Abstract
Synthesizing a genetic network which generates stable Turing patterns is one of the great challenges of synthetic biology, but a significant obstacle is the disconnect between the mathematical theory and the biological reality. Current mathematical understanding of patterning is typically restricted to systems of two or three chemical species, for which equations are tractable. However, when models seek to combine descriptions of intercellular signal diffusion and intracellular biochemistry, plausible genetic networks can consist of dozens of interacting species. In this paper, we suggest a method for reducing large biochemical systems that relies on removing the non-diffusible species, leaving only the diffusibles in the model. Such model reduction enables analysis to be conducted on a smaller number of differential equations. We provide conditions to guarantee that the full system forms patterns if the reduced system does, and vice versa. We confirm our technique with three examples: the Brusselator, an example proposed by Turing, and a biochemically plausible patterning system consisting of 17 species. These examples show that our method significantly simplifies the study of pattern formation in large systems where several species can be considered immobile.
Highlights
How cells coordinate with one another to form regular patterns of alternate differentiated states is a foundational question in developmental biology [1]
We investigate the question of whether model 2 reduction can be applied to a chemical reaction network (CRN) in a manner that preserves Turing pattern-forming behaviour
If the reduced model is stable for a region of parameter space, the full model cannot form patterns in that region. This is useful because the stable region is typically large, and unstable regions frequently correspond to physically impossible parameter values, and so model reduction can be an efficient way of eliminating systems incapable of pattern formation
Summary
How cells coordinate with one another to form regular patterns of alternate differentiated states is a foundational question in developmental biology [1]. While there are several mechanisms that are known to enable multicellular self-organization of regular patterns, such as the french flag model [7], we focus here on diffusion-driven instability (DDI) first described by Alan Turing [8] He proposed that two ‘morphogens’ (intercellular signalling molecules) could enable tissues to produce regular patterns, and introduced a framework based on the reaction–diffusion equations that can establish when a given chemical system has pattern-forming potential. These examples show that the method developed in this paper allows for quick and easy Turing pattern analysis of complex chemical systems with an arbitrarily large number of non-diffusible species
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