Computer control of gene expression: Robust setpoint tracking of - PowerPoint PPT Presentation

Introduction Mean Control Mean and variance control Examples Conclusion Computer control of gene expression: Robust setpoint tracking of protein mean and variance using integral feedback Corentin Briat and Mustafa Khammash 51st IEEE Conference on Decision and Control, Maui, Hawaii, 2012 Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 1/21

Introduction Mean Control Mean and variance control Examples Conclusion Outline • Introduction • Mean control • Mean and variance control • Conclusion and Future Works Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 2/21

Introduction Mean Control Mean and variance control Examples Conclusion Introduction Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 3/21

Introduction Mean Control Mean and variance control Examples Conclusion Stochastic chemical reaction network Variables • N molecular species S 1 , . . . , S N • M reactions R 1 , . . . , R M • Population of each species: random variables X 1 ( t ) , . . . , X N ( t ) Chemical Master Equation M ˙ � P ( κ , t ) = [ w k ( κ − s k ) P ( κ − s k , t ) − w k ( κ ) P ( κ , t )] (1) k =1 • P ( κ , t ) : probability to be in state κ at time t . • s k : stoichiometry vector associated to reaction R k . • w k : propensity function capturing the rate of the reaction R k . Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 4/21

Introduction Mean Control Mean and variance control Examples Conclusion Moments expression General case dE [ X ] = SE [ w ( X )] , dt (2) dE [ XX T ] SE [ w ( X ) X T ] + E [ w ( X ) X T ] T S T + S diag { E [ w ( X )] } S T = dt � s 1 ∈ R N × M : stoichiometry matrix. • S := � . . . s M � T ∈ R M : propensity vector. � w T w T • w ( X ) := . . . 1 M Affine propensity case w ( X ) = WX + w 0 dE [ X ] = SWE [ X ] + Sw 0 , dt (3) d Σ SW Σ + ( SW Σ) T + S diag( WE [ X ] + w 0 ) S T = dt • Σ : covariance matrix • Linear equations Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 5/21

Introduction Mean Control Mean and variance control Examples Conclusion Gene expression circuit k r R 1 : φ − → mRNA γ r R 2 : mRNA − → φ k p R 3 : mRNA − → protein+mRNA γ p R 4 : protein − → φ � 1 � k r � − 1 0 0 � T S = w ( X ) = γ r X 1 k p X 1 γ p X 2 0 0 1 − 1 Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 6/21

Introduction Mean Control Mean and variance control Examples Conclusion Moments dynamics − γ r 0 0 0 0 1     k p − γ p 0 0 0 0     x ( t ) ˙ = γ r 0 − 2 γ r 0 0  x ( t ) + 1  k r         0 0 k p − ( γ r + γ p ) 0 0   k p γ p 0 2 k p − 2 γ p 0 where the state variables are defined as � x 1 � x 3 � x 4 � := E [ X ] and := Σ x 2 x 4 x 5 • Asymptotically stable system with equilibrium point 1 = k r 2 = k p k r 3 = k r k p k r 5 = k p k r ( γ p + k p + γ r ) x ∗ , x ∗ , x ∗ , x ∗ 4 = γ r ( γ p + γ r ) , x ∗ . γ r γ p γ r γ r γ p γ r ( γ p + γ r ) Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 7/21

Introduction Mean Control Mean and variance control Examples Conclusion Mean Control Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 8/21

Introduction Mean Control Mean and variance control Examples Conclusion Problem statement Mean dynamics � − γ r � � x 1 ( t ) � 1 � � � � x 1 ( t ) ˙ 0 = + u ( t ) (4) x 2 ( t ) ˙ k p − γ p x 2 ( t ) 0 • Control input: transcription rate k r . • Controlled variable: mean number of proteins x 2 , [Klavins, 2010], [Milias-Argeitis et al. 2011] Positive PI Controller � t � � u ( t ) = ϕ k 1 ( µ ∗ − x 2 ( t )) + k 2 [ µ ∗ − x 2 ( s )] ds (5) 0 • µ ∗ : desired mean value. • k 1 , k 2 : gains of the PI controller • ϕ ( u ) := max { 0 , u } : nonnegativity constraint on the control input. Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 9/21

Introduction Mean Control Mean and variance control Examples Conclusion Global stability analysis Equilibrium point Given any µ ∗ ≥ 0 , the equilibrium point of the closed-loop system is given by 1 = µ ∗ γ p 2 = µ ∗ , u ∗ = µ ∗ γ p γ r , I ∗ = u ∗ x ∗ , x ∗ (6) k p k p k 2 where I ∗ is the equilibrium value of the integral term. Theorem - Global asymptotic stability Given system parameters k p , γ p , γ r > 0 and assume that k 2 k 1 > and k 2 > 0 . (7) γ p then the unique equilibrium point of the controlled system is globally asymptotically stable. • Straightforward extension to the robust case. Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 10/21

Introduction Mean Control Mean and variance control Examples Conclusion Proof sketch • LTI system + static nonlinearity in the sector [0 , 1] • Popov criterion can be used to infer stability of the closed-loop system: • Globally asymptotically stable if there exists q ≥ 0 such that ℜ [(1 + qjω ) H ( jω )] > − 1 (8) holds for all ω ∈ R and where k p ( k 1 s + k 2 ) H ( s ) = s ( s + γ r )( s + γ p ) . (9) • Equivalent to the positivity problem N 0 ( ω ) + qN 1 ( ω ) + D ( ω ) > 0 (10) • Descartes’ rule of signs yields the result. • Exact conditions can be obtained using Sturm series. Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 11/21

Introduction Mean Control Mean and variance control Examples Conclusion Mean and variance control Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 12/21

Introduction Mean Control Mean and variance control Examples Conclusion Fundamental limitations Mean vs. variance µ ∗ = k p u ∗ � k p � σ 2 1 ∗ = 1 + µ ∗ , (11) γ p + γ r γ p γ r • Need of a second control input → u 2 ≡ γ r Property The set of admissible reference values ( µ ∗ , σ 2 ∗ ) is given by � � 1 + k p � � ( x, y ) ∈ R 2 A := > 0 : x < y < x (12) γ p where k p , γ p > 0 . Independent of the controller ! Sketch • Lower bound ← C ν := σ ∗ /µ ∗ . • Upper bound ← nonnegativity of the control inputs. Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 13/21

Introduction Mean Control Mean and variance control Examples Conclusion System formulation Mean and variance dynamics x 1 ˙ = − u 2 x 1 + u 1 x 2 ˙ = k p x 1 − γ p x 2 x 3 ˙ = u 2 x 1 − 2 u 2 x 3 + u 1 x 4 ˙ = k p x 3 − γ p x 4 − u 2 x 4 (13) x 5 ˙ = k p x 1 + γ p x 2 + 2 k p x 4 − 2 γ p x 5 ˙ I 1 = µ ∗ − x 2 ˙ σ 2 I 2 = ∗ − x 5 • Bilinear system. Controller k 1 ( µ ∗ − x 2 ) + k 2 I 1 + k 3 ( σ 2 u 1 = ϕ � ∗ − x 5 ) + k 4 I 2 � k 5 ( µ ∗ − x 2 ) + k 6 I 1 + k 7 ( σ 2 u 2 = ϕ � ∗ − x 5 ) + k 8 I 2 � (14) ϕ ( u ) = max { u, 0 } • Multivariable positive PI controller. Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 14/21

Introduction Mean Control Mean and variance control Examples Conclusion Equilibrium Point Equilibrium point is unique Assume that k 2 k 8 − k 4 k 6 � = 0 , then the equilibrium point of the closed-loop system is unique and given by γ p σ 2 x ∗ µ ∗ = x ∗ x ∗ x ∗ = 3 , = µ ∗ , = ∗ , 1 2 5 k p γ p γ p k p µ ∗ x ∗ = µ ∗ , u ∗ = µ ∗ u ∗ 2 , u ∗ = − γ p + 4 1 2 γ p + u ∗ k p σ 2 ∗ − µ ∗ 2 and � I ∗ � k 2 � − 1 � u ∗ � k 4 � 1 1 = . I ∗ u ∗ k 6 k 8 2 2 Set of equilibrium points X ∗ := ( x ∗ , I ∗ ) ∈ R 7 : ( y ∗ , σ 2 � � ∗ ) ∈ A Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 15/21

Introduction Mean Control Mean and variance control Examples Conclusion Semiglobal stabilizability Theorem Given any k p , γ p > 0 , the mean/variance bilinear system is locally exponentially stabilizable around any equilibrium point in X ∗ using the PI control law. Moreover, there exists a PI control law that simultaneously locally exponentially stabilizes the mean/variance system around all the equilibrium points in X ∗ . Proof sketch • Open-loop system marginally stable • Two integrators (controller) • Difficulties: system large and structured controller. • Eigenvalue perturbation argument Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 16/21

Introduction Mean Control Mean and variance control Examples Conclusion Examples Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 17/21