Stochastic Modelling in Biology
Many processes in systems biology can be modelled as complex (biochemical) reaction systems in which different species interact via different reaction channels. In most applications the time evolution of such a system can be accurately described in terms of classical deterministic reaction kinetics: The reaction system is translated into a system of ordinary differential equations (ODEs) (the reaction-rate equations), and the solution of these ODEs indicates how the concentration or amount of each of the species changes in time.
This reaction-rate approach, however, is insufficient if some of the species are present in a very low number of particles and small-scale stochastic fluctuations can have large-scale effects. This is typically the case when gene regulatory networks are investigated. Here, the appropriate description is provided by the solution of the chemical master equation.
For example in gene regulatory networks the synthesis of proteins from the genetic information is governed by transcription factors which activate or inhibit the transcription of a gene. This complex interplay can be considered as a reaction system with several species interacting via a number of reaction channels. In contrast to traditional population models, however, a mathematical description with the deterministic mass-action kinetics usually yields qualitatively wrong results, because gene expression is an intrinsically stochastic process and involves populations with small, discrete particle numbers. Other examples where the reaction-rate approach leads to wrong results are viral kinetics where the fate of very few infectious individuals decides whether the infection spreads over large parts of the population, or in predator-prey systems where the presence of a few predators keeps the entire ecosystem in equilibrium. In all these examples, small changes in the population numbers of the critical species due to inherent stochastic noise can cause large-scale effects. Often the evolution of the entire system is crucially determined by one or two sub-populations which may consist of a very small number of individuals. In this situation, small changes in the critical sub-population due to inherent stochastic noise can cause large-scale effects. Hence, a reasonable mathematical model of such processes must respect both the stochasticity of the time evolution and the discreteness of population numbers.
For more information please see:
* Stochastic Modelling in Biology
The idea of hybrid deterministic-stochastic models is to interpolate between the accurate but computationally costly stochastic reaction kinetics and the simple but rather coarse deterministic description. In hybrid models, a part of the system (e.g. some of the species) is treated stochastically while the other part is represented in the deterministic setting. We use this approach to speed up stochastic simulations.
The main questions are
* how to couple the stochastic with the deterministic description, and
* how to derive error bounds for the modelling error caused by the hybrid description.
For more information please see:
* Hybrid models and dimension reduction
More information on these topics and to the other research field of our research group can be found on:
* Research group Numerical Methods for high-dimensional Systems
Tobias Jahnke and Michael Kreim
Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems
SIAM Multiscale Model. Simul., 10(4), 1119–1147, (2012)
Michael Henkel, David Mitzner, Peter Henklein, Franz-Josef Meyer-Almes, Anna Moroni, Mattia L. DiFrancesco, Leonhard M. Henkes, Michael Kreim, Stefan M. Kast, Ulrich Schubert, Gerhard Thiel
The Proapoptotic Influenza A Virus Protein PB1-F2 Forms a Nonselective Ion Channel
PLoS ONE 5(6), (2010)
Sabrina Gazzarrini, Ming Kang, Alessandra Abenavoli, Giulia Romani, Claudio Olivari, Daniele Gaslini, Giuseppina Ferrara, James L. Van Etten, Michael Kreim, Stefan M. Kast, Gerhard Thiel, Anna Moroni
Chlorella virus ATCV-1 encodes a functional potassium channel of 82 amino acids
Biochemical Journal, 420, 295-303 (2009)
Sascha Tayefeh, Thomas Kloss, Michael Kreim, Manuela Gebhardt, Dirk Baumeister, Brigitte Hertel, Christian Richter, Harald Schwalbe, Anna Moroni, Gerhard Thiel, Stefan M. Kast
Model Development for the Viral Kcv Potassium Channel
Biophysical Journal, 96, 485 - 498 (2009)
Michael Kreim and Christoph Giersch
Measuring in vivo elasticities of Calvin cycle enzymes: Network structure and patterns of modulations
Phytochemistry, 68, 2152 - 2162 (2007)
Simulation des Kaliumkanals Kcv in einem Doppel- Biomembran-System
Diplomarbeit, Technische Universität Darmstadt (2008)