Ordinary di erential equations with discontinuous right-hand sides … · 2018-03-07 · Ordinary...

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


One of the father of Nonsmooth Mechanics and Convex Analysis

Jean Jacques Moreau (1923 – 2014)

– 2/45


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


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


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


Introduction on gene regulatory network modeling

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





[−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

]




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.






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 .






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 .





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.





Outline









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



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.






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 ?




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)






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)






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




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.





Outline









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



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)






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 .






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)




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)




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)




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




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





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





Outline









Illustrations

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


Illustrations


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


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



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



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



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

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



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

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

Repressilator


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

Repressilator


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)



Extensions to 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


Extensions to general affine piecewise smooth systems



Extensions to general affine piecewise smooth systems – 42/45




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



Thank you for your attention.




Outline









References

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T. DeLuca, F. Facchinei, and C. Kanzow. A semismooth equation approach to thesolution of nonlinear complementarity problems. Mathematical Programming, 75:407–439, 1996.

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M.B. Elowitz and S. Leibler. A synthetic oscillatory network of transcriptionalregulators. Nature, 403(6767):335–338, 2000.

References – 45/45

http://www.sciencedirect.com/science/article/pii/S0167691110000332

citeseer.ist.psu.edu/billups97qpcomp.html



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