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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

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Introduction Mean Control Mean and variance control Examples Conclusion

Computer control of gene expression: Robust setpoint tracking

  • f 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

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Introduction Mean Control Mean and variance control Examples Conclusion

Outline

  • Introduction
  • Mean control
  • Mean and variance control
  • Conclusion and Future Works

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Introduction Mean Control Mean and variance control Examples Conclusion

Introduction

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Stochastic chemical reaction network

Variables

  • N molecular species S1, . . . , SN
  • M reactions R1, . . . , RM
  • Population of each species: random variables X1(t), . . . , XN(t)

Chemical Master Equation

˙ P(κ, t) =

M

  • k=1

[wk(κ − sk)P(κ − sk, t) − wk(κ)P(κ, t)] (1)

  • P(κ, t): probability to be in state κ at time t.
  • sk: stoichiometry vector associated to reaction Rk.
  • wk: propensity function capturing the rate of the reaction Rk.

Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 4/21

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Moments expression

General case

dE[X] dt = SE[w(X)], dE[XXT ] dt = SE[w(X)XT ] + E[w(X)XT ]T ST + S diag{E[w(X)]}ST (2)

  • S :=

s1 . . . sM

  • ∈ RN×M: stoichiometry matrix.
  • w(X) :=
  • wT

1

. . . wT

M

T ∈ RM: propensity vector.

Affine propensity case w(X) = WX + w0

dE[X] dt = SWE[X] + Sw0, dΣ dt = SWΣ + (SWΣ)T + S diag(WE[X] + w0)ST (3)

  • Σ: covariance matrix
  • Linear equations

Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 5/21

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Gene expression circuit

R1 : φ

kr

− → mRNA R2 : mRNA

γr

− → φ R3 : mRNA

kp

− → protein+mRNA R4 : protein

γp

− → φ S = 1 −1 1 −1

  • w(X) =

kr γrX1 kpX1 γpX2 T

Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 6/21

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Moments dynamics

˙ x(t) =      −γr kp −γp γr −2γr kp −(γr + γp) kp γp 2kp −2γp      x(t) +      1 1      kr where the state variables are defined as x1 x2

  • := E[X] and

x3 x4 x4 x5

  • := Σ
  • Asymptotically stable system with equilibrium point

x∗

1 = kr

γr , x∗

2 = kpkr

γpγr , x∗

3 = kr

γr , x∗

4 =

kpkr γr(γp + γr) , x∗

5 = kpkr(γp + kp + γr)

γpγr(γp + γr) .

Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 7/21

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Mean Control

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Problem statement

Mean dynamics

  • ˙

x1(t) ˙ x2(t)

  • =

−γr kp −γp x1(t) x2(t)

  • +

1

  • u(t)

(4)

  • Control input: transcription rate kr.
  • Controlled variable: mean number of proteins x2, [Klavins, 2010], [Milias-Argeitis

et al. 2011]

Positive PI Controller

u(t) = ϕ

  • k1(µ∗ − x2(t)) + k2

t [µ∗ − x2(s)]ds

  • (5)
  • µ∗: desired mean value.
  • k1, k2: gains of the PI controller
  • ϕ(u) := max{0, u}: nonnegativity constraint on the control input.

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Global stability analysis

Equilibrium point

Given any µ∗ ≥ 0, the equilibrium point of the closed-loop system is given by x∗

1 = µ∗γp

kp , x∗

2 = µ∗, u∗ = µ∗γpγr

kp , I∗ = u∗ k2 (6) where I∗ is the equilibrium value of the integral term.

Theorem - Global asymptotic stability

Given system parameters kp, γp, γr > 0 and assume that k1 > k2 γp and k2 > 0. (7) 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

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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 H(s) = kp(k1s + k2) s(s + γr)(s + γp) . (9)

  • Equivalent to the positivity problem

N0(ω) + qN1(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

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Mean and variance control

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Introduction Mean Control Mean and variance control Examples Conclusion

Fundamental limitations

Mean vs. variance

σ2

∗ =

  • 1 +

kp γp + γr

  • µ∗,

µ∗ = kpu∗

1

γpγr (11)

  • Need of a second control input → u2 ≡ γr

Property

The set of admissible reference values (µ∗, σ2

∗) is given by

A :=

  • (x, y) ∈ R2

>0 : x < y <

  • 1 + kp

γp

  • x
  • (12)

where kp, γp > 0. Independent of the controller !

Sketch

  • Lower bound ← Cν := σ∗/µ∗.
  • Upper bound ← nonnegativity of the control inputs.

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System formulation

Mean and variance dynamics

˙ x1 = −u2x1 + u1 ˙ x2 = kpx1 − γpx2 ˙ x3 = u2x1 − 2u2x3 + u1 ˙ x4 = kpx3 − γpx4 − u2x4 ˙ x5 = kpx1 + γpx2 + 2kpx4 − 2γpx5 ˙ I1 = µ∗ − x2 ˙ I2 = σ2

∗ − x5

(13)

  • Bilinear system.

Controller

u1 = ϕ

  • k1(µ∗ − x2) + k2I1 + k3(σ2

∗ − x5) + k4I2

  • u2

= ϕ

  • k5(µ∗ − x2) + k6I1 + k7(σ2

∗ − x5) + k8I2

  • ϕ(u)

= max{u, 0} (14)

  • Multivariable positive PI controller.

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Equilibrium Point

Equilibrium point is unique

Assume that k2k8 − k4k6 = 0, then the equilibrium point of the closed-loop system is unique and given by x∗

1

= γp kp µ∗ = x∗

3,

x∗

2

= µ∗, x∗

5

= σ2

∗,

x∗

4

= γp γp + u∗

2

µ∗, u∗

1

= γp kp µ∗u∗

2,

u∗

2

= −γp + kpµ∗ σ2

∗ − µ∗

and I∗

1

I∗

2

  • =

k2 k4 k6 k8 −1 u∗

1

u∗

2

  • .

Set of equilibrium points

X ∗ :=

  • (x∗, I∗) ∈ R7 : (y∗, σ2

∗) ∈ A

  • Corentin Briat and Mustafa Khammash

Computer control of gene expression: mean and variance control 15/21

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Semiglobal stabilizability

Theorem

Given any kp, γp > 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

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Examples

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Variance control

  • Model parameters taken from [Milias-Argeitis et al. 2011].
  • PI controller gains k1 = 1, k2 = 0.007, k7 = −0.2 and k8 = −0.0014 .

500 1000 1500 2000 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [min] Normalized mean and variance

Mean Variance Minimal variance Corentin Briat and Mustafa Khammash Computer control of gene expression: mean and variance control 18/21

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Conclusion

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Conclusion and Future Works

Conclusion

  • PI controller sufficient for mean and variance control.
  • PI control globally and simultaneously stabilizes the mean around any desired

equilibrium point.

  • PI control locally and simultaneously stabilizes the mean and variance around any

admissible equilibrium point.

Future Works

  • Implementation.
  • Generalization to more general networks (moment closure problem)

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Thank you for your attention

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