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An integrated framework for analysis of stochastic models of biochemical reactions

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Michal Komorowski

This document presents an integrated framework for analyzing stochastic models of biochemical reactions. It describes modeling biochemical reactions as stochastic processes, including representing the system state, reaction rates, and parameters. An example of modeling gene expression is also presented, showing the macroscopic rate equation, diffusion approximation, and linear noise approximation.

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An integrated framework for analysis of stochastic models of biochemical reactions Michał Komorowski Imperial College London Theoretical 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 Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
Fluorescent microscopy and flow cytometry A B 300 300 275 250 200 225 200 100 fluorescence (a.u.) 0 5 10 15 20 25 0 5 10 15 20 25 C D 300 300 250 250 200 200 150 150 100 100 0 5 10 15 20 25 0 5 10 15 20 25 time (hours) Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
Fluorescent microscopy and flow cytometry A B 300 300 275 250 200 225 200 100 fluorescence (a.u.) 0 5 10 15 20 25 0 5 10 15 20 25 C D 300 300 250 250 200 200 150 150 100 100 0 5 10 15 20 25 0 5 10 15 20 25 time (hours) Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
Fluorescent microscopy and flow cytometry A B 300 300 275 250 200 225 200 100 fluorescence (a.u.) 0 5 10 15 20 25 0 5 10 15 20 25 C D 300 300 250 250 200 200 150 150 100 100 0 5 10 15 20 25 0 5 10 15 20 25 time (hours) 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)) Parameters Θ = (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)) Parameters Θ = (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)) Parameters Θ = (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)) Parameters Θ = (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)) Parameters Θ = (θ1 , ..., θr ) x is a Poisson birth and death process Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
How about inference ? Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
How about inference ? Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(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) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(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) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation (least squares) dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(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) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation (least squares) dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation (data augmentation) dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(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) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation (least squares) dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt 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 = S ϕ F(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 equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds 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 equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds 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 equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds 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 equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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 data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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 data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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 data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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 data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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 data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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 data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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 likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
Hierarchical model for degradation rates: CHX experiment 40 30 fluorescence level 20 10 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 Rates differ between cells 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 Rates differ between cells 2 0 0 2 4 6 8 10 γP ∼ Gamma(µγp , σγp ) time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 Rates differ between cells 2 0 0 2 4 6 8 10 γP ∼ Gamma(µγp , σγp ) time (h) 8 6 density 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 degradation rate Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
DRB experiment 450 400 350 300 GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
DRB experiment 450 Model: 400 350 300 GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
DRB experiment 450 Model: 400 350 dr = (kr − γr r)dt+ kr + γr φr (t)dWr 300 dp = (kp r − γp p)dt + kp φr (t) + γp φr (t)dWp GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
DRB experiment 450 Model: 400 350 dr = (kr − γr r)dt+ kr + γr φr (t)dWr 300 dp = (kp r − γp p)dt + kp φr (t) + γp φr (t)dWp GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
DRB experiment 450 Model: 400 350 dr = (kr − γr r)dt+ kr + γr φr (t)dWr 300 dp = (kp r − γp p)dt + kp φr (t) + γp φr (t)dWp GFP Fluorescence 250 200 150 100 We can estimate 50 2 0 0 2 4 6 8 Time (hours) 10 12 14 16 γr ∼ Gamma(µγr , σγr ) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt dp = (kp r − γp p)dt Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt+ kr (t) + γr r dWr dp = (kp r − γp p)dt + kp r + γp pdWp Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt+ kr (t) + γr r dWr dp = (kp r − γp p)dt + kp r + γp pdWp Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt+ kr (t) + γr r dWr dp = (kp r − γp p)dt + kp r + γp pdWp 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 parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-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 parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-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 parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-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 parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-point stochastic model (TP) e.g. flow cytometry Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
Implications Sensitivity Robustness - global sensitivity analysis Information 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 and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can 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 and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can 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 and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can 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 and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can be evaluated via numerical integration of ODEs Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
Model equations - reminder Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Covariance matrix    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  (i,j) ΣQ (Θ) =   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 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 Stochastic Deterministic correlation=0.24218 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p 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 correlation=0.53838 Stochastic 0.1 0.1 0.1 0.1 Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p 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 correlation=0.92828 Stochastic 0.1 0.1 0.1 0.1 Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
Response to parameter perturbations: stochastic vs deterministic case Influence of temporal correlations Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
Response to parameter perturbations: stochastic vs deterministic case Influence of temporal correlations =30 0.1 0.1 0.1 0.1 Stochastic 0.05 0.05 0.05 0.05 Deterministic 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
Response to parameter perturbations: stochastic vs deterministic case Influence of temporal correlations =3 0.1 0.1 0.1 0.1 Stochastic Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
Response to parameter perturbations: stochastic vs deterministic case Influance of temporal correlations =0.3 Stochastic 0.1 0.1 0.1 0.1 Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) 80 set 1 set 2 70 set 3 set 4 60 Type TS TP DT 50 Stationary 4 2 1 det( FIM ) 40 Perturbation 4 4 3 30 20 Perturbation: 5-fold increased initial conditions 10 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
Role of parameters Eigen values normalized against model maximum 1 TS TP 0.8 DT 0.6 0.4 0.2 0 1 2 3 4 5 6 7 Eigen values normalized against total maximum 1 TS TP 0.8 DT 0.6 0.4 0.2 0 1 2 3 4 5 6 7 Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 27 / 31
Which parameters are involved in controlling stochastic effects? TS - heatmap, DT - contour plot Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 28 / 31
Fluorescent microscopy vs flow cytometry 23 x 10 12 TP TS 10 8 det( FIM ) 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 Number of TP measurements per time point 4 x 10 Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 29 / 31
Summary Efficient and simple inference framework for stochastic systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear 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 systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear 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 systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear 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 systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear 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 systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear noise approximation, BMC Bioinformatics, 10, 2009; Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
Acknowledgement Michael Stumpf Imperial College London ¨ ¨ Barbel Finkenstad Warwick University Dan Woodcock Warwick University David Rand Warwick 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

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An integrated framework for analysis of stochastic models of biochemical reactions

  • 1. An integrated framework for analysis of stochastic models of biochemical reactions Michał Komorowski Imperial College London Theoretical Systems Biology Group 21/03/11 Michał Komorowski Stochastic biochemical reactions 21/03/11 1 / 31
  • 2. 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
  • 3. Fluorescent reporter genes Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 3 / 31
  • 4. Fluorescent reporter genes Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 3 / 31
  • 5. Fluorescent microscopy and flow cytometry Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
  • 6. Fluorescent microscopy and flow cytometry A B 300 300 275 250 200 225 200 100 fluorescence (a.u.) 0 5 10 15 20 25 0 5 10 15 20 25 C D 300 300 250 250 200 200 150 150 100 100 0 5 10 15 20 25 0 5 10 15 20 25 time (hours) Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
  • 7. Fluorescent microscopy and flow cytometry A B 300 300 275 250 200 225 200 100 fluorescence (a.u.) 0 5 10 15 20 25 0 5 10 15 20 25 C D 300 300 250 250 200 200 150 150 100 100 0 5 10 15 20 25 0 5 10 15 20 25 time (hours) Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
  • 8. Fluorescent microscopy and flow cytometry A B 300 300 275 250 200 225 200 100 fluorescence (a.u.) 0 5 10 15 20 25 0 5 10 15 20 25 C D 300 300 250 250 200 200 150 150 100 100 0 5 10 15 20 25 0 5 10 15 20 25 time (hours) Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 4 / 31
  • 9. 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)) Parameters Θ = (θ1 , ..., θr ) x is a Poisson birth and death process Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
  • 10. 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)) Parameters Θ = (θ1 , ..., θr ) x is a Poisson birth and death process Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
  • 11. 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)) Parameters Θ = (θ1 , ..., θr ) x is a Poisson birth and death process Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
  • 12. 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)) Parameters Θ = (θ1 , ..., θr ) x is a Poisson birth and death process Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
  • 13. 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)) Parameters Θ = (θ1 , ..., θr ) x is a Poisson birth and death process Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31
  • 14. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 15. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 16. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 17. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 18. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 19. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 20. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 21. Example: gene expression Macroscopic rate equation ˙ φR = kR (t) − γR φR ˙ φP = kP φR − γP φP State x = (r, p) Diffusion approximation Stoichiometry dR = (kR (t) − γR R)dt + kR + γR RdWR 1 −1 0 0 S= dP = (kP R − γP P)dt + kP R + γP PdWP 0 0 1 −1 Rates Linear noise approximation R(t) = φR (t) + ξR (t) P(t) = φP (t) + ξP (t) F(x, Θ) = (kr , γr r, kp r, γp p) dξR = (−γR ξR )dt + kR (t) + γR φR dWξR , Parameters dξP = (kP ξR − γP ξP )dt + kP φP + γP φP dWξP Θ = (kr , γr , kp , γp ) Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 6 / 31
  • 22. Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
  • 23. Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
  • 24. Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
  • 25. Modelling chemical kinetics Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 7 / 31
  • 26. How about inference ? Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
  • 27. How about inference ? Chemical master equation l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
  • 28. How about inference ? Chemical master equation (likelihood-free methods, e.g. ABC) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
  • 29. How about inference ? Chemical master equation (likelihood-free methods, e.g. ABC) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation (least squares) dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
  • 30. How about inference ? Chemical master equation (likelihood-free methods, e.g. ABC) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation (least squares) dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt Diffusion approximation (data augmentation) dx = S F(x)dt + S diag F(x) dW Linear noise approximation x(t) = ϕ(t) + ξ(t) dξ = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
  • 31. How about inference ? Chemical master equation (likelihood-free methods, e.g. ABC) l dPt (x) = Pt (x − S·j )fj (x − S·j ) − Pt (x)fj (x) dt j=1 Macroscopic rate equation (least squares) dϕ = S F(ϕ) F(ϕ) = (f1 (ϕ), ..., fk (ϕ)) dt 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 = S ϕ F(ϕ)ξdt + S diag F(ϕ) dW Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 8 / 31
  • 32. Model equations LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
  • 33. Model equations LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
  • 34. Model equations LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
  • 35. Model equations LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31
  • 36. Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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
  • 37. Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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
  • 38. Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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
  • 39. Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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
  • 40. Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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
  • 41. Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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
  • 42. Distribution of data Vector of measurements xQ ≡ (xt1 , . . . , xtn ) for Q ∈ {TS, TP, DT} time-series (TS) e.g. fluorescent microscopy end-time-point (TP) e.g. fluorescent cytometry deterministic (DT) e.g. population data xQ ∼ MVN(µ(Θ), ΣQ (Θ)) µ(Θ) = (ϕ(t1 ), ..., ϕ(tn ))    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  ΣQ (Θ)(i,j) =   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
  • 43. Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
  • 44. Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
  • 45. Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
  • 46. Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
  • 47. Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
  • 48. Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
  • 49. Advantages of the framework Inference Explicit likelihood Time-series, end-time-point data Very low computational cost, compared to other methods Hidden variables Measurement error Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 11 / 31
  • 50. Hierarchical model for degradation rates: CHX experiment 40 30 fluorescence level 20 10 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
  • 51. Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
  • 52. Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
  • 53. Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 Rates differ between cells 0 0 2 4 6 8 10 time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
  • 54. Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 Rates differ between cells 2 0 0 2 4 6 8 10 γP ∼ Gamma(µγp , σγp ) time (h) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
  • 55. Hierarchical model for degradation rates: CHX experiment 40 Model: 30 fluorescence level dp = (kp − γp p)dt+ kp + γp φp (t)dW 20 10 Rates differ between cells 2 0 0 2 4 6 8 10 γP ∼ Gamma(µγp , σγp ) time (h) 8 6 density 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 degradation rate Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31
  • 56. DRB experiment 450 400 350 300 GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
  • 57. DRB experiment 450 Model: 400 350 300 GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
  • 58. DRB experiment 450 Model: 400 350 dr = (kr − γr r)dt+ kr + γr φr (t)dWr 300 dp = (kp r − γp p)dt + kp φr (t) + γp φr (t)dWp GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
  • 59. DRB experiment 450 Model: 400 350 dr = (kr − γr r)dt+ kr + γr φr (t)dWr 300 dp = (kp r − γp p)dt + kp φr (t) + γp φr (t)dWp GFP Fluorescence 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Time (hours) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
  • 60. DRB experiment 450 Model: 400 350 dr = (kr − γr r)dt+ kr + γr φr (t)dWr 300 dp = (kp r − γp p)dt + kp φr (t) + γp φr (t)dWp GFP Fluorescence 250 200 150 100 We can estimate 50 2 0 0 2 4 6 8 Time (hours) 10 12 14 16 γr ∼ Gamma(µγr , σγr ) Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 13 / 31
  • 61. Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
  • 62. Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
  • 63. Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
  • 64. Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
  • 65. Fluorescent proteins as transcriptional reporters in single cells Observed fluorescence and time-course of endogenous protein differ fluorescence intensity (a.u.) 200 400 600 800 GH3 rat pituitary cells with EGFP linked to prolactin gene promoter Trascription is triggered at the start of the experiment No data on mRNA level 0 0 5 10 15 20 25 Informative prior on mRNA and time (hours) protein degradation rate Experiment: Claire Harper, Mike White; Department of Biology, University of Liverpool Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 14 / 31
  • 66. Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
  • 67. Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
  • 68. Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
  • 69. Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt dp = (kp r − γp p)dt Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
  • 70. Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt+ kr (t) + γr r dWr dp = (kp r − γp p)dt + kp r + γp pdWp Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
  • 71. Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt+ kr (t) + γr r dWr dp = (kp r − γp p)dt + kp r + γp pdWp Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
  • 72. Fluorescent proteins as transcriptional reporters in single cells Calculating back to the transcription level Model: dr = (kr (t) − γr r)dt+ kr (t) + γr r dWr dp = (kp r − γp p)dt + kp r + γp pdWp p(obs) = λp Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 15 / 31
  • 73. 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
  • 74. 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
  • 75. 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
  • 76. Sensitivity for stochastic systems: motivation Difference in response to perturbations in parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-point stochastic model (TP) e.g. flow cytometry Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
  • 77. Sensitivity for stochastic systems: motivation Difference in response to perturbations in parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-point stochastic model (TP) e.g. flow cytometry Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
  • 78. Sensitivity for stochastic systems: motivation Difference in response to perturbations in parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-point stochastic model (TP) e.g. flow cytometry Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
  • 79. Sensitivity for stochastic systems: motivation Difference in response to perturbations in parameters Deterministic model ( DT) e.g. population average Time-series stochastic model (TS) e.g. fluorescent microscopy Time-point stochastic model (TP) e.g. flow cytometry Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31
  • 80. Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
  • 81. Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
  • 82. Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
  • 83. Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
  • 84. Implications Sensitivity Robustness - global sensitivity analysis Information content of data Optimal experimental design Idetifiability Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 18 / 31
  • 85. Sensitivity and Fisher Information Classical sensitivity coefficients for an observable X and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can be evaluated via numerical integration of ODEs Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
  • 86. Sensitivity and Fisher Information Classical sensitivity coefficients for an observable X and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can be evaluated via numerical integration of ODEs Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
  • 87. Sensitivity and Fisher Information Classical sensitivity coefficients for an observable X and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can be evaluated via numerical integration of ODEs Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
  • 88. Sensitivity and Fisher Information Classical sensitivity coefficients for an observable X and parameter θ ∂X ∂θ Stochastic case: observable X is drawn from a distribution ψ 2 ∂ log ψ(X, θ) I(θ) = E ∂θ For stochastic model of chemical reactions evaluated using Monte Carlo simulations Can be evaluated via numerical integration of ODEs Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 19 / 31
  • 89. Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
  • 90. Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
  • 91. Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
  • 92. Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
  • 93. Model equations - reminder LNA implies Gaussian distribution x(t) ∼ MVN(ϕ(t), V(t)) Mean ϕ(t) given as s solution of the rate equation Variances dV(t) = A(ϕ, Θ, t)V + VA(ϕ, Θ, t)T + E(ϕ, Θ, t)E(ϕ, Θ, t)T dt Covariances cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t dΦ(ti , s) = A(ϕ, Θ, s)Φ(ti , s), Φ(ti , ti ) = I ds Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 20 / 31
  • 94. Model equations - reminder Fisher information ∂µ T ∂µ 1 ∂Σ −1 ∂Σ I(θ) = Σ(θ) + trace(Σ−1 Σ ) ∂θ ∂θ 2 ∂θ ∂θ Covariance matrix    V(ti ) for i = j Q ∈ {TS, TP} σ2I for i = j Q ∈ {DT}  (i,j) ΣQ (Θ) =   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
  • 95. Example: expression of a gene Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 22 / 31
  • 96. Response to parameter perturbations: stochastic vs deterministic case Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
  • 97. Response to parameter perturbations: stochastic vs deterministic case Influence of correlation between RNA and protein Stochastic Deterministic correlation=0.24218 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
  • 98. Response to parameter perturbations: stochastic vs deterministic case Influence of correlation between RNA and protein correlation=0.53838 Stochastic 0.1 0.1 0.1 0.1 Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
  • 99. Response to parameter perturbations: stochastic vs deterministic case Influence of correlation between RNA and protein correlation=0.92828 Stochastic 0.1 0.1 0.1 0.1 Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 23 / 31
  • 100. Response to parameter perturbations: stochastic vs deterministic case Influence of temporal correlations Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
  • 101. Response to parameter perturbations: stochastic vs deterministic case Influence of temporal correlations =30 0.1 0.1 0.1 0.1 Stochastic 0.05 0.05 0.05 0.05 Deterministic 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
  • 102. Response to parameter perturbations: stochastic vs deterministic case Influence of temporal correlations =3 0.1 0.1 0.1 0.1 Stochastic Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
  • 103. Response to parameter perturbations: stochastic vs deterministic case Influance of temporal correlations =0.3 Stochastic 0.1 0.1 0.1 0.1 Deterministic 0.05 0.05 0.05 0.05 0 0 0 0 kr 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 kp 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0 0 0 0 r 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 p 0 0 0 0 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 0.1 0.05 0 0.05 0.1 k k r p r p Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 24 / 31
  • 104. Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
  • 105. Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
  • 106. Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
  • 107. Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
  • 108. Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
  • 109. Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) Type TS TP DT Stationary 4 2 1 Perturbation 4 4 3 Perturbation: 5-fold increased initial conditions Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
  • 110. Amount of information in the data Only protein level is measured Measurements are taken from a stationary state # of identifiable parameters optimal sampling frequency (non-zero eigenvalues) 80 set 1 set 2 70 set 3 set 4 60 Type TS TP DT 50 Stationary 4 2 1 det( FIM ) 40 Perturbation 4 4 3 30 20 Perturbation: 5-fold increased initial conditions 10 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 25 / 31
  • 111. p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
  • 112. p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
  • 113. p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
  • 114. p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
  • 115. p53 system p53 protein regulates cell cycle, response to DNA damage and it is a tumour repressor. x = (p, y0 , y). y0 - mdm2 precursor p - p53 y - mdm2 Deterministic version: ˙ φp φp = βx − αx φp − αk φy φp + k ˙ y = β y φp − α0 φy φ0 0 ˙ φy = α 0 φy0 − α y φy . Parameter vector Θ = (βx , αx , αk , k, βy , α0 , αy ). Role of parameters: which parameters control stochastic effects in the model? Fluorescent microscopy or flow cytometry? Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 26 / 31
  • 116. Role of parameters Eigen values normalized against model maximum 1 TS TP 0.8 DT 0.6 0.4 0.2 0 1 2 3 4 5 6 7 Eigen values normalized against total maximum 1 TS TP 0.8 DT 0.6 0.4 0.2 0 1 2 3 4 5 6 7 Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 27 / 31
  • 117. Which parameters are involved in controlling stochastic effects? TS - heatmap, DT - contour plot Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 28 / 31
  • 118. Fluorescent microscopy vs flow cytometry 23 x 10 12 TP TS 10 8 det( FIM ) 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 Number of TP measurements per time point 4 x 10 Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 29 / 31
  • 119. Summary Efficient and simple inference framework for stochastic systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear noise approximation, BMC Bioinformatics, 10, 2009; Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
  • 120. Summary Efficient and simple inference framework for stochastic systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear noise approximation, BMC Bioinformatics, 10, 2009; Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
  • 121. Summary Efficient and simple inference framework for stochastic systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear noise approximation, BMC Bioinformatics, 10, 2009; Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
  • 122. Summary Efficient and simple inference framework for stochastic systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear noise approximation, BMC Bioinformatics, 10, 2009; Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
  • 123. Summary Efficient and simple inference framework for stochastic systems Fisher Information Matrix for stochastic models can be represented as solutions of ODEs Substantial differences is sensitivities between stochastic and deterministic models may exist Applicability experimental design Matlab package for sensitivity of stochastic systems available www.theosysbio.bio.ic.ac.uk/resources/stns/ Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemical kinetics models, PNAS in press, 2011. ¨ Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise and other parameters of gene expression, Biophysical J., 98, 2010. ¨ Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using the linear noise approximation, BMC Bioinformatics, 10, 2009; Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31
  • 124. Acknowledgement Michael Stumpf Imperial College London ¨ ¨ Barbel Finkenstad Warwick University Dan Woodcock Warwick University David Rand Warwick University Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 31 / 31
  • 125. Thank you! Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 31 / 31