Final exam

The exams (oral, 30min) will take place on 17 March 2009, between 10 am and 1 pm, in my office.

Termine
Vorlesung:	Mittwoch 15:45-17:15	Seminarraum 33

Lehrende
Dozent	Prof. Dr. Tobias Jahnke
	Sprechstunde: Montag, 10:00 - 11:00 Uhr
	Zimmer 3.042 Kollegiengebäude Mathematik (20.30)
	Email: tobias.jahnke@kit.edu

Downloads

Numerical examples from Chapter III ("Stochastic vs. deterministic reaction kinetics")
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII

Plots from Chapter VII:
$Liouville equation,$
$Kramers-Moyal expansion,$
$Kurtz' Theorem (1),$
$Kurtz' Theorem (2),$
$Kurtz' Theorem (3)$

Summary

Chemical reactions are most often described by the traditional reaction-rate approach: a reaction
system of n interacting substances is translated into a system of n coupled ordinary
differential equations, and the corresponding solution indicates how the concentration of each
substance evolves in time. Although this approach is well-established and provides an accurate
model in many application areas, it may produce completely wrong results when it is applied
to reactions that occur in cells. The reason for this failure is twofold. In cellular reaction systems
such as, e.g., gene regulatory networks or viral replication, some of the interacting species
contain only such a small number of copies that it does not make sense to consider the concentrations
as continuously varying functions. Moreover, cells are very noisy in a stochastic
sense, because reactions occur randomly, and small stochastic fluctuations can produce largescale
effects. Therefore, the deterministic, continuous description of the reaction-rate approach
has to be replaced by a stochastic and discrete framework known as stochastic reaction kinetics.

At the beginning of the course, the reasons for the failure of the traditional reactionrate
approach will be investigated in somewhat more detail. Then, it will be shown how the
random evolution of a reaction system can be simulated by a stochastic simulation algorithm
using Monte-Carlo techniques. As an alternative to stochastic simulations, we will derive the
chemical master equation which describes the evolution of the probability distribution of the
system. The pros and cons of both alternatives will be discussed, and the relationships to the
traditional reaction-rate approach will be investigated extensively.

In realistic applications both alternatives computing realisations with the stochastic
simulation algorithm or solving the chemical master equation turn out to be computationally
expensive. Therefore, we will investigate the question how the computational costs can be
reduced if additional assumptions are made and/or a small approximation error is accepted.
This leads to a number of reduced or hybrid models in which different kinds of equations are
coupled. Moreover, several numerical methods devised for performing stochastic simulations or
solving the chemical master equation will be presented.

Preliminaries: Basic knowledge in analysis, lineare algebra, stochastics and ordinary differential
equations.

Prüfung

The exams (oral, 30min) will take place on 17 March 2009, between 10 am and 1 pm, in my office.