Abstract
In this lecture I will describe the lysogeny maintenance
switch in phage lambda. The standard picture leads naturally
to a mathematical model, which is to predict the first exit time
of motion governed by a system of coupled stochastic
differential equations, when the deterministic parts of these
equations have an equilibrium point (steady state).
We discuss robustness properties of the model, and what
one can actually predict with it, or with similar types of
models. A major problem in getting quantitative predictions
for genetic control networks is that, even in this paradigmatic
example, not all basic biochemical data are firmly established.
Thermal escape from a potential well is a classical model
of reaction kinetics. Although there is no potential field
in the lambda model, nevertheless, around each steady state
of the drift field, there is a something usually called
quasi-potential - or Wentzel-Freidlin action -, which plays
much the same role as the potential in Kramers' thermal
exit problem. To compute this quasi-potential is however
not entirely trivial, in contrast to the potential-field
exit problem, where it can be read off directly from the equations.
Furthermore, the quasi-potential is only locally defined,
around each steady state. To the extent that gene regulatary
networks can be modelled by models of this kind, that still
leaves more possibilities than motion in potential landscapes,
e.g. cyclic phenomena.