an integrated framework for analysis of stochastic models of biochemical reactions

An integrated framework for analysis of stochasticmodels of biochemical reactions

Michał Komorowski

Imperial College LondonTheoretical Systems Biology Group

21/03/11

Michał Komorowski Stochastic biochemical reactions 21/03/11 1 / 31

Outline

1 Motivation: models and data

2 Modeling framework

3 Inference: examples

4 Sensitivity, Fisher Information, statistical model analysis

Michał Komorowski Stochastic biochemical reactions 21/03/11 2 / 31

Fluorescent reporter genes

Michał Komorowski Stochastic biochemical reactions Motivation 21/03/11 3 / 31

Fluorescent reporter genes

Fluorescent microscopy and flow cytometry

0 5 10 15 20 25

time (hours)

0 5 10 15 20 25

time (hours)

0 5 10 15 20 25

time (hours)

0 5 10 15 20 25

time (hours)

Chemical kinetics model

System’s state

x = (x1, . . . , xN)T

Stoichiometry matrix

S = {Sij}i=1,2...N; j=1,2...l

(x1, ...., xN)→ (x1 + S1j, ...., xN + SNj)

Reaction rates

F(x,Θ) = (f1(x,Θ), ..., fl(x,Θ))

ParametersΘ = (θ1, ..., θr)

x is a Poisson birth and death process

Michał Komorowski Stochastic biochemical reactions Modelling 21/03/11 5 / 31

System’s state

x = (x1, . . . , xN)T

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))

System’s state

x = (x1, . . . , xN)T

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))

System’s state

x = (x1, . . . , xN)T

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))

System’s state

x = (x1, . . . , xN)T

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))

Example: gene expression

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

F(x,Θ) = (kr, γrr, kpr, γpp)

ParametersΘ = (kr, γr, kp, γp)

Macroscopic rate equation

φR = kR(t)− γRφR

φP = kPφR − γPφP

Diffusion approximation

dR = (kR(t)− γRR)dt +√

kR + γRRdWR

dP = (kPR− γPP)dt +√

kPR + γPPdWP

Linear noise approximationR(t) = φR(t) + ξR(t) P(t) = φP(t) + ξP(t)

dξR = (−γRξR)dt +√

kR(t) + γRφRdWξR ,

dξP = (kPξR − γPξP)dt +√

kPφP + γPφPdWξP

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

kR + γRRdWR

kPR + γPPdWP

kPφP + γPφPdWξP

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

kR + γRRdWR

kPR + γPPdWP

kPφP + γPφPdWξP

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

kR + γRRdWR

kPR + γPPdWP

kPφP + γPφPdWξP

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

kR + γRRdWR

kPR + γPPdWP

kPφP + γPφPdWξP

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

kR + γRRdWR

kPR + γPPdWP

kPφP + γPφPdWξP

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

kR + γRRdWR

kPR + γPPdWP

kPφP + γPφPdWξP

State x = (r, p)

Stoichiometry

(1 −1 0 00 0 1 −1

)Rates

kR + γRRdWR

kPR + γPPdWP

kPφP + γPφPdWξP

Modelling chemical kineticsChemical master equation

dPt(x)

l∑j=1

Pt(x− S·j)fj(x− S·j)− Pt(x)fj(x)

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

dx = S F(x)dt + S(

diag{√

F(x)})

Linear noise approximation

x(t) = ϕ(t) + ξ(t)

dξ = SOϕF(ϕ)ξdt + S(

diag{√

F(ϕ)})

dPt(x)

l∑j=1

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

dx = S F(x)dt + S(

diag{√

F(x)})

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

dPt(x)

l∑j=1

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

dx = S F(x)dt + S(

diag{√

F(x)})

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

dPt(x)

l∑j=1

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

dx = S F(x)dt + S(

diag{√

F(x)})

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

How about inference ?

Chemical master equation

(likelihood-free methods, e.g. ABC)

dPt(x)

l∑j=1

(least squares)

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

(data augmentation)

dx = S F(x)dt + S(

diag{√

F(x)})

(explicite likelihood)

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

How about inference ?Chemical master equation

(likelihood-free methods, e.g. ABC)

dPt(x)

l∑j=1

(least squares)

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

(data augmentation)

dx = S F(x)dt + S(

diag{√

F(x)})

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

How about inference ?Chemical master equation (likelihood-free methods, e.g. ABC)

dPt(x)

l∑j=1

(least squares)

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

(data augmentation)

dx = S F(x)dt + S(

diag{√

F(x)})

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

dPt(x)

l∑j=1

Macroscopic rate equation (least squares)

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

(data augmentation)

dx = S F(x)dt + S(

diag{√

F(x)})

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

dPt(x)

l∑j=1

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

Diffusion approximation (data augmentation)

dx = S F(x)dt + S(

diag{√

F(x)})

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

dPt(x)

l∑j=1

= S F(ϕ) F(ϕ) = (f1(ϕ), ..., fk(ϕ))

Diffusion approximation (data augmentation)

dx = S F(x)dt + S(

diag{√

F(x)})

Linear noise approximation (explicite likelihood)

x(t) = ϕ(t) + ξ(t)

diag{√

F(ϕ)})

Model equations

LNA implies Gaussian distribution

x(t) ∼ MVN(ϕ(t),V(t))

Mean ϕ(t) given as s solution of the rate equationVariances

dV(t)dt

= A(ϕ,Θ, t)V + VA(ϕ,Θ, t)T + E(ϕ,Θ, t)E(ϕ,Θ, t)T

Covariances

cov(x(s), x(t)) = V(s)Φ(s, t)T for s ≥ t

dΦ(ti, s)ds

= A(ϕ,Θ, s)Φ(ti, s), Φ(ti, ti) = I

Michał Komorowski Stochastic biochemical reactions Inference 21/03/11 9 / 31

Model equations

dV(t)dt

Covariances

dΦ(ti, s)ds

Model equations

dV(t)dt

Covariances

dΦ(ti, s)ds

Model equations

dV(t)dt

Covariances

dΦ(ti, s)ds

Distribution of dataVector of measurements

xQ ≡ (xt1 , . . . , xtn) for Q ∈ {TS,TP,DT}

time-series (TS) e.g. fluorescent microscopyend-time-point (TP) e.g. fluorescent cytometrydeterministic (DT) e.g. population data

xQ ∼ MVN(µ(Θ),ΣQ(Θ))

µ(Θ) = (ϕ(t1), ..., ϕ(tn))

ΣQ(Θ)(i,j) =

V(ti) for i = j Q ∈ {TS,TP}σ2ε I for i = j Q ∈ {DT}0 for i < j Q ∈ {TP,DT}

V(ti)Φ(ti, tj)T for i < j Q ∈ {TS}

µ(Θ) = (ϕ(t1), ..., ϕ(tn))

ΣQ(Θ)(i,j) =

µ(Θ) = (ϕ(t1), ..., ϕ(tn))

ΣQ(Θ)(i,j) =

µ(Θ) = (ϕ(t1), ..., ϕ(tn))

ΣQ(Θ)(i,j) =

µ(Θ) = (ϕ(t1), ..., ϕ(tn))

ΣQ(Θ)(i,j) =

µ(Θ) = (ϕ(t1), ..., ϕ(tn))

ΣQ(Θ)(i,j) =

µ(Θ) = (ϕ(t1), ..., ϕ(tn))

ΣQ(Θ)(i,j) =

Advantages of the framework

Inference

Explicit likelihoodTime-series, end-time-point dataVery low computational cost, compared to other methodsHidden variablesMeasurement error

Inference

Hierarchical model for degradation rates: CHXexperiment

0 2 4 6 8 10

time (h)

0.0 0.2 0.4 0.6 0.8 1.0

degradation rate

Model:dp = (kp − γpp)dt+

√kp + γpφp(t)dW

Rates differ between cells

γP ∼ Gamma(µγp , σ2γp)

Michał Komorowski Stochastic biochemical reactions Examples 21/03/11 12 / 31

0 2 4 6 8 10

time (h)

0.0 0.2 0.4 0.6 0.8 1.0

degradation rate

√kp + γpφp(t)dW

0 2 4 6 8 10

time (h)

0.0 0.2 0.4 0.6 0.8 1.0

degradation rate

√kp + γpφp(t)dW

0 2 4 6 8 10

time (h)

0.0 0.2 0.4 0.6 0.8 1.0

degradation rate

√kp + γpφp(t)dW

0 2 4 6 8 10

time (h)

0.0 0.2 0.4 0.6 0.8 1.0

degradation rate

√kp + γpφp(t)dW

0 2 4 6 8 10

time (h)

0.0 0.2 0.4 0.6 0.8 1.0

degradation rate

√kp + γpφp(t)dW

DRB experiment

0 2 4 6 8 10 12 14 160

Time (hours)

Model:

dr = (kr − γrr)dt+√

kr + γrφr(t)dWr

dp = (kpr − γpp)dt +√

kpφr(t) + γpφr(t)dWp

We can estimate

γr ∼ Gamma(µγr , σ2γr )

DRB experiment

0 2 4 6 8 10 12 14 160

Time (hours)

Model:

kr + γrφr(t)dWr

We can estimate

DRB experiment

0 2 4 6 8 10 12 14 160

Time (hours)

Model:

kr + γrφr(t)dWr

We can estimate

DRB experiment

0 2 4 6 8 10 12 14 160

Time (hours)

Model:

kr + γrφr(t)dWr

We can estimate

DRB experiment

0 2 4 6 8 10 12 14 160

Time (hours)

Model:

kr + γrφr(t)dWr

We can estimate

Fluorescent proteins as transcriptional reporters insingle cells

0 5 10 15 20 25

time (hours)

Experiment: Claire Harper, Mike White;Department of Biology, University of Liverpool

Observed fluorescence andtime-course of endogenous proteindifferGH3 rat pituitary cells with EGFPlinked to prolactin gene promoterTrascription is triggered at the start ofthe experimentNo data on mRNA levelInformative prior on mRNA andprotein degradation rate

0 5 10 15 20 25

time (hours)

0 5 10 15 20 25

time (hours)

0 5 10 15 20 25

time (hours)

0 5 10 15 20 25

time (hours)

Calculating back to the transcription level

Model:

dr = (kr(t)− γrr)dt+√

kr(t) + γrr dWr

kpr + γppdWp

p(obs) = λp

Model:

kr(t) + γrr dWr

kpr + γppdWp

p(obs) = λp

Model:

kr(t) + γrr dWr

kpr + γppdWp

p(obs) = λp

Model:

dr = (kr(t)− γrr)dt

kr(t) + γrr dWr

dp = (kpr − γpp)dt

kpr + γppdWp

p(obs) = λp

Model:

kr(t) + γrr dWr

kpr + γppdWp

p(obs) = λp

Model:

kr(t) + γrr dWr

kpr + γppdWp

p(obs) = λp

Model:

kr(t) + γrr dWr

kpr + γppdWp

p(obs) = λp

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

Inference results

Sensitivity for stochastic systems: motivation

Difference in response to perturbations in parametersDeterministic model ( DT) e.g. population averageTime-series stochastic model (TS) e.g. fluorescent microscopyTime-point stochastic model (TP) e.g. flow cytometry

Michał Komorowski Stochastic biochemical reactions Fisher Information 21/03/11 17 / 31

Implications

SensitivityRobustness - global sensitivity analysisInformation content of data

Optimal experimental design

Idetifiability

Implications

Idetifiability

Implications

Idetifiability

Implications

Idetifiability

Implications

Idetifiability

Sensitivity and Fisher Information

Classical sensitivity coefficients for an observable X andparameter θ

∂X∂θ

Stochastic case: observable X is drawn from a distribution ψ

I(θ) = E(∂ logψ(X, θ)

For stochastic model of chemical reactions evaluated using MonteCarlo simulationsCan be evaluated via numerical integration of ODEs

∂X∂θ

Model equations - reminderLNA implies Gaussian distribution

dV(t)dt

Covariances

dΦ(ti, s)ds

Fisher information

I(θ) =∂µ

TΣ(θ)

∂θ+

trace(Σ−1∂Σ

∂θΣ−1∂Σ

∂θ)

dV(t)dt

Covariances

dΦ(ti, s)ds

Fisher information

I(θ) =∂µ

TΣ(θ)

∂θ+

trace(Σ−1∂Σ

∂θΣ−1∂Σ

∂θ)

dV(t)dt

Covariances

dΦ(ti, s)ds

Fisher information

I(θ) =∂µ

TΣ(θ)

∂θ+

trace(Σ−1∂Σ

∂θΣ−1∂Σ

∂θ)

dV(t)dt

Covariances

dΦ(ti, s)ds

Fisher information

I(θ) =∂µ

TΣ(θ)

∂θ+

trace(Σ−1∂Σ

∂θΣ−1∂Σ

∂θ)

dV(t)dt

Covariances

dΦ(ti, s)ds

Fisher information

I(θ) =∂µ

TΣ(θ)

∂θ+

trace(Σ−1∂Σ

∂θΣ−1∂Σ

∂θ)

Model equations - reminder

Fisher information

I(θ) =∂µ

TΣ(θ)

∂θ+

trace(Σ−1∂Σ

∂θΣ−1∂Σ

∂θ)

Covariance matrix

ΣQ(Θ)(i,j) =

Example: expression of a gene

Response to parameter perturbations:stochastic vs deterministic case

Influence of correlation between RNA and protein

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

correlation=0.24218

0.1 0.05 0 0.05 0.10.1

StochasticDeterministic

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

correlation=0.24218

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

correlation=0.53838

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

correlation=0.92828

0.1 0.05 0 0.05 0.10.1

Influence of temporal correlations

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

0.1 0.05 0 0.05 0.10.1

Influance of temporal correlations

0.1 0.05 0 0.05 0.10.1

r0.1 0.05 0 0.05 0.1

0.1 0.05 0 0.05 0.10.1

Amount of information in the data

Only protein level is measuredMeasurements are taken from a stationary state

# of identifiable parameters(non-zero eigenvalues)

optimal sampling frequency

Type TS TP DTStationary 4 2 1Perturbation 4 4 3

Perturbation: 5-fold increased initial conditions

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

set 1set 2set 3set 4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

p53 systemp53 protein regulates cell cycle, response to DNA damage and it is atumour repressor.

x = (p, y0, y).

p - p53y0 - mdm2 precursory - mdm2

Deterministic version:

φp = βx − αxφp − αkφyφp

φp + k

φy0 = βyφp − α0φy0

φy = α0φy0 − αyφy.

Parameter vector

Θ = (βx, αx, αk, k, βy, α0, αy).

Role of parameters: which parameters control stochastic effects inthe model?Fluorescent microscopy or flow cytometry?

x = (p, y0, y).

φp + k

Parameter vector

x = (p, y0, y).

φp + k

Parameter vector

x = (p, y0, y).

φp + k

Parameter vector

x = (p, y0, y).

φp + k

Parameter vector

Role of parameters

1 2 3 4 5 6 70

1Eigen values normalized against model maximum

TSTPDT

1 2 3 4 5 6 70

1Eigen values normalized against total maximum

TSTPDT

Which parameters are involved in controllingstochastic effects?

TS - heatmap, DT - contour plot

Fluorescent microscopy vs flow cytometry

0 1 2 3 4 5 6 7 8 9 10x 104

12x 1023

Number of TP measurements per time point

Summary

Efficient and simple inference framework for stochastic systemsFisher Information Matrix for stochastic models can berepresented as solutions of ODEsSubstantial differences is sensitivities between stochastic anddeterministic models may existApplicability experimental designMatlab package for sensitivity of stochastic systems availablewww.theosysbio.bio.ic.ac.uk/resources/stns/

Komorowski M.,Costa M.J., Rand D., Stumpf M.P.H. Sensitivity, robustness and identifiability in stochastic chemicalkinetics models, PNAS in press, 2011.

Komorowski M.,Finkenstadt B., Rand D. Using single fluorescent reporter gene to infer half-life of extrinsic noise andother parameters of gene expression, Biophysical J., 98, 2010.

Komorowski M.,Finkenstadt B., Harper C., Rand D. Bayesian estimation of the biochemical kinetics parameters using thelinear noise approximation, BMC Bioinformatics, 10, 2009;

Michał Komorowski Stochastic biochemical reactions Summary 21/03/11 30 / 31

Summary

Acknowledgement

Michael StumpfImperial College London

Barbel FinkenstadWarwick University

Dan WoodcockWarwick University

David RandWarwick University

Thank you!

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