Interaction of Turing patterns with an external linear morphogen gradient

Abstract

We investigate a simple generic model of a reaction-diffusion system consisting of an activator and an inhibitor molecule in the presence of a linear morphogen gradient. We assume that this morphogen gradient is established independently of the reaction-diffusion system and acts by increasing the production of the activator proportional to the morphogen concentration. The model is motivated by several existing models in developmental biology in which a Turing patterning mechanism is proposed and various chemical gradients are known to be important for development. Mathematically, this leads to reactiondiffusion equations with explicit spatial dependence. We investigate how the Turing pattern is affected, if it exists. We also show that in the parameter range where a Turing pattern is not possible, the system may nevertheless produce 'Turing-like' patterns.

Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case, with an Application to Skeletal Pattern Formation

“…The above equations were studied by some of the authors of the present paper in one spatial dimension in Glimm et al (2009). Similar reaction-diffusion equations with space-dependent coefficients were studied by other authors before, for instance by Page et al (2003Page et al ( , 2005 and by Borckmans et al (1992).…”

“…In Glimm et al (2009), we set out to investigate the interaction of reactiondiffusion models with a modulating chemical gradient. In a generic model, there are two chemicals, an inhibitor, and an activator molecule, which are secreted by cells, diffuse, decay, and react with each other.…”

“…The results of Glimm et al (2009) can be summarized as follows: In the parameter region where a Turing pattern is possible, adding a "small" chemical signal essentially does not change the wave number of the pattern. It changes the baseline of the pattern to a linear function which increases toward the source of the external signal, as one would expect.…”

“…We also considered in Glimm et al (2009) the parameter region where the system (1) cannot produce Turing patterns in the absence of an external morphogen (that is when ε c = 0). Maybe surprisingly, for a large subregion of this parameter region, the system can produce Turing-like patterns if the external morphogen is present, that is for nonzero ε c .…”

“…Indeed, this is the case when the activator diffuses faster than the inhibitor. However, in the example involving an external morphogen considered in Glimm et al (2009), the system can produce "Turing-like" oscillatory patterns for values of d up to approximately 5.48. A caveat is that the amplitude is very small for larger values of d. This result is related to one obtained in Page et al (2003), where it was shown that a spatially discontinuous coefficient in the reaction kinetics can lead to similar patterning outside of the Turing space.…”