Size matters: An analytical study on the role of tissue size in spatiotemporal distribution of morphogens unveils a transition between different Reaction-Diffusion regimes

• DOI: 10.1101/2021.02.16.431401
• Publication Date: 2021-02-17T08:04:30Z
• Authors: Alberto S. Ceccar...; Augusto Borges; Osvaldo Chara
• Keywords: Artificial intelligence; Global Diversity of Microbial Eukaryotes and Their Evolution; Mathematical analysis; Biochemistry; Gene; Fluorescence recovery after photobleaching; Fluorescence; Agricultural and Biological Sciences; Diffusion; 03 medical and health sciences; Biochemistry, Genetics and Molecular Biology; FOS: Mathematics; Genetics; Molecular Biology; Biology; Ecology, Evolution, Behavior and Systematics; Mathematical Modeling of Cancer Growth and Treatment; 0303 health sciences; Domain (mathematical analysis); Physics; Life Sciences; Optics; Computer science; 3. Good health; Chemistry; Evolution and Ecology of Endophyte-Grass Symbiosis; Reaction–diffusion system; Modeling and Simulation; Biological system; FOS: Biological sciences; Physical Sciences; Crossover; Pattern formation; Multiscale Model; Thermodynamics; Statistical physics; Mathematics; Morphogen

AbstractThe 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.