Size matters: an analytical study on the role of tissue size in spatiotemporal distribution of morphogens unveils a transition between different reaction-diffusion regimes

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 spatiotemporal morphogen distributions has not been completely elucidated. In this study, we analytically investigated the spatiotemporal distributions of morphogens predicted by a reaction-diffusion model in a finite 1D 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 [~]3.3 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.