Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
Robustness and Entropy of
Biological Networks
Thomas Manke
Max Planck Institute for
Molecular Genetics, Berlin
Berlin, 02.03.2006
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
Cellular Robustness
Empirical observation:
• Reproducible phenotype
• Cells are resilient against
molecular perturbations
picture from Forsburg lab, USC
maintenance of (non-equilibrium) steady state
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
Perturbation Experiments
Knockouts in yeast:
(Winzeler,1999)
only few essential proteins !
resilience of steady state
March 2-3, 2006 Thomas Manke
f i
xMax-Planck-Institut
Workshop „Systems Biology“
j
für molekulare Genetik
Understanding robustness
Dynamical analysis:
increasing data on molecular species and processes
microscopic description: x(t+1) = f( x(t) , p)
Topological analysis: f i
qualitative data on molecular relations: x j
network structure determines key properties.
An emerging dogma:
STRUCTURE DYNAMICS FUNCTION
March 2-3, 2006 Thomas Manke
f i
xMax-Planck-Institut
Workshop „Systems Biology“
j
für molekulare Genetik
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
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
Quantifying microscopic uncertainty
Network relational data
Consider stochastic process
Network characterisation
characterisation of dynamical process
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
Network entropy
The stationary distribution pi is defined as:
pP =p
Entropy Definition (Kolmogorov-Sinai invariant)
H(P) = - Si pi Sj pij log pij
= average uncertainty about future state
= pathway diversity
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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 …
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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)
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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:
Entropy rank 1 2 3 4 N-1 N
Lethality index 1 1 0 1 1 0
Systematically check whether the top k nodes
show an enriched amount of lethal proteins
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
Systematic checks
… false positives/negatives
… compartmental bias
… similar for yeast
… proteins with high contribution to network resilience
are preferentially essential !
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
Thank you !
Collaborators:
• Lloyd Demetrius
• Martin Vingron
Funding:
• EU-grant “TEMBLOR” QLRI-CT-2001-00015
• National Genome Research Network (NGFN)
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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 p ij =1
Network characterisation
characterisation of dynamical process
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
A variational principle
Perron-Frobenius eigenvalue (topological invariant)
log l =
sup {-Sij pi pij log pij + Sij pi aij log pij }
P
• corresponding eigenvector vi is strictly positive for
irreducible matrices aij (strongly connected graphs)
• for Boolean matrices: entropy maximisation
March 2-3, 2006 Thomas Manke
Max-Planck-Institut
für molekulare Genetik Workshop „Systems Biology“
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
March 2-3, 2006 Thomas Manke