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Robustness and Entropy of Biological NetworksPowerPoint Presentation

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

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

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

Perturbation Experiments

Knockouts in yeast:

(Winzeler,1999)

only few essential proteins !

resilience of steady state

Thomas 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 DYNAMICSFUNCTION

Thomas 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

Thomas 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

Thomas Manke

Quantifying microscopic uncertainty

Network relational data

Consider stochastic process

Network characterisation

characterisation of dynamical process

Thomas Manke

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

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

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

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

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

Systematic checks

… false positives/negatives

… compartmental bias

… similar for yeast

… proteins with high contribution to network resilience

are preferentially essential !

Thomas 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

Thomas 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

Thomas Manke

Thank you !

- Collaborators:
- Lloyd Demetrius
- Martin Vingron

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

Thomas Manke

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

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

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

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