Estimation of parameter sensitivities for stochastic reactionnetworks

Ankit GuptaJoint work with Mustafa Khammash

Department of Biosystems Science and EngineeringETH Zürich

Uncertainty Quantification WorkshopKing Abdullah University of Science and Technology

January 7, 2016.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The problem

Consider a stochastic process pXθptqqtě0 which depends on some modelparameter θ.

Such a process may represent a phenomenon with dynamical uncertainties.These uncertainties affect the distribution of any state XθpTq.

These effects can be quantified by estimating an output of the form

Epf pXθpTqqq.

If the parameter θ is also uncertain, its effect on output-variability can bemeasured by the localised sensitivity value

Sθpf , Tq “B

BθEpf pXθpTqqq.

If θ is a random variable with distribution Pα, we may also be interested in a moreglobal sensitivity value

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

We mostly focus on local sensitivity but we later explore extensions to the globalcase.

Main difficulty is that the dynamics is itself stochastic – an extra level of samplingis required to obtain Epf pXθpTqqq.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The problem

Consider a stochastic process pXθptqqtě0 which depends on some modelparameter θ.

Such a process may represent a phenomenon with dynamical uncertainties.These uncertainties affect the distribution of any state XθpTq.

These effects can be quantified by estimating an output of the form

Epf pXθpTqqq.

If the parameter θ is also uncertain, its effect on output-variability can bemeasured by the localised sensitivity value

Sθpf , Tq “B

BθEpf pXθpTqqq.

If θ is a random variable with distribution Pα, we may also be interested in a moreglobal sensitivity value

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

We mostly focus on local sensitivity but we later explore extensions to the globalcase.

Main difficulty is that the dynamics is itself stochastic – an extra level of samplingis required to obtain Epf pXθpTqqq.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The problem

Consider a stochastic process pXθptqqtě0 which depends on some modelparameter θ.

Such a process may represent a phenomenon with dynamical uncertainties.These uncertainties affect the distribution of any state XθpTq.

These effects can be quantified by estimating an output of the form

Epf pXθpTqqq.

If the parameter θ is also uncertain, its effect on output-variability can bemeasured by the localised sensitivity value

Sθpf , Tq “B

BθEpf pXθpTqqq.

If θ is a random variable with distribution Pα, we may also be interested in a moreglobal sensitivity value

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

We mostly focus on local sensitivity but we later explore extensions to the globalcase.

Main difficulty is that the dynamics is itself stochastic – an extra level of samplingis required to obtain Epf pXθpTqqq.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The problem

Consider a stochastic process pXθptqqtě0 which depends on some modelparameter θ.

Such a process may represent a phenomenon with dynamical uncertainties.These uncertainties affect the distribution of any state XθpTq.

These effects can be quantified by estimating an output of the form

Epf pXθpTqqq.

If the parameter θ is also uncertain, its effect on output-variability can bemeasured by the localised sensitivity value

Sθpf , Tq “B

BθEpf pXθpTqqq.

If θ is a random variable with distribution Pα, we may also be interested in a moreglobal sensitivity value

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

We mostly focus on local sensitivity but we later explore extensions to the globalcase.

Main difficulty is that the dynamics is itself stochastic – an extra level of samplingis required to obtain Epf pXθpTqqq.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The problem

Consider a stochastic process pXθptqqtě0 which depends on some modelparameter θ.

Such a process may represent a phenomenon with dynamical uncertainties.These uncertainties affect the distribution of any state XθpTq.

These effects can be quantified by estimating an output of the form

Epf pXθpTqqq.

If the parameter θ is also uncertain, its effect on output-variability can bemeasured by the localised sensitivity value

Sθpf , Tq “B

BθEpf pXθpTqqq.

If θ is a random variable with distribution Pα, we may also be interested in a moreglobal sensitivity value

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

We mostly focus on local sensitivity but we later explore extensions to the globalcase.

Main difficulty is that the dynamics is itself stochastic – an extra level of samplingis required to obtain Epf pXθpTqqq.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The problem

Consider a stochastic process pXθptqqtě0 which depends on some modelparameter θ.

Such a process may represent a phenomenon with dynamical uncertainties.These uncertainties affect the distribution of any state XθpTq.

These effects can be quantified by estimating an output of the form

Epf pXθpTqqq.

If the parameter θ is also uncertain, its effect on output-variability can bemeasured by the localised sensitivity value

Sθpf , Tq “B

BθEpf pXθpTqqq.

If θ is a random variable with distribution Pα, we may also be interested in a moreglobal sensitivity value

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

We mostly focus on local sensitivity but we later explore extensions to the globalcase.

Main difficulty is that the dynamics is itself stochastic – an extra level of samplingis required to obtain Epf pXθpTqqq.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The problem

Consider a stochastic process pXθptqqtě0 which depends on some modelparameter θ.

Such a process may represent a phenomenon with dynamical uncertainties.These uncertainties affect the distribution of any state XθpTq.

These effects can be quantified by estimating an output of the form

Epf pXθpTqqq.

If the parameter θ is also uncertain, its effect on output-variability can bemeasured by the localised sensitivity value

Sθpf , Tq “B

BθEpf pXθpTqqq.

If θ is a random variable with distribution Pα, we may also be interested in a moreglobal sensitivity value

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

We mostly focus on local sensitivity but we later explore extensions to the globalcase.

Main difficulty is that the dynamics is itself stochastic – an extra level of samplingis required to obtain Epf pXθpTqqq.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Biological context

A cell is a very noisy place!

There is randomness in the intracellular dynamics. This leads to heterogeneityamong identical cells.

Often cells pool their random outputs f pXθpTqq at the tissue-level or in thebloodstream to form the population-level output Epf pXθpTqqq.

f(X(1)(T))

f(X(8)(T)) f(X(4)(T))

f(X(2)(T))

f(X(6)(T))

f(X(5)(T))

f(X(3)(T))

f(X(9)(T))

f(X(10)(T)) f(X(7)(T))

Central Question: How does uncertainty in θ affect such an output?

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Biological context

A cell is a very noisy place!

There is randomness in the intracellular dynamics. This leads to heterogeneityamong identical cells.

Often cells pool their random outputs f pXθpTqq at the tissue-level or in thebloodstream to form the population-level output Epf pXθpTqqq.

f(X(1)(T))

f(X(8)(T)) f(X(4)(T))

f(X(2)(T))

f(X(6)(T))

f(X(5)(T))

f(X(3)(T))

f(X(9)(T))

f(X(10)(T)) f(X(7)(T))

Central Question: How does uncertainty in θ affect such an output?

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Biological context

A cell is a very noisy place!

There is randomness in the intracellular dynamics. This leads to heterogeneityamong identical cells.

Often cells pool their random outputs f pXθpTqq at the tissue-level or in thebloodstream to form the population-level output Epf pXθpTqqq.

f(X(1)(T))

f(X(8)(T)) f(X(4)(T))

f(X(2)(T))

f(X(6)(T))

f(X(5)(T))

f(X(3)(T))

f(X(9)(T))

f(X(10)(T)) f(X(7)(T))

Central Question: How does uncertainty in θ affect such an output?

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Biological context

A cell is a very noisy place!

There is randomness in the intracellular dynamics. This leads to heterogeneityamong identical cells.

Often cells pool their random outputs f pXθpTqq at the tissue-level or in thebloodstream to form the population-level output Epf pXθpTqqq.

f(X(1)(T))

f(X(8)(T)) f(X(4)(T))

f(X(2)(T))

f(X(6)(T))

f(X(5)(T))

f(X(3)(T))

f(X(9)(T))

f(X(10)(T)) f(X(7)(T))

Central Question: How does uncertainty in θ affect such an output?

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Experimental evidence for randomness

Randomness in gene expression can cause identical cells to have different proteinlevels.

Credit: Elowitz et al. Science 2002 Credit: Jeff Hasty.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Experimental evidence for randomness

Cellular randomness can lead to phenotypic variation.

Bacillus subtilis

Heterogenous Phenotypes from Identical Genotype

Credit: Rocky Mountain Laboratories

Salmonella

Ph

Ph

J.B. Kaper et al, Nature Rev. Microbiol. (2004)

E. coli

Ph

Credit: Michael Ellowitz

Credit: Mustafa Khammash

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

If the random variable θ has mean Epθq “ θ0 and its fluctuations around this meanare small then we can write

Var pEpfθpTqqq “„

B

BθEpf pXθpTqqq

2

θ“θ0

Varpθq “`

Sθ0 pf , Tq˘2 Varpθq.

Suppose that intracellular dynamics is described by a reaction network with dspecies S1, . . . , Sd and K reactions of the form

dÿ

i“1

νkiSi ÝÑ

dÿ

i“1

ρkiSi.

Deterministic models assume that all reactions fire continuously.

In many networks, some species have small copy-numbers and the associatedreactions fire intermittently.

This intermittency generates dynamical uncertainties (intrinsic noise) which canprofoundly affect the system’s behavior.

To quantify these effects, we model the reaction dynamics as a continuous timeMarkov chain (CTMC).

The state of the system is the vector of molecular counts.

Let ζk “ pρk ´ νkq. When the state is x “ px1, . . . , xdq, the k-th reaction fires at rateλkpx, θq and moves the state to x ` ζk .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

Let pXθptqqtě0 be the Markov chain describing the reaction dynamics. Thisprocess can be represented by the random time change representation:

Xθptq “ Xθp0q `Kÿ

k“1

Yk

ˆż t

0λkpXθpsq, θqds

˙

ζk,

where Yk-s are independent unit rate Poisson processes.

Here θ is some parameter of the system (rate constant, cell volume etc.).Parameter sensitivities of the form

Sθpf , Tq “B

BθE pf pXθpTqqq

are useful for determining important network parameters, analyzing robustness,fine-tuning outputs etc.

The mapping θ ÞÑ E pf pXθpTqqq is generally unknown, but we can generatesamples of XθpTq using Gillespie’s Algorithm (SSA).

Since we cannot compute Sθpf , Tq directly, we need to estimate this quantity usingsamples of XθpTq.

This causes problems as we now discuss.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

Let pXθptqqtě0 be the Markov chain describing the reaction dynamics. Thisprocess can be represented by the random time change representation:

Xθptq “ Xθp0q `Kÿ

k“1

Yk

ˆż t

0λkpXθpsq, θqds

˙

ζk,

where Yk-s are independent unit rate Poisson processes.

Here θ is some parameter of the system (rate constant, cell volume etc.).Parameter sensitivities of the form

Sθpf , Tq “B

BθE pf pXθpTqqq

are useful for determining important network parameters, analyzing robustness,fine-tuning outputs etc.

The mapping θ ÞÑ E pf pXθpTqqq is generally unknown, but we can generatesamples of XθpTq using Gillespie’s Algorithm (SSA).

Since we cannot compute Sθpf , Tq directly, we need to estimate this quantity usingsamples of XθpTq.

This causes problems as we now discuss.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

Let pXθptqqtě0 be the Markov chain describing the reaction dynamics. Thisprocess can be represented by the random time change representation:

Xθptq “ Xθp0q `Kÿ

k“1

Yk

ˆż t

0λkpXθpsq, θqds

˙

ζk,

where Yk-s are independent unit rate Poisson processes.

Here θ is some parameter of the system (rate constant, cell volume etc.).Parameter sensitivities of the form

Sθpf , Tq “B

BθE pf pXθpTqqq

are useful for determining important network parameters, analyzing robustness,fine-tuning outputs etc.

The mapping θ ÞÑ E pf pXθpTqqq is generally unknown, but we can generatesamples of XθpTq using Gillespie’s Algorithm (SSA).

Since we cannot compute Sθpf , Tq directly, we need to estimate this quantity usingsamples of XθpTq.

This causes problems as we now discuss.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

Let pXθptqqtě0 be the Markov chain describing the reaction dynamics. Thisprocess can be represented by the random time change representation:

Xθptq “ Xθp0q `Kÿ

k“1

Yk

ˆż t

0λkpXθpsq, θqds

˙

ζk,

where Yk-s are independent unit rate Poisson processes.

Here θ is some parameter of the system (rate constant, cell volume etc.).Parameter sensitivities of the form

Sθpf , Tq “B

BθE pf pXθpTqqq

are useful for determining important network parameters, analyzing robustness,fine-tuning outputs etc.

The mapping θ ÞÑ E pf pXθpTqqq is generally unknown, but we can generatesamples of XθpTq using Gillespie’s Algorithm (SSA).

Since we cannot compute Sθpf , Tq directly, we need to estimate this quantity usingsamples of XθpTq.

This causes problems as we now discuss.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

The Markov Chain model

Let pXθptqqtě0 be the Markov chain describing the reaction dynamics. Thisprocess can be represented by the random time change representation:

Xθptq “ Xθp0q `Kÿ

k“1

Yk

ˆż t

0λkpXθpsq, θqds

˙

ζk,

where Yk-s are independent unit rate Poisson processes.

Here θ is some parameter of the system (rate constant, cell volume etc.).Parameter sensitivities of the form

Sθpf , Tq “B

BθE pf pXθpTqqq

are useful for determining important network parameters, analyzing robustness,fine-tuning outputs etc.

The mapping θ ÞÑ E pf pXθpTqqq is generally unknown, but we can generatesamples of XθpTq using Gillespie’s Algorithm (SSA).

Since we cannot compute Sθpf , Tq directly, we need to estimate this quantity usingsamples of XθpTq.

This causes problems as we now discuss.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Finite-difference schemes

Many methods for estimating Sθpf , Tq use a finite-difference approximation

Sθ,hpf , Tq “1hE pf pXθ`hpTqq ´ f pXθpTqqq .

The processes Xθ and Xθ`h can be intelligently coupled using the random timechange representation (Rathinam et. al. 2010, Anderson 2012).

Such a coupling reduces the estimator variance but it is still biased.

Trade off: Bias is proportional to h but estimator variance is proportional to 1{h. Itbecomes difficult to control the approximation error.

x

x

x

Computationalburden

Finite incrementh

xTrue sensitivity Estimated sensitivity

1

2

3

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

Estimator distribution

Such a problem does not arise if the dynamics pXθptqqtě0 is deterministic. In thiscase, approximation error can be controlled using h.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Finite-difference schemes

Many methods for estimating Sθpf , Tq use a finite-difference approximation

Sθ,hpf , Tq “1hE pf pXθ`hpTqq ´ f pXθpTqqq .

The processes Xθ and Xθ`h can be intelligently coupled using the random timechange representation (Rathinam et. al. 2010, Anderson 2012).

Such a coupling reduces the estimator variance but it is still biased.

Trade off: Bias is proportional to h but estimator variance is proportional to 1{h. Itbecomes difficult to control the approximation error.

x

x

x

Computationalburden

Finite incrementh

xTrue sensitivity Estimated sensitivity

1

2

3

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

Estimator distribution

Such a problem does not arise if the dynamics pXθptqqtě0 is deterministic. In thiscase, approximation error can be controlled using h.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Finite-difference schemes

Many methods for estimating Sθpf , Tq use a finite-difference approximation

Sθ,hpf , Tq “1hE pf pXθ`hpTqq ´ f pXθpTqqq .

The processes Xθ and Xθ`h can be intelligently coupled using the random timechange representation (Rathinam et. al. 2010, Anderson 2012).

Such a coupling reduces the estimator variance but it is still biased.

Trade off: Bias is proportional to h but estimator variance is proportional to 1{h. Itbecomes difficult to control the approximation error.

x

x

x

Computationalburden

Finite incrementh

xTrue sensitivity Estimated sensitivity

1

2

3

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

Estimator distribution

Such a problem does not arise if the dynamics pXθptqqtě0 is deterministic. In thiscase, approximation error can be controlled using h.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Finite-difference schemes

Many methods for estimating Sθpf , Tq use a finite-difference approximation

Sθ,hpf , Tq “1hE pf pXθ`hpTqq ´ f pXθpTqqq .

The processes Xθ and Xθ`h can be intelligently coupled using the random timechange representation (Rathinam et. al. 2010, Anderson 2012).

Such a coupling reduces the estimator variance but it is still biased.

Trade off: Bias is proportional to h but estimator variance is proportional to 1{h. Itbecomes difficult to control the approximation error.

x

x

x

Computationalburden

Finite incrementh

xTrue sensitivity Estimated sensitivity

1

2

3

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

Estimator distribution

Such a problem does not arise if the dynamics pXθptqqtě0 is deterministic. In thiscase, approximation error can be controlled using h.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Finite-difference schemes

Many methods for estimating Sθpf , Tq use a finite-difference approximation

Sθ,hpf , Tq “1hE pf pXθ`hpTqq ´ f pXθpTqqq .

The processes Xθ and Xθ`h can be intelligently coupled using the random timechange representation (Rathinam et. al. 2010, Anderson 2012).

Such a coupling reduces the estimator variance but it is still biased.

Trade off: Bias is proportional to h but estimator variance is proportional to 1{h. Itbecomes difficult to control the approximation error.

x

x

x

Computationalburden

Finite incrementh

xTrue sensitivity Estimated sensitivity

1

2

3

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

543210-1-2-3-4-5

Normal Distribution

Estimator distribution

Such a problem does not arise if the dynamics pXθptqqtě0 is deterministic. In thiscase, approximation error can be controlled using h.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Our Main Result

We propose a new formula for sensitivity computation.

This formula can provide unbiased estimates of parameter sensitivity using simulation ofthe process pXθptqqtě0 at a fixed θ.

We can also use this formula as a tool for deriving sensitivity approximations.

Let φθpx, f , tq be defined by

φθpx, f , tq “ E pf pXθptqqq ,

where pXθptqqtě0 represents the reaction dynamics with initial state Xθp0q “ x.

Parameter sensitivity can be expressed as

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

The integrand has a simple interpretation:

Due to an infinitesimal perturbation of θ, the probability of having and extra firing ofreaction k at time t is proportional to

BλkpXθptq, θqBθ

.

The change in φθpx, f , Tq due to this extra firing of reaction k at time t is simplypφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq .

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Proof

Proof is based on split coupling1:

Xθptq “ x0 `

Kÿ

k“1

ζkYk

ˆż t

0λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hqds

˙

`

Kÿ

k“1

ζkYp1qk

ˆż t

0

`

λkpXθpsq, θq ´ λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hq˘

ds˙

Xθ`hptq “ x0 `

Kÿ

k“1

ζkYk

ˆż t

0λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hqds

˙

`

Kÿ

k“1

ζkYp2qk

ˆż t

0

`

λkpXθ`hpsq, θ ` hq ´ λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hq˘

ds˙

.

With this coupling it is possible to compute the limit2:

Sθpf , Tq “ limhÑ0

E`

f pXθ`hpTqq˘

´ E`

f pXθpTqq˘

h.

Rearranging the limit gives us the sensitivity formula.

1D. Anderson. “An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time MarkovChains”. In: SIAM: Journal on Numerical Analysis (2012).

2A. Gupta and M. Khammash. “Unbiased Estimation of Parameter Sensitivities for Stochastic Chemical ReactionNetworks”. In: SIAM Journal on Scientific Computing 35.6 (2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Proof

Proof is based on split coupling1:

Xθptq “ x0 `

Kÿ

k“1

ζkYk

ˆż t

0λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hqds

˙

`

Kÿ

k“1

ζkYp1qk

ˆż t

0

`

λkpXθpsq, θq ´ λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hq˘

ds˙

Xθ`hptq “ x0 `

Kÿ

k“1

ζkYk

ˆż t

0λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hqds

˙

`

Kÿ

k“1

ζkYp2qk

ˆż t

0

`

λkpXθ`hpsq, θ ` hq ´ λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hq˘

ds˙

.

With this coupling it is possible to compute the limit2:

Sθpf , Tq “ limhÑ0

E`

f pXθ`hpTqq˘

´ E`

f pXθpTqq˘

h.

Rearranging the limit gives us the sensitivity formula.

1D. Anderson. “An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time MarkovChains”. In: SIAM: Journal on Numerical Analysis (2012).

2A. Gupta and M. Khammash. “Unbiased Estimation of Parameter Sensitivities for Stochastic Chemical ReactionNetworks”. In: SIAM Journal on Scientific Computing 35.6 (2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Proof

Proof is based on split coupling1:

Xθptq “ x0 `

Kÿ

k“1

ζkYk

ˆż t

0λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hqds

˙

`

Kÿ

k“1

ζkYp1qk

ˆż t

0

`

λkpXθpsq, θq ´ λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hq˘

ds˙

Xθ`hptq “ x0 `

Kÿ

k“1

ζkYk

ˆż t

0λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hqds

˙

`

Kÿ

k“1

ζkYp2qk

ˆż t

0

`

λkpXθ`hpsq, θ ` hq ´ λkpXθpsq, θq ^ λkpXθ`hpsq, θ ` hq˘

ds˙

.

With this coupling it is possible to compute the limit2:

Sθpf , Tq “ limhÑ0

E`

f pXθ`hpTqq˘

´ E`

f pXθpTqq˘

h.

Rearranging the limit gives us the sensitivity formula.

1D. Anderson. “An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time MarkovChains”. In: SIAM: Journal on Numerical Analysis (2012).

2A. Gupta and M. Khammash. “Unbiased Estimation of Parameter Sensitivities for Stochastic Chemical ReactionNetworks”. In: SIAM Journal on Scientific Computing 35.6 (2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Unbiased estimation of parameter sensitivity

Recall the formula

Sθpf , Tq “Kÿ

k“1

Eˆż T

0

BλkpXθptq, θqBθ

pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq dt˙

.

Generally we do not know an explicit formula for the difference

DθpXθptq, f , T ´ t, kq “ pφθpXθptq ` ζk, f , T ´ tq ´ φθpXθptq, f , T ´ tqq.

These differences must be estimated on the run using auxiliary paths.

Only a handful of such estimations are possible.

We mark a small collection of (random) times T “ tt1, . . . , tnu Ă r0, Ts.

For each t P T , we estimate the difference DθpXθptq, f , T ´ t, kq using twoindependent auxiliary paths tied according to the split coupling.

For efficiency we make sure that the decision t P T is based only on Xθptq.

In3 we obtain such a marking T using Poisson random variables and show thatthe resulting estimator is still unbiased.

3Ankit Gupta and Mustafa Khammash. “An efficient and unbiased method for sensitivity analysis of stochasticreaction networks”. In: Journal of The Royal Society Interface 11.101 (2014).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We consider the network given in Gardner et al.4.

This network has two species X1 and X2 that interact through the following fourreactions

Hλ1ÝÑ X1, X1

λ2ÝÑ H, H

λ3ÝÑ X2 and X2

λ4ÝÑ H,

The propensity functions λi-s are given by

λ1px1, x2q “α1

1` xβ2, λ2px1, x2q “ x1, λ3px1, x2q “

α2

1` xγ1and λ4px1, x2q “ x2.

Note that λ1 and λ3 are nonlinear Hill functions which show mutual repression.

The stochastic trajectories look like:

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

20

40

60

80

X1(t)[M

olecules]

Time

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

X2(t)[M

olecules]

Time

4Timothy S. Gardner, Charles R. Cantor, and James J. Collins. “Construction of a genetic toggle switch inEscherichia coli”. In: Nature 403.6767 (2000), pp. 339–342.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We consider the network given in Gardner et al.4.

This network has two species X1 and X2 that interact through the following fourreactions

Hλ1ÝÑ X1, X1

λ2ÝÑ H, H

λ3ÝÑ X2 and X2

λ4ÝÑ H,

The propensity functions λi-s are given by

λ1px1, x2q “α1

1` xβ2, λ2px1, x2q “ x1, λ3px1, x2q “

α2

1` xγ1and λ4px1, x2q “ x2.

Note that λ1 and λ3 are nonlinear Hill functions which show mutual repression.

The stochastic trajectories look like:

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

20

40

60

80

X1(t)[M

olecules]

Time

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

X2(t)[M

olecules]

Time

4Timothy S. Gardner, Charles R. Cantor, and James J. Collins. “Construction of a genetic toggle switch inEscherichia coli”. In: Nature 403.6767 (2000), pp. 339–342.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We consider the network given in Gardner et al.4.

This network has two species X1 and X2 that interact through the following fourreactions

Hλ1ÝÑ X1, X1

λ2ÝÑ H, H

λ3ÝÑ X2 and X2

λ4ÝÑ H,

The propensity functions λi-s are given by

λ1px1, x2q “α1

1` xβ2, λ2px1, x2q “ x1, λ3px1, x2q “

α2

1` xγ1and λ4px1, x2q “ x2.

Note that λ1 and λ3 are nonlinear Hill functions which show mutual repression.

The stochastic trajectories look like:

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

20

40

60

80

X1(t)[M

olecules]

Time

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

X2(t)[M

olecules]

Time

4Timothy S. Gardner, Charles R. Cantor, and James J. Collins. “Construction of a genetic toggle switch inEscherichia coli”. In: Nature 403.6767 (2000), pp. 339–342.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We consider the network given in Gardner et al.4.

This network has two species X1 and X2 that interact through the following fourreactions

Hλ1ÝÑ X1, X1

λ2ÝÑ H, H

λ3ÝÑ X2 and X2

λ4ÝÑ H,

The propensity functions λi-s are given by

λ1px1, x2q “α1

1` xβ2, λ2px1, x2q “ x1, λ3px1, x2q “

α2

1` xγ1and λ4px1, x2q “ x2.

Note that λ1 and λ3 are nonlinear Hill functions which show mutual repression.

The stochastic trajectories look like:

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

20

40

60

80

X1(t)[M

olecules]

Time

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

X2(t)[M

olecules]

Time

4Timothy S. Gardner, Charles R. Cantor, and James J. Collins. “Construction of a genetic toggle switch inEscherichia coli”. In: Nature 403.6767 (2000), pp. 339–342.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We consider the network given in Gardner et al.4.

This network has two species X1 and X2 that interact through the following fourreactions

Hλ1ÝÑ X1, X1

λ2ÝÑ H, H

λ3ÝÑ X2 and X2

λ4ÝÑ H,

The propensity functions λi-s are given by

λ1px1, x2q “α1

1` xβ2, λ2px1, x2q “ x1, λ3px1, x2q “

α2

1` xγ1and λ4px1, x2q “ x2.

Note that λ1 and λ3 are nonlinear Hill functions which show mutual repression.

The stochastic trajectories look like:

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

20

40

60

80

X1(t)[M

olecules]

Time

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

10

20

30

40

X2(t)[M

olecu

les]

Time

4Timothy S. Gardner, Charles R. Cantor, and James J. Collins. “Construction of a genetic toggle switch inEscherichia coli”. In: Nature 403.6767 (2000), pp. 339–342.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We estimate the sensitivity of EpX1ptqq w.r.t. various parameters θ “ α1, α2, β andγ.

For a given accuracy (p-value), our method (PPA) is far more efficient than theother unbiased method (Girsanov) and also the other finite-difference methods(CFD and CRP).

100

101

102

103

104Genetic toggle switch : Confidence level p = 0.95

CPU

Tim

e (s

econ

ds)

_1 _2 ` a

170

5.81

63

11.815.6

5.898.85

2.84

149 144

313

33.6

h =

0.1

h =

0.1

h =

0.1

h =

0.01

312 277

1.47e+03

86.9

h =

0.1

h =

0.1

h =

0.1

h =

0.01

100

101

102

103

104Genetic toggle switch : Confidence level p = 0.99

CPU

Tim

e (s

econ

ds)

_1 _2 ` a

297

13.6

99.2

19.523.7

11.3 14.9

6.34

165 179

933

178

h =

0.1

h =

0.1

h =

0.1

h =

0.01

835 8601.65e+03

156

h =

0.1

h =

0.1

h =

0.1

h =

0.01

GirsanovPPACFDCRP

GirsanovPPACFDCRP

In this example, PPA outperforms other methods by factors of 8 (Girsanov), 20(CFD) and 64 (CRP) respectively.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We estimate the sensitivity of EpX1ptqq w.r.t. various parameters θ “ α1, α2, β andγ.

For a given accuracy (p-value), our method (PPA) is far more efficient than theother unbiased method (Girsanov) and also the other finite-difference methods(CFD and CRP).

100

101

102

103

104Genetic toggle switch : Confidence level p = 0.95

CPU

Tim

e (s

econ

ds)

_1 _2 ` a

170

5.81

63

11.815.6

5.898.85

2.84

149 144

313

33.6

h =

0.1

h =

0.1

h =

0.1

h =

0.01

312 277

1.47e+03

86.9

h =

0.1

h =

0.1

h =

0.1

h =

0.01

100

101

102

103

104Genetic toggle switch : Confidence level p = 0.99

CPU

Tim

e (s

econ

ds)

_1 _2 ` a

297

13.6

99.2

19.523.7

11.3 14.9

6.34

165 179

933

178

h =

0.1

h =

0.1

h =

0.1

h =

0.01

835 8601.65e+03

156

h =

0.1

h =

0.1

h =

0.1

h =

0.01

GirsanovPPACFDCRP

GirsanovPPACFDCRP

In this example, PPA outperforms other methods by factors of 8 (Girsanov), 20(CFD) and 64 (CRP) respectively.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Example: Genetic Toggle Switch

We estimate the sensitivity of EpX1ptqq w.r.t. various parameters θ “ α1, α2, β andγ.

For a given accuracy (p-value), our method (PPA) is far more efficient than theother unbiased method (Girsanov) and also the other finite-difference methods(CFD and CRP).

100

101

102

103

104Genetic toggle switch : Confidence level p = 0.95

CPU

Tim

e (s

econ

ds)

_1 _2 ` a

170

5.81

63

11.815.6

5.898.85

2.84

149 144

313

33.6

h =

0.1

h =

0.1

h =

0.1

h =

0.01

312 277

1.47e+03

86.9

h =

0.1

h =

0.1

h =

0.1

h =

0.01

100

101

102

103

104Genetic toggle switch : Confidence level p = 0.99

CPU

Tim

e (s

econ

ds)

_1 _2 ` a

297

13.6

99.2

19.523.7

11.3 14.9

6.34

165 179

933

178

h =

0.1

h =

0.1

h =

0.1

h =

0.01

835 8601.65e+03

156

h =

0.1

h =

0.1

h =

0.1

h =

0.01

GirsanovPPACFDCRP

GirsanovPPACFDCRP

In this example, PPA outperforms other methods by factors of 8 (Girsanov), 20(CFD) and 64 (CRP) respectively.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:

Parametrise the original dynamics pXNθ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:

Parametrise the original dynamics pXNθ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:

Parametrise the original dynamics pXNθ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:

Parametrise the original dynamics pXNθ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:Parametrise the original dynamics pXN

θ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:Parametrise the original dynamics pXN

θ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:Parametrise the original dynamics pXN

θ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:Parametrise the original dynamics pXN

θ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:Parametrise the original dynamics pXN

θ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Simulations of the process pXθptqqtě0 can be computationally demanding.

Rate constants and species copy-numbers may vary over several orders of magnitude.

Reactions have different time-scales. Fast reactions take up most of computational time.

To counter this problem we need dynamical approximations or model reductions5:Parametrise the original dynamics pXN

θ ptqqtě0 by a scaling factor N.

Under some assumptions we have the process-level convergence XNθ ÝÑ Xθ as

N Ñ 8.

For large N we can assume XNθ « Xθ and simulate Xθ instead of XN

θ .

The process-level convergence is generally in the Skorohod space DRd r0,8q.Note that the following function F : DRd r0,8q Ñ R is continuous

Fpxq “ż T

0f pxptqqdt.

Therefore if XNθ ÝÑ Xθ then we must have SN

θ pf , Tq Ñ Sθpf , Tq where Sθpf , Tq isbased only on the limiting process Xθ (reduced model).

Note that for large N we have SNθ pf , Tq « Sθpf , Tq.

5Hye-Won Kang and Thomas G. Kurtz. “Separation of time-scales and model reduction for stochastic reactionnetworks”. In: Ann. Appl. Probab. 23.2 (Apr. 2013).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Estimation of sensitivity values using model reductions based on quasi-stationaryapproximation (QSA) has been considered in our recent paper6.

Consider the heat-shock model7

X1c1ÝÑ X2, X2

c2ÝÑ X1 and X2

c3ÝÑ X3

with c1 “ 1, c2 “ 2 and c3 “ 5ˆ 10´4.

Let θ “ c1 “ 1 . We speed up the process by the factor N “ 104.

The first two reactions become fast. Applying QSA we can reduce the model to

X2

´

5θ2`θ

¯

ÝÑ X3.

With this reduced model we can easily estimate the sensitivity of EpXN3 ptqq w.r.t. θ:

Sensitivity Value Number of Samples CPU time (s)Original Model 4.2138˘ 0.2107 34932 1663.34Reduced Model 4.2017˘ 0.2100 35056 0.2333

6Ankit Gupta and Mustafa Khammash. “Sensitivity analysis for stochastic chemical reaction networks withmultiple time-scales”. In: Electron. J. Probab. 19 (2014).

7H. El-Samad et al. “Surviving heat shock: Control strategies for robustness and performance”. In: PNAS 102.8(2005).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Estimation of sensitivity values using model reductions based on quasi-stationaryapproximation (QSA) has been considered in our recent paper6.

Consider the heat-shock model7

X1c1ÝÑ X2, X2

c2ÝÑ X1 and X2

c3ÝÑ X3

with c1 “ 1, c2 “ 2 and c3 “ 5ˆ 10´4.

Let θ “ c1 “ 1 . We speed up the process by the factor N “ 104.

The first two reactions become fast. Applying QSA we can reduce the model to

X2

´

5θ2`θ

¯

ÝÑ X3.

With this reduced model we can easily estimate the sensitivity of EpXN3 ptqq w.r.t. θ:

Sensitivity Value Number of Samples CPU time (s)Original Model 4.2138˘ 0.2107 34932 1663.34Reduced Model 4.2017˘ 0.2100 35056 0.2333

6Ankit Gupta and Mustafa Khammash. “Sensitivity analysis for stochastic chemical reaction networks withmultiple time-scales”. In: Electron. J. Probab. 19 (2014).

7H. El-Samad et al. “Surviving heat shock: Control strategies for robustness and performance”. In: PNAS 102.8(2005).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Estimation of sensitivity values using model reductions based on quasi-stationaryapproximation (QSA) has been considered in our recent paper6.

Consider the heat-shock model7

X1c1ÝÑ X2, X2

c2ÝÑ X1 and X2

c3ÝÑ X3

with c1 “ 1, c2 “ 2 and c3 “ 5ˆ 10´4.

Let θ “ c1 “ 1 . We speed up the process by the factor N “ 104.

The first two reactions become fast. Applying QSA we can reduce the model to

X2

´

5θ2`θ

¯

ÝÑ X3.

With this reduced model we can easily estimate the sensitivity of EpXN3 ptqq w.r.t. θ:

Sensitivity Value Number of Samples CPU time (s)Original Model 4.2138˘ 0.2107 34932 1663.34Reduced Model 4.2017˘ 0.2100 35056 0.2333

6Ankit Gupta and Mustafa Khammash. “Sensitivity analysis for stochastic chemical reaction networks withmultiple time-scales”. In: Electron. J. Probab. 19 (2014).

7H. El-Samad et al. “Surviving heat shock: Control strategies for robustness and performance”. In: PNAS 102.8(2005).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Estimation of sensitivity values using model reductions based on quasi-stationaryapproximation (QSA) has been considered in our recent paper6.

Consider the heat-shock model7

X1c1ÝÑ X2, X2

c2ÝÑ X1 and X2

c3ÝÑ X3

with c1 “ 1, c2 “ 2 and c3 “ 5ˆ 10´4.

Let θ “ c1 “ 1 . We speed up the process by the factor N “ 104.

The first two reactions become fast. Applying QSA we can reduce the model to

X2

´

5θ2`θ

¯

ÝÑ X3.

With this reduced model we can easily estimate the sensitivity of EpXN3 ptqq w.r.t. θ:

Sensitivity Value Number of Samples CPU time (s)Original Model 4.2138˘ 0.2107 34932 1663.34Reduced Model 4.2017˘ 0.2100 35056 0.2333

6Ankit Gupta and Mustafa Khammash. “Sensitivity analysis for stochastic chemical reaction networks withmultiple time-scales”. In: Electron. J. Probab. 19 (2014).

7H. El-Samad et al. “Surviving heat shock: Control strategies for robustness and performance”. In: PNAS 102.8(2005).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Tool for analysis and estimation

Estimation of sensitivity values using model reductions based on quasi-stationaryapproximation (QSA) has been considered in our recent paper6.

Consider the heat-shock model7

X1c1ÝÑ X2, X2

c2ÝÑ X1 and X2

c3ÝÑ X3

with c1 “ 1, c2 “ 2 and c3 “ 5ˆ 10´4.

Let θ “ c1 “ 1 . We speed up the process by the factor N “ 104.

The first two reactions become fast. Applying QSA we can reduce the model to

X2

´

5θ2`θ

¯

ÝÑ X3.

With this reduced model we can easily estimate the sensitivity of EpXN3 ptqq w.r.t. θ:

Sensitivity Value Number of Samples CPU time (s)Original Model 4.2138˘ 0.2107 34932 1663.34Reduced Model 4.2017˘ 0.2100 35056 0.2333

6Ankit Gupta and Mustafa Khammash. “Sensitivity analysis for stochastic chemical reaction networks withmultiple time-scales”. In: Electron. J. Probab. 19 (2014).

7H. El-Samad et al. “Surviving heat shock: Control strategies for robustness and performance”. In: PNAS 102.8(2005).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Forthcoming Work

Some model reductions yield a Piecewise Deterministic Markov process (PDMP):

Some species evolve continuously while others evolve discretely.

Some reactions fire continuously while others fire intermittently.

In a forthcoming paper, we show that sensitivity values for the original network canbe efficiently estimated using the PDMP approximation.

Exact simulation methods (like SSA) account for each reaction firing, which can becumbersome.

Approximate schemes like τ -leaping account for multiple reaction firings in asingle step.

In a forthcoming work, we show that sensitivity values for the original network canbe efficiently estimated using τ -leaping schemes:

We exploit known convergence results8,9,10 that show that τ -leap dynamics Xτθ

converges to the actual dynamics Xθ as τ Ñ 0.

We compute the sensitivity value Sτθ pf , Tq in such a way that Sτθ pf , Tq Ñ Sθpf , Tq asτ Ñ 0.

8David F. Anderson, Arnab Ganguly, and Thomas G. Kurtz. “Error analysis of tau-leap simulation methods”. In:Ann. Appl. Probab. 21.6 (2011).

9Muruhan Rathinam et al. “Consistency and stability of tau-leaping schemes for chemical reaction systems”. In:Multiscale Model. Simul. 4.3 (2005).

10Tiejun Li. “Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems”. In:Multiscale Modeling & Simulation 6.2 (2007), pp. 417–436.

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Future Directions

Randomness in parameter θ can also be seen as extrinsic noise.

If θ has distribution Pα, we would like to estimate

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

to quantify the contribution of α-uncertainty in output-uncertainty.

In a recent work, Zechner et al.11 show that we can marginalize the process toobtain an equivalent CTMC pXαptqqtě0 that does not involve θ.

We are trying to exploit these ideas to find efficient estimator for Sαpf , Tq.

We can also combine our method with other methods to form novel multi-levelschemes for sensitivity estimation12.

Finally we would like to generalise these ideas to higher-order sensitivities.

We also want to use our results to find efficient methods for directly estimatingsteady-state sensitivities.

11Christoph Zechner and Heinz Koeppl. “Uncoupled Analysis of Stochastic Reaction Networks in FluctuatingEnvironments”. In: PLoS Comput Biol 10 (Dec. 2014).

12Elizabeth Skubak Wolf and David F. Anderson. “Hybrid pathwise sensitivity methods for discrete stochasticmodels of chemical reaction systems”. In: The Journal of Chemical Physics 142 (2015).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Future Directions

Randomness in parameter θ can also be seen as extrinsic noise.

If θ has distribution Pα, we would like to estimate

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

to quantify the contribution of α-uncertainty in output-uncertainty.

In a recent work, Zechner et al.11 show that we can marginalize the process toobtain an equivalent CTMC pXαptqqtě0 that does not involve θ.

We are trying to exploit these ideas to find efficient estimator for Sαpf , Tq.

We can also combine our method with other methods to form novel multi-levelschemes for sensitivity estimation12.

Finally we would like to generalise these ideas to higher-order sensitivities.

We also want to use our results to find efficient methods for directly estimatingsteady-state sensitivities.

11Christoph Zechner and Heinz Koeppl. “Uncoupled Analysis of Stochastic Reaction Networks in FluctuatingEnvironments”. In: PLoS Comput Biol 10 (Dec. 2014).

12Elizabeth Skubak Wolf and David F. Anderson. “Hybrid pathwise sensitivity methods for discrete stochasticmodels of chemical reaction systems”. In: The Journal of Chemical Physics 142 (2015).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Future Directions

Randomness in parameter θ can also be seen as extrinsic noise.

If θ has distribution Pα, we would like to estimate

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

to quantify the contribution of α-uncertainty in output-uncertainty.

In a recent work, Zechner et al.11 show that we can marginalize the process toobtain an equivalent CTMC pXαptqqtě0 that does not involve θ.

We are trying to exploit these ideas to find efficient estimator for Sαpf , Tq.

We can also combine our method with other methods to form novel multi-levelschemes for sensitivity estimation12.

Finally we would like to generalise these ideas to higher-order sensitivities.

We also want to use our results to find efficient methods for directly estimatingsteady-state sensitivities.

11Christoph Zechner and Heinz Koeppl. “Uncoupled Analysis of Stochastic Reaction Networks in FluctuatingEnvironments”. In: PLoS Comput Biol 10 (Dec. 2014).

12Elizabeth Skubak Wolf and David F. Anderson. “Hybrid pathwise sensitivity methods for discrete stochasticmodels of chemical reaction systems”. In: The Journal of Chemical Physics 142 (2015).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Future Directions

Randomness in parameter θ can also be seen as extrinsic noise.

If θ has distribution Pα, we would like to estimate

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

to quantify the contribution of α-uncertainty in output-uncertainty.

In a recent work, Zechner et al.11 show that we can marginalize the process toobtain an equivalent CTMC pXαptqqtě0 that does not involve θ.

We are trying to exploit these ideas to find efficient estimator for Sαpf , Tq.

We can also combine our method with other methods to form novel multi-levelschemes for sensitivity estimation12.

Finally we would like to generalise these ideas to higher-order sensitivities.

We also want to use our results to find efficient methods for directly estimatingsteady-state sensitivities.

11Christoph Zechner and Heinz Koeppl. “Uncoupled Analysis of Stochastic Reaction Networks in FluctuatingEnvironments”. In: PLoS Comput Biol 10 (Dec. 2014).

12Elizabeth Skubak Wolf and David F. Anderson. “Hybrid pathwise sensitivity methods for discrete stochasticmodels of chemical reaction systems”. In: The Journal of Chemical Physics 142 (2015).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Future Directions

Randomness in parameter θ can also be seen as extrinsic noise.

If θ has distribution Pα, we would like to estimate

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

to quantify the contribution of α-uncertainty in output-uncertainty.

In a recent work, Zechner et al.11 show that we can marginalize the process toobtain an equivalent CTMC pXαptqqtě0 that does not involve θ.

We are trying to exploit these ideas to find efficient estimator for Sαpf , Tq.

We can also combine our method with other methods to form novel multi-levelschemes for sensitivity estimation12.

Finally we would like to generalise these ideas to higher-order sensitivities.

We also want to use our results to find efficient methods for directly estimatingsteady-state sensitivities.

11Christoph Zechner and Heinz Koeppl. “Uncoupled Analysis of Stochastic Reaction Networks in FluctuatingEnvironments”. In: PLoS Comput Biol 10 (Dec. 2014).

12Elizabeth Skubak Wolf and David F. Anderson. “Hybrid pathwise sensitivity methods for discrete stochasticmodels of chemical reaction systems”. In: The Journal of Chemical Physics 142 (2015).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Future Directions

Randomness in parameter θ can also be seen as extrinsic noise.

If θ has distribution Pα, we would like to estimate

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

to quantify the contribution of α-uncertainty in output-uncertainty.

In a recent work, Zechner et al.11 show that we can marginalize the process toobtain an equivalent CTMC pXαptqqtě0 that does not involve θ.

We are trying to exploit these ideas to find efficient estimator for Sαpf , Tq.

We can also combine our method with other methods to form novel multi-levelschemes for sensitivity estimation12.

Finally we would like to generalise these ideas to higher-order sensitivities.

We also want to use our results to find efficient methods for directly estimatingsteady-state sensitivities.

11Christoph Zechner and Heinz Koeppl. “Uncoupled Analysis of Stochastic Reaction Networks in FluctuatingEnvironments”. In: PLoS Comput Biol 10 (Dec. 2014).

12Elizabeth Skubak Wolf and David F. Anderson. “Hybrid pathwise sensitivity methods for discrete stochasticmodels of chemical reaction systems”. In: The Journal of Chemical Physics 142 (2015).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Future Directions

Randomness in parameter θ can also be seen as extrinsic noise.

If θ has distribution Pα, we would like to estimate

Sαpf , Tq “B

Bα

„ż

Epf pXθpTqq|θqPαpdθq

.

to quantify the contribution of α-uncertainty in output-uncertainty.

In a recent work, Zechner et al.11 show that we can marginalize the process toobtain an equivalent CTMC pXαptqqtě0 that does not involve θ.

We are trying to exploit these ideas to find efficient estimator for Sαpf , Tq.

We can also combine our method with other methods to form novel multi-levelschemes for sensitivity estimation12.

Finally we would like to generalise these ideas to higher-order sensitivities.

We also want to use our results to find efficient methods for directly estimatingsteady-state sensitivities.

11Christoph Zechner and Heinz Koeppl. “Uncoupled Analysis of Stochastic Reaction Networks in FluctuatingEnvironments”. In: PLoS Comput Biol 10 (Dec. 2014).

12Elizabeth Skubak Wolf and David F. Anderson. “Hybrid pathwise sensitivity methods for discrete stochasticmodels of chemical reaction systems”. In: The Journal of Chemical Physics 142 (2015).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks

Acknowledgements

Mustafa KhammashProfessor of Control Theory and Systems BiologyHead, Department of Biosystems Science and Engineering (D-BSSE)Swiss Federal Institute of Technology, Zurich (ETHZ).

Muruhan RathinamProfessor of MathematicsDepartment of Mathematics and StatisticsUniversity of Maryland Baltimore County, Baltimore, USA.

Christoph ZechnerPostdoctoral Research FellowDepartment of Biosystems Science and Engineering (D-BSSE)Swiss Federal Institute of Technology, Zurich (ETHZ).

Ankit Gupta Estimation of parameter sensitivities for stochastic reaction networks