Realization theory for systems biology

Transcription

1 Realization theory for systems biology Mihály Petreczky CNRS Ecole Central Lille, France February 3, 2016

2 Outline Control theory and its role in biology. Realization problem: an abstract formulation. Realization theory of polynomial/rational/nash systems. Realization theory of interconnection structure.

3 What is control theory about? Plant u y Controller Plant dynamical system (behavior changes with time) Controller dynamical system

4 What is control theory about? Plant u y Controller Task: find controller u = C(y) such that the y has the desired properties: y 0, 0 y 2 (s)ds minimal, etc.

5 Example: thermostat Plant heating on ẋ = x + 18 ẋ = x + 22 heating off Outputs: temperature x Control input: heating on and heating off. Control objective: maintain the temperature between C Controller heating on if x < 19.5 heating off if x > 20.5

6 How do we solve control problems? Find a mathematical model (state-space representation) ẋ(t) = f (x(t),u(t)) y(t) = h(x(t),u(t)) of the input-output behavior of the plant u y. Compute a controller ξ(t) = M(ξ(t),y(t)) u(t) = C(ξ(t),y(t)) such that has the desired properties. y(.)

7 Mathematical tools for control Tools for computing controllers: Stability theory of dynamical systems, Lyapunov s theory: the plant + controller ξ(t) = M(ξ(t),y(t)) u(t) = C(ξ(t),y(t)) ẋ(t) = f (x(t),u(t)) y(t) = h(x(t)) should at least be (asymptotically) stable around a trajectory. Optimal control (calculus of variations): the control law u( ) should optimize a cost functional J (x,u) Controllers have to be computed: numerical methods, optimization.

8 Mathematical tools for control: cont Making controllers requires models (differential/difference equations) How to estimate parameters of differential/difference equations from measured data (system identification: statistics, stochastic processes, optimization). How to simplify models without loosing too much of their observed behavior (model reduction). What is the relationship among various models which are observationally equivalent (realization theory)?

9 Control theory and systems biology Feedback control theory cybernetics. Living organisms are control systems: plenty of feedback loops. Control theory tell us how to design feedback. Biologists want to understand why a particular feedback is there. Systems biology is about reverse engineering of feedback interconnection

10 Realization theory: reverse engineering of plant models Realization theory: problem statement We observe the input-output behavior (black-box) u(t) y(t) of a physical process u(t) y(t)

11 Realization theory: reverse engineering of plant models Realization theory: problem statement We observe the input-output behavior (black-box) u(t) y(t) Which mathematical models (fixed structure) u(t) T ω(t) + ω(t) = Ku(t), y(t) = ω(t) y(t) can describe the observed behavior of the black-box?

12 Realization theory for biological systems We can fix either the algebraic structure of the models. u(t) Ṡ = α 1 (ES) g 12 β 1 S h 11 E h 14 ES = α 2 S g 21 E g 24 β 2 (ES) h 22 Ṗ = α 3 (ES) g 32 E g 34 E = const y(t)

13 or the interconnection structure u ẋ 1 = 6x u u(t) u u x 1 ẋ 2 = x 1 11x 3 12u x 2 y(t) ẋ 3 = x 2 + 6x 3 + 3u x 3 y

14 Polynomial/rational/Nash systems: biochemical reactions k 1 E + S ES E + P k 1 k 2 mass-action kinetics: polynomial equations Ṡ = k 1 E S + k 1 ES ES = Ė = k 1E S (k 1 + k 2 )ES Ṗ = k 2 ES

15 Polynomial/rational/Nash systems: biochemical reactions Michaelis-Menten kinetics: rational equations ( Ṡ = k 1 E t E t S E S + k t S 1 Ṗ = v max S S+K m S+K m ) S+K m power-function models: Nash systems Ṡ = α 1 (ES) g 12 β 1 S h 11 E h 14 ES = α 2 S g 21 E g 24 β 2 (ES) h 22 Ṗ = α 3 (ES) g 32 E g 34 E = const

16 Systems input space U R k output space R r Σ = (X,f,h,x 0 ) state-space X = R n dynamics ẋ(t) = f (x(t),u(t)) output function y(t) = h(x(t)) initial state x(0) = x 0 X ( α U : f α,i = f i (x( ),α))

17 Framework Irreducible variety X = X({f 1,...,f s } R[X 1,...,X n ]) = {a R n 1 i s : f i (a) = 0} Polynomial functions A(X) and rational functions Q(X) A(X) = {p : X R f R[X 1,...,X n ] a X : p(a) = f (a)} Q(X) = {p/q p,q A,q 0} Nash manifold d m i X = {a R n p ij (a) ε ij 0} p ij R[X 1,...,X n ], ε ij {<,=} i=1 j=1 Nash functions N(X) analytic f : X R s.t. {(x,y) R n+1 f (x) = y} is semi-algebraic

18 Polynomial and rational systems Σ = (X,f,h,x 0 ) - polynomial / rational system X - irreducible variety ẋ(t) = f (x(t),u(t)) α U : f α,i = f i (x( ),α) A(X) / Q(X) h : X R r - output map with h i A(X) / Q(X) x 0 X - initial state X = R 2, h(x 1,x 2,x 3 ) = x 2 X = R 2, h(x 1,x 2 ) = x 2 ẋ 1 = ax 1 u + bx 3 ẋ 1 = ax 1 + cx 1+bx1 2 x 1 +d ẋ 2 = cx 3 ẋ 2 = ex 1 x 1 +d ẋ 3 = ax 1 u (b + c)x 3 x 0 = (1,1) x 0 = (1,1,1) Realization theory of polynomial/rational systems: [Sontag 1970 s, Bartuszewicz 1980 s, Nemcova & Van Schuppen 2000 s]:w

19 Nash systems Σ = (X,f,h,x 0 ) - Nash system X - semi-alg. connected Nash manifold ẋ(t) = f (x(t),u(t)) α U : f α,i = f i (x( ),α) N(X) h : X R r - output map with h i N(X) x 0 X - initial state X = R 3 +, h(x 1,x 2,x 3 ) = x 3 ẋ 1 = x 2 1 x ẋ 2 = x x x x ẋ 3 = x x x x 0 = (1,1,1)

20 Admissible controls Inputs: piecewise-constant u : 0,T u Ω 0,T u = {s T : 0 s t} Constant inputs: T = [0,+ ) : 0,T u = [0,T u ] [ω,t] : 0,t s ω Ω Concatenation of inputs: u : 0,T u Ω, v : 0,T v Ω { u(t) t t T u v : 0,T u + T v t u v(t T u ) t > T u

21 Admissible controls: continued Set of admissible control inputs U pc a set of piecewise-constant controls such that constant inputs belong U pc ω Ω t T : [ω,t] U pc U pc is closed under restricting inputs to an interval u U pc t 0,T u : u 0,t U pc Inputs from U pc can be extended on a small time interval with any constant. u U pc ω Ω ε > 0 : and u [ω,ε] U pc

22 Problem formulation Response maps p : U pc R r is a response map if (p j A(U pc R)) p j (u) = p j ((α 1,t 1 ) (α k,t k )) = p j α1,...,α k (t 1,...,t k ) = j 1,...,j k =0 a j1,...,j k Π k i=1 tj i i u U pc Realization problem - existence Given a response map p : U pc R r Find a system Σ = (X,f,h,x 0 ) such that p(u) = h(x Σ (T u ;x 0,u)) for all u U pc U pc (Σ)

23 Rational systems: some definitions Σ is reachable if the set of reachable states x(t) is Zariski dense. The observation algebra Q(Σ) is the smallest algebra which contains h and which is closed under the Lie-derivative L fα, α R m. Σ is observable, if Q(Σ) equals the ring of rational functions. For simplicity: output dimension 1. D α derivation on the space of input-output maps D α ϕ(u)(s) = d dt ϕ(u (α,t))(t + s) t=0 + A(p) be the smallest algebra which contains p and which is closed under derivation D α. Q(p) the quotient field of A(p).

24 Realization theory of rational systems [Nemcova, Van Schuppen] Σ rational system If Σ is observable and reachable, then it is minimal. The converse is true under further conditions. Σ is minimal if and only if trdegq(σ) = dima(p). If two rational systems are both realizations of p, they are both reachable and observable, then they are birationally isomorphic. Any rational system can be converted to a reachable and observable one, while preserving the input-output behavior. p has a realization by a rational system if and only if Q(p) is finitely generated.

25 Application of realization theory: identifiability Parametrized system Σ(P) = {Σ(θ) = (X θ,f θ,h θ,x0 θ) θ P} P R s an irreducible variety - parameter set the same input spaces U R m and output spaces R r A parametrization P : P Σ(P) is globally identifiable if each θ can be determined uniquely from the input-output map of Σ(θ). structurally identifiable if each θ outside an algebraic set (of measure zero) can be determined uniquely from the input-output map of Σ(θ) Identifiability ensures that the parameter estimation problem is well posed.!

26 Application of realization theory: identifiability Theorem Σ(P) - structured rational system: structurally canonical ( Σ(θ) reachable and observable for almost all θ) structurally distinguishes parameters (Σ(θ) is injective except on a set of measure zero) Then the following are equivalent P : θ Σ(θ) is structurally identifiable For almost all θ 1,θ 2 the only isomorphism between Σ(θ 1 ) and Σ(θ 2 ) is the identity.... can be extended to global identifiability.

27 Existence of Nash realizations Theorem (Nemcova,Petreczky,Van Schuppen) p has a Nash realization trdeg A obs (p) < + trdeg A obs (p) < + p has a Nash realization - open problem

28 Local realizations Theorem (Nemcova, Petreczky) trdeg A obs (p) < +, U < + u U pc : p u has a local Nash realization local Nash realization Σ = (X,f,h,x 0 ): p(u) = h(x Σ (T u ;x 0,u)) u U pc U pc (Σ) small enough shifted response map p u : p u (v) = p(u v) v U pc s.t. u v U pc Proof relies on implicit function theorem for Nash functions.

29 Semialgebraic reachability Σ = (X,f,h,x 0 ) Nash system Σ semialgebraically reachable, if Σ semi-algebraically reachable if any Nash functions which vanishes on the reachable set equals zero, i.e. g N(X) : (g = 0 on R (x 0 ) g = 0) R (x 0 ) = {x Σ (T u ;x 0,u) u U pc (Σ)} Reachability reduction Every Nash system can be converted to a semi-algebraically reachable one, while remaining a local realization of the same input-output map. The procedure relies on implicit function theorem for Nash functions.

30 Semialgebraic observability Σ = (X,f,h,x 0 ) Nash system A obs (Σ) algebra generated by h i, L ω1 L ωk h i i = 1,...,r,ω 1,...,ω k Ω,k N. L ω g Lie derivative Σ semialgebraically observable trdeg A obs (Σ) = trdeg N(X) Observability reduction Every Nash system can be converted to a semi-algebraically observable one, while remaining a local realization of the same input-output map. The procedure relies on implicit function theorem for Nash functions.

31 Minimality Σ is minimal local realization of p dimσ dimσ for all local realizations Σ of p Main results Σ local Nash realization of p u Σ minimal dimσ = trdeg A obs (p) Σ reachable + observable Σ 1,Σ 2 reachable + observable local Nash realization of p u,v 1 X Σ1,V 2 X Σ2 : Σ V 1 1,ΣV 2 2 isomorphic local realizations of p u

32 Summary Conditions for existence of a Nash system realizing the given input-output behavior Conditions for minimality, minimal systems are locally unique. We can convert any realization to a minimal one.

33 Problem formulation: interconnection structure Interconnected system, Σ 1,Σ 2,Σ 3 subsystems u x 1 Σ 1 Σ 2 u(t) x 3 x 2 y(t) Σ 3 y Its interconnection structure is the graph u x 1 x 2 x 3 y

34 Realization with interconnection structure We observe the input-output behavior (black-box) u(t) y(t) Problem: Given a graph find a complex system which reproduces the input-output behavior and has the same interconnection structure as this graph:

35 Example: interconnection structure u x 1 x 2 x 3 y = we are looking for systems u x 1 Σ 1 Σ 2 u(t) x 3 x 2 y(t) Σ 3 y

36 Realization of connectivity structure: motivation Discovering the topology of gene regulatory networks. Drug design: if we know the topology, we know which link to cut. Discovering interaction between regions of brain using fmri.

37 Reverse engineering of interconnection structure It would be tempting to find the interconnection structure of a complete black-box: problem is not well posed. u ẋ 1 = 2x 1 + u y = u(t) u u ẋ 2 = 3x 2 + u 3 i=1 x i y(t) ẋ 3 = 3x 3 + u

38 has the the same input-output behavior from zero initial state as u ẋ 1 = 6x u u(t) u u x 1 ẋ 2 = x 1 11x 3 12u x 2 y(t) ẋ 3 = x 2 + 6x 3 + 3u x 3 y but the connectivity structures are totally different!

39 Connectivity structure for linear system A linear system is a diff. equation ẋ = Ax + Bu, y = Cx, x(0) = 0 A R n n,c R 1 n,b R n 1. Connectivity structure is a directed graph G = (V,E) V = {x 1,...,x n,u,y} vertices, x 1,...,x n,u,y symbols. E edges: e E e = (x j,x i ) A i,j 0 e = (y,x i ) C i 0 e = (x i,u) B i 0 Condensed graph GS: graph formed by strongly connected components of G

40 Connectivity structure for linear system Suppose H(s) = b(s) a(s) and dega(s) = n Theorem (Bras,Petreczky,Westra,Roebroeck,Peeters) A condensed subgraph cannot have more components than the number of divisors of a(s) over reals. Extension to several outputs, further results exist.

41 Example: model of fmri signal 0.5u 1.25x 1 2.5(x 2 1) x 1 ẋ = x 2 x ( 3 5 ) x 1.25x x3 4x 4 (1) y = 0.04 x 4 x x x (2) y MRI signal x 1,...,x 4 neuronal activity u cognitive input.

42 Example: model of fmri signal Linearization: H(s) = ż = z u (3) y = [ ] z (4) s s s 3 +15s s+12.5 Divisors of the denominator: s + 1,s + 2 and s s H cannot be realized by a system whose graph has more than 3 components. H can be realized by a system whose graph has exactly 3 components.

43 Coordinated stochastic linear systems: discussion We characterized connectivity in terms of output processes. Old tool: Granger noncausality. In neuroscience connectivity of brain regions is investigated, using: Recursive models relating future outputs to past ones, using Granger causality Difference equations in state-space form, using the graph of the system The results above are the first step to reconcile these two approaches.

44 Conclusions Control theory could be useful for systems biology. Realization theory is important for biological modelling: sanity check. Open problems: Is it relevant for biology? We looked at the algebraic structure and the network topology: how to combine the two worlds? For mathematicians: a lot of non-trivial (at least for control theorists) mathematics.