Robustness and Entropy of Biological Networks

1. Robustness and Entropy of Biological Networks Thomas Manke Max Planck Institute for Molecular Genetics, Berlin

2. 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 from structure to function: predicting essential proteins Thomas Manke

3. Cellular Robustness • Empirical observation: • Reproducible phenotype • Cells are resilient against • molecular perturbations picture from Forsburg lab, USC  maintenance of (non-equilibrium) steady state Thomas Manke

4. Perturbation Experiments Knockouts in yeast: (Winzeler,1999) only few essential proteins !  resilience of steady state Thomas Manke

5. 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 Thomas Manke

6. 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 Thomas Manke

7. 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 Thomas Manke

8. Quantifying microscopic uncertainty Network relational data Consider stochastic process Network characterisation  characterisation of dynamical process Thomas Manke

9. Network entropy The stationary distributionpi is defined as: p P =p Entropy Definition (Kolmogorov-Sinai invariant) H(P) = - Si pi Sj pij log pij = average uncertainty about future state = pathway diversity Thomas Manke

10. Network Entropy and structural observables scale-free star circular random H=2.3 H=2.0 H=2.9 H=4.0 L=3.5 L=12.9 L=3.0 L=2.0 Entropy is correlated with many other properties: Distances, degree distribution, degree-degree correlations … Thomas Manke

11. Network Entropy and Robustness same number of nodes/edges differentwiring 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) Thomas Manke

12. From Structure to Function An application: protein interaction network (C.elegans) global network characterisation  characterisation of individual proteins ? Hypothesis: Proteins with higher contributions to topological robustness are preferentially lethal (cf. Structure Function paradigm) only 10% show lethal phenotype Thomas Manke

13. Entropic ranking and essential proteins Entropy decomposition H = Si pi Hi Proposal: rank nodes according to their value of pi Hi (and not by local connectivity !) Ranked list of N proteins: Systematically check whether the top k nodes show an enriched amount of lethal proteins Thomas Manke

14. Thomas Manke

15. Systematic checks … false positives/negatives … compartmental bias … similar for yeast … proteins with high contribution to network resilience are preferentially essential ! Thomas Manke

16. 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 Thomas Manke

17. 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 Thomas Manke

18. Thank you ! • Collaborators: • Lloyd Demetrius • Martin Vingron • Funding: • EU-grant “TEMBLOR” QLRI-CT-2001-00015 • National Genome Research Network (NGFN) Thomas Manke

19. Processes on Networks • Consider a simple random walk on a network defined by • adjacency matrix A = (aij) • permissble processes P = (pij): • aij = 0 pij = 0 • Sj pij = 1 Network characterisation  characterisation of dynamical process Thomas Manke

20. A variational principle Perron-Frobenius eigenvalue (topological invariant) logl = sup {-Sij pi pij log pij +Sij pi aij log pij } P • corresponding eigenvectorvi is strictly positive for • irreducible matrices aij (strongly connected graphs) • for Boolean matrices:  entropy maximisation Thomas Manke

21. A unique process ... pij = aij vj / l vi Arnold, Gundlach, Demetrius; Ann. Prob. (2004):  pij satisfies the variational principle uniquely !  non-equilibrium extension of Gibbs principle  “Gibbs distribution” Network Entropy = KS-entropy of this process Thomas Manke