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Modelling Stochastic Dynamics in Complex Biological Networks

Modelling Stochastic Dynamics in Complex Biological Networks. Andrea Rocco Department of Statistics University of Oxford (21 February 2006). COMPLEX ADAPTIVE SYSTEMS GROUP SEMINARS Sa ïd Business School, University of Oxford. Outline. General approach of Systems Biology

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Modelling Stochastic Dynamics in Complex Biological Networks

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  1. Modelling Stochastic Dynamics in Complex Biological Networks Andrea Rocco Department of Statistics University of Oxford (21 February 2006) COMPLEX ADAPTIVE SYSTEMS GROUP SEMINARS Saïd Business School, University of Oxford

  2. Outline • General approach of Systems Biology • Different types of stochasticity Internal & external fluctuations • Metabolic Networks • Stochastic kinetic modelling Non-trivial effects of external noise • Stochastic system-level modelling Stochastic Metabolic Control Analysis New Summation Theorems • Concluding remarks

  3. Systems Biology approach NATURE µ - world M - world small scales(complex) large scales(maybe simple) Are we able to understand it, and reproduce it? Example: Reduction of complexity in -phage epigenetic switch [Ptashne (1992), Sneppen (2002-2003)]

  4. Modelling Complex Systems Complex Systems: All microscopic constituents are equally dynamically relevant • Modelling (meant as reduction) may fail • Mathematical Replicas may need to be invoked [Westerhoff (2005)]

  5. Dynamics in Networks • Option 1: Large scale (statistical) analysis [Albert & Barabási (1999)] • Option 2: Dynamical descriptions Dynamical descriptions µ-scopic (kinetics) M-scopic (MCA) link

  6. Two fundamental ingredients: 1. Spatial dependencies Within the Replica approach: • Segmentation in Drosophila During embryonic development cells differentiate according to their position in the embryo [Driever and Nusslein-Volhard (1988)] • MinCDE Protein system in E. coli Determination of midcell point before division: Dynamical compartmentalization as an emergent property [Howard et al. (2001-2003)]

  7. Two fundamental ingredients:2. Stochasticity • Thermal fluctuations Coupling with a heat bath – internal • Statistical fluctuations Low copy number of biochemical species – internal • Parameter fluctuations pH, temperature, etc, … – external

  8. Metabolic networks: kinetic description

  9. Adding external noise By Taylor - expanding: Stochastic Differential Equation (SDE)

  10. Multiplicative-noise SDEs Multiplicative noise: Stochastic Integral ill-defined Ito vs Stratonovich Dilemma…

  11. Stratonovich Ito Ito vs Stratonovich Assuming -correlated noise is “physical” Stratonovich Prescription In other words: is equivalent to: where:

  12. Implications for the steady state New contribution to “deterministic” dynamics Steady state: c c c steady + noise c steady time time

  13. System-level modelling: Metabolic Control Analysis Local variables(enzymes) control Global (system) variables(fluxes, concentrations) • Procedure: • Let the system relax to its steady state • Apply small local perturbation (enzyme) • Wait for relaxation onto new steady state • Measure the change in global variables (fluxes & concentrations) Flux control coefficients: Concentration control coefficients:

  14. Summation Theorems (concentrations) Euler’s Theorem for homogeneous functions: Steady state concentrations:

  15. Stochastic Metabolic Control Analysis Control based on noise !!!

  16. Concluding remarks • Implemented external noise on kinetics • Non-trivial effects: fluctuations do not average out • Implications on MCA • Stochastic MCA • Extension of Summation Theorem for concentrations • Control based on noise • To do: • Extension of Summation Theorem for fluxes • Extension to include spatial dependencies • Experimental validation

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