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

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Robustness and entropy of biological networks

Robustness and Entropy of Biological Networks

Thomas Manke

Max Planck Institute for Molecular Genetics, Berlin


Outline

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

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

Perturbation Experiments

Knockouts in yeast:

(Winzeler,1999)

only few essential proteins !

 resilience of steady state

Thomas Manke


Understanding robustness

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


A thermodynamic approach

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

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

Quantifying microscopic uncertainty

Network relational data

Consider stochastic process

Network characterisation 

characterisation of dynamical process

Thomas Manke


Network entropy

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

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

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

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

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


Robustness and entropy of biological networks

Thomas Manke


Systematic checks

Systematic checks

… false positives/negatives

… compartmental bias

… similar for yeast

… proteins with high contribution to network resilience

are preferentially essential !

Thomas Manke


Skipped

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

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

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

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

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

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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