Scaling of morphogenetic patterns in continuous and discrete models

In biological systems, individuals which belong to the same species can have different sizes. However, the ratios between the different parts of their bodies remain the same for individuals of different sizes. For example, for fully developed organism with segmented structure (i.e. insects), the number of segment across the size range of the individuals does not change. This morphological scaling plays a major role in the development of the organism and it has been the object of biological studies (Cooke 1981, Day and Lawrence 2000, Parker 2011) and mathematical modelling (Othmer and Pate 1980, Gregor, Tank et al. 2007, Kerszberg and Wolpert 2007) for many decades. Such scaling involves adjusting intrinsic scale of spatial patterns of gene expression that are set up during the development to the size of the system (Umulis and Othmer 2013). On the biological side, the evidence of scaling has been demonstrated experimentally on various objects including embryos. (Spemann 1938, Gregor, Bialek et al. 2005). For example, a Xenopus embryo was physically cut into dorsal and ventral halves in experimental conditions. The dorsal half which contains the “Spemann organizer” developed into a small embryo with normal proportion (Spemann 1938). Similar experiments carried out for the case of the sea urchin embryo lead to a smaller size of individuals (Khaner 1993). Also for flies of different species, the number of stripes on their embryos during their development remains the same although they are of different sizes. These stripes, which are visible at an earlier stage of the embryonic development, correspond to the spatial pattern of gene expressions and are the origin of the segmented body of the flies (Jaeger, Surkova et al. 2004, Gregor, Bialek et al. 2005, Arias 2008). On the mathematical side, Turing introduced the term “morphogens” for protein which is a key factor for pattern formation and he derived a model involving morphogens in which spatial patterns arise under certain conditions. Since then, various mathematical models of pattern formations have been developed. For the diffusion-based models, the spatial patterns do not scale with size. For models using reaction-diffusion equations, (combination of diffusion and biochemical reactions) a characteristic length scale is determined by the diffusion constant and reaction rate. Thus, when the size of the embryo changes, the spacing of the patterning remains fixed. This means that solutions of mathematical models based on reaction-diffusion do not show scaling (Tomlin and Axelrod 2007). The motivation of this work is to introduce possible mechanisms of scaling in biological systems and demonstrate those using mathematical models. After a discussion on how the scaling is considered in a few continuous models, we introduce our definition of scaling. We apply our definition of scaling to analyse properties of concentration profiles arising in various continuous models. Upon analysis of these profiles, we introduce modifications of mathematical models, in particular, two famous continuous models (Turing and Fitzhugh-Nagumo) to achieve scaling of their solutions. Following a presentation of continuous models, a discrete model of pattern formation based on a chain of logical elements (cellular automata) is also presented. This is more appropriate to represent the discreteness of biological systems with respect to their scaling properties: for a number of problems the issue of scaling doesn’t appear in the discrete formulation. This model has been developed to take account of local interactions between cells resulting into stationary pattern formation. We conclude this thesis by comparing our results with results obtained on other models and with experimental data particularly related to the different stages of the development of the fly embryo.