Modeling and analysis of reaction networks Control of the mean population Example Conclusion Integral population control of a quadratic dimerization process Corentin Briat and Mustafa Khammash Swiss Federal Institute of Technology – Zürich Department of Biosystems Science and Engineering 52nd IEEE Conference on Decision and Control, Florence, Italy, 2013 Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 1/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Outline • Setup • Modeling and analysis of reaction networks • Mean control of a dimerization process • Example • Conclusion Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 2/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Setup • Considered in the gene expression context • Mean control 1 and mean+variance control in 2 1 A. Milias-Argeitis, et al. In silico feedback for in vivo regulation of a gene expression circuit, Nature Biotechnology , 2011 2 C. Briat et al. Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback, 51st IEEE Conference on Decision and Control , 2012 Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 3/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Modeling and analysis of reaction networks Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 4/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Modeling biochemical networks Variables • N molecular species S 1 , . . . , S N • M reactions R 1 , . . . , R M k γ k b φ − − − → S 1 , S 1 − − − → φ, S 1 + S 2 − − − → S 3 Dynamics • Deterministic (ODEs) → concentrations x ( t ) ∈ R N ≥ 0 • Stochastic (jump processes) → molecule counts X ( t ) ∈ N N 0 Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 5/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Stochastic chemical reaction network Randomness in biology 1 • Intrinsic noise (variability inside a cell) • Extrinsic noise (cell-to-cell variability) • External noise (environment) Chemical Master Equation M ˙ � P ( κ , t ) = [ λ k ( κ − ζ k ) P ( κ − ζ k , t ) − λ k ( κ ) P ( κ , t )] (1) k =1 • P ( κ , t ) : probability to be in state κ at time t . • ζ k : stoichiometry vector associated to reaction R k . • λ k : propensity function capturing the rate of the reaction R k . 1 M. B. Elowitz, et al. Stochastic gene expression in a single cell, Science , 2002 Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 6/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Moments expression General case d E [ X ] = SE [ λ ( X )] , d t (2) d E [ XX ⊺ ] SE [ λ ( X ) X ⊺ ] + E [ λ ( X ) X ⊺ ] ⊺ S ⊺ + S diag { E [ λ ( X )] } S ⊺ = d t � ζ 1 ∈ R N × M : stoichiometry matrix. • S := � . . . ζ M � λ ⊺ � ⊺ ∈ R M : propensity vector. • λ ( X ) := λ ⊺ . . . 1 M Affine propensity case λ ( X ) = WX + λ 0 dE [ X ] = SWE [ X ] + Sλ 0 , dt (3) d Σ SW Σ + ( SW Σ) ⊺ + S diag( WE [ X ] + λ 0 ) S ⊺ = dt • Σ : covariance matrix • Linear equations Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 7/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Moments expression General case d E [ X ] = SE [ λ ( X )] , d t (2) d E [ XX ⊺ ] SE [ λ ( X ) X ⊺ ] + E [ λ ( X ) X ⊺ ] ⊺ S ⊺ + S diag { E [ λ ( X )] } S ⊺ = d t � ζ 1 ∈ R N × M : stoichiometry matrix. • S := � . . . ζ M � λ ⊺ � ⊺ ∈ R M : propensity vector. λ ⊺ • λ ( X ) := . . . 1 M Polynomial propensity case • Moment closure problem → first-order moments depend on the second-order ones, and so forth. . . • Infinite set of linear ODEs (unstructured, non-necessarily Metzler. . . ) • Closure methods Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 7/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Dimerization process Process k 1 b : − → : S 1 + S 1 − → R 1 φ S 1 R 2 S 2 γ 1 γ 2 R 3 : S 1 − → φ R 4 : S 2 − → φ • Stoichiometry matrix: � 1 − 2 − 1 0 � S = 0 1 0 − 1 • Propensity function: � � ⊺ b λ ( X ) = . k 1 2 X 1 ( X 1 − 1) γ 1 X 1 γ 2 X 2 Motivations • Goes beyond the affine case (e.g. gene expression 1 ) and introduce dimerization • Check whether the moments equations framework still applicable (closure problem) 1 C. Briat et al. Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback, 51st IEEE Conference on Decision and Control , 2012 Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 8/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Moments equations Dynamical model k 1 + ( b − γ 1 ) x 1 ( t ) − bx 1 ( t ) 2 − bv ( t ) x 1 ( t ) ˙ = − b 2 x 1 ( t ) − γ 2 x 2 ( t ) + b 2 x 1 ( t ) 2 + b x 2 ( t ) ˙ = 2 v ( t ) where • x i ( t ) := E [ X i ( t )] , i = 1 , 2 , • v ( t ) := V ( X 1 ( t )) is the variance of the random variable X 1 ( t ) . Difficulties • Nonlinear system (as opposed to linear for networks with affine propensities) • Unknown “input” variance v ( t ) := V ( X 1 ( t )) (closure problem). Is it bounded? Is it converging? • If v → v ∗ , we have an infinite number of non-isolated equilibrium points (model artefact since the first-order moments may have a unique stationary value) � x ∗ � v ∗ → 1 locally continuous x ∗ 2 Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 9/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Statement of the problem Objective Find k c such that the integral control law ˙ I ( t ) = µ − E [ X 2 ( t )] (3) u ( t ) = k c · max { I ( t ) , 0 } locally steers E [ X 2 ( t )] to µ (stability and attractivity of the corresponding equilibrium point). Subproblems • Are the moments bounded and converging for some parameter values? (stability) • Choose a control input that can drive E [ X 2 ] to µ asymptotically • Find conditions on the controller gain k c such that we have local asymptotic stability of the first order moments. Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 10/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Ergodicity of the dimerization process Reaction network k 1 b R 1 : φ − → S 1 R 2 : S 1 + S 1 − → S 2 γ 1 γ 2 : − → : − → R 3 S 1 φ R 4 S 2 φ Theorem For any positive value of the network parameters k 1 , b, γ 1 and γ 2 , the dimerization process is ergodic and has all its moments bounded and converging. ⇒ ( x 1 ( t ) , x 2 ( t ) , v ( t )) → ( x ∗ 1 , x ∗ 2 , v ∗ ) globally and exponentially • Proof relying on an ergodicity result developed in the paper 1 1 C. Briat, A. Gupta and M. Khammash, " A scalable computational framework for establishing long-term behavior of stochastic reaction networks ", submitted to PLOS computational biology Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 11/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Control of the mean population Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 12/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Choice of the control input Assumption The function S ∗ := x ∗ 2 1 − x ∗ 1 + v ∗ , where x ∗ 1 is the unique equilibrium solution for x 1 and v ∗ is the equilibrium variance, verifying the equation 1 − bS ∗ = 0 , k 1 − γx ∗ (4) is a continuous function of k 1 . Proposition For any µ > 0 , there exists a constant k 1 > 0 such that x 2 ( t ) → µ . • The birth rate k 1 can then be chosen as control input • Good for us, this is also the simplest case! • Other rates could have been also chosen Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 13/22
Modeling and analysis of reaction networks Control of the mean population Example Conclusion Nominal stabilization Theorem For any finite positive constants γ 1 , γ 2 , b, µ and any controller gain k c satisfying � � � 0 < k c < 2 γ 2 2 γ 1 + γ 2 + 2 γ 1 ( γ 1 + γ 2 ) , (5) the closed-loop system has a unique locally stable equilibrium point ( x ∗ 1 , x ∗ 2 , I ∗ ) in the positive orthant such that x ∗ 2 = µ . The equilibrium variance moreover satisfies � 0 , 2 γ 2 µ + 1 � v ∗ ∈ . (6) b 4 Corentin Briat and Mustafa Khammash Integral population control of a quadratic dimerization process 14/22
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