Bayesian Experimental Design for Stochastic KineticModels

Colin Gillespie, Richard Boys, Nina Wilkinson

Newcastle University, UK

April 1, 2014

Overview

Stochastic kinetic models

Simulation and inference

Optimal design problem

Future directions

Stochastic kinetic models

A biochemical network is represented as a set of pseudo-biochemicalreactions: for i = 1, . . . , v

Ri : pi1X1 + pi2X2 + · · ·+ piuXuθi−→ qi1X1 + qi2X2 + · · ·+ qiuXu

Stochastic rate constant θi

Hazard/instantaneous rate: hi(Xt , θi) where Xt = (X1,t , . . . ,Xu,t) is thecurrent state of the system

Under mass-action stochastic kinetics, the hazard function is proportionalto a product of binomial coefficients, with

hi(Xt , θi) = θi

u∏j=1

(Xj,t

pij

)

Stochastic kinetic models

Describe the SKM by a Markov jump process (MJP)

The effect of reaction Rk is to change the value of each species Xi byqki − pki

The time to the next reaction is

t ∼ Exp{h0(Xt ,θ)}

where h0(Xt ,θ) =∑v

i=1 hi(Xi , θi)

The reaction is of type i with probability hi(Xt , θi)/h0(Xt ,θ)

The process is easily simulated using the Direct method (Gillespiealgorithm)

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(Xt , θi)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(Xt , θi)

4 Reaction time: τ = −ln{U(0, 1)}/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to its hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically a simulated experiment has many simulated reactions

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(Xt , θi)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(Xt , θi)

4 Reaction time: τ = −ln{U(0, 1)}/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to its hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically a simulated experiment has many simulated reactions

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(Xt , θi)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(Xt , θi)

4 Reaction time: τ = −ln{U(0, 1)}/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to its hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically a simulated experiment has many simulated reactions

The direct method

1 Initialisation: initial conditions, reactions constants, and random numbergenerators

2 Propensities update: Update each of the v hazard functions, hi(Xt , θi)

3 Propensities total: Calculate the total hazard h0 =∑v

i=1 hi(Xt , θi)

4 Reaction time: τ = −ln{U(0, 1)}/h0 and t = t + τ

5 Reaction selection: A reaction is chosen proportional to its hazard6 Reaction execution: Update species7 Iteration: If the simulation time is exceeded stop, otherwise go back to

step 2

Typically a simulated experiment has many simulated reactions

Parameter estimation: the inverse problem

Given observations on the chemical species, can we infer the rateconstants, θ?

Typically we only partially observe species, at discrete times

0 5 10 15 20 25 30

0

100

200

300

400

Time

Pop

ulat

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Parameter estimation: the inverse problem

Given observations on the chemical species, can we infer the rateconstants, θ?

Typically we only partially observe species, at discrete times

0 5 10 15 20 25 30

0

100

200

300

400

Time

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ulat

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Parameter estimation: the inverse problem

Given observations on the chemical species, can we infer the rateconstants, θ?

Typically we only partially observe species, at discrete times

0 5 10 15 20 25 30

0

100

200

300

400

Time

Pop

ulat

ion

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Parameter estimation: the inverse problem

Given observations on the chemical species, can we infer the rateconstants, θ?

Typically we only partially observe species, at discrete times

0 5 10 15 20 25 30

0

100

200

300

400

Time

Pop

ulat

ion

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Parameter inference

Forward simulation approachesApproximate Bayesian computation (ABC)

Particle MCMC

Simulator approximationsMoment closure (2MA)

Linear noise approximation (LNA)

Langevin equation (SDE)

Gaussian processes (GP)

Example: The death model

A single reaction

Nθ−→ ∅

This model is sufficiently simple that we canobtain the probability of n individuals at timet analytically

Initialising with N(0) = n0 individuals at timet = 0, we have, for n = n0, n0 − 1, . . . , 0

pn(t) =

(n0

n

)e−θnt(1− e−θt)n0−n

0

10

20

30

40

50

0.0 2.5 5.0 7.5 10.0Time

Pop

ulat

ion

What would an optimal design look like?

Suppose we want to observe the process at k time points. What time pointsshould we use?

●

● ●0

10

20

30

40

50

0.0 2.5 5.0 7.5 10.0Time

Pop

ulat

ion

Time = 1 Time = 3 Time = 5

0.00

0.01

0.02

0.03

0 1 2 3 0 1 2 3 0 1 2 3θ

π(θ|

d)

What would an optimal design look like?

Suppose we want to observe the process at k time points. What time pointsshould we use?

●

● ●0

10

20

30

40

50

0.0 2.5 5.0 7.5 10.0Time

Pop

ulat

ion

Time = 1 Time = 3 Time = 5

0.00

0.01

0.02

0.03

0 1 2 3 0 1 2 3 0 1 2 3θ

π(θ|

d)

What would an optimal design look like?

Suppose we want to observe the process at k time points. What time pointsshould we use?

●

● ●0

10

20

30

40

50

0.0 2.5 5.0 7.5 10.0Time

Pop

ulat

ion

Time = 1 Time = 3 Time = 5

0.00

0.01

0.02

0.03

0 1 2 3 0 1 2 3 0 1 2 3θ

π(θ|

d)

Choosing the best design

A typical design is d = (t1, t2, . . . , tk)

Since the design parameter has to be chosen before doing theexperiment we need to maximise

u(d) = Eθ,y [u(d, y ,θ)] =∫

y

∫θ

u(d, y ,θ)π(y |d,θ)π(θ)dθdy

The best design d∗ maximises u(d)

Typically u(d) is analytically intractable

The utility we choose is

u(d, y ,θ) ≡ u(d, y) =1

det{Var(θ|d, y)}

which does not depend on θ

Choosing the best design

The utility of a design d = (t1, t2, . . . , tk) is

u(d) = Eθ,y [u(d, y ,θ)] = · · · =∫θπ(θ)u(d,θ)dθ

which can be approximated by

u(d) = Eθ [u(d,θ)] '1m

m∑i=1

u(d,θi)

where the θi are a random sample of size m from the prior π(θ)

Typically, u(d,θ) is computationally expensive to evaluate

Fast approximation u(d ,θ)

1 Using a Latin hyper-cube design, estimate u(d,θ) at multiple (d,θ)points

2 Fit a GP emulator to u(d,θ), with mean function

m(d,θ) = β0 +∑

i

βiθi +∑

j

βjdj

and a squared exponential covariance function

K (x i , x j |a, r) = a exp{(x i − x j)

T diag(r)−2(x i − x j)}

where x = (d,θ)3 Use the GP emulators within an MCMC scheme to estimate

d∗ = argmax u(d)

Toy model: Pure death process

Prior: θ ∼ LN(−0.005, 0.01)

Minimax Latin hypercube design using400 training points

Since we can solve the Master equation,we can calculate the utility functionexactly

For a 1-d design, our approximation isexcellent

For multiple design points, theapproximation is very close to the exact

How many design points should we use:what value for k?

0 2 4 6 8 10

100

110

120

130

140

Time

Exp

ecte

d U

tility

#Design points

Exp

ecte

d U

tility

1 2 3 4

130

135

140

145

150

Results: 2-point design d = (t1, t2), t1 < t2

Exact GP

0.0

2.5

5.0

7.5

0.0 2.5 5.0 7.5 0.0 2.5 5.0 7.5Design Point 1

Des

ign

Poi

nt 2

0 50 100u(t1, t2)

Exact GP

0

1

2

3

0 1 2 3 0 1 2 3Design Point 1

Des

ign

Poi

nt 2

Future directions

Current approaches for estimating u(d) of SKM do not scale

We need to be able to handle larger, more complex models

Surface of the utility function is very flat. May need to implement asequential strategy to gradually zoom into a region around the optimaldesign d∗

May be useful to use approximations of the stochastic kinetic model tofocus into the region around the optimal design quickly

Source code (R and LATEX of these slides):https://github.com/csgillespie/talks/