Modelling Stochastic Dynamics in Complex Biological Networks

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Modelling Stochastic Dynamics in Complex
Biological Networks
Andrea Rocco Department of Statistics University
of Oxford (21 February 2006)
COMPLEX ADAPTIVE SYSTEMS GROUP SEMINARS Saïd
Business School, University of Oxford
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Outline

• General approach of Systems Biology
• Different types of stochasticity
• Internal external fluctuations
• Metabolic Networks
• Stochastic kinetic modelling
• Non-trivial effects of external noise
• Stochastic system-level modelling
• Stochastic Metabolic Control
  Analysis
• New Summation Theorems
• Concluding remarks

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Systems Biology approach
NATURE
µ - world
M - world
small scales(complex)
large scales(maybe simple)
Are we able to understand it, and reproduce it?
Example Reduction of complexity in -phage
epigenetic switch
Ptashne (1992), Sneppen (2002-2003)
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Modelling Complex Systems
Complex Systems
All microscopic constituents are equally
dynamically relevant

• Modelling (meant as reduction) may fail
• Mathematical Replicas may need to be
  invoked Westerhoff (2005)

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Dynamics in Networks

• Option 1 Large scale (statistical) analysis

Albert Barabási (1999)

• Option 2 Dynamical descriptions

Dynamical descriptions
µ-scopic (kinetics)
M-scopic (MCA)
link
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Two fundamental ingredients 1. Spatial
dependencies

• Within the Replica approach
• Segmentation in Drosophila
• During embryonic development cells
  differentiate according to their position in the
  embryo
• Driever and Nusslein-Volhard (1988)
• MinCDE Protein system in E. coli
• Determination of midcell point before division
  
• Dynamical compartmentalization as an emergent
  property
• Howard et al. (2001-2003)

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Two fundamental ingredients2. Stochasticity

• Thermal fluctuations
• Coupling with a heat bath internal
• Statistical fluctuations
• Low copy number of biochemical species
  internal
• Parameter fluctuations
• pH, temperature, etc, external

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Metabolic networks kinetic description
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Adding external noise
By Taylor - expanding
Stochastic Differential Equation (SDE)
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Multiplicative-noise SDEs
Multiplicative noise Stochastic Integral
ill-defined Ito vs Stratonovich Dilemma
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Ito vs Stratonovich
Assuming -correlated noise is physical
Stratonovich Prescription
In other words
is equivalent to
where
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Implications for the steady state
New contribution to deterministic dynamics
Steady state
c
c
c steady noise
c steady
time
time
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System-level modelling Metabolic Control Analysis
Local variables(enzymes)
Global (system) variables(fluxes, concentrations)
control

• Procedure
• Let the system relax to its steady state
• Apply small local perturbation (enzyme)
• Wait for relaxation onto new steady state
• Measure the change in global variables (fluxes
  concentrations)

Flux control coefficients
Concentration control coefficients
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Summation Theorems (concentrations)
Eulers Theorem for homogeneous functions
Steady state concentrations
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Stochastic Metabolic Control Analysis
Control based on noise !!!
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Concluding remarks

• Implemented external noise on kinetics
• Non-trivial effects fluctuations do not average
  out
• Implications on MCA
• Stochastic MCA
• Extension of Summation Theorem for
  concentrations
• Control based on noise
• To do
• Extension of Summation Theorem for fluxes
• Extension to include spatial dependencies
• Experimental validation