Realization theory for systems biology Mihály Petreczky CNRS Ecole Central Lille, France February 3, 2016
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.
What is control theory about ? Plant y u Controller ◮ Plant – dynamical system (behavior changes with time) ◮ Controller – dynamical system
What is control theory about ? Plant y u Controller Task: find controller u = C ( y ) such that the y has the desired properties: ◮ y → 0, ◮ � ∞ 0 y 2 ( s ) ds minimal, etc.
Example: thermostat Plant heating on ˙ ˙ x = − x + 18 x = − x + 22 heating off ◮ Outputs: temperature x ◮ Control input: ‘heating on’ and ‘heating off’. Control objective: maintain the temperature between 19 . 5 − 20 . 5 ◦ C Controller heating on if x < 19 . 5 heating off if x > 20 . 5
How do we solve control problems ? Find a mathematical model (state-space representation) ˙ x ( 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 y ( . ) has the desired properties.
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 )) ˙ x ( 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.
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) ?
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
Realization theory: reverse engineering of plant models Realization theory: problem statement We observe the input-output behavior (black-box) y ( t ) u ( t ) of a physical process y ( t ) u ( t )
Realization theory: reverse engineering of plant models Realization theory: problem statement We observe the input-output behavior (black-box) y ( t ) u ( t ) Which mathematical models (fixed structure) y ( t ) T ˙ ω ( t ) + ω ( t ) = Ku ( t ) , y ( t ) = ω ( t ) u ( t ) can describe the observed behavior of the black-box ?
Realization theory for biological systems We can fix either the algebraic structure of the models. ˙ S = α 1 ( ES ) g 12 − β 1 S h 11 E h 14 y ( t ) ES = α 2 S g 21 E g 24 − β 2 ( ES ) h 22 ˙ u ( t ) P = α 3 ( ES ) g 32 E g 34 ˙ E = const
or the interconnection structure ˙ x 1 = 6 x 3 + 11 u u x 1 u x 2 = x 1 − 11 x 3 − 12 u ˙ y ( t ) u u ( t ) x 2 x ′ 3 ˙ x 3 = x 2 + 6 x 3 + 3 u y
Polynomial/rational/Nash systems: biochemical reactions k 1 k 2 E + S ES E + P ✲ ✲ ✛ k − 1 mass-action kinetics: polynomial equations ˙ S = − k 1 E · S + k − 1 ES ES = − ˙ ˙ E = k 1 E · S − ( k − 1 + k 2 ) ES ˙ P = k 2 ES Slides of Jana Nemcova
Polynomial/rational/Nash systems: biochemical reactions Michaelis-Menten kinetics: rational equations � � ˙ E t S E t S S = − k 1 E t − S + k − 1 S + K m S + K m ˙ P = v max S S + K m power-function models: Nash systems S = 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 ˙ ˙ P = α 3 ( ES ) g 32 E g 34 E = const Slides of Jana Nemcova
Systems U ⊆ R k input space R r output space Σ = ( X , f , h , x 0 ) ◮ state-space X = R n ◮ dynamics ˙ x ( t ) = f ( x ( t ) , u ( t )) ( ∀ α ∈ U : f α , i = f i ( x ( · ) , α )) ◮ output function y ( t ) = h ( x ( t )) ◮ initial state x ( 0 ) = x 0 ∈ X Slides of Jana Nemcova
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 m i d { a ∈ R n | p ij ( a ) ε ij 0 } � � X = 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 Slides of Jana Nemcova
Polynomial and rational systems Σ = ( X , f , h , x 0 ) - polynomial / rational system ◮ X - irreducible variety ◮ ˙ x ( 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 x 1 = − ax 1 + cx 1 + bx 2 ˙ ˙ x 1 = − ax 1 u + bx 3 1 x 1 + d ex 1 ˙ ˙ x 2 = cx 3 x 2 = x 1 + d x 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 Slides of Jana Nemcova
Nash systems Σ = ( X , f , h , x 0 ) - Nash system ◮ X - semi-alg. connected Nash manifold ◮ ˙ ∀ α ∈ U : f α , i = f i ( x ( · ) , α ) ∈ N ( X ) x ( t ) = f ( x ( t ) , u ( t )) ◮ 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 x 1 = 1 . 75817 . 10 − 2 . 37 − 1 . 4489 x 2 1 x − 1 . 05 ˙ 2 x 2 = 5 0 . 5125 6 . 04276 . 10 − 2 x 0 . 75 − 1 . 93417 . 10 − 4 x 4 . 65 x − 0 . 45625 x − 4 . 29 ˙ 1 2 2 3 − 3 . 4657 . 10 − 2 x 0 . 3 x − 4 . 29 x 3 = 1 . 93417 x 4 . 65 ˙ 2 3 3 x 0 = ( 1 , 1 , 1 ) Slides of Jana Nemcova
Admissible controls Inputs: piecewise-constant u : � 0 , T u � → Ω � 0 , T u � = { s ∈ T : 0 ≤ s ≤ t } T = [ 0 , + ∞ ) : � 0 , T u � = [ 0 , T u ] Constant inputs: [ 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 u ⊔ v : � 0 , T u + T v � ∋ t �→ v ( t − T u ) t > T u Slides of Jana Nemcova
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
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 α 1 ,..., α k ( t 1 ,..., t k ) ∞ i = 1 t j i a j 1 ,..., j k Π k ∑ = ∀ u ∈ U pc i j 1 ,..., j k = 0 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) Slides of Jana Nemcova
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 f α , 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 ) .
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 trdeg Q (Σ) = dim A ( 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.
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