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

Akiko M Nakamasu

doi:10.3389/fphy.2021.805659

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.

Highlights

In 1952, Alan Turing postulated that two substrates interacting with each other show instability when they diffuse at different speeds
Previous laser ablation experiments reveal that the presence of mutual interactions between two types of pigment cells are necessary to generate Turing patterns (Figure 1E)
Two types of pigment cell inhibit each other at a one-cell distance even though the inhibition from xanthophore is inapparent until melanophores in adjacent stripes are eliminated

Summary

Introduction

In 1952, Alan Turing postulated that two substrates interacting with each other show instability when they diffuse at different speeds. He explained this diffusion-driven instability by utilizing a linear reaction-diffusion (RD) model. This model demonstrates that spatial inhomogeneity (i.e., the Turing pattern) could be generated by such conditions. This relationship is known to generate patterns though the components remain to be explored. Zebrafish have a pattern of stripes on their body and fins (Figure 1A). Numerous zebrafish pigment-pattern mutants were artificially generated [5], and the corresponding genes were identified. One of the most important mutants is leopard, which produces a spotted pattern that is representative of Turing patterns (Figure 1B) [6] identifies

Methods

Results

Conclusion