1. Illia Horenko & Wilhelm Huisinga Einführungsvortrag zum Seminar Modellierung dynamischer Prozesse in der Zellbiologie: Deterministischer Zugang Freie Universität Berlin, 08. May 2003

2. Math. Modellierung in der Zellbiologie Contents Homogeneous deterministic model –Derivation of reaction equation –Matlab-application (Michelis-Menten) Heterogeneous deterministic model –Derivation of reaction-diffusion equation –Numerical realisation (Michaelis-Menten)

3. Math. Modellierung in der Zellbiologie Mass Action Law Bimolecular reaction: Model assumptions: M, V constant; distribution of particles is homogeneous

4. Math. Modellierung in der Zellbiologie Mass Action Law Change of particles number due to the reactions: :

5. Math. Modellierung in der Zellbiologie ODEs: Solution Linear autonomous ODE:Analytical solution: Numerical methods Implicit(ode15s) Rational representation: Polynomial representation: Explicit(ode23, ode45) where n n AAAA...

6. Math. Modellierung in der Zellbiologie ODEs: Matlab Numerics Bimolecular kinetics: Timesteps are adapted to absolute error tolerance AbsTol odeset(…)

7. Math. Modellierung in der Zellbiologie ODEs: Stiffness Michaelis-Menten enzyme kinetics: Models with very differrent timescales are stiff implicit integrators such as ode15s are recommended

8. Math. Modellierung in der Zellbiologie Efficient Modelling of heterogeneity compartment models (partially heterogeneous): (1) well-stirred within compartment (ODE,MP) & (2) interaction between compartment (consistent coupling) photo: http://genome-www.stanford.edu/Saccharomyces/yeast_images.shtml Note: the more complex the model, the more parameters it needs! reaction-diffusion models (fully heterogeneous) (a) concentration in time and space (deterministic PDE) or (b) 3d-molecular positions (random walk and reaction probabilities)

9. Math. Modellierung in der Zellbiologie Heterogeneous deterministic Model Bimolecular reaction: Model assumptions: M, V constant; distribution of particles is inhomogeneous (dependent on coordinate R) Change of concentration i due to the reactions and diffusion:

10. Math. Modellierung in der Zellbiologie Species i is described through concentration and diffusion constant Modelling heterogeneity (PDE) Modell reduction: use spherical symmetry

11. Math. Modellierung in der Zellbiologie PDE: Numerics When is large w. r. to reaction constatnts => homogenous modelling is sufficient! 1.Spatial discretisation 2.Time discretisation System of ODEs for concentrations at grid points Finite differences: Method of lines

12. Math. Modellierung in der Zellbiologie PDE: Michaelis-Menten S, ES and P are fixed on the membrane (D=0) E diffuse freely from nucleus to membrane