an integrated framework for analysis of stochastic models of biochemical reactions
TRANSCRIPT
![Page 1: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/1.jpg)
An integrated framework for analysis of stochasticmodels of biochemical reactions
Michał Komorowski
Imperial College LondonTheoretical Systems Biology Group
21/03/11
Michał Komorowski Stochastic biochemical reactions 21/03/11 1 / 31
![Page 2: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/2.jpg)
Outline
1 Motivation: models and data
2 Modeling framework
3 Inference: examples
4 Sensitivity, Fisher Information, statistical model analysis
Michał Komorowski Stochastic biochemical reactions 21/03/11 2 / 31
![Page 3: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/3.jpg)
Fluorescent reporter genes
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 3 / 31
![Page 4: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/4.jpg)
Fluorescent reporter genes
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 3 / 31
![Page 5: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/5.jpg)
Fluorescent microscopy and flow cytometry
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Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
![Page 6: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/6.jpg)
Fluorescent microscopy and flow cytometry
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Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
![Page 7: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/7.jpg)
Fluorescent microscopy and flow cytometry
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Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
![Page 8: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/8.jpg)
Fluorescent microscopy and flow cytometry
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Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
![Page 9: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/9.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...l
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fl(x,Θ))
ParametersΘ = (θ1, ..., θr)
x is a Poisson birth and death process
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
![Page 10: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/10.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...l
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fl(x,Θ))
ParametersΘ = (θ1, ..., θr)
x is a Poisson birth and death process
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
![Page 11: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/11.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...l
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fl(x,Θ))
ParametersΘ = (θ1, ..., θr)
x is a Poisson birth and death process
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
![Page 12: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/12.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...l
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fl(x,Θ))
ParametersΘ = (θ1, ..., θr)
x is a Poisson birth and death process
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
![Page 13: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/13.jpg)
Chemical kinetics model
System’s state
x = (x1, . . . , xN)T
Stoichiometry matrix
S = {Sij}i=1,2...N; j=1,2...l
(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)
Reaction rates
F(x,Θ) = (f1(x,Θ), ..., fl(x,Θ))
ParametersΘ = (θ1, ..., θr)
x is a Poisson birth and death process
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
![Page 14: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/14.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 15: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/15.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 16: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/16.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 17: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/17.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 18: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/18.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 19: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/19.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 20: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/20.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 21: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/21.jpg)
Example: gene expression
State x = (r, p)
Stoichiometry
S =
(1 −1 0 00 0 1 −1
)Rates
F(x,Θ) = (kr, γrr, kpr, γpp)
ParametersΘ = (kr, γr, kp, γp)
Macroscopic rate equation
φR = kR(t)− γRφR
φP = kPφR − γPφP
Diffusion approximation
dR = (kR(t)− γRR)dt +√
kR + γRRdWR
dP = (kPR− γPP)dt +√
kPR + γPPdWP
Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)
dξR = (−γRξR)dt +√
kR(t) + γRφRdWξR ,
dξP = (kPξR − γPξP)dt +√
kPφP + γPφPdWξP
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
![Page 22: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/22.jpg)
Modelling chemical kineticsChemical master equation
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
![Page 23: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/23.jpg)
Modelling chemical kineticsChemical master equation
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
![Page 24: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/24.jpg)
Modelling chemical kineticsChemical master equation
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
![Page 25: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/25.jpg)
Modelling chemical kineticsChemical master equation
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
![Page 26: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/26.jpg)
How about inference ?
Chemical master equation
(likelihood-free methods, e.g. ABC)
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation
(least squares)
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
(data augmentation)
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
(explicite likelihood)
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
![Page 27: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/27.jpg)
How about inference ?Chemical master equation
(likelihood-free methods, e.g. ABC)
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation
(least squares)
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
(data augmentation)
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
(explicite likelihood)
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
![Page 28: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/28.jpg)
How about inference ?Chemical master equation (likelihood-free methods, e.g. ABC)
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation
(least squares)
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
(data augmentation)
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
(explicite likelihood)
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
![Page 29: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/29.jpg)
How about inference ?Chemical master equation (likelihood-free methods, e.g. ABC)
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation (least squares)
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation
(data augmentation)
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
(explicite likelihood)
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
![Page 30: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/30.jpg)
How about inference ?Chemical master equation (likelihood-free methods, e.g. ABC)
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation (least squares)
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation (data augmentation)
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation
(explicite likelihood)
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
![Page 31: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/31.jpg)
How about inference ?Chemical master equation (likelihood-free methods, e.g. ABC)
dPt(x)
dt=
l∑j=1
Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)
Macroscopic rate equation (least squares)
dϕdt
= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))
Diffusion approximation (data augmentation)
dx = S F(x)dt + S(
diag{√
F(x)})
dW
Linear noise approximation (explicite likelihood)
x(t) = ϕ(t) + ξ(t)
dξ = SOϕF(ϕ)ξdt + S(
diag{√
F(ϕ)})
dW
Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
![Page 32: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/32.jpg)
Model equations
LNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
![Page 33: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/33.jpg)
Model equations
LNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
![Page 34: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/34.jpg)
Model equations
LNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
![Page 35: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/35.jpg)
Model equations
LNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
![Page 36: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/36.jpg)
Distribution of dataVector of measurements
xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}
time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data
xQ ∼ MVN(µ(Θ),ΣQ(Θ))
µ(Θ) = (ϕ(t1), ..., ϕ(tn))
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 10 / 31
![Page 37: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/37.jpg)
Distribution of dataVector of measurements
xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}
time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data
xQ ∼ MVN(µ(Θ),ΣQ(Θ))
µ(Θ) = (ϕ(t1), ..., ϕ(tn))
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 10 / 31
![Page 38: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/38.jpg)
Distribution of dataVector of measurements
xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}
time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data
xQ ∼ MVN(µ(Θ),ΣQ(Θ))
µ(Θ) = (ϕ(t1), ..., ϕ(tn))
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 10 / 31
![Page 39: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/39.jpg)
Distribution of dataVector of measurements
xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}
time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data
xQ ∼ MVN(µ(Θ),ΣQ(Θ))
µ(Θ) = (ϕ(t1), ..., ϕ(tn))
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 10 / 31
![Page 40: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/40.jpg)
Distribution of dataVector of measurements
xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}
time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data
xQ ∼ MVN(µ(Θ),ΣQ(Θ))
µ(Θ) = (ϕ(t1), ..., ϕ(tn))
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 10 / 31
![Page 41: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/41.jpg)
Distribution of dataVector of measurements
xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}
time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data
xQ ∼ MVN(µ(Θ),ΣQ(Θ))
µ(Θ) = (ϕ(t1), ..., ϕ(tn))
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 10 / 31
![Page 42: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/42.jpg)
Distribution of dataVector of measurements
xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}
time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data
xQ ∼ MVN(µ(Θ),ΣQ(Θ))
µ(Θ) = (ϕ(t1), ..., ϕ(tn))
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 10 / 31
![Page 43: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/43.jpg)
Advantages of the framework
Inference
Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
![Page 44: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/44.jpg)
Advantages of the framework
Inference
Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
![Page 45: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/45.jpg)
Advantages of the framework
Inference
Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
![Page 46: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/46.jpg)
Advantages of the framework
Inference
Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
![Page 47: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/47.jpg)
Advantages of the framework
Inference
Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
![Page 48: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/48.jpg)
Advantages of the framework
Inference
Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
![Page 49: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/49.jpg)
Advantages of the framework
Inference
Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error
Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
![Page 50: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/50.jpg)
Hierarchical model for degradation rates: CHXexperiment
0 2 4 6 8 10
010
2030
40
time (h)
fluor
esce
nce
leve
l
0.0 0.2 0.4 0.6 0.8 1.0
02
46
8
degradation rate
dens
ity
Model:dp = (kp − γpp)dt+
√kp + γpφp(t)dW
Rates differ between cells
γP ∼ Gamma(µγp , σ2γp)
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
![Page 51: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/51.jpg)
Hierarchical model for degradation rates: CHXexperiment
0 2 4 6 8 10
010
2030
40
time (h)
fluor
esce
nce
leve
l
0.0 0.2 0.4 0.6 0.8 1.0
02
46
8
degradation rate
dens
ity
Model:dp = (kp − γpp)dt+
√kp + γpφp(t)dW
Rates differ between cells
γP ∼ Gamma(µγp , σ2γp)
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
![Page 52: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/52.jpg)
Hierarchical model for degradation rates: CHXexperiment
0 2 4 6 8 10
010
2030
40
time (h)
fluor
esce
nce
leve
l
0.0 0.2 0.4 0.6 0.8 1.0
02
46
8
degradation rate
dens
ity
Model:dp = (kp − γpp)dt+
√kp + γpφp(t)dW
Rates differ between cells
γP ∼ Gamma(µγp , σ2γp)
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
![Page 53: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/53.jpg)
Hierarchical model for degradation rates: CHXexperiment
0 2 4 6 8 10
010
2030
40
time (h)
fluor
esce
nce
leve
l
0.0 0.2 0.4 0.6 0.8 1.0
02
46
8
degradation rate
dens
ity
Model:dp = (kp − γpp)dt+
√kp + γpφp(t)dW
Rates differ between cells
γP ∼ Gamma(µγp , σ2γp)
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
![Page 54: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/54.jpg)
Hierarchical model for degradation rates: CHXexperiment
0 2 4 6 8 10
010
2030
40
time (h)
fluor
esce
nce
leve
l
0.0 0.2 0.4 0.6 0.8 1.0
02
46
8
degradation rate
dens
ity
Model:dp = (kp − γpp)dt+
√kp + γpφp(t)dW
Rates differ between cells
γP ∼ Gamma(µγp , σ2γp)
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
![Page 55: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/55.jpg)
Hierarchical model for degradation rates: CHXexperiment
0 2 4 6 8 10
010
2030
40
time (h)
fluor
esce
nce
leve
l
0.0 0.2 0.4 0.6 0.8 1.0
02
46
8
degradation rate
dens
ity
Model:dp = (kp − γpp)dt+
√kp + γpφp(t)dW
Rates differ between cells
γP ∼ Gamma(µγp , σ2γp)
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
![Page 56: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/56.jpg)
DRB experiment
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
GFP
Flu
ores
cenc
e
Time (hours)
Model:
dr = (kr − γrr)dt+√
kr + γrφr(t)dWr
dp = (kpr − γpp)dt +√
kpφr(t) + γpφr(t)dWp
We can estimate
γr ∼ Gamma(µγr , σ2γr )
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
![Page 57: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/57.jpg)
DRB experiment
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
GFP
Flu
ores
cenc
e
Time (hours)
Model:
dr = (kr − γrr)dt+√
kr + γrφr(t)dWr
dp = (kpr − γpp)dt +√
kpφr(t) + γpφr(t)dWp
We can estimate
γr ∼ Gamma(µγr , σ2γr )
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
![Page 58: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/58.jpg)
DRB experiment
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
GFP
Flu
ores
cenc
e
Time (hours)
Model:
dr = (kr − γrr)dt+√
kr + γrφr(t)dWr
dp = (kpr − γpp)dt +√
kpφr(t) + γpφr(t)dWp
We can estimate
γr ∼ Gamma(µγr , σ2γr )
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
![Page 59: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/59.jpg)
DRB experiment
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
GFP
Flu
ores
cenc
e
Time (hours)
Model:
dr = (kr − γrr)dt+√
kr + γrφr(t)dWr
dp = (kpr − γpp)dt +√
kpφr(t) + γpφr(t)dWp
We can estimate
γr ∼ Gamma(µγr , σ2γr )
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
![Page 60: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/60.jpg)
DRB experiment
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
450
GFP
Flu
ores
cenc
e
Time (hours)
Model:
dr = (kr − γrr)dt+√
kr + γrφr(t)dWr
dp = (kpr − γpp)dt +√
kpφr(t) + γpφr(t)dWp
We can estimate
γr ∼ Gamma(µγr , σ2γr )
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
![Page 61: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/61.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
0 5 10 15 20 25
020
040
060
080
0
time (hours)
fluor
esce
nce
inte
nsity
(a.
u.)
Experiment: Claire Harper, Mike White;Department of Biology, University of Liverpool
Observed fluorescence andtime-course of endogenous proteindifferGH3 rat pituitary cells with EGFPlinked to prolactin gene promoterTrascription is triggered at the start ofthe experimentNo data on mRNA levelInformative prior on mRNA andprotein degradation rate
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
![Page 62: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/62.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
0 5 10 15 20 25
020
040
060
080
0
time (hours)
fluor
esce
nce
inte
nsity
(a.
u.)
Experiment: Claire Harper, Mike White;Department of Biology, University of Liverpool
Observed fluorescence andtime-course of endogenous proteindifferGH3 rat pituitary cells with EGFPlinked to prolactin gene promoterTrascription is triggered at the start ofthe experimentNo data on mRNA levelInformative prior on mRNA andprotein degradation rate
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
![Page 63: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/63.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
0 5 10 15 20 25
020
040
060
080
0
time (hours)
fluor
esce
nce
inte
nsity
(a.
u.)
Experiment: Claire Harper, Mike White;Department of Biology, University of Liverpool
Observed fluorescence andtime-course of endogenous proteindifferGH3 rat pituitary cells with EGFPlinked to prolactin gene promoterTrascription is triggered at the start ofthe experimentNo data on mRNA levelInformative prior on mRNA andprotein degradation rate
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
![Page 64: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/64.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
0 5 10 15 20 25
020
040
060
080
0
time (hours)
fluor
esce
nce
inte
nsity
(a.
u.)
Experiment: Claire Harper, Mike White;Department of Biology, University of Liverpool
Observed fluorescence andtime-course of endogenous proteindifferGH3 rat pituitary cells with EGFPlinked to prolactin gene promoterTrascription is triggered at the start ofthe experimentNo data on mRNA levelInformative prior on mRNA andprotein degradation rate
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
![Page 65: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/65.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
0 5 10 15 20 25
020
040
060
080
0
time (hours)
fluor
esce
nce
inte
nsity
(a.
u.)
Experiment: Claire Harper, Mike White;Department of Biology, University of Liverpool
Observed fluorescence andtime-course of endogenous proteindifferGH3 rat pituitary cells with EGFPlinked to prolactin gene promoterTrascription is triggered at the start ofthe experimentNo data on mRNA levelInformative prior on mRNA andprotein degradation rate
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
![Page 66: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/66.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
Calculating back to the transcription level
Model:
dr = (kr(t)− γrr)dt+√
kr(t) + γrr dWr
dp = (kpr − γpp)dt +√
kpr + γppdWp
p(obs) = λp
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
![Page 67: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/67.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
Calculating back to the transcription level
Model:
dr = (kr(t)− γrr)dt+√
kr(t) + γrr dWr
dp = (kpr − γpp)dt +√
kpr + γppdWp
p(obs) = λp
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
![Page 68: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/68.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
Calculating back to the transcription level
Model:
dr = (kr(t)− γrr)dt+√
kr(t) + γrr dWr
dp = (kpr − γpp)dt +√
kpr + γppdWp
p(obs) = λp
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
![Page 69: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/69.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
Calculating back to the transcription level
Model:
dr = (kr(t)− γrr)dt
+√
kr(t) + γrr dWr
dp = (kpr − γpp)dt
+√
kpr + γppdWp
p(obs) = λp
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
![Page 70: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/70.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
Calculating back to the transcription level
Model:
dr = (kr(t)− γrr)dt+√
kr(t) + γrr dWr
dp = (kpr − γpp)dt +√
kpr + γppdWp
p(obs) = λp
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
![Page 71: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/71.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
Calculating back to the transcription level
Model:
dr = (kr(t)− γrr)dt+√
kr(t) + γrr dWr
dp = (kpr − γpp)dt +√
kpr + γppdWp
p(obs) = λp
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
![Page 72: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/72.jpg)
Fluorescent proteins as transcriptional reporters insingle cells
Calculating back to the transcription level
Model:
dr = (kr(t)− γrr)dt+√
kr(t) + γrr dWr
dp = (kpr − γpp)dt +√
kpr + γppdWp
p(obs) = λp
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
![Page 73: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/73.jpg)
Inference results
We estimated scaling factor λ = 2.11 (1.24 - 3.56)Translation in absolute units kp =0.46 (0.14 - 1.51)Transcription profile in absolute units
Finkenstadt B., Heron E.,Komorowski M. et al.Reconstruction of transcriptional dynamics, Bioinformatics 24, 2008
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 16 / 31
![Page 74: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/74.jpg)
Inference results
We estimated scaling factor λ = 2.11 (1.24 - 3.56)Translation in absolute units kp =0.46 (0.14 - 1.51)Transcription profile in absolute units
Finkenstadt B., Heron E.,Komorowski M. et al.Reconstruction of transcriptional dynamics, Bioinformatics 24, 2008
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 16 / 31
![Page 75: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/75.jpg)
Inference results
We estimated scaling factor λ = 2.11 (1.24 - 3.56)Translation in absolute units kp =0.46 (0.14 - 1.51)Transcription profile in absolute units
Finkenstadt B., Heron E.,Komorowski M. et al.Reconstruction of transcriptional dynamics, Bioinformatics 24, 2008
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 16 / 31
![Page 76: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/76.jpg)
Sensitivity for stochastic systems: motivation
Difference in response to perturbations in parametersDeterministic model ( DT) e.g. population averageTime-series stochastic model (TS) e.g. fluorescent microscopyTime-point stochastic model (TP) e.g. flow cytometry
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
![Page 77: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/77.jpg)
Sensitivity for stochastic systems: motivation
Difference in response to perturbations in parametersDeterministic model ( DT) e.g. population averageTime-series stochastic model (TS) e.g. fluorescent microscopyTime-point stochastic model (TP) e.g. flow cytometry
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
![Page 78: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/78.jpg)
Sensitivity for stochastic systems: motivation
Difference in response to perturbations in parametersDeterministic model ( DT) e.g. population averageTime-series stochastic model (TS) e.g. fluorescent microscopyTime-point stochastic model (TP) e.g. flow cytometry
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
![Page 79: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/79.jpg)
Sensitivity for stochastic systems: motivation
Difference in response to perturbations in parametersDeterministic model ( DT) e.g. population averageTime-series stochastic model (TS) e.g. fluorescent microscopyTime-point stochastic model (TP) e.g. flow cytometry
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
![Page 80: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/80.jpg)
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
![Page 81: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/81.jpg)
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
![Page 82: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/82.jpg)
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
![Page 83: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/83.jpg)
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
![Page 84: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/84.jpg)
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
![Page 85: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/85.jpg)
Sensitivity and Fisher Information
Classical sensitivity coefficients for an observable X andparameter θ
∂X∂θ
Stochastic case: observable X is drawn from a distribution ψ
I(θ) = E(∂ logψ(X, θ)
∂θ
)2
For stochastic model of chemical reactions evaluated using MonteCarlo simulationsCan be evaluated via numerical integration of ODEs
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
![Page 86: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/86.jpg)
Sensitivity and Fisher Information
Classical sensitivity coefficients for an observable X andparameter θ
∂X∂θ
Stochastic case: observable X is drawn from a distribution ψ
I(θ) = E(∂ logψ(X, θ)
∂θ
)2
For stochastic model of chemical reactions evaluated using MonteCarlo simulationsCan be evaluated via numerical integration of ODEs
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
![Page 87: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/87.jpg)
Sensitivity and Fisher Information
Classical sensitivity coefficients for an observable X andparameter θ
∂X∂θ
Stochastic case: observable X is drawn from a distribution ψ
I(θ) = E(∂ logψ(X, θ)
∂θ
)2
For stochastic model of chemical reactions evaluated using MonteCarlo simulationsCan be evaluated via numerical integration of ODEs
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
![Page 88: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/88.jpg)
Sensitivity and Fisher Information
Classical sensitivity coefficients for an observable X andparameter θ
∂X∂θ
Stochastic case: observable X is drawn from a distribution ψ
I(θ) = E(∂ logψ(X, θ)
∂θ
)2
For stochastic model of chemical reactions evaluated using MonteCarlo simulationsCan be evaluated via numerical integration of ODEs
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
![Page 89: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/89.jpg)
Model equations - reminderLNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Fisher information
I(θ) =∂µ
∂θ
TΣ(θ)
∂µ
∂θ+
12
trace(Σ−1∂Σ
∂θΣ−1∂Σ
∂θ)
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
![Page 90: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/90.jpg)
Model equations - reminderLNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Fisher information
I(θ) =∂µ
∂θ
TΣ(θ)
∂µ
∂θ+
12
trace(Σ−1∂Σ
∂θΣ−1∂Σ
∂θ)
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
![Page 91: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/91.jpg)
Model equations - reminderLNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Fisher information
I(θ) =∂µ
∂θ
TΣ(θ)
∂µ
∂θ+
12
trace(Σ−1∂Σ
∂θΣ−1∂Σ
∂θ)
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
![Page 92: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/92.jpg)
Model equations - reminderLNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Fisher information
I(θ) =∂µ
∂θ
TΣ(θ)
∂µ
∂θ+
12
trace(Σ−1∂Σ
∂θΣ−1∂Σ
∂θ)
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
![Page 93: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/93.jpg)
Model equations - reminderLNA implies Gaussian distribution
x(t) ∼ MVN(ϕ(t),V(t))
Mean ϕ(t) given as s solution of the rate equationVariances
dV(t)dt
= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T
Covariances
cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t
dΦ(ti, s)ds
= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I
Fisher information
I(θ) =∂µ
∂θ
TΣ(θ)
∂µ
∂θ+
12
trace(Σ−1∂Σ
∂θΣ−1∂Σ
∂θ)
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
![Page 94: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/94.jpg)
Model equations - reminder
Fisher information
I(θ) =∂µ
∂θ
TΣ(θ)
∂µ
∂θ+
12
trace(Σ−1∂Σ
∂θΣ−1∂Σ
∂θ)
Covariance matrix
ΣQ(Θ)(i,j) =
V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}
V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 21 / 31
![Page 95: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/95.jpg)
Example: expression of a gene
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 22 / 31
![Page 96: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/96.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influence of correlation between RNA and protein
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
correlation=0.24218
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
![Page 97: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/97.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influence of correlation between RNA and protein
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
correlation=0.24218
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
![Page 98: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/98.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influence of correlation between RNA and protein
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
correlation=0.53838
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
![Page 99: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/99.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influence of correlation between RNA and protein
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
correlation=0.92828
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
![Page 100: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/100.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influence of temporal correlations
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
=30
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
![Page 101: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/101.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influence of temporal correlations
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
=30
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
![Page 102: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/102.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influence of temporal correlations
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
=3
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
![Page 103: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/103.jpg)
Response to parameter perturbations:stochastic vs deterministic case
Influance of temporal correlations
k r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
k p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
p
kr
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
kp
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
r0.1 0.05 0 0.05 0.1
0.1
0.05
0
0.05
0.1
=0.3
p
0.1 0.05 0 0.05 0.10.1
0.05
0
0.05
0.1
StochasticDeterministic
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
![Page 104: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/104.jpg)
Amount of information in the data
Only protein level is measuredMeasurements are taken from a stationary state
# of identifiable parameters(non-zero eigenvalues)
optimal sampling frequency
Type TS TP DTStationary 4 2 1Perturbation 4 4 3
Perturbation: 5-fold increased initial conditions
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
det(
FIM
)
set 1set 2set 3set 4
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
![Page 105: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/105.jpg)
Amount of information in the data
Only protein level is measuredMeasurements are taken from a stationary state
# of identifiable parameters(non-zero eigenvalues)
optimal sampling frequency
Type TS TP DTStationary 4 2 1Perturbation 4 4 3
Perturbation: 5-fold increased initial conditions
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
det(
FIM
)
set 1set 2set 3set 4
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
![Page 106: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/106.jpg)
Amount of information in the data
Only protein level is measuredMeasurements are taken from a stationary state
# of identifiable parameters(non-zero eigenvalues)
optimal sampling frequency
Type TS TP DTStationary 4 2 1Perturbation 4 4 3
Perturbation: 5-fold increased initial conditions
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
det(
FIM
)
set 1set 2set 3set 4
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
![Page 107: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/107.jpg)
Amount of information in the data
Only protein level is measuredMeasurements are taken from a stationary state
# of identifiable parameters(non-zero eigenvalues)
optimal sampling frequency
Type TS TP DTStationary 4 2 1Perturbation 4 4 3
Perturbation: 5-fold increased initial conditions
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
det(
FIM
)
set 1set 2set 3set 4
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
![Page 108: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/108.jpg)
Amount of information in the data
Only protein level is measuredMeasurements are taken from a stationary state
# of identifiable parameters(non-zero eigenvalues)
optimal sampling frequency
Type TS TP DTStationary 4 2 1Perturbation 4 4 3
Perturbation: 5-fold increased initial conditions
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
det(
FIM
)
set 1set 2set 3set 4
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
![Page 109: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/109.jpg)
Amount of information in the data
Only protein level is measuredMeasurements are taken from a stationary state
# of identifiable parameters(non-zero eigenvalues)
optimal sampling frequency
Type TS TP DTStationary 4 2 1Perturbation 4 4 3
Perturbation: 5-fold increased initial conditions
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
det(
FIM
)
set 1set 2set 3set 4
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
![Page 110: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/110.jpg)
Amount of information in the data
Only protein level is measuredMeasurements are taken from a stationary state
# of identifiable parameters(non-zero eigenvalues)
optimal sampling frequency
Type TS TP DTStationary 4 2 1Perturbation 4 4 3
Perturbation: 5-fold increased initial conditions
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
10
20
30
40
50
60
70
80
det(
FIM
)
set 1set 2set 3set 4
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
![Page 111: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/111.jpg)
p53 systemp53 protein regulates cell cycle, response to DNA damage and it is atumour repressor.
x = (p, y0, y).
p - p53y0 - mdm2 precursory - mdm2
Deterministic version:
φp = βx − αxφp − αkφyφp
φp + k
φy0 = βyφp − α0φy0
φy = α0φy0 − αyφy.
Parameter vector
Θ = (βx, αx, αk, k, βy, α0, αy).
Role of parameters: which parameters control stochastic effects inthe model?Fluorescent microscopy or flow cytometry?
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
![Page 112: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/112.jpg)
p53 systemp53 protein regulates cell cycle, response to DNA damage and it is atumour repressor.
x = (p, y0, y).
p - p53y0 - mdm2 precursory - mdm2
Deterministic version:
φp = βx − αxφp − αkφyφp
φp + k
φy0 = βyφp − α0φy0
φy = α0φy0 − αyφy.
Parameter vector
Θ = (βx, αx, αk, k, βy, α0, αy).
Role of parameters: which parameters control stochastic effects inthe model?Fluorescent microscopy or flow cytometry?
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
![Page 113: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/113.jpg)
p53 systemp53 protein regulates cell cycle, response to DNA damage and it is atumour repressor.
x = (p, y0, y).
p - p53y0 - mdm2 precursory - mdm2
Deterministic version:
φp = βx − αxφp − αkφyφp
φp + k
φy0 = βyφp − α0φy0
φy = α0φy0 − αyφy.
Parameter vector
Θ = (βx, αx, αk, k, βy, α0, αy).
Role of parameters: which parameters control stochastic effects inthe model?Fluorescent microscopy or flow cytometry?
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
![Page 114: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/114.jpg)
p53 systemp53 protein regulates cell cycle, response to DNA damage and it is atumour repressor.
x = (p, y0, y).
p - p53y0 - mdm2 precursory - mdm2
Deterministic version:
φp = βx − αxφp − αkφyφp
φp + k
φy0 = βyφp − α0φy0
φy = α0φy0 − αyφy.
Parameter vector
Θ = (βx, αx, αk, k, βy, α0, αy).
Role of parameters: which parameters control stochastic effects inthe model?Fluorescent microscopy or flow cytometry?
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
![Page 115: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/115.jpg)
p53 systemp53 protein regulates cell cycle, response to DNA damage and it is atumour repressor.
x = (p, y0, y).
p - p53y0 - mdm2 precursory - mdm2
Deterministic version:
φp = βx − αxφp − αkφyφp
φp + k
φy0 = βyφp − α0φy0
φy = α0φy0 − αyφy.
Parameter vector
Θ = (βx, αx, αk, k, βy, α0, αy).
Role of parameters: which parameters control stochastic effects inthe model?Fluorescent microscopy or flow cytometry?
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
![Page 116: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/116.jpg)
Role of parameters
1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1Eigen values normalized against model maximum
TSTPDT
1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1Eigen values normalized against total maximum
TSTPDT
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 27 / 31
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Which parameters are involved in controllingstochastic effects?
TS - heatmap, DT - contour plot
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 28 / 31
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Fluorescent microscopy vs flow cytometry
0 1 2 3 4 5 6 7 8 9 10x 104
0
2
4
6
8
10
12x 1023
Number of TP measurements per time point
det(
FIM
)
TPTS
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 29 / 31
![Page 119: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/119.jpg)
Summary
Efficient and simple inference framework for stochastic systemsFisher Information Matrix for stochastic models can berepresented as solutions of ODEsSubstantial differences is sensitivities between stochastic anddeterministic models may existApplicability experimental designMatlab package for sensitivity of stochastic systems availablewww.theosysbio.bio.ic.ac.uk/resources/stns/
Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemicalkinetics models, PNAS in press, 2011.
Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise andother parameters of gene expression, Biophysical J., 98, 2010.
Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using thelinear noise approximation, BMC Bioinformatics, 10, 2009;
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
![Page 120: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/120.jpg)
Summary
Efficient and simple inference framework for stochastic systemsFisher Information Matrix for stochastic models can berepresented as solutions of ODEsSubstantial differences is sensitivities between stochastic anddeterministic models may existApplicability experimental designMatlab package for sensitivity of stochastic systems availablewww.theosysbio.bio.ic.ac.uk/resources/stns/
Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemicalkinetics models, PNAS in press, 2011.
Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise andother parameters of gene expression, Biophysical J., 98, 2010.
Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using thelinear noise approximation, BMC Bioinformatics, 10, 2009;
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
![Page 121: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/121.jpg)
Summary
Efficient and simple inference framework for stochastic systemsFisher Information Matrix for stochastic models can berepresented as solutions of ODEsSubstantial differences is sensitivities between stochastic anddeterministic models may existApplicability experimental designMatlab package for sensitivity of stochastic systems availablewww.theosysbio.bio.ic.ac.uk/resources/stns/
Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemicalkinetics models, PNAS in press, 2011.
Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise andother parameters of gene expression, Biophysical J., 98, 2010.
Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using thelinear noise approximation, BMC Bioinformatics, 10, 2009;
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
![Page 122: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/122.jpg)
Summary
Efficient and simple inference framework for stochastic systemsFisher Information Matrix for stochastic models can berepresented as solutions of ODEsSubstantial differences is sensitivities between stochastic anddeterministic models may existApplicability experimental designMatlab package for sensitivity of stochastic systems availablewww.theosysbio.bio.ic.ac.uk/resources/stns/
Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemicalkinetics models, PNAS in press, 2011.
Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise andother parameters of gene expression, Biophysical J., 98, 2010.
Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using thelinear noise approximation, BMC Bioinformatics, 10, 2009;
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
![Page 123: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/123.jpg)
Summary
Efficient and simple inference framework for stochastic systemsFisher Information Matrix for stochastic models can berepresented as solutions of ODEsSubstantial differences is sensitivities between stochastic anddeterministic models may existApplicability experimental designMatlab package for sensitivity of stochastic systems availablewww.theosysbio.bio.ic.ac.uk/resources/stns/
Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemicalkinetics models, PNAS in press, 2011.
Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise andother parameters of gene expression, Biophysical J., 98, 2010.
Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using thelinear noise approximation, BMC Bioinformatics, 10, 2009;
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
![Page 124: An integrated framework for analysis of stochastic models of biochemical reactions](https://reader033.vdocument.in/reader033/viewer/2022060120/55933d2f1a28abca748b45d3/html5/thumbnails/124.jpg)
Acknowledgement
Michael StumpfImperial College London
Barbel FinkenstadWarwick University
Dan WoodcockWarwick University
David RandWarwick University
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 31 / 31
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Thank you!
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 31 / 31