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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

An integrated framework for analysis of stochastic models of biochemical reactions

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An integrated framework for analysis of stochastic models of biochemical reactions