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Robustness and Entropy of Biological Networks

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Robustness and Entropy of Biological Networks

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Robustness and Entropy of Biological Networks

Thomas Manke

Max Planck Institute for Molecular Genetics, Berlin

- 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

- Empirical observation:
- Reproducible phenotype
- Cells are resilient against
- molecular perturbations

picture from Forsburg lab, USC

maintenance of (non-equilibrium) steady state

Thomas Manke

Knockouts in yeast:

(Winzeler,1999)

only few essential proteins !

resilience of steady state

Thomas Manke

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 DYNAMICSFUNCTION

Thomas Manke

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

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

Network relational data

Consider stochastic process

Network characterisation

characterisation of dynamical process

Thomas Manke

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

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

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

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

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

Thomas Manke

… false positives/negatives

… compartmental bias

… similar for yeast

… proteins with high contribution to network resilience

are preferentially essential !

Thomas Manke

- 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

- 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

- Collaborators:
- Lloyd Demetrius
- Martin Vingron

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

Thomas Manke

- 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

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

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