstract

Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifest as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models demonstrate that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish reveal apparent interactions among pigment cells, which allow for a three-component RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system remain unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formation. Here, molecular-level evidence derived from connexin analyses is linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells are generalized as “gates,” and the effects of respective gates were deduced. The model uses partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model, including nonlinear reaction terms, enables the consideration of the behavior of components realistically.

Convolution Representation of Traveling Pulses in Reaction-Diffusion Systems

“…We consider the pulse solution of the KT model (17) in the interval ½−L, L under periodic boundary conditions. To clarify the mechanism of the pulse solution occurrence in the KT model (17), we consider the linear stability of the spatially homogeneous solution. The growth rate of instability is obtained through linear stability analysis.…”

“…In this appendix, following similar procedure used in ref. [38], we derive the linear growth rate of instability of the spatially homogeneous solution in the KT model (17). In our choice of parameters, there is a spatially homogeneous solution of the RD system (12); u = v = 0.…”

“…Among many Turing patterns in nature, the pigmentation on the skin of zebrafish has been well-reported [14][15][16][17]. Fish pigmentation shows many patterns such as spot, stripe, and labyrinth.…”

Spot–Ladder Selection of Dislocation Patterns in Metal Fatigue

“…Existing studies on diffusion-induced instability have focused primarily on soft matter, including biological [7][8][9][10][11] and chemical [7,[12][13][14][15] systems, in which the typical periodicity in length ranges from cm to mm or slightly less. Conversely, Turing patterns that emerge in "hard" matter show far smaller length scales [16][17][18][19].…”