Illia Horenko Wilhelm Huisinga & Einführungsvortrag zum Seminar Modellierung dynamischer Prozesse in der Zellbiologie Freie Universität Berlin, 17. April 2003

Math. Modellierung in der Zellbiologie Biological processes signal pathways, metabolism, cell cycle, membrane transport and pumps, excitability of ion channels intercellular communication, pheromone response A few examples: reference for graphics below

Math. Modellierung in der Zellbiologie Common modelling strategies detailed description: (1) positional information of every molecule (2) interaction with others molecules  only suitable for (very) small subsystems most common simplifying assumptions: (1’) well-stired mixture=spatial homogeneity (2’) reaction rates & law of mass action or reaction probabilities & combinatorics representation of species according to (1’) by (a) concentrations  deterministic differential equations (ODEs) (b) number of molecules  stochastic simulation algorithm (Markov process)

Math. Modellierung in der Zellbiologie Example Conversion of substrate to product catalysed by enzymes: deterministicstochastic state: concentration [X](t) of species X at time t state: number of molecules X(t) (random variable) of species S at time t equation of motion: ODEequation of motion: Markov Process …

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) reaction-diffusion models (fully heterogeneous) (a) concentration in time and space (deterministic PDE) or (b) 3d-molecular positions (random walk and reaction probabilities) photo: Note: the more complex the model, the more parameters it needs!

Math. Modellierung in der Zellbiologie For example yeast Saccharomyces cerevisiae photo: „Yeast have many genes with homologs in humans. Has our understanding of these genes helped our understanding of human biology or disease? In his Perspective, Botstein argues "yes„ […]“ Botstein, Chervitz&Cherry, GENETICS: Yeast as a Model Organism

Math. Modellierung in der Zellbiologie Saccharomyces cerevisiae photo: “MAP (Mitogen Activated protein) kinase pathways play key roles in cellular response towards extracellular signals.” van Drogen & Peter Biology of the Cell 93 (2001) model of the MAPK signalling pathways of yeast:

Math. Modellierung in der Zellbiologie Saccharomyces cerevisiae photo: Question: how does yeast adapt to different osmotic conditions?  signalling pathway based on osmosensors “Yeast cells in their natural habitats must adapt to extremes of osmotic conditions such as the saturating sugar of drying fruits and the nearly pure water of rain” (Posas et al., Cell 86, 1996)

Math. Modellierung in der Zellbiologie yeast’s two osmosensors de Nadal, Alepus & Posas, EMBO reports 3 (2002) “Our current knowledge confirms that many principles of osmoadaptation are conserved across eukaryotes, and therefore the use of yeast as basic model system has been of great value elucidating the signal transduction mechanisms underlying the response to high osmolarity.”

Math. Modellierung in der Zellbiologie osmotic shock Phosphorelay module Sln1 Ypd1PYpd1 Ssk1 Ssk1P Sln1AP Sln1HP in cooperation with Edda Klipp&group

Math. Modellierung in der Zellbiologie osmotic shock stochastic Markov process model Sln1 Ypd1PYpd1 Ssk1 Ssk1P Sln1AP Sln1HP in cooperation with Edda Klipp&group

Math. Modellierung in der Zellbiologie osmotic shock deterministic ODE model Sln1 Ypd1PYpd1 Ssk1 Ssk1P Sln1AP Sln1HP in cooperation with Edda Klipp&group

Math. Modellierung in der Zellbiologie Comparison of results stochastic Markov Process model deterministic ODE model

Math. Modellierung in der Zellbiologie Species i is described through concentration and diffusion constant Modelling heterogeneity (PDE) 1.Spatial discretisation 2.Time discretisation System of ODEs for concentrations at grid points Finite differences: When is large than homogenous modelling is sufficient! Method of lines

Math. Modellierung in der Zellbiologie Modelling heterogeneity (PDE) Sln1, Sln1AP, Sln1HP are fixed on the membrane Ypd1, Ypd1P, Ssk1, Ssk1P diffuse freely in the cytoplasm  minor influence of heterogeneity due to fast diffusion

Math. Modellierung in der Zellbiologie Comparison of results deterministic ODE model deterministic PDE model

Math. Modellierung in der Zellbiologie Conclusion What are the benefits of computational biology? What are the conclusions to draw? What are the problems encountered?