Abstract
The reaction–diffusion model constitutes one of the most influential mathematical models to study distribution of morphogens in tissues. Despite its widespread use, the effect of finite tissue size on model-predicted spatio-temporal morphogen distributions has not been completely elucidated. In this study, we analytically investigated the spatio-temporal distributions of morphogens predicted by a reaction–diffusion model in a finite one-dimensional domain, as a proxy for a biological tissue, and compared it with the solution of the infinite-domain model. We explored the reduced parameter, the tissue length in units of a characteristic reaction–diffusion length, and identified two reaction–diffusion regimes separated by a crossover tissue size estimated in approximately three characteristic reaction–diffusion lengths. While above this crossover the infinite-domain model constitutes a good approximation, it breaks below this crossover, whereas the finite-domain model faithfully describes the entire parameter space. We evaluated whether the infinite-domain model renders accurate estimations of diffusion coefficients when fitted to finite spatial profiles, a procedure typically followed in fluorescence recovery after photobleaching (FRAP) experiments. We found that the infinite-domain model overestimates diffusion coefficients when the domain is smaller than the crossover tissue size. Thus, the crossover tissue size may be instrumental in selecting the suitable reaction–diffusion model to study tissue morphogenesis.
Highlights
In their transition towards maturity, tissues are crucially regulated by molecules known as morphogens, whose precise spatio-temporal distribution triggers the downstream changes in protein expression responsible for the exact differentiation patterns
The problem of how a morphogen propagates over a tissue while it is being eliminated was mathematically encoded in the exquisite reaction–diffusion model by the great Alan Turing, who coined the ‘morphogen’ neologism to illustrate its character of ‘form generator’ [1]
We present the analytical solution of a reaction–diffusion model describing de novo formation of a morphogen and its spread within a finite domain, as a proxy for a tissue
Summary
In their transition towards maturity, tissues are crucially regulated by molecules known as morphogens, whose precise spatio-temporal distribution triggers the downstream changes in protein expression responsible for the exact differentiation patterns. From Turing’s aforementioned seminal article and the study from Gierer & Meinhardt [2], a progressive wealth of reaction–diffusion models were developed, paving the way to become an essential and pivotal concept to understand tissue morphogenesis [3,4,5,6]. The model was extensively used to investigate distributions of morphogens in a variety of tissues and organisms such as Drosophila melanogaster wing imaginal disc [7], chick limb [8] and the stripe pattern of Danio rerio [9], among other examples
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