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## Modellansatz: Embryonic Patterns

In March 2018 Gudrun visited University College London and recorded three conversations with mathematicians working there.

Her first partner was Karen Page. She works in Mathematical Biology and is interested in mathematical models for pattern formation. An example would be the question why (and how) a human embryo develops five fingers on each hand. The basic information for that is coded into the DNA but how the pattern develops over time is a very complicated process which we understand only partly. Another example is the patterning of neurons within the vertebrate nervous system. The neurons are specified by levels of proteins. Binding of other proteins at the enhancer region of DNA decides whether a gene produces protein or not. This type of work needs a strong collaboration with biologists who observe certain behaviours and do experiments. Ideally they are interested in the mathematical tools as well.

One focus of Karen's work is the development of the nervous system in its embryonic form as the neural tube. She models it with the help of dynamical systems. At the moment they contain three ordinary differential equations for the temporal changes in levels of three proteins. Since they influence each other the system is coupled. Moreover a fourth protein enters the system as an external parameter. It is called sonic hedgehog (Shh). It plays a key role in regulating the growth of digits on limbs and organization of the brain. It has different effects on the cells of the developing embryo depending on its concentration.

Concerning the mathematical theory the Poincaré Bendixson theorem completely characterizes the long-time behaviour of two-dimensional dynamical systems. Working with three equations there is room for more interesting long-term scenarios. For example it is possible to observe chaotic behaviour.

Karen was introduced to questions of Mathematical Biology when starting to work on her DPhil. Her topic was Turing patterns. These are possible solutions to systems of Partial differential equations that are thermodynamically non-equilibrium. They develop from random perturbations about a homogeneous state, with the help of an input of energy.

Prof. Page studied mathematics and physics in Cambridge and did her DPhil in Oxford in 1999. After that she spent two years at the Institute for Advanced Study in Princeton and has been working at UCL since 2001.

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