Mathematical models of complexity

Ovidiu Radulescu, IRMAR and Symbiose project IRISA

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SummaryCV

Brief state of the art: complex systems, systems biology

Contributions in biology:

Markov processes in molecular biology

Qualitative equations for functional genomics

PDE models for pattern formation

Conclusion

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CVEducation: 1989 Diplôme d’Ingénieur Physique des Solides, Bucarest1994 Doctorat Physique des Solides, Orsay (félicitations)1996 DEA probabilités, Marne-la-Vallée

Recherche: interdisciplinarité, transversalité2 post-docs (Pays Bas et Angleterre) 27 articles acceptés, 12 proceedings conf.

Enseignement:1991-1993, moniteur physique Orsay, vacataire Ecole Centrale de Paris1993-1996 ATER et PRAG physique, Marne la Valléedepuis 1999 MC en mathématiques à Rennes 1encadrement d’une thése (en mathématiques) et d’une dizaine de stages

Responsabilités:membre commission informatique, animation d’un groupe de travailcoordinateur d’une ACI

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Complex systems

Quasicrystals(Orsay)

Cellular physiology(Rennes) Development

(Rennes)

Incommensurate composites(Nimégue)

Wormlike micelles(Leeds)

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What complex systems have in common

• Order as framework for transformation: crystals, dissipative structures, patterns

• Defects as motors for transformation: points, lines, interfaces

• Hierarchical organisation

• Nonlinearity

• Stability, robustness

• Universality

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Systems biology

• Mathematical modeling of physiology

• Transversal field, imports methods from physics, control theory, automata, chemical kinetics

• After rapid evolution, critical stage: obstacle raised by the complexity of higher organisms (models are scarce or weakly predictive)

• There is a need for new methodsanalysis methods for massive datamodel reductionmore realistic models using physico-chemistry

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Gene regulation is the result of many interactions

8Network models unify various processes

Gap genes, first 3 hours of DrosophilaLactose operon, E.Coli

Nutritional stress, E.Coli

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Chromatin ImmunoPrecipitation on Chip

DNA Chip

Various kind of data: differences of concentrations, direct test of qualitative interaction

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StrategyAims:• Model construction• Model analysis• Biological predictions

Difficulties:• Data collection is massive but unguided• Reverse engineering is difficult• Models are non-linear and in very high dimension• Interpretation of computer simulations is difficult

My solutions:• Guide data collection (experiment design)• Do not start reverse engineering from scratch (model correction)• Develop new mathematical techniques for model analysis• Look for network design principles

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Jump Markov processes

Ordinary differential equations

Qualitative equations

Partial differential equations

Thermodynamic limitPiecewise deterministic

Discretisation

Partial thermodynamic limit

Averaging

My mathematical garden

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My contributions

My contributions to this field:1) Modeling stochasticity of molecular biology processes by

piecewise deterministic Markov processes2) Qualitative equations for analysis of massive data3) Carr-Pego type model reduction for pattern formation4) Measure concentration as framework for robustness

CollaborationsComputer scientists: A.Siegel, M.LeBorgne (IRISA Symbiose),

M.Samsonova(St.Petersburg)Biologists: N.Theret (INSERM), S.Lagarrigue (INRA), A.Lilienbaum

(CNRS), J.Reinitz (Stony Brook)Mathematicians: S.Vakulenko(St.Petersburg), A.Gorban(Leicester),

E.Pécou(Nice)

Research project MathResoGen (2003-2006)

Modeling stochasticity in molecular biology by Markov processes

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Markov jump processes: Renyi, Bartholomay, 50’

Modeling stochastic effects

are n chemical species

rninZiii ,1, =∈−= αβθ

nininini AAAA ββαα ++++ ←→ ...... 1111

nAA ,...,1

biochemical reaction

nZX ∈ is the state

∑=

−−+ +=nr

iXiXi

iiXqXqX

1(.)])((.))([,.)( θθ δδμ

∑=

−+=nr

iii XVXVX

1)]()([)(λ

∑=

−+=nr

jjjii XVXVXVXq

1)]()([)/()(

intensity

distribution of jumps

jump probability

jump vector

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Suppose that the mass action law is satisfied

∑=

==nr

iii xxFsxFds

sdx1

)(v)()),(()( θ

Thermodynamic (deterministic) limit

Rescale the process ΩiXix /=

∏=

=Ω=n

s

issiiii XXXV

1xk)(v),(v)(

α

∏=

−−−− =Ω=n

s

issiiii XXXV

1xk)(v),(v)(

β

For the Markov jump processes converges in probability to the solution of a system of ordinary differential equations (Kurtz, 70)

∞→Ω ix

Ω : reaction volume

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Some species are in small numbers!

Piecewise deterministic limit

0 , ),,()(~

≠∀ΩΩ= iri

XXVXV

rf

ii θε

,∞→Ω

O.Radulescu, A.Muller, A.Crudu (TSI in press)

,0→ε

use frequent/rare species decomposition

),( rXfXX =

1→Ωε

0 , ),,(v)(~

=∀ΩΩΩ= iri

XXXV

rf

ii θε

mass action law is not applicable and should be replaced by

concentration of rare species

reactions acting on rare species

reactions not acting on rare species

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Piecewise deterministic limit result

For the Markov jump process converges to a piecewise deterministic process:

1,0, →→∞→ εε ΩΩ ),/( rf XXX Ω=

O.Radulescu, A.Muller, A.Crudu (TSI in press)

i0i

i

~

v))(),(()(

θθ∑

=

==r

ff

srXsxfFdssdx

)(sx fis discrete and jumps with intensity

Between two jumps is continuous and satisfies:

)(srX ),(~

rfi XxV

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Application: hybrid stochastic simulation algorithm

O.Radulescu, A.Muller, A.Crudu (TSI in press)

0,, 00 === tXXxx rrff

)](exp[~ , rXx fλτ)()( τ+→ txtx ff

rXτ+→tt

1. Initialize2. Generate exponential random time3. Use deterministic solver to propagate4. Change to a new discrete value5. Increment time6. If t<tmax goto 2

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Application to happloinsufficiency

Biological problem:

Syndrom due to deficient genotype : insufficient copy numberPhenotype: heterogenous cell populations

Aim:

Find the simplest model that reproduces this situation

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Model for haploinsufficiency

dP/dt= - k3PdP/dt=k2-k3P

k-1 k1

Markov jump model (Cook 99)

*1

1GG

k

k−←→

PGGk

+→ ** 2

3kP→

Result: If the model allows a piecewise deterministic approximation

)(2 Ω=Ok

⎪⎩

⎪⎨⎧

=−

=+−=

0if,1if,

*3

*23

GPkGkPk

dtdP

O.Radulescu, A.Muller, A.Crudu (TSI in press)

Study intermittency of trajectories and the invariant distribution

1*=+GG

Ω= /1ε

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Conclusion

Results:• The protein production is intermittent• The heterogeneity of the phenotype can be described by a Beta

distribution

The same method will be applied to larger, more complex models; in project NFκB signaling

Qualitative equations

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Qualitative equationsBiological problem:

Following a perturbation (stress, signal) the state of the cell changes. Variations of hundreds or thousands of variables can be monitored. How to use this information?

Steps:

• develop an “elasticity” theory of graphs (O.Radulescu et al. J.R.Soc.Interface 2006)

• translate this theory into qualitative equations (with A.Siegel et al. Biosystems 2006)

• polynomial algorithms for solving systems of qualitative equations (with Ph.Veber, M.leBorgne et al. Complexus 2006)

• application to huge networks (with C.Vargas et al., proc. RIAMS 2006)

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Elasticity of graphs),( PXFdt

dX = 0),( =PXFSteady state is perturbed

∑∑→ →

→

∂∈=

ijj

ij

ij

Gji X

C

aX δδ

G∂ Perturbations propagate along pathsin the directed graph

0 iff arcan is ),( ≠∂

∂=

i

j

X

Fjiaij

XP δδ →

graph boundary

(O.Radulescu et al. J.R.Soc.Interface 2006)

Steady state equationdynamics

Dirichlet solution:

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Qualitative equations

Li=Le+LacY-LacZ

∑∈∂

∂ −⎟⎠

⎞⎜⎝

⎛−=

)(

1ipredj

jjiiX

iF

i XaX δδ

subgraph

∑∈

=)(

)) ()((ipredj

jjii XsignasignXsign δδ

?,,

Dirichlet solution for subgraph

Qualitative equation

+−∈sign

(with A.Siegel et al.Biosystems 2006, with Ph.Veber, M.LeBorgne Complexus 2006)

Sign algebra

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Algorithm for solving qualitative equations•Map signs to elements of the finite field Z/3Z

•Map qualitative equations to polynomial equations over Z/3Z

•NP complete problem

•Ternary Decision Trees contracted to directed acyclic graphs andsystematic use of cache memory for non-redundant computation

•Obtain exhaustive lists of solutions within minutes for 1000 nodes

(with Ph.Veber, M.LeBorgne Complexus 2006)

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Predictions of a model

hard components: variables whose values are the same (+ or -) in any solution the hard components are the predictions of the model

Le, cAMP, A are observed

Li,G,LacZ,LacY, LacI are hard components

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Experiment design

Any value of the triplet(Le,G,A) can be extendedto a solution

These variables have no validation power

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Use validation power for experiment design

Only 2 values (out of 8) of (LacI,A,LacZ), namely(+,−, −) (−, +,+) can be extendedto a solution

Define validation power as:

Choose high validation power sets for optimal design

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Large scale application: nutritional stress of E.Coli1258 nodes, 2526 interactions, 10600 states, 1016 solutions

We have obtained both:• a set of predictions: from 40 observations in the stationary phase,

401 hard components, 26% of the network• a set of corrections to the model: necessarily include σ factors

Partial differential equations

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Pattern formationProblem:• Patterns form in very different complex systems (Drosophila

embryo before gastrulation, shear banding of complex fluids). • The examples are of Wolpert type, less studied in mathematics.

Can we find an unified approach?

Cornerstones:• of complex fluids: understand the relation between structure and

flow properties• of developmental biology: understand canalization, stability of

development

Collaborations:P.D.Olmsted (Physics,Leeds), JP.Decruppe(Physics,Metz), JF.Berret, G.Porte

(Physics,Montpellier) on wormlike micellesS.Vakulenko (Maths,St.Petersburg), J.Reinitz(Appl.Maths and Biology, Stony Brook) on

Drosophila

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Problem 1:

Syncitial blastoderm, before gastrulation

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Synt

hesi

sTr

ansp

ort

Dec

ay

∑=

++=∂∂ N

baababaa

a hxmTtxuTgRttxu

1))(),((),(

),(2 txuD aa∇+

),( txuaaλ−

Model Reaction-diffusion equations

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Synt

hesi

s

Genetic Interconnectivity Matrix (T):

T parameters:

positive: negative: zero:

activationrepressionno interaction

1

2

…

N

1 2 … NabGene

T11 T12 …

…

…

T22T21

TN1 TN2

… …

T1N

T2N

TNN

…

∑=

++=N

baababaa

a hxmTtxuTgRdttxdu

1))(),((),(

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Synt

hesi

s

Action of maternal gradient (bicoid)

Bicoid profile m(x) develops in 1h after fertilizationand remains constant during the blastoderm

∑=

++=N

baababaa

a hxmTtxuTgRdttxdu

1))(),((),(

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Synt

hesi

sTr

ansp

ort

Dec

ay

+Da (n) vi−1a − vi

a( )− vi+1a − vi

a( )[ ]

−λavia

The regulation-expression function g(u):1.0

0.0-6.0 0.0 6.0

Rel

ativ

e A

ctiv

atio

n: g

(u)

Total Regulatory Input: u

∑=

++=N

baababaa

a hxmTtxuTgRdttxdu

1))(),((),(

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Problem 2: Shear banding of wormlike micellesHadamard instability

O.Radulescu et al. Rheol.Acta 1999, with PD.Olmsted J.Rheol. 1999

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Model: Fluid-structure coupling

σρ .v)v.( ∇=∇+∂t

Navier-Stokes

ττμ //2)()()v.( 2 Σ−Δ+Σ∇=ΣΔ+ΔΣ−ΣΩ−ΩΣ−Σ∇+∂ Dat

Johnson-Segalman constitutive model + stress diffusion

Re=0 approximation

.,0..constS =+==∇ γεσσ )1(

.

2

2

WSyS

DtS −+−

∂

∂=∂

∂ γτ

SWyW

DtW .

2

2

γτ +−∂

∂=∂

∂

principal flow equations

Stress dynamics is described by a reaction-diffusion system

O.Radulescu, PDOlmsted J.Non-Newtonian.Fl.Mech. 2000

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Common framework: R-D PDE with small diffusion

),,(22 txufuDut εε +∇=

Ω⊂Ω∈ ,qRxnRtxuu ∈= ),(

dndddiagD ,...,2,1=

)()0,( 0 xuxu =

Ω∂∈=∇ xxnxu ,0)().(

Cauchy problem for the PDE system

is compact with smooth frontier

initial data

no flux boundary conditions

idea : consider the following shorted equation

),,v(v txft ε=

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Result 1: Classification of patterning mechanisms

0when0,t,in uniformly ,0t)-v(x,t)(x,u →>Ω∈→ εε x

t)(x,uε solution of the full system

v(x,t) solution of the shorted equation

Patterning is diffusion neutral if for vanishing diffusion, the solution of the full system converges uniformly to the solution of the shorted equation

If not, patterning is diffusion dependent

O.Radulescu and S.Vakulenko, arXiv qBio 2006

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Result 2: Classification of interfaces

For a given x, the shorted equation has only one attractor

Type 1 interface

O.Radulescu and S.Vakulenko, arXiv qBio 2006

Type 2 interface

For a given x, the shorted equation has severalattractors, here 2:

(x)φ

(x)(x), 21 φφ

Patterning with type 1 interfaces is diffusion neutral

Patterning with type 2 interfaces is diffusion dependent

The width of type 2 interfaces can be arbitrarily small

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Theorem on the diffusion neutral patterning

O.Radulescu and S.Vakulenko, arXiv qBio 2006

Consider the time autonomous situation and the shorted equation

The patterning is diffusion neutral under the following conditions on the shortedequation:

i) uniform dissipativity

ii) strong linear stability

calculated at the attractor

iii) attraction basin condition

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Theorem on the movement of type II interfaces in the bistable case

O.Radulescu and S.Vakulenko, arXiv qBio 2006

Invariant manifold decomposition for

Travelling wave solution for the space homogeneous eq.

The solution of space inhomogeneous equation is of the moving interface type

Equation for the position q(t) of the interface

This extends results of Carr-Pego(90) and Fife (89)

The velocity of a Type II interface is proportionalto the square root of the diffusion coefficient

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Application1: stress diffusion coefficient from interface kinetics

Diffusion is smallD~0.003-0.011 μm2s-1

w~30-40 nm

O.Radulescu et al. Europhys.Lett. 2003

10s-130s-1 .γ

60s-1

σ.γ

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Application2: Diffusion dependent patterning of Drosophila

1) parameter fit of Reinitz model from time dependent data by simulated annealing

2) compute attractors of shorted equation

Result: patterning is diffusion dependentImprovement of model fit: Rapid method of parameter identification using

interface kinetics

with Gursky, Manu, Vakulenko, unpublished

presented at Nanobio’06, St.Petersburg

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Comment on the impact in biology

Compared to Turing models, the gene circuit model is realistic:• the pattern is not a periodic modulation of a homogeneous state• the pattern results from the interaction of development genes, is

guided by maternal gradients and has aperiodic transients

Treating the set of segmentation genes as a dynamical system allows to understand:

• The logic of interactions (open problem) and transformations • The stability of the result (open problem)• The possible errors in mutants (open problem)

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Conclusion and future projectsStart of a long term project: produce powerful mathematical tools for analysis of complex systems.

Strategy: Model simplification *The invariant manifold technique of Carr-Pego*piecewise deterministic approach for Markov processes *graph theory methods for chemical kinetics models

An intrinsic relation exists between model reduction, stochasticityand robustness: concentration phenomena!

Physical chemistry for diffusion and transport in physiology.

CollaborationUpi Bhalla NCBS Bangalore, planned co-tutored phD.A.Gorban (Leicester) Egide/Alliance sponsorship J.Reinitz (Stony Brook) and Samsonova (St.Petersburg)ASC project with INRA on modeling lipid metabolismproject ANR SITCON with CurieSymbiose team IRISA

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Project 3: Robustness of biological systems

Cube concentration

Distributed robustness

r-robustness

Simplex concentration

Stony Brook (Reinitz), Leicester (Gorban), St.Petersburg(Samsonova, Vakulenko)

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Project 4: LIPID METABOLISM

Hierachical modeling:1)Extended model2) Abstract model

Multiorgans, multispecies

Heterogeneous dataMicroarraysBiochemical dosages

Symbiose, INRA Rennes and Toulouse

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Project 5: SITCONModeling signal transduction induced by a

chimeric oncogene

CELL PROLIFERATIO

N

Νfκb pathway

TGFβ pathway

IGF pathway

…

EWS/FLI1

Ewing network

APOPTOSIS

EWS/FLI1Institut Curie, Symbiose

Farey sets and spectra of incommensurate structures

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Number theory and Incommensurate compounds

Related problems:Hofstadter Butterfly

Bellissard’s gap labellingArnold tongues