max-planck-institut für molekulare genetik workshop „systems biology“ berlin, 02.03.2006...

Berlin, 02.03.2006

Max-Planck-Institut für molekulare Genetik Workshop „Systems Biology“

Robustness and Entropy of Biological Networks

Thomas Manke

Max Planck Institute for Molecular Genetics, Berlin

March 2-3, 2006 Thomas Manke


Outline Cellular Resilience

steady states and perturbation experiments

A thermodynamic frameworka fluctuation theorem (role of microscopic uncertainty)

Network Entropynetwork data and pathway diversity

a global network characterisation Applications

from structure to function: predicting essential proteins



Cellular Robustness

Empirical observation:

• Reproducible phenotype

• Cells are resilient against molecular perturbations

maintenance of (non-equilibrium) steady state

picture from Forsburg lab, USC



Perturbation Experiments

Knockouts in yeast:(Winzeler,1999)only few essential proteins !

resilience of steady state



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

ij

f

x

i

j

f

x



A thermodynamic approach

Key idea:macroscopic properties follow simple rules,despite our ignorance about microscopic complexity

ij

f

x

Key tool: Statistical mechanics (Gibbs-Boltzmann):Entropy links microscopic and macroscopic world

Key result: Microscopic uncertainties macroscopic resilience



Fluctuation theorems

Equilibrium: Kubo 1950The return rate to equilibrium state (dissipation) is determined by correlation functions (fluctuations) at equilibrium

Ergodic systems at steady-state: Demetrius et al. 2004Changes in robustness are positively correlated with changes in dynamical entropy

“robustness” = return rate to steady state



Quantifying microscopic uncertainty

Network characterisation characterisation of dynamical process

Consider stochastic processNetwork relational data



Network entropy

The stationary distribution i is defined as:

P = Entropy Definition (Kolmogorov-Sinai invariant)

H(P) = - i i j pij log pij

= average uncertainty about future state= pathway diversity



Network Entropy and structural observables

circular random scale-free star

H=2.0 H=2.3 H=2.9 H=4.0

L=12.9 L=3.5 L=3.0 L=2.0

Entropy is correlated with many other properties:Distances, degree distribution, degree-degree correlations …



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)



From Structure to FunctionAn 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)



Entropic ranking and essential proteins

Entropy decomposition

H = i i Hi

Proposal: rank nodes according to their value of i Hi

(and not by local connectivity !)

Ranked list of N proteins:Entropy rank 1 2 3 4 N-1 N

Lethality index 1 1 0 1 1 0

Systematically check whether the top k nodesshow an enriched amount of lethal proteins



Systematic checks

… false positives/negatives

… compartmental bias

… similar for yeast

… proteins with high contribution to network resilience are preferentially essential !



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



Summary Cellular Resilience

Structure Dynamics FunctionThermodynamic approach

Network Entropyglobal network characterization

measure of pathway diversitycorrelates with structural resilience

Functional Analysis entropy correlates with lethality



Thank you !

Collaborators:• Lloyd Demetrius• Martin Vingron

Funding:• EU-grant “TEMBLOR” QLRI-CT-2001-00015• National Genome Research Network (NGFN)



Processes on NetworksConsider a simple random walk on a network defined by

adjacency matrix A = (aij)

permissble processes P = (pij):

• aij = 0 pij = 0

• j pij = 1

Network characterisation characterisation of dynamical process



A variational principle

log =

sup {-ij i pij log pij + ij i aij log pij } P

Perron-Frobenius eigenvalue (topological invariant)

• corresponding eigenvector vi is strictly positive for

irreducible matrices aij (strongly connected graphs)

• for Boolean matrices: entropy maximisation



A unique process ...

pij = aij vj / 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

max-planck-institut für molekulare genetik workshop „systems biology“ berlin, 02.03.2006...

Documents

thomas manke network

berlin slide

network structure

usc slide

equilibrium ergodic

dynamical entropy robustness

entropy links microscopic

pathway diversity slide