Mathematical framework for studying biological
robustness
Ovidiu Radulescu
IRMAR UMR 6025, University of Rennes 1, France
Matbrac meeting, November 20, 2008
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Outline
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Complex biological systems and Laplacian determinism
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Robustness by dimension compression
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Multiscale robustness
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Organism as a computable system
Sidney Brenner, DNA is self-sufficient: given the DNA sequence of
an organism, we can compute the organism.
Functional genomics: protein A binds to promoter region of gene B
which can be methylated or not, in the presence of protein C,
which can be phosphorylated or not, ...
Developmental biology: identify all developmental genes, compute
the network of interactions (an oriented graph with +/-) take a
Teraflop computer, determine fate.
Morphogenesis is unfolding information on genes and interaction
between genes stored on DNA.
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First epigenetic amendment: environment
The fate is not specified only by genes, but also by the sequence of
environments in which morphogenesis takes place.
Environment = external conditions (resulting eventually from
interacting organisms: ecosystem), but also state of other internal
variables resulting from the history of the processes.
However, the organism and parts of organism are continuously
reacting to the outer world, creating their own, homeostatic
environment.
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Second epigenetic amendment: noise
thermal noise: all interaction energies are of order kT
mesoscopic noise in multi-scale systems: low number species, slow
variables
noisy input: separating a subsystem from the rest of the world
generically demands replacing the rest of the world by noise
Q1: What guarantees reliable functioning in epigenetic noisy
landscape?
Q2: Can we think at a mathematical framework for the study of
robustness?
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Systems approach
Complexity theory (cybernetics, synergetics, control theory,
catastrophe theory, etc.)
Law of requisite variety
Order from noise
W.Ross Ashby: the variety in the
control system must be equal to or
larger than the variety of the
perturbations in order to achieve
control.
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Heinz von Foerster: noise or random
perturbations will help a
self-organizing system to find more
stable states in its fitness landscape.
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Systems approach
Principle of asymmetric
transitions: is variety possible?
transitions go from unstable to
stable : state reduction,
Waddington’s chreods.
spontaneous decrease of variety is
possible (canalization)
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Catastrophes and structural
stability
Structural stability of attracting sets.
Flexibility: the set of attractors
change at bifurcations.
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Beyond René Thom: Prédire n’est pas expliquer
Nowadays, we feel more confident. The state of an organism (or
part of it) is a point in a high dimensional space of genes and gene
products concentrations.
This point satisfies some dynamics (differential equations,
finite-state cellular automata), eventually stochastic (Gillespie
dynamics). If all parameters are known, the dynamics is
computable.
Biological systems are open, multi-scale, heterogeneous systems.
Information can be very detailed on some parts, extremely scarce
on others. Parameters are unknown.
Q0: What is computable?
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Prolegomena for a theory of robustness
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Robustness by dimension reduction : from Gromov to von
Dassow
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Model reduction : a framework for studying robustness
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Multiscale robustness : a lesson from the fly
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Robustness by dimension compression
Chain of catalysed
transformations
Transcription models
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Gromov concentration
Objects in high-dimension look small in projection.
Levy theorem (cube concentration): F (k1, k2, ..., kn) with F
1-Lipschitzian, concentrates Var (F ) ∼ 1/N.
P
x
Examples of 1-Lipschitzian functions: N1 N
i=1 xi , f (x) = K +x .
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Simplex concentration
order statistics K(1) >> K(2) >> ...K(r ) >> ...K(n) , M = K(r )
log-uniform parameters with average spacing δ in log-scale
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Var (log K ) << δ 2 , no overlap Var (log M) ∼ Var (log K )
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Simplex concentration
order statistics K(1) >> K(2) >> ...K(r ) >> ...K(n) , M = K(r )
log-uniform parameters with average spacing δ in log-scale
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Var (log K ) << δ 2 , no overlap Var (log M) ∼ Var (log K )
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Var (log K ) ∼ δ , saturation Var (log M) = δ 2
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Simplex concentration
order statistics K(1) >> K(2) >> ...K(r ) >> ...K(n) , M = K(r )
log-uniform parameters with average spacing δ in log-scale
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Var (log K ) << δ 2 , no overlap Var (log M) ∼ Var (log K )
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Var (log K ) ∼ δ , saturation Var (log M) = δ 2
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Var (log K ) >> δ 2 , simplex concentration
Var (log M) ∼ Var (log K )/N 2
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Simplex concentration
order statistics K(1) >> K(2) >> ...K(r ) >> ...K(n) , M = K(r )
log-uniform parameters with average spacing δ in log-scale
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Var (log K ) << δ 2 , no overlap Var (log M) ∼ Var (log K )
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Var (log K ) ∼ δ , saturation Var (log M) = δ 2
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Var (log K ) >> δ 2 , simplex concentration
Var (log M) ∼ Var (log K )/N 2
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Min - max combinations
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Von Dassow’s robustness
von Dassow et al Nature 00
Hypothesis: robustness by dimension compression.
To prove this: a) dynamics is low dimensional (possible). b) there
are only a few critical parameters (true). c) construct the coarse
graining mapping (difficult).
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DARPP-32 pathway: dynamics is low dimensional
Barbano et al., PNAS 2007.
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Linear network of chemical reactions
Ai are reagents, ci is concentration of Ai .
All the reactions are of the type Ai → Aj .
kji > 0 is the reaction Ai → Aj rate constant.
The reaction rates: wji = kji ci .
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Linear network of chemical reactions
Ai are reagents, ci is concentration of Ai .
All the reactions are of the type Ai → Aj .
kji > 0 is the reaction Ai → Aj rate constant.
The reaction rates: wji = kji ci .
Kinetic equation
X
X
dci
= ki0 +
kij cj − (
kji )ci ,
dt
j≥1
j≥0
or in vector form: ċ = K0 + Kc.
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(1)
Hierarchical models
Systems biology models need constants and these are most of the
time unknown.
We have some ideas about the network structure: reaction graph,
influence graph, etc.
Usually, something is big, and something is small enough,
we can guess the constant ordering (I = (i, j)):
kI1 kI2 kI3 ...
We say that the system has separated constants.
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Limiting step
Linear chain of reactions A1 → A2 → ...An with reaction rate
constants ki (for Ai → Ai+1 )
Let kq be the smallest constant: kq ki (i 6= q)
In time scale ∼ 1/kq :
A1 , ...Aq−1 transform fast into Aq ,
Aq+1 , ...An−1 transform fast into An ,
only two components, Aq and An , are present,
kq A
the whole chain behaves as a single reaction Aq →
n
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Limitation theory for linear, hierarchical models: an
example
Gorban and Radulescu Adv.Chem.Eng.08, Radulescu et al BMC Systems Biol. 08
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NFκB pathway: testing concentration for nonlinear models
Gorban and Radulescu, IET Systems Biol. 07
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Model reduction and critical parameters
Radulescu et al BMC Systems Biol. 08
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Model reduction and robustness
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Model reduction provides mathematical framework for
robustness.
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Complex biological systems have simple, robust dynamics.
Robust dynamics is computable. No need for full parameter
identification.
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There are remaining critical parameters allowing control.
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Generic response to statistical perturbations (cube or simplex
concentration) : need high compression, not necessarily high
dimension.
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Robust system design: network topology is not all. Need also
order relations among parameters.
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Multiscale robustness
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A lesson from the fly
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Complex, but not bottom level: the gene circuit model
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Reproduces canalization properties
Manu et al. 08, in review Plos
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Redundancy effect
Manu et al. 08, in review Plos
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Interacting kink model
Vakulenko and Radulescu, manuscript
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The 2-scale robustness picture
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Multi-scale robustness
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Robustness can exist at many scales. Multi-scale model
reduction allows to focus on a particular scale.
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Biological dynamical systems can pass from one dynamical
simplification to another one. All these simplifications can be
robust.
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The two simplifications for Drosophila gap gene system
dynamics, the diffusionless approximation and the interacting
kinks approximation explain stability at various times.
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Alternating cushion ensures robust design of Drosophila gap
gene system.
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Acknowledgements
Alexander Gorban, University of Leicester
Andrei Zinovyev, Curie Institute
Alain Lilienbaum, Stress et Pathologies du Cytosquelette, Paris 7
Sergei Vakulenko, Institute of Print, St. Petersburg
Manu, John Reinitz, Stony Brook University
Maria Samsonova, St.Petersburg State Polytechnical University
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