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Introduction In-vivo control - Theory In-vivo control - Example Conclusion A Control Theory for Stochastic Biomolecular Regulation Corentin Briat, Ankit Gupta and Mustafa Khammash SIAM Conference on Control and its Applications - 08/07/2015 Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 0/14
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A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

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Page 1: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

A Control Theory for Stochastic Biomolecular Regulation

Corentin Briat, Ankit Gupta and Mustafa Khammash

SIAM Conference on Control and its Applications - 08/07/2015

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 0/14

Page 2: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Introduction

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 0/14

Page 3: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Reaction networks

A reaction network is. . .• A set of d distinct species X1, . . . ,Xd

• A set of K reactions R1, . . . , RK specifying how species interact with each otherand for each reaction we have

• A stoichiometric vector ζk ∈ Zd describing how reactions change the state value• A propensity function λk ∈ R≥0 describing the "strength" of the reaction

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 1/14

Page 4: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Reaction networks

A reaction network is. . .• A set of d distinct species X1, . . . ,Xd

• A set of K reactions R1, . . . , RK specifying how species interact with each otherand for each reaction we have

• A stoichiometric vector ζk ∈ Zd describing how reactions change the state value• A propensity function λk ∈ R≥0 describing the "strength" of the reaction

Deterministic networks• Large populations (concentrations are well-defined), e.g. as in chemistry• Lots of analytical tools, e.g. reaction network theory, dynamical systems theory,

Lyapunov theory of stability, nonlinear control theory, etc.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 1/14

Page 5: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Reaction networks

A reaction network is. . .• A set of d distinct species X1, . . . ,Xd

• A set of K reactions R1, . . . , RK specifying how species interact with each otherand for each reaction we have

• A stoichiometric vector ζk ∈ Zd describing how reactions change the state value• A propensity function λk ∈ R≥0 describing the "strength" of the reaction

Deterministic networks• Large populations (concentrations are well-defined), e.g. as in chemistry• Lots of analytical tools, e.g. reaction network theory, dynamical systems theory,

Lyapunov theory of stability, nonlinear control theory, etc.

Stochastic networks• Low populations (concentrations are NOT well defined)• Biological processes where key molecules are in low copy number (mRNA '10

copies per cell)• No well-established theory for biology, “analysis" often based on simulations. . .• No well-established control theory

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 1/14

Page 6: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Chemical master equation

State and dynamics

• The state X ∈ Nd0 is vector of random variables representing molecules count• The dynamics of the process is described by a jump Markov process (X(t))t≥0

Chemical Master Equation (Forward Kolmogorov equation)

px0 (x, t) =K∑k=1

λk(x− ζk)px0 (x− ζk, t)− λk(x)px0 (x, t), x ∈ Nd0

where px0 (x, t) = P[X(t) = x|X(0) = x0] and px0 (x, 0) = δx0 (x).

Solving the CME

• Infinite countable number of linear time-invariant ODEs• Exactly solvable only in very simple cases• Some numerical schemes are available (FSP, QTT, etc) but limited by the curse of

dimensionality; if X ∈ {0, . . . , x− 1}d, then we have xd states

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 2/14

Page 7: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Chemical master equation

State and dynamics

• The state X ∈ Nd0 is vector of random variables representing molecules count• The dynamics of the process is described by a jump Markov process (X(t))t≥0

Chemical Master Equation (Forward Kolmogorov equation)

px0 (x, t) =K∑k=1

λk(x− ζk)px0 (x− ζk, t)− λk(x)px0 (x, t), x ∈ Nd0

where px0 (x, t) = P[X(t) = x|X(0) = x0] and px0 (x, 0) = δx0 (x).

Solving the CME

• Infinite countable number of linear time-invariant ODEs• Exactly solvable only in very simple cases• Some numerical schemes are available (FSP, QTT, etc) but limited by the curse of

dimensionality; if X ∈ {0, . . . , x− 1}d, then we have xd states

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 2/14

Page 8: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Ergodicity of reaction networks

ErgodicityA given stochastic reaction network is ergodic if there is a probability distribution πsuch that for all x0 ∈ Nd0 , we have that px0 (x, t)→ π as t→∞.

Theorem (Condition for ergodicity1)Assume that

(a) the state-space of the network is irreducible; and

(b) there exists a norm-like function V (x) such that the drift conditionK∑i=1

λi(x)[V (x+ ζi)− V (x)] ≤ c1 − c2V (x)

holds for some c1, c2 > 0 and for all x ∈ Nd0 .Then, the stochastic reaction networkis ergodic.

Choosing V (x) = 〈v, x〉, v > 0, allows to establish the ergodicity of a wide class ofexisting reaction networks2

1 S. P. Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 19932 A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks,

PLOS Computational Biology, 2014

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 3/14

Page 9: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Ergodicity of reaction networks

ErgodicityA given stochastic reaction network is ergodic if there is a probability distribution πsuch that for all x0 ∈ Nd0 , we have that px0 (x, t)→ π as t→∞.

Theorem (Condition for ergodicity1)Assume that

(a) the state-space of the network is irreducible; and

(b) there exists a norm-like function V (x) such that the drift conditionK∑i=1

λi(x)[V (x+ ζi)− V (x)] ≤ c1 − c2V (x)

holds for some c1, c2 > 0 and for all x ∈ Nd0 .Then, the stochastic reaction networkis ergodic.

Choosing V (x) = 〈v, x〉, v > 0, allows to establish the ergodicity of a wide class ofexisting reaction networks2

1 S. P. Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 19932 A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks,

PLOS Computational Biology, 2014

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 3/14

Page 10: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Ergodicity of reaction networks

ErgodicityA given stochastic reaction network is ergodic if there is a probability distribution πsuch that for all x0 ∈ Nd0 , we have that px0 (x, t)→ π as t→∞.

Theorem (Condition for ergodicity1)Assume that

(a) the state-space of the network is irreducible; and

(b) there exists a norm-like function V (x) such that the drift conditionK∑i=1

λi(x)[V (x+ ζi)− V (x)] ≤ c1 − c2V (x)

holds for some c1, c2 > 0 and for all x ∈ Nd0 .Then, the stochastic reaction networkis ergodic.

Choosing V (x) = 〈v, x〉, v > 0, allows to establish the ergodicity of a wide class ofexisting reaction networks2

1 S. P. Meyn and R. L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. , 19932 A. Gupta, C. Briat, and M. Khammash. A scalable computational framework for establishing long-term behavior of stochastic reaction networks,

PLOS Computational Biology, 2014

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 3/14

Page 11: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Control problems

In-silico control• Controllers are implemented outside cells• Single cell1 or population control2

In-vivo control• Controllers are implemented inside cells• Single cell and population control3

1 J. Uhlendorf, et al. Long-term model predictive control of gene expression at the population and single-cell levels, Proceedings of the National

Academy of Sciences of the United States of America, 20122 A. Milias-Argeitis, et al. In silico feedback for in vivo regulation of a gene expression circuit, Nature Biotechnology, 20113 C. Briat, A. Gupta, and M. Khammash. Integral feedback generically achieves perfect adaptation in stochastic biochemical networks, ArXiV, 2015

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 4/14

Page 12: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Control problems

In-silico control• Controllers are implemented outside cells• Single cell1 or population control2

In-vivo control• Controllers are implemented inside cells• Single cell and population control3

1 J. Uhlendorf, et al. Long-term model predictive control of gene expression at the population and single-cell levels, Proceedings of the National

Academy of Sciences of the United States of America, 20122 A. Milias-Argeitis, et al. In silico feedback for in vivo regulation of a gene expression circuit, Nature Biotechnology, 20113 C. Briat, A. Gupta, and M. Khammash. Integral feedback generically achieves perfect adaptation in stochastic biochemical networks, ArXiV, 2015

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 4/14

Page 13: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

In-vivo population control - Theory

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 4/14

Page 14: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Setup

Open-loop reaction network

• d molecular species: X1, . . . ,Xd

• X1 is the actuated species: ∅ u−−−→X1

• Measured/controlled species: Y = X`

Problem

Find a controller such that the closed-loop network is ergodic and such that we haveE[Y (t)]→ µ as t→∞, for some reference value µ

The controller• Two species Z1 and Z2.

∅ µ−−−→ Z1︸ ︷︷ ︸reference

, ∅ Y−−−→ Z2︸ ︷︷ ︸measurement

, Z1 + Z2η−−−→ ∅︸ ︷︷ ︸

comparison

, ∅ kZ1−−−→X1︸ ︷︷ ︸actuation

.

where k, η > 0 are control parameters.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 5/14

Page 15: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Setup

Open-loop reaction network

• d molecular species: X1, . . . ,Xd

• X1 is the actuated species: ∅ u−−−→X1

• Measured/controlled species: Y = X`

Problem

Find a controller such that the closed-loop network is ergodic and such that we haveE[Y (t)]→ µ as t→∞, for some reference value µ

The controller• Two species Z1 and Z2.

∅ µ−−−→ Z1︸ ︷︷ ︸reference

, ∅ Y−−−→ Z2︸ ︷︷ ︸measurement

, Z1 + Z2η−−−→ ∅︸ ︷︷ ︸

comparison

, ∅ kZ1−−−→X1︸ ︷︷ ︸actuation

.

where k, η > 0 are control parameters.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 5/14

Page 16: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Setup

Open-loop reaction network

• d molecular species: X1, . . . ,Xd

• X1 is the actuated species: ∅ u−−−→X1

• Measured/controlled species: Y = X`

Problem

Find a controller such that the closed-loop network is ergodic and such that we haveE[Y (t)]→ µ as t→∞, for some reference value µ

The controller• Two species Z1 and Z2.

∅ µ−−−→ Z1︸ ︷︷ ︸reference

, ∅ Y−−−→ Z2︸ ︷︷ ︸measurement

, Z1 + Z2η−−−→ ∅︸ ︷︷ ︸

comparison

, ∅ kZ1−−−→X1︸ ︷︷ ︸actuation

.

where k, η > 0 are control parameters.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 5/14

Page 17: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

The hidden integral action1

Moments equations

d

dtE[Z1(t)] = µ− ηE[Z1(t)Z2(t)]

d

dtE[Z2(t)] = E[Y (t)]− ηE[Z1(t)Z2(t)].

Integral action

• We have thatd

dtE[Z1(t)− Z2(t)] = µ− E[Y (t)],

so we have an integral action on the mean• Closed-loop ergodic⇒ E[Y (t)]→ µ as t→∞• No need for solving moments equations→ no closure problem :)

1 K. Oishi and E. Klavins. Biomolecular implementation of linear I/O systems, IET Systems Biology, 2010

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 6/14

Page 18: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

The hidden integral action1

Moments equations

d

dtE[Z1(t)] = µ− ηE[Z1(t)Z2(t)]

d

dtE[Z2(t)] = E[Y (t)]− ηE[Z1(t)Z2(t)].

Integral action

• We have thatd

dtE[Z1(t)− Z2(t)] = µ− E[Y (t)],

so we have an integral action on the mean• Closed-loop ergodic⇒ E[Y (t)]→ µ as t→∞• No need for solving moments equations→ no closure problem :)

1 K. Oishi and E. Klavins. Biomolecular implementation of linear I/O systems, IET Systems Biology, 2010

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 6/14

Page 19: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

General stabilization result

TheoremLet V (x) = 〈v, x〉 with v ∈ Rd>0 and W (x) = 〈w, x〉 with w ∈ Rd≥0, w1, w` > 0.Assume that

(a) the state-space of the open-loop reaction network is irreducible; and

(b) there exist c1, c3 > 0 and c2 ≥ 0 such that

K∑k=1

λk(x)[V (x+ ζk)− V (x)] ≤ −c1V (x),

K∑k=1

λk(x)[W (x+ ζk)−W (x)] ≥ −c2 − c3x`,(1)

hold for all x ∈ Nd0 (together with some other technical conditions).

Then, the closed-loop network is ergodic and we have that E[Y (t)]→ µ as t→∞.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 7/14

Page 20: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Unimolecular networks

TheoremLet us consider a unimolecular reaction network with irreducible state-space. Assumethat its first-order moments system

d

dtE[X(t)] = AE[X(t)] + e1u(t)

y(t) = eT` E[X(t)](2)

is

(a) asymptotically stable, i.e A Hurwitz stable (LP)

(b) output controllable, i.e. rank[eT` e1 eT` Ae1 . . . eT` A

d−1e1]

= 1 (LP)

Then, for all control parameters k, η > 0,

(a) the closed-loop reaction network (system + controller) is ergodic

(b) all the first and second order moments of the random variables X1, . . . , Xd areuniformly bounded and globally converging

(c) E[Y (t)]→ µ as t→∞.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 8/14

Page 21: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Unimolecular networks

TheoremLet us consider a unimolecular reaction network with irreducible state-space. Assumethat its first-order moments system

d

dtE[X(t)] = AE[X(t)] + e1u(t)

y(t) = eT` E[X(t)](2)

is

(a) asymptotically stable, i.e A Hurwitz stable (LP)

(b) output controllable, i.e. rank[eT` e1 eT` Ae1 . . . eT` A

d−1e1]

= 1 (LP)

Then, for all control parameters k, η > 0,

(a) the closed-loop reaction network (system + controller) is ergodic

(b) all the first and second order moments of the random variables X1, . . . , Xd areuniformly bounded and globally converging

(c) E[Y (t)]→ µ as t→∞.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 8/14

Page 22: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Unimolecular networks

TheoremLet us consider a unimolecular reaction network with irreducible state-space. Assumethat its first-order moments system

d

dtE[X(t)] = AE[X(t)] + e1u(t)

y(t) = eT` E[X(t)](2)

is

(a) asymptotically stable, i.e A Hurwitz stable (LP)

(b) output controllable, i.e. rank[eT` e1 eT` Ae1 . . . eT` A

d−1e1]

= 1 (LP)

Then, for all control parameters k, η > 0,

(a) the closed-loop reaction network (system + controller) is ergodic

(b) all the first and second order moments of the random variables X1, . . . , Xd areuniformly bounded and globally converging

(c) E[Y (t)]→ µ as t→∞.

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 8/14

Page 23: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Properties

Closed-loop system

• Robust ergodicity, tracking and disturbance rejection• Population control is achieved

Controller• Innocuous: open-loop ergodic & output controllable⇒ closed-loop ergodic• Decentralized: use only local information (single-cell control)• Implementable: small number of reactions

Additional remarks• No moment closure problem• Expected to work on a wide class of networks (even though the theory is not there

yet)

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 9/14

Page 24: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

In-vivo population control - Example

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 9/14

Page 25: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Gene expression network d = 2, K = 4

R1 : ∅ kr−−−→ mRNA (X1)

R2 : mRNAγr−−−→ ∅

R3 : mRNAkp−−−→ mRNA+protein (X1 + X2)

R4 : proteinγp−−−→ ∅

S =[ζ1 ζ2 ζ3 ζ4

]λ(x) = [ λ1(x) λ2(x) λ3(x) λ4(x) ]T

=

[1 −1 0 00 0 1 −1

]= [ kr γrx1 kpx1 γpx2 ]T

We want to control the average number of proteins by suitably acting on thetranscription rate kr

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 10/14

Page 26: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Gene expression network d = 2, K = 4

R1 : ∅ kr−−−→ mRNA (X1)

R2 : mRNAγr−−−→ ∅

R3 : mRNAkp−−−→ mRNA+protein (X1 + X2)

R4 : proteinγp−−−→ ∅

S =[ζ1 ζ2 ζ3 ζ4

]λ(x) = [ λ1(x) λ2(x) λ3(x) λ4(x) ]T

=

[1 −1 0 00 0 1 −1

]= [ kr γrx1 kpx1 γpx2 ]T

We want to control the average number of proteins by suitably acting on thetranscription rate kr

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 10/14

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Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Gene expression control

TheoremFor any values of the system parameters kp, γr, γp > 0 and the control parametersµ, k, η > 0, the closed-loop network is ergodic and we have that E[X2(t)]→ µ ast→∞ globally.

0 10 20 30 40 50 60 700

2

4

6

8

10

12

14

16

18

Time t

Pop

ulation

[Molecules]

X1(t)X2(t)Z1(t)Z2(t)

0 10 20 30 40 50 60 700

1

2

3

4

5

6

7

8

9

10

Time

Pop

ulation

averages

[Molecules]

E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 11/14

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Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Comparison with deterministic control

Deterministic

x1 = kz1 − γrx1x2 = kpx1 − γpx2z1 = µ− ηz1z2z2 = x2 − ηz1z2

0 5 10 15 20 25 300

1

2

3

4

5

6

7

Time

Pop

ulation

concentrations

x1(t)x2(t)z1(t)z2(t)

Stochastic

E[X1] = kE[Z1]− γrE[X1]

E[X2] = kpE[X1]− γpE[X2]

E[Z1] = µ− ηE[Z1]E[Z2]−ηV (Z1, Z2)

E[Z2] = E[X2]− ηE[Z1]E[Z2]−ηV (Z1, Z2)

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

Time

Pop

ulation

averages

[Molecules]

E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 12/14

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Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Comparison with deterministic control

Deterministic

x1 = kz1 − γrx1x2 = kpx1 − γpx2z1 = µ− ηz1z2z2 = x2 − ηz1z2

0 5 10 15 20 25 300

1

2

3

4

5

6

7

Time

Pop

ulation

concentrations

x1(t)x2(t)z1(t)z2(t)

Stochastic

E[X1] = kE[Z1]− γrE[X1]

E[X2] = kpE[X1]− γpE[X2]

E[Z1] = µ− ηE[Z1]E[Z2]−ηV (Z1, Z2)

E[Z2] = E[X2]− ηE[Z1]E[Z2]−ηV (Z1, Z2)

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

Time

Pop

ulation

averages

[Molecules]

E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 12/14

Page 30: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Robustness - Perfect adaptation

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averages

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E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]

(a) Perturbation of the controller gain k

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E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]

(b) Perturbation of the translation rate kp

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(c) Perturbation of the mRNA degradation rate

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E[X1(t)]E[X2(t)]E[Z1(t)]E[Z2(t)]

(d) Perturbation of the protein degradation rate

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Page 31: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Concluding statements

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Page 32: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Concluding statements

What has been done• In-vivo (integral) control motif seems promising• Population control• Perfect adaptation

What needs to be done• Implementation• Extensions: bimolecular networks, different inputs, multiple inputs/outputs,

different control motifs→ biomolecular control theory

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 14/14

Page 33: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Concluding statements

What has been done• In-vivo (integral) control motif seems promising• Population control• Perfect adaptation

What needs to be done• Implementation• Extensions: bimolecular networks, different inputs, multiple inputs/outputs,

different control motifs→ biomolecular control theory

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 14/14

Page 34: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Thank you for your attention

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Page 35: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Computational results

TheoremThe following statements are equivalent:

(a) The matrix A is Hurwitz and the triplet (A, e1, eT` ) is output-controllable.

(b) There exist v ∈ Rd>0 and w ∈ Rd≥0 with wT e1 > 0, wT e` > 0, such that

vTA < 0 and wTA+ eT` = 0.

Comments• Linear program• Can be robustified→ if A ∈ [A−, A+], then vT+A

+ < 0 and wT−A− + eT` = 0.

• Can be made structural→ A ∈ {, 0,⊕}d×d

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 14/14

Page 36: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Computational results

TheoremThe following statements are equivalent:

(a) The matrix A is Hurwitz and the triplet (A, e1, eT` ) is output-controllable.

(b) There exist v ∈ Rd>0 and w ∈ Rd≥0 with wT e1 > 0, wT e` > 0, such that

vTA < 0 and wTA+ eT` = 0.

Comments• Linear program• Can be robustified→ if A ∈ [A−, A+], then vT+A

+ < 0 and wT−A− + eT` = 0.

• Can be made structural→ A ∈ {, 0,⊕}d×d

Corentin Briat A Control Theory for Stochastic Biomolecular Regulation 14/14

Page 37: A Control Theory for Stochastic Biomolecular Regulation · Lots of analytical tools, e.g. reaction network theory, dynamical systems theory, Lyapunov theory of stability, nonlinear

Introduction In-vivo control - Theory In-vivo control - Example Conclusion

Implementation

Bacterial DNA Plasmids

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