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Multiscale mathematical modelling of dynamics of structure formation in cell systems

Final Report Summary - BIOSTRUCT (Multiscale mathematical modelling of dynamics of structure formation in cell systems)

The project resulted in the establishment of the new research group "Applied Analysis and Modelling in Biosciences" at the Institute of Applied Mathematics, Interdisciplinary Center of Scientific Computing (IWR) and BIOQUANT Center, University of Heidelberg.

Our work was devoted to mathematical modelling, analysis and simulation of transport processes and regulatory feedback in multicellular systems. We used rigorous mathematical techniques, innovative mathematical modelling and computation, and maintained close collaborations with experimentalists. The main lines of our analytical research were devoted to: (1) pattern formation mechanisms in the systems of reaction-diffusion type; (2) nonlinear structured population models; linking continuous and discrete structures.

In the first area of focus we developed analytical and numerical approach to study basic systems of reaction-diffusion equations coupled with ordinary differential equations (ODE), which we previously derived using homogenisation techniques. For the homogenised receptor-based model we showed strong two-scale convergence of the correctors describing the approximation errors for a fixed scaling parameter. We investigated how the structure of nonlinearities determines model dynamics. In particularly, we analysed the systems with multiple constant steady states and showed that bistability is not sufficient for the existence of stable patterns. We identified and analysed three types of pattern formation mechanisms: (1) discontinuous patterns in the systems with hysteresis in the quasi-stationary ODE subsystem, (2) nonstationary spike patterns in the systems with diffusion-driven instability (DDI), and (3) stable discontinuous patterns in the models with DDI and hysteresis. We provided a systematic description of stable hysteresis-driven discontinuous stationary solutions, which may be monotone, periodic or irregular. Furthermore, we showed that there exist no stable Turing patterns in such models, i.e. the same mechanism which destabilizes constant solutions, destabilizes also all continuous spatially heterogenous stationary solutions. It may lead to a diffusion-driven unbounded growth and mass concentration, either in finite or infinite time, which yields a novel class of pattern formation phenomenon in reaction-diffusion systems.

The second, related, area of focus concerned the models of dynamics of structured cell populations. We introduced and compared different mathematical descriptions of such processes: (1) a multi-compartmental model of a discrete set of populations, (2) a nonlinear structured population equation to describe the effects of continuous cell transitions, (3) a related state-dependent delay differential equation, (4) a structured population model with non-Lipschitz zeroes in the velocity to account for both, discrete and continuous, types of transitions. We highlighted the differences and links between the approaches and discussed them in biological context. Analysis of the models led to development of a novel approach to study structured population models in the spaces of positive Radon measures with a suitable metric, adjusted to the nonconservative character of the problem.

Mathematical methods and models developed in the project were applied to specific problems of developmental and cell biology. Models of hysteresis-driven pattern formation led to the design of the experiments, which indicated new aspects of Wnt signalling in Hydra, related to the hysteresis effect or biomechanical interactions. Applications of the cell differentiation models resulted in a better understanding of the role of asymmetric cell divisions, regulatory feedbacks and clonal selection in hematopoiesis and leukemogenesis. Additionally, we started working on the derivation of effective models starting from the first principle modelling to describe transport of cells and molecules through a heterogeneoues tissue.