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Non-linear dynamics of biological systems
Non-linear dynamics is the study of dynamical processes in nature. Because usually they do not obey linear laws, the superposition principle does not hold. On the one hand, this means that small perturbations can decay again, which is an important prerequisite to obtain stable limit cycles (oscillations). On the other hand, small perturbations need not to stay small, thus small variations in initial conditions can lead to very different results (deterministic chaos). Non-linear dynamics can be studied through non-linear differential or difference equations and in both cases, graphical methods are very helpful. The range of typical behaviour of non-linear systems includes bistability, switch-like behaviour and oscillations, which occur in many natural and man-made systems. This course offers an introduction to the basic tools to understand these responses as well as to different applications in biology, including molecular processes like enzyme kinetics, cellular processes like hearing and evolutionary processes like coexistence of competing species.
The course is designed for students of physics and related disciplines after the Vordiplom / Bachelor and will be given in English. A basic understanding of physics and differential equations is sufficient to attend. The course takes place every Tuesday from 2.15 - 3.45 pm at HS2 at the Hoersaalgebaeude Physik (INF 308). Every two weeks on Wednesday from 4 - 6 pm the solutions to the exercises will be discussed in a tutorial run by Dr. Jakob Schluttig and taken place in seminar room 44 in the Bioquant-building (INF 267). Click here for a schedule for the exercises.
Exercises
Additional material
- Introduction
- Membrane excitation
- Physics of muscle
- Network motifs
- John Tyson et al., Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell, Current Opinion in Cell Biology 2003
- Script from earlier course on NLD (in German)
- Handout fixed points in 2d linear systems
Software
- pplane, a great phase plane analysis tool for Matlab from John Polking at Rice University
- xppaut, a great stand-alone phase plane analysis and bifurcation tool from Bard Ermentrout at the University of Pittsburgh
- Here you find information on dynamic systems tools in Mathematica
- Matlab program for the glycolysis oscillator
- Matlab program for the van der Pol oscillator
- Matlab program for a simple model for epidemics
Recommended reading
- SH Strogatz, Nonlinear dynamics and chaos, Westview 1994
- JD Murray, Mathematical biology, 3rd edition (now in volumes I and II), Springer 2002
Additional reading
- L Edelstein-Keshet, Mathematical Models in Biology, Random House 1988
- J Keener and J Sneyd, Mathematical Physiology, Springer 1998
- C Fall et al, eds, Computational Cell Biology, Springer 2002
- M Nowak, Evolutionary Dynamics, Harvard University Press 2006
- U Alon, An Introduction to systems biology, Chapman and Hall/CRC 2007
- M Cross and H Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press 2009
