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Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Ordinary differential equations with discontinuous right-handsides as complementarity systems.
Application to gene regulatory networks.
Vincent AcaryJoint work with Hidde de Jong and Bernard Brogliato
INRIA Rhone–Alpes, Grenoble.
CMM, Santiago de Chile, March 2nd 2014
– 1/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
One of the father of Nonsmooth Mechanics and Convex Analysis
Jean Jacques Moreau (1923 – 2014)
– 2/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Bio.
Team-Project BIPOP. INRIA. Centre de Grenoble Rhone–Alpes“One of the Jean Jacques Moreau’s fan club”. Convex Analysis and NonsmoothMechanics.
I Scientific leader : Bernard Brogliato
I Nonsmooth dynamical systems : Modeling, analysis, simulation and control.
I Nonsmooth Optimization : Analysis & algorithms. (Claude Lemarechal & JeromeMalick)
Personal research themes
I Nonsmooth Dynamical systems. Higher order Moreau’s sweeping process.Complementarity systems and Filippov systems
I Modeling and simulation of switched electrical circuits
I Discretization method for sliding mode control and optimal control.
I Formulation and numerical solvers for Coulomb’s friction and Signorini’s problem.Second order cone programming.
I Time–integration techniques for nonsmooth mechanical systems
– 3/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Outline
Outline
Outline
I Basic facts on piecewise linear modeling of gene regulatory networkI Notion of Filippov solutions
I Filippov extensionI Aizerman & Pyatnitskii extension
I Mixed Complementarity Systems formulation
I Numerical time–integration
I Illustrations
I Conclusions & perspectives.
Outline – 4/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Outline
Outline
Introduction on gene regulatory network modelingA first simple network of two genesDefinition of Piece-Wise Linear (PWL) models
Filippov’s solutionsConcept of Filippov’s solutionsFilippov’s extension of PWL modelsAizerman & Pyatnitskii’s extension of PWL modelsRelations between the extensions
Numerical Methods for the AP-extensionPrinciplesReformulation as MCS/DVIThe general time-discretization framework.Solution methods for MCP
IllustrationsThe first simple network of two genesSynthetic oscillator with positive feedbackRepressilator
Extensions to general affine piecewise smooth systemsGeneral affine piecewise smooth systems
Conclusions & Perspectives
Outline – 5/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Introduction on gene regulatory network modeling
A first simple network of two genes
A first simple network of two genes
Protein 1
gene 1
Protein 2
gene 2
PWL model corresponding to this network.x1 = −γ1 x1 + κ1 s+(x2, θ
12) s−(x1, θ
21)
x2 = −γ2 x2 + κ2 s+(x1, θ11) s−(x2, θ
22)
(1)
where
I x1, x2 are cellular protein or RNA concentrations
I κ1, κ2 and γ1, γ2 are positive synthesis and degradation constants, respectively,
I θ11 , θ
21 , θ
12 , θ
22 are constant strictly positive threshold concentrations of regulation
I s+ and s− are step functions
s+(xj , θkj )=
1 if xj > θk
j
0 if xj < θkj
and s−(xj , θkj )=
0 if xj > θk
j
1 if xj < θkj
, (2)
Introduction on gene regulatory network modeling – 6/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Introduction on gene regulatory network modeling
A first simple network of two genes
A first simple network of two genes
[−γ1x1−γ2x2 + κ2
]θ12
θ22
x2
θ11 θ21x1
[−γ1x1 + κ1−γ2x2 + κ2
][−γ1x1 + κ1−γ2x2
] [−γ1x1−γ2x2 + κ2
]
[−γ1x1 + κ1−γ2x2
] [−γ1x1−γ2x2
][−γ1x1 + κ1−γ2x2
]
[−γ1x1−γ2x2
] [−γ1x1−γ2x2 + κ2
]
Introduction on gene regulatory network modeling – 7/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Introduction on gene regulatory network modeling
Definition of Piece-Wise Linear (PWL) models
Definition of Piece-Wise Linear (PWL) models
Notation
I x = (x1, . . . , xn)T ∈ Ω a vector of cellular protein or RNA concentrations, whereΩ ⊂ Rn
+ is a bounded n-dimensional hyperrectangular subspace of Rn+
I For each concentration variable xi , i ∈ 1, . . . , n, we distinguish a set ofconstant, strictly positive threshold concentrations θ1
i , . . . , θpii , pi > 0.
I Θ =⋃
i∈1,...,n,k∈1,...,pix ∈ Ω | xi = θki the subspace of Ω defined by the
threshold hyperplanes.
Introduction on gene regulatory network modeling – 8/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Introduction on gene regulatory network modeling
Definition of Piece-Wise Linear (PWL) models
Definition of Piece-Wise Linear (PWL) models
Definition 1 (PWL model)A PWL model of a gene regulatory network is defined by a set of coupled differentialequations
xi = fi (x) = −γi xi + bi (x) = −γi xi +∑l∈Li
κli bl
i (x), i ∈ 1, . . . , n, (3)
where
I κli and γi are positive synthesis and degradation constants, respectively,
I Li ⊂ N are sets of indices of regulation terms,
I bli : Ω \Θ→ 0, 1 are so-called Boolean regulation functions.
Boolean functions and step functionsStep functions can be associated with Boolean variables X k
j such that
X kj (x) = (xj > θk
j ) = s+(xj , θkj )
X kj (x) = (xj < θk
j ) = s−(xj , θkj ),
(4)
where X denotes the complemented variables of X .
Introduction on gene regulatory network modeling – 9/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Introduction on gene regulatory network modeling
Definition of Piece-Wise Linear (PWL) models
Definition of Piece-Wise Linear (PWL) models
Generically, any Boolean function bli (x) can be rewritten in minterm disjunctive
normal form (DNF):
bli (x) =
2p−1∑α=0
c li,αmα(x), (5)
with c li,α ∈ 0, 1. For the set of variables X k
j , j ∈ 1, . . . , n, k ∈ 1, . . . , pj, we have
2p minterms, with p =∑
j∈1,...,n pj ,
mα(x) =n∏
j=1
pj∏k=1
X kj (x), α ∈ 0, . . . , 2p − 1. (6)
where X kj (x) is a literal defined either as the Boolean variable X k
j or its negation X kj .
Introduction on gene regulatory network modeling – 10/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Introduction on gene regulatory network modeling
Definition of Piece-Wise Linear (PWL) models
Assumption 1The regulation functions bl
i (·) are multiaffine functions, that is, they are affine with
respect to each s+(xj , θkj ), for j ∈ 1, . . . n and k ∈ 1, . . . , pj.
Assumption 1 can be shown to be generic for all regulation functions corresponding toBoolean functions written in minterm disjunctive normal form.
Assumption 2Every step function s+(xj , θ
kj ), with j ∈ 1, . . . n and k ∈ 1, . . . , pj, occurs in at
most one bi (·), i ∈ 1, . . . , n.Assumption 2 is a rather weak modeling assumption, in the sense that there is usuallyno compelling biological reason for two genes to be regulated at exactly the samethreshold.
Introduction on gene regulatory network modeling – 11/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Introduction on gene regulatory network modeling
Definition of Piece-Wise Linear (PWL) models
Outline
Introduction on gene regulatory network modelingA first simple network of two genesDefinition of Piece-Wise Linear (PWL) models
Filippov’s solutionsConcept of Filippov’s solutionsFilippov’s extension of PWL modelsAizerman & Pyatnitskii’s extension of PWL modelsRelations between the extensions
Numerical Methods for the AP-extensionPrinciplesReformulation as MCS/DVIThe general time-discretization framework.Solution methods for MCP
IllustrationsThe first simple network of two genesSynthetic oscillator with positive feedbackRepressilator
Extensions to general affine piecewise smooth systemsGeneral affine piecewise smooth systems
Conclusions & Perspectives
Introduction on gene regulatory network modeling – 12/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Concept of Filippov’s solutions
Concept of Filippov’s solutions
ODE with discontinuous R.H.S.
I Step functions s±(xj , θkj ) gives rise to mathematical complications, because the
step functions are undefined and discontinuous at xj = θkj .
I Reformulation as differential inclusions
x ∈ F (x)
Numerous options for defining the set–valued function F .
I Filippov’s definition of solutions as a absolutely-continuous function x(·) suchthat x(t) ∈ F (x(t)) holds almost everywhere on [t0,T ] with x(t0) = x0.
Filippov’s solutions – 13/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Filippov’s extension of PWL models
Filippov’s extension of PWL models
Discontinuous Dynamics in Ω \Θ
xi = fi (x) = −γi xi + bi (x), i ∈ 1, . . . , n. (3)
Definition 3 (F-extension of PWL models)The F-extension of the PWL model (3) is defined by the differential inclusion
x ∈ F (x), with F (x) = co
( lim
y→x, y 6∈Θf (y)
), x ∈ Ω, (7)
where co(P) denotes the closed convex hull of the set P, and limy→x, y 6∈Θ f (y) theset of all limit values of f (y), for y 6∈ Θ and y → x.
Filippov’s solutions – 14/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Filippov’s extension of PWL models
Filippov’s extension of PWL models
Properties
I Classical definition of Filippov’s extension
I Existence of solutions under mild assumptions
I Uniqueness is not ensured.
Issues: Hardly tractable formulation for Numerics
x ∈ F (x)
General time–discretization scheme. (Dontchev and Lempio, 1992)
xk+1 − xk
h∈ F (xk )
I How to practically build F (x) ?
I How to compute and to choose a selection σ ∈ F (x) ?
I How to avoid numerical chattering ?
Filippov’s solutions – 15/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Aizerman & Pyatnitskii’s extension of PWL models
Aizerman & Pyatnitskii’s extension of PWL models
Discontinuous Dynamics in Ω \Θ
xi = fi (x) = −γi xi + bi (x), i ∈ 1, . . . , n. (3)
Introduction of selection σ = (σ11 , . . . , σ
p1
1 , . . . , σ1n, . . . , σ
pnn )T ∈ [0, 1]p.
Let us define the function g : Rp → Rn by
gi (σ) =∑l∈Ln
κli bl
i (σ), j ∈ 1, . . . , n. (8)
where bli (·) are obtained from bl
i (·) by replacing every occurrence of s+(xj , θkj ) and
s−(xj , θkj ) by σk
j and 1− σkj , respectively.
Reformulation of the PWL model (3)
xi = fi (x) = −γi xi + gi (σ), i ∈ 1, . . . , n, (9)
Filippov’s solutions – 16/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Aizerman & Pyatnitskii’s extension of PWL models
Aizerman & Pyatnitskii’s extension of PWL models
Multi-valued step function
S+(xj , θkj ) =
1 xj > θk
j
[0,1] xj = θkj
0 xj < θkj
and S−(xj , θkj ) =
0 xj > θk
j
[0,1] xj = θkj
1 xj < θkj
. (10)
Interesting equivalence. ,
σkj ∈ S+(xj , θ
kj )⇐⇒ (xj − θk
j ) ∈ N[0,1](σkj ) (11)
Filippov’s solutions – 17/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Aizerman & Pyatnitskii’s extension of PWL models
Aizerman & Pyatnitskii’s extension of PWL models
Definition 4 (AP-extension of PWL models)The AP-extension of a PWL model (3) is defined by the following differential inclusion
x ∈
G1(x)...
Gn(x)
=
−γ1 x1 + g1(σ)
...−γn xn + gn(σ)
∣∣∣∣∣σkj ∈ S+(xj , θ
kj ), j ∈ 1, . . . , n, k ∈ 1, . . . , pj
.
(12)
Properties
I Definition related to the Utkin concept of equivalent control method
I Existence of solutions is not so generic. G is not convex !
I Promising extension for Numerics.
Ü Mixed complementarity systems or Differential Variational Inequalities
Filippov’s solutions – 18/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Relations between the extensions
Relations between the extensions
Proposition 5 ((Machina and Ponosov, 2011))Under Assumption 1 (multiaffine R.H.S.), F (x) = co(G(x)) for all x ∈ Ω.
Comments
I Every Filippov solution with the AP-extension is a Filippov solution with theF-extension
I Not sufficient to ensure the existence of a solution.
Proposition 6Under Assumptions 1 and 2, F (x) = G(x) for all x ∈ Ω.
Comments
I We retrieve the existence of solutions.
Filippov’s solutions – 19/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Filippov’s solutions
Relations between the extensions
Outline
Introduction on gene regulatory network modelingA first simple network of two genesDefinition of Piece-Wise Linear (PWL) models
Filippov’s solutionsConcept of Filippov’s solutionsFilippov’s extension of PWL modelsAizerman & Pyatnitskii’s extension of PWL modelsRelations between the extensions
Numerical Methods for the AP-extensionPrinciplesReformulation as MCS/DVIThe general time-discretization framework.Solution methods for MCP
IllustrationsThe first simple network of two genesSynthetic oscillator with positive feedbackRepressilator
Extensions to general affine piecewise smooth systemsGeneral affine piecewise smooth systems
Conclusions & Perspectives
Filippov’s solutions – 20/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Principles
Principles
1. A reformulation of the set–valued relation
σ ∈ S+(x , θ), (13)
into inclusion into normal cones, Complementarity Problems (CP) andfinite–dimensional Variational Inequalities (VI)
2. An implicit event-capturing time–stepping scheme, mainly based on the backwardEuler scheme which allows to deal with the switch–like behaviour and the slidingmotion
3. The use of efficient numerical solvers for the one-step problem which results fromthe time–discretization of the CP/VI formulation of the problem
Numerical Methods for the AP-extension – 21/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Reformulation as MCS/DVI
Reformulation as MCS/DVI
The relationσ ∈ S+(x , θ), (14)
can be equivalently reformulated in the form of an inclusion as
(x − θ) ∈ N[0,1](σ). (15)
In turn, the relation (15) are equivalent to the complementarity conditions0 6 1− σ ⊥ (x − θ)+ > 00 6 σ ⊥ (x − θ)− > 0,
(16)
where the symbol x ⊥ y means xT y = 0 and y+, y− respectively stand for thepositive and negative parts of y . Finally, an equivalent formulation of (15) is given bythe following VI : find σ ∈ [0, 1] such that
(θ − x)T (σ − σ′) > 0 for all σ′ ∈ [0, 1]. (17)
Numerical Methods for the AP-extension – 22/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Reformulation as MCS/DVI
Reformulation as MCS/DVI
Let us now define the affine function y : Rn → Rp such that
y(x) = Cx − θ =
x1 − θ11
...x1 − θp1
1...
xn − θn1
...xn − θpn
1
T
∈ Rp (18)
where C ∈ Rp×n with Cij ∈ 0, 1 and θ = [θ11 , . . . , θ
p11 , . . . , θ
1n, . . . , θ
pnn ]T .
Numerical Methods for the AP-extension – 23/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Reformulation as MCS/DVI
Reformulation as MCS/DVI
The AP-extension of the PWL system in Definition 4 can be written compactly asx = −diag(γ)x + g(σ)
y(x) = Cx − θ ∈ N[0,1]p (σ)(19)
where diag(γ) ∈ Rn×n is the diagonal matrix made of the components γi , i = 1 . . . n.
Ü We get
I Mixed Complementarity Systems (MCS)or
I Differential Variational Inequalities (DVI)
Numerical Methods for the AP-extension – 24/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
The general time-discretization framework.
An event-capturing scheme. The one–step problem
The proposed time discretization of (19) over a time–interval [tk , tk+1] of length h:xk+1 = xk − h diag(γ)xk+τ + h g(σk+1),yk+1 = Cxk+1 − θ,yk+1 ∈ N[0,1]p (σk+1).
(20)
with the initial condition x0 = x(t0). xk+τ = τxk+1 + (1− τ)xk for τ ∈ [0, 1]
Formulation into MCP/VILet us define the vector
zk+1 =
[xk+1
σk+1
]∈ Rn+m, (21)
and the function H : Rn+m → Rn+m as
H(zk+1) =
[xk+1 − xk + h diag(γ)xk+τ − h g(σk+1)
θ − Cxk+1
]. (22)
Then the problem (20) can be recast into the following inclusion
− H(zk+1) ∈ NRn×[0,1]p (zk+1). (23)
Numerical Methods for the AP-extension – 25/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
The general time-discretization framework.
Existence of solutions
Proposition 7Let H : Rn+m → Rn+m be the function defined in (22). Under Assumption 1, theproblem to find z ∈ Rn × [0, 1]p such that
− H(z) ∈ NRn×[0,1]p (z), (24)
has a nonempty and compact solution set.
Sketch of the proof
I Substitute xk+1 in the inclusion to get a reduced inclusion
− h(σ) ∈ N[0,1]p (σ) (25)
with
h(σ) = θ − Cx = θ − C diag(1/(1 + hτγ)) [(In − h(1− τ)diag(γ))xk + h g(σ)](26)
I h is continuous and [0, 1]p is compact convex
Apply Corollary 2.2.5 (Facchinei and Pang, 2003, page 148)
Numerical Methods for the AP-extension – 26/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Solution methods for MCP
Solution methodsDefinition 8 (Mixed complementarity Problem(MCP) (Dirkse and Ferris,1995))Given a function H : Rn+m → Rn+m and lower and upper bounds l , u ∈ Rn+m
, theMixed complementarity Problem (MCP) is to find z ∈ Rn+m and w , v ∈ Rn+m
+ suchthat
(MCP)
H(z) = w − v
l 6 z 6 u(u − z)T v = 0(z − l)T w = 0
. (27)
Numerical algorithms
I MILES (Rutherford, 1993) classical Newton–Josephy method,
I PATH (Ralph, 1994 ; Dirkse and Ferris, 1995)
I NE/SQP (Gabriel and Pang, 1992 ; Pang and Gabriel, 1993) generalizedNewton’s method based on the minimum function
I QPCOMP (Billups and Ferris, 1995) NE/SQP
I SMOOTH (Chen and Mangasarian, 1996) smooth approximations of the NCP,
I SEMISMOOTH (DeLuca et al., 1996) semismooth Newton withFischer–Burmeister function,
I ...Numerical Methods for the AP-extension – 27/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Solution methods for MCP
Enumerative solution methods
I With the classical Newton–Josephy method linearization, we get a MLCPyα+1 = Wα+1σα+1 + qα+1
yα+1 ∈ N[0,1]p (σα+1), (28)
where
Wα+1 = hC M−1 B(σα),
qα+1 = CM−1 [(In − h(1− τ)diag(γ))xk + hg(σα) + hB(σα)σα]− θ.M = In + hτdiag(γ)B(σ) = ∇σg(σ)
I Efficient enumerative solvers (see for instance (Al-Khayyal, 1987 ; Sherali et al.,1998 ; Judice et al., 2002) )Enumerating several solutions corresponding to various modesQualitative insight on the nature of solutions
Numerical Methods for the AP-extension – 28/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Solution methods for MCP
Stationary Points
Finding stationary points of the AP-extension of PWL systems is equivalent to solvethe following MCP
0 = −diag(γ)x + g(σ)
y(x) = Cx − θ ∈ N[0,1]p (σ)(29)
or more compactly
Cdiag(1/(1 + γ))g(σ)− θ ∈ N[0,1]p (σ) (30)
With the same reasoning as in the proof of Proposition 7, the VI/CP (30) has anonempty compact set of solutions.We have just to checked that some solutions belong to Ω
Numerical Methods for the AP-extension – 29/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Numerical Methods for the AP-extension
Solution methods for MCP
Outline
Introduction on gene regulatory network modelingA first simple network of two genesDefinition of Piece-Wise Linear (PWL) models
Filippov’s solutionsConcept of Filippov’s solutionsFilippov’s extension of PWL modelsAizerman & Pyatnitskii’s extension of PWL modelsRelations between the extensions
Numerical Methods for the AP-extensionPrinciplesReformulation as MCS/DVIThe general time-discretization framework.Solution methods for MCP
IllustrationsThe first simple network of two genesSynthetic oscillator with positive feedbackRepressilator
Extensions to general affine piecewise smooth systemsGeneral affine piecewise smooth systems
Conclusions & Perspectives
Numerical Methods for the AP-extension – 30/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
The first simple network of two genes
The first simple network of two genes
Protein 1
gene 1
Protein 2
gene 2
PWL model corresponding to this network.x1 = −γ1 x1 + κ1 s+(x2, θ
12) s−(x1, θ
21)
x2 = −γ2 x2 + κ2 s+(x1, θ11) s−(x2, θ
22)
(31)
The parameters are: θ11 = θ1
2 = 4, θ21 = θ2
2 = 8, κ1 = κ2 = 40, γ1 = 4.5 and γ2 = 1.5.
Illustrations – 31/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
The first simple network of two genes
Different trajectories of system (1) depicting the nature of the threeequilibria.
0
2
4
6
8
10
12
0 2 4 6 8 10 12
x2
x1
θ12
θ22
θ11 θ21
Illustrations – 32/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
The first simple network of two genes
Reduced study around the stationary point x1 = θ21 and x2 = θ2
2
Restriction of the domain of interest to Ω = Ω ∩ (θ11 ,+∞)× (θ2
1 ,+∞). The originalsystem (19) can be then reduced to
x = −diag(γ)x + Bσ + κ
Cx − θ ∈ N[0,1]2 (σ)(32)
with
σ =
[σ2
σ4
], C =
[1 00 1
], θ =
[θ2
1θ2
2
], B =
[−κ1 0
0 −κ2
], κ =
[κ1
κ2
].
(33)
Illustrations – 33/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
The first simple network of two genes
Reduced study around the stationary point x1 = θ21 and x2 = θ2
2
Properties
1. The one–step VI
θ − Cdiag(1/(1 + hτγ))[(In − h(1− τ)diag(γ))xk + h Bσ + hκ)
]∈ N[0,1]2 (σ)
(34)is strongly monotone and has an unique solution (see (Facchinei and Pang, 2003,Theorem 2.3.3))
2. No numerical chattering in comparison with explicit schemes (Dontchev andLempio, 1992)Application of Lemma 3 in (Acary and Brogliato, 2010).
Illustrations – 34/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
The first simple network of two genes
Reduced study around the stationary point x1 = θ21 and x2 = θ2
2
5
6
7
8
9
10
5 6 7 8 9 10
x2
x1
θ22
θ21
(a) Explicit scheme
6.4
6.6
6.8
7
7.2
7.4
7.6
7.8
8
8.2
7.9 8 8.1 8.2 8.3 8.4 8.5x2
x1
θ22
θ21
(b) Explicit scheme (zoom 1)
7.95
7.96
7.97
7.98
7.99
8
8.01
7.985 7.99 7.995 8 8.005 8.01 8.015 8.02 8.025
x2
x1
θ22
θ21
(c) Explicit scheme (zoom 2)
5
6
7
8
9
10
5 6 7 8 9 10
x2
x1
θ22
θ21
(d) Implicit scheme
6.4
6.6
6.8
7
7.2
7.4
7.6
7.8
8
8.2
7.9 8 8.1 8.2 8.3 8.4 8.5
x2
x1
θ22
θ21
(e) Implicit scheme (zoom 1)
7.95
7.96
7.97
7.98
7.99
8
8.01
7.985 7.99 7.995 8 8.005 8.01 8.015 8.02 8.025
x2
x1
θ22
θ21
(f) Implicit scheme (zoom 2)
Figure : Illustration of the numerical “chattering” effect on the system (1) starting from
x(t0) = [10, 5]T . Different levels of details. (a)-(b)-(c): Explicit Euler scheme with h = 10−3.
(d)-(e)-(f): The proposed implicit Euler scheme with h = 10−2.
Illustrations – 35/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
The first simple network of two genes
Reduced study around the stationary point x1 = θ21 and x2 = θ2
2
Lemma 9The equilibrium point θ = (θ2
1 , θ22) is finite-time stable in Ω.
Sketch of the proof
I Recast into monotone DI, z ∈ T (z)
I Use Lyapunov function V (z) = 12
zT z
I Since 0 ∈ T (0) and Br (0) ∈ T (0), r > 0, then we have
V (z)− r√
V (z) 6 0
Illustrations – 36/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
Synthetic oscillator with positive feedback
Synthetic oscillator with positive feedback
LacI
lacI
NRI
glnG
NRIp
x1 = −γ1 x1 + κ1 s+(x2, θ
22)
x2 = −γ2 x2 + κ2 s−(x1, θ1) s+(x2, θ12)
(35)
The variables x1 and x2 represent the concentrations of the proteins LacI and GlnG,respectively.The following parameter values have been used in the simulations: γ1 = γ2 = 0.032,κ1 = 0.08, κ2 = 0.16, and θ1 = 1, θ1
2 = 1, and θ22 = 4.
Illustrations – 37/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
Synthetic oscillator with positive feedback
Synthetic oscillator with positive feedback
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1 1.2
x2
(Gln
G)
x1 (LacI)
θ12
θ22
θ1
Illustrations – 38/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
Repressilator
Repressilator consisting of three genes (Elowitz and Leibler, 2000)
LacI
lacI
TetR
tetR cI
CI
(a) x1 = −γ1 x1 + κ11 + κ2
1 s−(x3, θ3)x2 = −γ2 x2 + κ1
2 + κ22 s−(x1, θ1)
x3 = −γ3 x3 + κ13 + κ2
3 s−(x2, θ2)(36)
The variables x1, x2, and x3 represent the concentrations of the proteins LacI, TetR,and CI, respectively.The following parameter values have been used in the simulations:γ1 = γ2 = γ3 = 0.2, κ1
1 = κ12 = κ1
3 = 4.8 10−4, κ21 = κ2
2 = κ23 = 4.8 10−1, and
θ1 = θ2 = θ3 = 1.
Illustrations – 39/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
Repressilator
Repressilator consisting of three genes (Elowitz and Leibler, 2000)
00.5
11.5
22.5
x1
(Lac
I)
θ1
00.5
11.5
22.5
x2
(tet
R)
θ2
00.5
11.5
22.5
0 20 40 60 80 100
x3
(cI)
t (time)
θ3
Illustrations – 40/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Illustrations
Repressilator
Repressilator consisting of three genes (Elowitz and Leibler, 2000)
00.2
0.40.6
0.81
1.21.4
1.61.8
2
00.2
0.40.6
0.81
1.21.4
1.61.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x3 (cI)
x1 (LacI)x2 (TetR)
x3 (cI)
Illustrations – 40/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Extensions to general affine piecewise smooth systems
General affine piecewise smooth systems
General affine piecewise smooth systems
χk
χi
x = fi(x , t)
χj
x = fj(x , t)
x = fk(x , t)
Figure : Partitioning
Extensions to general affine piecewise smooth systems – 41/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Extensions to general affine piecewise smooth systems
General affine piecewise smooth systems
General affine piecewise smooth systems
Extensions to general affine piecewise smooth systems – 42/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Conclusions & Perspectives
Conclusions & Perspectives
Conclusions
1. Development of numerical simulation methods for AP-extensions of PWL models,integrated in existing simulation platform.
2. Equivalence of different solution concepts for PWL models (F- andAP-extensions) under reasonable biological assumptions, which means thatsimulation methods are broadly applicable.
3. MCS/DVI formulation permits theoretical investigations (stability, finite–timeconvergence, ...)
4. Demonstrate practical usefulness of approach by numerical simulation of dynamicsof three synthetic networks (good qualitative correspondence with data).
Perspectives
1. Simulation of real networks with 100 up to 1000 genes.
2. Extension to general polyhedral switching affine system (PSAS) and to piecewisesmooth systems
3. Convergence without monotony, convex RHS, or one–sided Lipschitz condition.
4. Higher order event-driven schemes
5. General stability theory when only attractive surfaces are involved in a subset ofinterest.
Conclusions & Perspectives – 43/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Conclusions & Perspectives
Thank you for your attention.
Conclusions & Perspectives – 44/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
Conclusions & Perspectives
Outline
Introduction on gene regulatory network modelingA first simple network of two genesDefinition of Piece-Wise Linear (PWL) models
Filippov’s solutionsConcept of Filippov’s solutionsFilippov’s extension of PWL modelsAizerman & Pyatnitskii’s extension of PWL modelsRelations between the extensions
Numerical Methods for the AP-extensionPrinciplesReformulation as MCS/DVIThe general time-discretization framework.Solution methods for MCP
IllustrationsThe first simple network of two genesSynthetic oscillator with positive feedbackRepressilator
Extensions to general affine piecewise smooth systemsGeneral affine piecewise smooth systems
Conclusions & Perspectives
Conclusions & Perspectives – 45/45
Ordinary differential equations with discontinuous right-hand sides as complementarity systems. Application to gene regulatory networks.
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Conclusions & Perspectives
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Conclusions & Perspectives – 45/45