An integrated framework for analysis of stochastic models of biochemical reactions Michal Komorowski Imperial College London Theoretical Systems Biology Group 21/03/11 Michal Komorowski Stochastic biochemical reactions 21/03/11 1 / 31
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
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
Fluorescent reporter genes
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 3 / 31
Fluorescent reporter genes
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 3 / 31
Fluorescent microscopy and flow cytometry
0 5 10 15 20 25
200
225
250
275
300 A
0 5 10 15 20 25
100
200
300
B
fluo
resc
ence
(a.
u.)
0 5 10 15 20 25
100
150
200
250
300
C
0 5 10 15 20 25
100
150
200
250
300
time (hours)
D
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
Fluorescent microscopy and flow cytometry
0 5 10 15 20 25
200
225
250
275
300 A
0 5 10 15 20 25
100
200
300
B
fluo
resc
ence
(a.
u.)
0 5 10 15 20 25
100
150
200
250
300
C
0 5 10 15 20 25
100
150
200
250
300
time (hours)
D
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
Fluorescent microscopy and flow cytometry
0 5 10 15 20 25
200
225
250
275
300 A
0 5 10 15 20 25
100
200
300
B
fluo
resc
ence
(a.
u.)
0 5 10 15 20 25
100
150
200
250
300
C
0 5 10 15 20 25
100
150
200
250
300
time (hours)
D
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
Fluorescent microscopy and flow cytometry
0 5 10 15 20 25
200
225
250
275
300 A
0 5 10 15 20 25
100
200
300
B
fluo
resc
ence
(a.
u.)
0 5 10 15 20 25
100
150
200
250
300
C
0 5 10 15 20 25
100
150
200
250
300
time (hours)
D
Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications
SensitivityRobustness - global sensitivity analysisInformation content of data
Optimal experimental design
Idetifiability
Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
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
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
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
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
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
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
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
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
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
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
Example: expression of a gene
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 22 / 31
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Which parameters are involved in controllingstochastic effects?
TS - heatmap, DT - contour plot
Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 28 / 31
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
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
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
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
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
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
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
Thank you!
Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 31 / 31