1. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ Berlin, 02.03.2006 Robustness and Entropy of Biological Networks Thomas Manke Max Planck Institute for Molecular Genetics, Berlin

2. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Outline Cellular Resilience steady states and perturbation experiments A thermodynamic framework a fluctuation theorem (role of microscopic uncertainty) Network Entropy network data and pathway diversity a global network characterisation Applications f rom structure to function: predicting essential proteins

3. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Cellular Robustness Empirical observation: Reproducible phenotype Cells are resilient against molecular perturbations  maintenance of (non-equilibrium) steady state picture from Forsburg lab, USC

4. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Perturbation Experiments Knockouts in yeast: (Winzeler,1999) only few essential proteins !  resilience of steady state

5. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Understanding robustness Dynamical analysis: increasing data on molecular species and processes microscopic description: x(t+1) = f( x(t), p) Topological analysis: qualitative data on molecular relations: network structure determines key properties. An emerging dogma: STRUCTURE  DYNAMICS  FUNCTION

6. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke A thermodynamic approach Key idea: macroscopic properties follow simple rules, despite our ignorance about microscopic complexity Key tool: Statistical mechanics (Gibbs-Boltzmann): Entropy links microscopic and macroscopic world Key result: Microscopic uncertainties  macroscopic resilience

7. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Fluctuation theorems Equilibrium: Kubo 1950 The return rate to equilibrium state (dissipation) is determined by correlation functions (fluctuations) at equilibrium Ergodic systems at steady-state: Demetrius et al. 2004 Changes in robustness are positively correlated with changes in dynamical entropy “robustness” = return rate to steady state

8. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Quantifying microscopic uncertainty Network characterisation  characterisation of dynamical process Consider stochastic process Network relational data

9. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Network entropy The stationary distribution  i is defined as: P =  Entropy Definition (Kolmogorov-Sinai invariant) H(P) = -  i  i  j p ij log p ij = average uncertainty about future state = pathway diversity

10. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Network Entropy and structural observables circularrandom scale-freestar H=2.0 H=2.3 H=2.9H=4.0 L=12.9 L=3.5 L=3.0L=2.0 Entropy is correlated with many other properties: Distances, degree distribution, degree-degree correlations …

11. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Network Entropy and Robustness same number of nodes/edges different wiring schemes  different entropy Observation: Topological resilience increases with entropy ! Network entropy = proxy for resilience against random perturbations L.Demetrius, T.Manke; Physica A 346 (2005). L. Demetrius,V. Gundlach, G. Ochs; Theor. Biol. 65 (2004)

12. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke From Structure to Function An application: protein interaction network (C.elegans) global network characterisation  characterisation of individual proteins ? only 10% show lethal phenotype Hypothesis: Proteins with higher contributions to topological robustness are preferentially lethal (cf. Structure  Function paradigm)

13. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Entropic ranking and essential proteins Entropy decomposition H =  i  i H i Proposal: rank nodes according to their value of  i H i (and not by local connectivity !) Ranked list of N proteins: Entropy rank1234N-1N Lethality index110110 Systematically check whether the top k nodes show an enriched amount of lethal proteins

14. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke

15. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Systematic checks … false positives/negatives … compartmental bias … similar for yeast … proteins with high contribution to network resilience are preferentially essential !

16. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Skipped Which Stochastic Process ?  from variational principle Network selection & evolution  Demetrius & Manke, 2003 Correlation with structural observables  emerge as effective correlates of entropy  can go beyond

17. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Summary Cellular Resilience Structure  Dynamics  Function Thermodynamic approach Network Entropy global network characterization measure of pathway diversity correlates with structural resilience Functional Analysis entropy correlates with lethality

18. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Thank you ! Collaborators: Lloyd Demetrius Martin Vingron Funding: EU-grant “TEMBLOR” QLRI-CT-2001-00015 National Genome Research Network (NGFN)

19. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke Processes on Networks Consider a simple random walk on a network defined by adjacency matrix A = (a ij ) permissble processes P = (p ij ): a ij = 0 p ij = 0  j p ij = 1 Network characterisation  characterisation of dynamical process

20. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke A variational principle log = sup {-  ij  i p ij log p ij +  ij  i a ij log p ij } P Perron-Frobenius eigenvalue (topological invariant) corresponding eigenvector v i is strictly positive for irreducible matrices a ij (strongly connected graphs) for Boolean matrices:  entropy maximisation

21. Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“ March 2-3, 2006Thomas Manke A unique process... p ij = a ij v j / v i Arnold, Gundlach, Demetrius; Ann. Prob. (2004) :  p ij satisfies the variational principle uniquely !  non-equilibrium extension of Gibbs principle  “Gibbs distribution” Network Entropy = KS-entropy of this process