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

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

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

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

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

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

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