Abstract
The Turing instability in the reaction-diffusion system is a widely recognized mechanism of the morphogen gradient self-organization during the embryonic development. One of the essential conditions for such self-organization is sharp difference in the diffusion rates of the reacting substances (morphogens). In classical models this condition is satisfied only for significantly different values of diffusion coefficients which cannot hold for morphogens of similar molecular size. One of the most realistic explanations of the difference in diffusion rate is the difference between adsorption of morphogens to the extracellular matrix (ECM). Basing on this assumption we develop a novel mathematical model and demonstrate its effectiveness in describing several well-known examples of biological patterning. Our model consisting of three reaction-diffusion equations has the Turing-type instability and includes two elements with equal diffusivity and immobile binding sites as the third reaction substance. The model is an extension of the classical Gierer-Meinhardt two-components model and can be reduced to it under certain conditions. Incorporation of ECM in the model system allows us to validate the model for available experimental parameters. According to our model introduction of binding sites gradient, which is frequently observed in embryonic tissues, allows one to generate more types of different spatial patterns than can be obtained with two-components models. Thus, besides providing an essential condition for the Turing instability for the system of morphogen with close values of the diffusion coefficients, the morphogen adsorption on ECM may be important as a factor that increases the variability of self-organizing structures.
Highlights
Nonequilibrium or dynamic self-organization is supposed to play a central role in the embryonic patterning [1,2,3]
Self-organizing processes can be described by discrete models based on cellular automata approach [8] or by continuous models based on reaction-diffusion partial differential equations (PDE) approach
In the present work we demonstrate how the adsorption could be incorporated to the simple two-component model
Summary
Nonequilibrium (dissipative) or dynamic self-organization is supposed to play a central role in the embryonic patterning [1,2,3]. The most simple models, which demonstrate Turing instability, consist of two reaction-diffusion differential equations and describe the formation of stable gradients of two hypothetical substances called “activator” and “inhibitor”. These substances have nonlinear interactions with each other and diffuse with sharply different rates: the activator slowly and the inhibitor fast. Morphogene adsorption as a possible Turing instability regulator interaction of activator with ECM Such modification allows us to obtain stable dissipative structures for morphogens with similar diffusion coefficients. As well as other plots are built with python using libraries numpy, scipy and matplotlib
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