STOPATT · Stochastic pattern formation in biochemical systems

Stochastic pattern formation in biochemical systems

What is the problem/issue being addressed? Chemotaxis is defined as the oriented movement of cells (or an organism) in response to a chemical gradient. Many sorts of motile cells undergo chemotaxis. For example, bacteria and many amoeboid cells can move in the direction of a food source. In our bodies, immune cells like macrophages and neutrophils can move towards invading cells. Other cells, connected with the immune response and wound healing, are attracted to areas of inflammation by chemical signals. This macroscopic system of equations is usually derived from microscopic behavior by studying the limit behavior. From the microscopic perspective, one interprets the movements of the cells as a result of microscopic irregular movement of the single members of the population of cells. Taking the limit and passing from the microscopic to the macroscopic equation, one is neglecting the fluctuations. Another source of randomness is due to the environment. Here, biologists distinguish between internal (or intrinsic) noise caused by the irregular movement of the cells and external noise caused by a random environment. An appropriate mathematical approach to establishing more realistic models is the incorporation of stochastic processes. To model the randomness, one adds a random forcing term to the system. Adding a stochastic driving term has a highly non–trivial impact on the behavior of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behavior, which is most helpful in understanding the real processes and is also often more realistic. E.g., if the system starts to switch between two quasi-stable states, the pattern may change or starts to wander around. Bifurcations are smeared out, and the importance of tipping points may change. Although the project is of theoretical nature, the findings may be used to develop a more realistic model in biology and medicine and to explain phenomena that appear but can not be explained by purely deterministic systems. Improving models in biology or medicine have a natural impact on society. Understanding, e.g., the self-organization of cells, means, e.g., understanding biological processes like wound healing or cancer in a better way. This can lead to the design of new strategies against cancer. The aim of the project was first to investigate the mathematical existence, uniqueness, and dynamical behavior of these processes. Then, to investigate its long-time dynamical behavior, and, finally, the modelisation of these processes. Why is it important for society? Model organisms are species that are extensively studied to understand biological phenomena that provide insight into the functions of other organisms. Researching these organisms explores the basic mathematics, biology, and chemistry of life. As a next question, we ask ourselves, why choose to study model organisms instead of the organism of interest itself? Research of biological systems, development, or genetics on humans is complicated and poses ethical challenges in many situations. In biological systems, it is very difficult to separate the effects of a gene from the effects of the environment. However, with models, variables of interest can be engineered and confounding variables can be controlled accordingly. What are the overall objectives? The overall objective of the project was first to investigate the mathematical solvability theory (existence, uniqueness), and then the dynamical behavior of these processes. As a next step, we investigated its long-time dynamical behavior, and, finally, the modelization of these processes. This mathematical analysis establishes important discoveries in the world of mathematical biological models in presence of random perturbations.

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