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.

References

A. Turing: The Chemical Basis of Morphogenesis Philosophical
Transactions of the Royal Society of London B. 237 (641):
37–72.1952

J.D. Murray: Mathematical Biology. Springer Science &
Business Media, 2013. ISBN 978-3-662-08539-4

M. Cohen, K.M. Page e.a: A theoretical framework for the
regulation of Shh morphogen-controlled gene expression.
Development, 141(20), 3868-3878, 2014.

N. Balaskas e.a.: Gene regulatory logic for reading the Sonic
Hedgehog signaling gradient in the vertebrate neural tube. Cell.
148, 273-284, 2012.

J. Panovska-Griffiths e.a.: A gene regulatory motif that
generates oscillatory or multiway switch outputs. J. Roy. Soc.
Interface. 10.79, 2013.


Podcasts

L. Adlung: Systembiologie, Gespräch mit G. Thäter und S.
Ritterbusch im Modellansatz Podcast, Folge 39, Fakultät für
Mathematik, Karlsruher Institut für Technologie (KIT), 2016.

Omega Tau-Podcast 069: Grundlagen der Zellbiologie

Omega Tau-Podcast 072: Forschung in der Zellbiologie

Konscience-Podcast 024, Kapitel 5: Das Hochlandgen aus "Wie
kam das bloß durch die Ethikkommission?"