Stochastic Models for Chemical Reactionskurtz/Lectures/BASEL17M.pdf · Stochastic Models for Chemical Reactions Specifying Markov chains Specifying inﬁnitesimal behavior Chemical

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Stochastic Models for Chemical Reactions

Specifying Markov chains

• Specifying infinitesimal behavior

• Chemical reactions

• Poisson processes

• Counting processes

• Markov chains

• Martingale properties

• Master equation

• Equivalence of formulations

• Simulation

• Classical scaling

• Central limit theorem

• Diffusion approximation

Multiscale limits

• Multiscale models

• Time scales

• Model of viral infection

• Enzyme reaction

• Michaelis-Menten equation



• Heat shock model


See References, in particular Anderson and Kurtz (2015).


Specifying infinitesimal behaviorAn ordinary differential equation is specified by describing how afunction should vary over a small period of time

X(t+ ∆t)−X(t) ≈ F (X(t))∆t

A more precise description (consider a telescoping sum)

X(t) = X(0) +

∫ t

0

F (X(s))ds

The first fundamental question: Does this infinitesimal descriptionuniquely determine the function? (Of course, we must specify X(0)along with F .)


Infinitesimal behavior for jump processes

We are interested in functions that are piecewise constant and ran-dom. Changes, when they occur, won’t be small

X(t+ ∆t)−X(t) ≈ ζ

What is small?

P{X(t+ ∆t)−X(t) = ζ|Ft} ≈ λζ(t)∆t

The probability of seeing a jump of a particular size.

Can we specify the λζ in some way that determines X? For the ODE,F depended on X . Maybe λζ should depend on X .

P{X(t+ ∆t)−X(t) = ζ|Ft} ≈ λζ(X(t))∆t


Chemical networksModeling counts of individual chemical species: X(t) will be the vec-tor whose components give the numbers of molecules of each chem-ical species.

For a binary reaction S1 + S2 ⇀ S3 or S1 + S2 ⇀ S3 + S4

λk(x) = κ′kx1x2 ζk =

−1−110

or

−1−111

For S1 ⇀ S2 or S1 ⇀ S2 + S3,

λk(x) = κ′kx1.

For 2S1 ⇀ S2,

λk(x) = κ′kx1(x1 − 1) ζk =

(−21

)


Poisson processesSimplest case:

Only one possible jump: ζ = 1

λ is constant

P{Yλ(t+ ∆t)− Yλ(t) = 1|FYλt } ≈ λ∆t

A more precise description:

• Yλ is constant except for jumps of +1.

• Disjoint increments of Yλ are independent (λ is not random).

• The distribution of Yλ(t+ a)−Yλ(t) does not depend on t (λ doesnot depend on t).


Characterization of the Poisson process

These assumptions imply

P{Yλ(t+ a)− Yλ(t) = k} = e−λa(λa)k

k!,

soP{Yλ(t+ ∆t)− Yλ(t) = 1|FYλ

t } = λ∆te−λ∆t ≈ λ∆t.

Note that if Y = Y1 is a Poisson process with parameter 1, thenYλ(t) ≡ Y (λt) is a Poisson process with parameter λ. So, λ is a rate.


Counting processesNext simplest case:

Only one possible jump ζ = 1.

P{X(t + ∆t) − X(t) = 1|Ft} ≈ λ(t)∆t where λ depends on t and israndom. λ is called the intensity of X .

To specify a model, specify λ as a function of X and perhaps anotherstochastic process ξ (e.g. λ(t) = β(X(t), ξ(t)).

To make this more precise (take X(0) = 0)

X(t) = Y (

∫ t

0

β(X(s), ξ(s))ds)

where Y is a unit Poisson process.


Markov chainsA process X is Markov if

E[f(X(t+ a))|FXt ] = E[f(X(t+ a))|X(t)], t, a ≥ 0.

Infinitesimal specification of a Markov chain:

P{X(t+ ∆t)−X(t) = ζk|FXt } ≈ βk(X(t))∆t

(For simplicity assume only finitely many possible jumps ζk.)

X(t) = X(0) +∑k

Rk(t)ζk,

Rk a counting process counting the times X takes jump ζk.

P{Rk(t+ ∆t)−Rk(t) = 1|FXt } ≈ βk(X(t))∆t

so, for independent unit Poisson processes Yk,

X(t) = X(0) +∑k

Yk(

∫ t

0

βk(X(s))ds)ζk


Enzyme reaction

A+ E AE ⇀ E +B

XE(t) = XE(0)− Y1(

∫ t

0

κ1XA(s)XE(s)ds) + Y2(κ2

∫ t

0

XAE(s)ds)

+Y3(κ3

∫ t

0

XAE(s)ds)

XA(t) = XA(0)− Y1(

∫ t

0

κ1XA(s)XE(s)ds)

+Y2(κ2

∫ t

0

XAE(s)ds)

The remainder of the system is determined by

XAE(t) = M−XE(t) XB(t) = XB(0)+XA(0)+XAE(0)−XA(t)−XAE(t)


Martingale properties

X(t) = X(0) +∑k

Rk(t)ζk = X(0) +∑k

Yk(

∫ t

0

βk(X(s))ds)ζk

Under moment conditions, setting Yk(u) = Yk(u)− u

Rk(t) ≡ Rk(t)−∫ t

0

βk(X(s))ds = Yk(

∫ t

0

βk(X(s))ds)

is a martingale. In particular E[Rk(t)] = 0.


Martingale propertiesSince

f(X(t))− f(X(0)) =∑k

∫ t

0

(f(X(s−) + ζk)− f(X(s−))dRk(s),

defining Af(x) =∑

k βk(x)(f(x+ ζk)− f(x)),

f(X(t))− f(X(0))−∫ t

0

Af(X(s))ds

=∑k

∫ t

0

(f(X(s−) + ζk)− f(X(s−))dRk(s)

is a martingale. In particular

E[f(X(t))− f(X(0))−∫ t

0

Af(X(s))ds] = 0,


Master equationSince

E[f(X(t))− f(X(0))−∫ t

0

Af(X(s))ds] = 0,

letting f(x) = 1{m}(x), E[f(X(t))] = P{X(t) = m} ≡ pm(t) satisfies

pm(t) = pm(0) +

∫ t

0

(∑k

βk(m− ζk)pm−ζk(s)− (∑k

βk(m))pm(s)

)giving the Kolmogorov forward or master equation.


Equivalence of formulationsWe now have three ways of making the infinitesimal specification

P{X(t+ ∆t)−X(t) = ζk|FXt } ≈ βk(X(t))∆t

precise:

1. The stochastic equation: X(t) = X(0) +∑

k Yk(∫ t

0 βk(X(s))ds)ζk

2. The requirement that f(X(t)) − f(X(0)) −∫ t

0 Af(X(s))ds be amartingale for Af(x) =

∑k βk(x)(f(x+ ζk)− f(x))

3. The master equation for the one-dimensional distributions:

pm(t) = pm(0)+

∫ t

0

(∑k

βk(m− ζk)pm−ζk(s)− (∑k

βk(m))pm(s)

)Fortunately, if the solution of the stochastic equation doesn’t blowup, the three are equivalent.


Holding timesIf X(t) = x, then assuming that none of the other reactions fire, thatis none of the Rl jump before before Rk, the time τk(t) until the nextjump of

Rk(t) = Yk(

∫ t

0

βk(X(s))ds)

is exponentially distributed

P{τk(t) > s|X(t) = x} = e−βk(x)s.

The time of the next jump is τ(t) = mink τk(t) so

P{τ(t) > s|X(t) = x} =∏k

P{τk(t) > s|X(t) = x} = e−∑βk(x)s

The calculation implies the length of time X stays in a state x (theholding time in x) before jumping is exponentially distributed withparameter

∑k βk(x).


Jump distribution

A similar calculations gives

P{τk(t) = τ(t)|X(t) = x} =βk(x)∑l βl(x)

.

Note that this expression does not depend on the value of τ(t).


Gillespie stochastic simulation algorithm Gillespie (1976,1977)

Suppose X(0) = Y0. Generate an expontial random variable ∆0 withparameter

∑k βk(Y0).

∆0 = − 1∑k βk(Y0)

logU01

where U01 is uniform [0, 1].

Suppose the βk are β1, . . . , βm and define γ0 = 0 and

γl(y) =

∑lk=1 βk(y)∑mk=1 βk(y)

With an independent uniform [0, 1], U02, generate the next state by

X(∆0) = Y1 =m∑l=1

1(γl−1(Y0),γl(Y0)](U02)(Y0 + ζl).

Repeat.


Algorithms suggested by the stochastic equationSimulating the unit Poisson processes and simply solving the time-change equation

X(t) = X(0) +∑k

Yk(

∫ t

0

βk(X(s))ds)ζk

gives the next reaction (next jump) method as defined by Gibson andBruck (2000). (See also Anderson (2007).)

Doing an Euler type approximation for the integrals gives the τ -leapmethod of Gillespie (2001). Compare

X(tm) = X(0) +m∑i=1

F (X(ti−1))(ti − ti−1)

to

X(tm) = X(0) +∑k

Yk(m∑i=1

βk(X(ti−1))(ti − ti−1))ζk


Properties of Poisson process

Law of large numbers

Y (Nu)

N≈ u lim

N→∞

Y (Nu)

N= u

Central limit theorem

Y (Nu)−Nu√N

=Y (Nu)√

N≈ W (u)

Y (Nu)−Nu√N

⇒ W (u)

W a standard Brownian motion.


Reaction networksWe consider a network of reactions involving s0 chemical species,S1, . . . , Ss0.

s0∑i=1

νikSi ⇀

s0∑i=1

ν ′ikSi

where the νik and ν ′ik are nonnegative integers.

νk the vector whose ith element is νik.

ν ′k the vector whose ith element is ν ′ik.

ζk = ν ′k − νk.


Equations for the system state

The state of the system satisfies

X(t) = X(0) +∑k

Rk(t)(ν′k − νk)

= X(0) +∑k

Yk(

∫ t

0

λk(X(s))ds)ζk

For a binary reaction S1 + S2 ⇀ S3 or S1 + S2 ⇀ S3 + S4

λk(x) = κ′kx1x2

For S1 ⇀ S2 or S1 ⇀ S2 + S3,

λk(x) = κ′kx1.

For 2S1 ⇀ S2,λk(x) = κ′kx1(x1 − 1).


Classical scalingBasic assumption: Binary reaction rate constants scale inversely withsome measure of the volume: κ′k = κkN

−10 .

N0 Avogadro’s number times the volume in liters

Ci(t) = N−10 Xi(t) is the chemical concentration and

κ′kxi = N0κkci, κ′kxixj = N0κkcicj.

so

Ci(t) = Ci(0) +∑k

N−10 Yk(N0

∫ t

0

λk(C(s))ds)(ν ′ki − νki).

Applying the law of large numbers, the concentrations can be ap-proximated by the solution of

Ci(t) = Ci(0) +∑k

∫ t

0

λk(C(s))dsζki.


Scaling argumentsConsider a sequence of equations

CNi (t) = CN

i (0) +∑k

N−1Yk(N

∫ t

0

λk(CN(s))ds)ζki.

CN0 is the original model.

N0 is large, so CN0 might reasonably be approximated by

Ci = limN→∞

CNi


Central limit theorem/Van Kampen ApproximationFor F (c) =

∑k λk(c)ζk where ζk = ν ′k − νk

V N(t) ≡√N(CN(t)− C(t))

≈ V N(0) +√N(∑

k

1

NYk(N

∫ t

0

λk(CN(s))ds)(ν ′k − νk)

−∫ t

0

F (C(s))ds))

= V N(0) +∑k

1√NYk(N

∫ t

0


+

∫ t

0

√N(F (CN(s))− F (C(s)))ds

≈ V N(0) +∑k

Wk(

∫ t

0


+

∫ t

0

∇F (C(s)))V N(s)ds


Gaussian limit

V N converges to the solution of

V (t) = V (0) +∑k

Wk(

∫ t

0

λk(C(s))ds)(ν ′k− νk) +

∫ t

0

∇F (C(s)))V (s)ds

CN(t) ≈ C(t) +1√NV (t)


Diffusion approximation

CN(t) = CN(0) +∑k

N−1Yk(N

∫ t

0


≈ CN(0) +∑k

N−1/2Wk(

∫ t

0


+

∫ t

0

F (CN(s))ds,

whereF (c) =

∑k

λk(c)(ν′k − νk).

The diffusion approximation is given by the equation

CN(t) = CN(0)+∑k

N−1/2Wk(

∫ t

0

λk(CN(s))ds)(ν ′k−νk)+

∫ t

0

F (CN(s))ds.


Ito formulation

The time-change formulation is equivalent to the Ito equation

CN(t) = CN(0) +∑k

N−1/2

∫ t

0

√λk(CN(s))dWk(s)(ν

′k − νk)

+

∫ t

0

F (CN(s))ds

= CN(0) +∑k

N−1/2

∫ t

0

σ(CN(s))dW (s) +

∫ t

0

F (CN(s))ds,

where σ(c) is the matrix with columns√λk(c)(ν

′k − νk).

See Kurtz (1977/78), Ethier and Kurtz (1986), Chapter 10, Gardiner(2004), Chapter 7, and van Kampen (1981).


Multiscale limits

• Multiscale models

• Time scales

• Model of viral infection

• Enzyme reaction

• Michaelis-Menten equation



• Heat shock model


Multiscale models Kang and Kurtz (2013)

Fix N0 >> 1. For each species i, define the normalized abundances (orsimply, the abundances) by

Zi(t) = N−αi0 Xi(t),

where αi ≥ 0 should be selected so that Zi = O(1). Note that theabundance may be the species number (αi = 0) or the species con-centration or something else.

The rate constants may also vary over several orders of magnitudeso scale the rate constants κ′k = κkN

βk0 so that κk = O(1).

Then

κ′kxixj = Nβk+αi+αj0 κkzizj κ′kxi(xi − 1) = Nβk+2αi

0 κkzi(zi −N−αi0 )

κ′kxi = κkNβk+αi0 zi

Note that the exponent on N0 is ρk = βk + α · νk.


A parameterized family of models

Then, noting that νk · α =∑

i νikαi,

Zi(t) = Zi(0) +∑k

N−αi0 Yk(

∫ t

0

Nβk+νk·α0 λk(Z(s))ds)(ν ′ik − νik).

Let

ZNi (t) = Zi(0) +

∑k

N−αiYk(

∫ t

0

Nβk+νk·αλk(ZN(s))ds)(ν ′ik − νik).

Then the “true” model is Z = ZN0.


Time-scale parameter

Let

ZN,γi (t) ≡ ZN

i (tNγ)

= Zi(0) +∑k

N−αiYk(

∫ t

0

Nγ+βk+νk·αλk(ZN,γ(s))ds)ζik.

Equation is “balanced” if

max{βk + νk · α : ζik > 0} = max{βk + νk · α : ζik < 0}

If the equation is not balanced then we need

γ + βk + νk · α ≤ αi (1)

for all i and k such that ζik 6= 0.

The time-scale of species i: γi = αi −max{βk + νk · α : ζik 6= 0}


Example: Model of a viral infectionSrivastava, You, Summers, and Yin (2002), Haseltine and Rawlings(2002), Ball, Kurtz, Popovic, and Rempala (2006)

Three time-varying species, the viral template, the viral genome, andthe viral structural protein (indexed, 1, 2, 3 respectively).

The model involves six reactions,

S1 + stuffκ′1⇀ S1 + S2

S2κ′2⇀ S1

S1 + stuffκ′3⇀ S1 + S3

S1κ′4⇀ ∅

S3κ′5⇀ ∅

S2 + S3κ′6⇀ S4


Stochastic system

X1(t) = X1(0) + Y2(

∫ t

0

κ′2X2(s)ds)− Y4(

∫ t

0

κ′4X1(s)ds)

X2(t) = X2(0) + Y1(

∫ t

0

κ′1X1(s)ds)− Y2(

∫ t

0

κ′2X2(s)ds)

−Y6(

∫ t

0

κ′6X2(s)X3(s)ds)

X3(t) = X3(0) + Y3(

∫ t

0

κ′3X1(s)ds)− Y5(

∫ t

0

κ′5X3(s)ds)

−Y6(

∫ t

0

κ′6X2(s)X3(s)ds)

κ′1 1 κ′4 0.25κ′2 0.025 κ′5 2κ′3 1000 κ′6 7.5× 10−6


Figure 1: Simulation (Haseltine and Rawlings 2002)


Balance equations for the viral model

ZN1 (t) = Z1(0) +N−α1Y2(

∫ t

0

κ2Nβ2+α2ZN

2 (s)ds)−N−α1Y4(

∫ t

0

κ4Nβ4+α1ZN

1 (s)ds)

ZN2 (t) = Z2(0) +N−α2Y1(

∫ t

0

κ1Nβ1+α1ZN

1 (s)ds)−N−α2Y2(

∫ t

0

κ2Nβ2+α2ZN

2 (s)ds)

−N−α2Y6(

∫ t

0

κ6Nβ6+α2+α3ZN

2 (s)ZN3 (s)ds)

ZN3 (t) = Z3(0) +N−α3Y3(

∫ t

0

κ3Nβ3+α1ZN

1 (s)ds)−N−α3Y5(

∫ t

0

κ5Nβ5+α3ZN

3 (s)ds)

−N−α3Y6(

∫ t

0

κ6Nβ6+α2+α3ZN

2 (s)ZN3 (s)ds)

β2 + α2 = β4 + α1

β1 + α1 = (β2 + α2) ∨ (β6 + α2 + α3)

β3 + α1 = (β5 + α3) ∨ (β6 + α2 + α3)

β3 ≥ β5 ≥ β1 ≥ β4 ≥ β2 ≥ β6


An example

β2 + α2 = β4 + α1

β1 + α1 = (β2 + α2) ∨ (β6 + α2 + α3)

β3 + α1 = (β5 + α3) ∨ (β6 + α2 + α3)

β3 ≥ β5 ≥ β1 ≥ β4 ≥ β2 ≥ β6

β1 0 α1 0β2 −2

3 α223

β3 1 α3 1β4 0 γ1 0β5 0 γ2

23

β6 −53 γ3 0


Scaling parameters Ball, Kurtz, Popovic, and Rempala (2006)

Each Xi is scaled according to its abundance in the system.

For N0 = 1000, X1 = O(N 00 ), X2 = O(N

2/30 ), and X3 = O(N0) and we

take Z1 = X1, Z2 = X2N−2/30 , and Z3 = X3N

−10 .

Expressing the rate constants in terms of N0 = 1000

κ′1 1 1

κ′2 0.025 2.5N−2/30

κ′3 1000 N0

κ′4 0.25 .25κ′5 2 2

κ′6 7.5× 10−6 .75N−5/30


Normalized system

With the scaled rate constants and abundances, we have

ZN1 (t) = ZN

1 (0) + Y2(

∫ t

0

2.5ZN2 (s)ds)− Y4(

∫ t

0

.25ZN1 (s)ds)

ZN2 (t) = ZN

2 (0) +N−2/3Y1(

∫ t

0

ZN1 (s)ds)−N−2/3Y2(

∫ t

0

2.5ZN2 (s)ds)

−N−2/3Y6(

∫ t

0

.75ZN2 (s)ZN

3 (s)ds)

ZN3 (t) = ZN

3 (0) +N−1Y3(

∫ t

0

NZN1 (s)ds)−N−1Y5(

∫ t

0

2NZN3 (s)ds)

−N−1Y6(

∫ t

0

.75ZN2 (s)ZN

3 (s)ds),


Limiting system

Passing to the limit, we have

Z1(t) = Z1(0) + Y2(

∫ t

0

2.5Z2(s)ds)− Y4(

∫ t

0

.25Z1(s)ds)

Z2(t) = Z2(0)

Z3(t) = Z3(0) +

∫ t

0

Z1(s)ds−∫ t

0

2Z3(s)ds


Fast time-scaleDefine ZN,γ

i (t) = ZNi (Nγt). For γ = 2

3 ,

ZN,γ1 (t) = Z1(0) + Y2(

∫ t

0

2.5N 2/3ZN,γ2 (s)ds)− Y4(

∫ t

0

.25N 2/3ZN,γ1 (s)ds)

ZN,γ2 (t) = Z2(0) +N−2/3Y1(

∫ t

0

N 2/3ZN,γ1 (s)ds)

−N−2/3Y2(

∫ t

0

2.5N 2/3ZN,γ2 (s)ds)

−N−2/3Y6(N2/3

∫ t

0

.75ZN,γ2 (s)ZN,γ

3 (s)ds)

ZN,γ3 (t) = Z3(0) +N−1Y3(

∫ t

0

N 5/3ZN,γ1 (s)ds)−N−1Y5(

∫ t

0

2N 5/3ZN,γ3 (s)ds)

−N−1Y6(

∫ t

0

.75N 2/3ZN,γ2 (s)ZN,γ

3 (s)ds)


Averaging

As N → ∞, dividing the equations for ZN,γ1 and ZN,γ

3 by N 2/3 showsthat ∫ t

0

ZN,γ1 (s)ds− 10

∫ t

0

ZN,γ2 (s)ds→ 0∫ t

0

ZN,γ3 (s)ds− 5

∫ t

0

ZN,γ2 (s)ds→ 0.

The assertion for ZN,γ3 and the fact that ZN,γ

2 is asymptotically regularimply ∫ t

0

ZN,γ2 (s)ZN,γ

3 (s)ds− 5

∫ t

0

ZN,γ2 (s)2ds→ 0.

It follows that ZN,γ2 converges to the solution of (2).


Law of large numbers

Theorem 1 Let γ = 23 . For each δ > 0 and t > 0,

limN→∞

P{ sup0≤s≤t

|ZN,γ2 (s)− Z∞,γ2 (s)| ≥ δ} = 0,

where Z∞,γ2 is the solution of

Z∞,γ2 (t) = Z2(0) +

∫ t

0

7.5Z∞,γ2 (s)ds)−∫ t

0

3.75Z∞,γ2 (s)2ds. (2)


Approximate modelsWe have a family of models indexed by N for which N = N0 givesthe “correct” model.

Other values of N and any limits as N →∞ (perhaps with a changeof time-scale) give approximate models. The challenge is to select theαi, but once that is done, the intial condition for index N is give by

ZNi (0) = N−αii Xi(0),

where the Xi(0) are the initial species numbers in the correct model.

If limN→∞ ZNi (·Nγ) = Z∞,γi , then we should have

Xi(t) ≈ Nαi0 Z

∞i (tN−γ0 ).

For example, in the virus model

X2(t) ≈ (1000)2/3Z∞,γ2 (t(1000)−2/3)


Enzyme reaction (cf. Darden (1982))

Basic assumption: A small number of enzymes interacts with a largenumber O(N) of substrate molecules.

A+ E AE ⇀ E +B

XE(t) = XE(0)− Y1(

∫ t

0

κ1XA(s)XE(s)ds) + Y2(κ2

∫ t

0

XAE(s)ds)

+Y3(κ3

∫ t

0

XAE(s)ds)

XA(t) = XA(0)− Y1(

∫ t

0

κ1XA(s)XE(s)ds)

+Y2(κ2

∫ t

0

XAE(s)ds)


Scaled model

Assume that production and dissociation reactions are fast, and thesubstrate is present in large numbers. ZN

A = N−1XNA .

XNE (t) = XE(0)− Y1(N

∫ t

0

κ1ZNA (s)XN

E (s)ds) + Y2(Nκ2

∫ t

0

XNAE(s)ds)

+Y3(Nκ3

∫ t

0

XNAE(s)ds)

ZNA (t) = ZN

A (0)−N−1Y1(N

∫ t

0

κ1ZNA (s)XN

E (s)ds)

+N−1Y2(Nκ2

∫ t

0

XNAE(s)ds)

m ≡ XNE (t) +XN

AE(t) does not depend on t.


Averaging by the equation

Dividing the first equation by N and taking the limit

N−1XNE (t) = N−1XE(0)−N−1Y1(N

∫ t

0

κ1ZNA (s)XN

E (s)ds)

+N−1Y2(Nκ2

∫ t

0

XNAE(s)ds)

+N−1Y3(Nκ3

∫ t

0

XNAE(s)ds)

0 = limN→∞

∫ t

0

(−κ1ZNA (s)XN

E (s) + κ2XNAE(s) + κ3X

NAE(s))ds

and

limN→∞

∫ t

0

XNE (s)ds = lim

N→∞

∫ t

0

(m−XNAE(s))ds =

∫ t

0

m(κ2 + κ3)

κ1ZA(s) + κ2 + κ3ds.


Michaelis-Menten equation

ZNA (t) = ZN

A (0)−N−1Y1(N

∫ t

0

κ1ZNA (s)XN

E (s)ds)

+N−1Y2(Nκ2

∫ t

0

XNAE(s)ds)

which converges to

ZA(t) = ZA(0)−∫ t

0

κ1ZA(s)m(κ2 + κ3)

κ1ZA(s) + κ2 + κ3ds

+κ2

∫ t

0

mκ1ZA(s)


= ZA(0)−∫ t

0

mκ1κ3ZA(s)



Gaussian limit Kang, Kurtz, and Popovic (2014)

Call the limiting quadratic variation∫ t

0 σ2(ZA(s))ds.∫ t

0

σ2(ZA(s))ds

=

∫ t

0

((h(ZA(s))− 1)2(κ2 + κ3) + (h(ZA(s)) + 1)2κ2 + h(ZA(s))2κ3)

× mκ1ZA(s)


ThenUN(t) =

√N(ZN

A (t)− Z(t))

converges to the solution of

U(t) = U(0) +W (

∫ t

0

σ2(ZA(s))ds)−∫ t

0

mκ1κ3(κ2 + κ3)

(κ2 + κ3 + κ1ZA(s))2U(s)ds


Diffusion approximation

Rationale: Find a sequence of diffusion processes DNA such that

√N(DN

A − ZA)⇒ U.

So

DNA (t) = ZN

A (0) +1√N

∫ t

0

σ(DNA (s))dW (s)−

∫ t

0

mκ1κ3DNA (s)

κ2 + κ3 + κ1DNA (s)

ds


Heat shock model

The following reaction network is a given as a model for the heatshock response in E. Coli by Srivastava, Peterson, and Bentley (2001).The analysis is from Kang (2012).

Reaction Intensity Reaction Intensity∅ → S8 4.00× 100 S6 + S8 → S9 3.62× 10−4XS6XS8

S2 → S3 7.00× 10−1XS2 S8 → ∅ 9.99× 10−5XS8

S3 → S2 1.30× 10−1XS3 S9 → S6 + S8 4.40× 10−5XS9

∅ → S2 7.00× 10−3XS1 ∅ → S1 1.40× 10−5

stuff + S3 → S5 + S2 6.30× 10−3XS3 S1 → ∅ 1.40× 10−6XS1

stuff + S3 → S4 + S2 4.88× 10−3XS3 S7 → S6 1.42× 10−6XS4XS7

stuff + S3 → S6 + S2 4.88× 10−3XS3 S5 → ∅ 1.80× 10−8XS5

S7 → S2 + S6 4.40× 10−4XS7 S6 → ∅ 6.40× 10−10XS6

S2 + S6 → S7 3.62× 10−4XS2XS6 S4 → ∅ 7.40× 10−11XS4


Exponents

ρ1 = β1

ρ2 = α2 + β2

ρ3 = α3 + β3

ρ4 = α1 + β4

ρ5 = α3 + β5

ρ6 = α3 + β6

ρ7 = α3 + β7

ρ8 = α7 + β8

ρ9 = α2 + α6 + β9

ρ10 = α6 + α8 + β10

ρ11 = α8 + β11

ρ12 = α9 + β12

ρ13 = β13

ρ14 = α1 + β14

ρ15 = α4 + α7 + β15

ρ16 = α5 + β16

ρ17 = α6 + β17

ρ18 = α4 + β18


Balance equations

{Z1} ρ13 = ρ14{Z2} max{ρ3, ρ4, ρ5, ρ6, ρ7, ρ8} = ρ2 ∨ ρ9{Z3} ρ2 = max{ρ3, ρ5, ρ6, ρ7}{Z4} ρ6 = ρ18{Z5} ρ5 = ρ16{Z6} max{ρ7, ρ8, ρ12, ρ15} = ρ9 ∨ ρ17{Z7} ρ9 = ρ8 ∨ ρ15{Z8} ρ1 ∨ ρ12 = ρ10 ∨ ρ11{Z9} ρ10 = ρ12{Z2 + Z3 + Z7} ρ4 = ρ15{Z2 + Z3} ρ4 ∨ ρ8 = ρ9{Z2 + Z7} max{ρ3, ρ4, ρ5, ρ6, ρ7} = ρ2 ∨ ρ15{Z6 + Z7 + Z9} ρ7 = ρ17{Z6 + Z9} max{ρ7, ρ8, ρ15} = ρ9 ∨ ρ17{Z6 + Z7} ρ7 ∨ ρ12 = ρ17 ∨ ρ10{Z8 + Z9} ρ1 = ρ17


θ · Z First scale Second scale Third scale{Z1} γ ≤ 2 balanced balanced{Z2} balanced balanced balanced{Z3} balanced balanced balanced{Z4} γ ≤ 2 γ ≤ 2 balanced{Z5} γ ≤ 2 γ ≤ 2 balanced{Z6} γ ≤ 1 balanced balanced{Z7} γ ≤ 1 γ ≤ 1 balanced{Z8} γ ≤ 0 γ ≤ 1 balanced{Z9} balanced balanced balanced{Z2 + Z3 + Z7} γ ≤ 0 balanced balanced{Z2 + Z3} γ ≤ 0 γ ≤ 1 balanced{Z2 + Z7} balanced balanced balanced{Z6 + Z7 + Z9} γ ≤ 1 γ ≤ 2 γ ≤ 2{Z6 + Z9} γ ≤ 1 γ ≤ 2 γ ≤ 2{Z6 + Z7} γ ≤ 1 balanced balanced{Z8 + Z9} γ ≤ 0 γ ≤ 1 balanced


ZN,γ1 (t) = ZN,γ

1 (0) +N−α1,γY13(

∫ t

0

κ13Nγ−2 ds)−N−α1,γY14(

∫ t

0

κ14Nγ−2+α1,γZN,γ

1 (s) ds)

ZN,γ2 (t) = ZN,γ

2 (0) +N−α2,γY3(

∫ t

0

κ3NγZN,γ

3 (s) ds) +N−α2,γY4(

∫ t

0

κ4Nγ−1ZN,γ

1 (s) ds)

+N−α2,γY5(

∫ t

0


3 (s) ds) +N−α2,γY6(

∫ t

0


3 (s) ds)

+N−α2,γY7(

∫ t

0


3 (s) ds) +N−α2,γY8(

∫ t

0

κ8Nγ−2ZN,γ

7 (s) ds)

−N−α2,γY2(

∫ t

0

κ2Nγ+α2,γZN,γ

2 (s) ds)−N−α2,γY9(

∫ t

0


2 (s)ZN,γ6 (s) ds)

ZN,γ3 (t) = ZN,γ

3 (0) +N−α2,γY2(

∫ t

0

κ2Nγ+α2,γZN,γ

2 (s) ds)

−N−α2,γY3(

∫ t

0

κ3Nγ+α2,γZN,γ

3 (s) ds)−N−α2,γY5(

∫ t

0


3 (s) ds)

−N−α2,γY6(

∫ t

0


3 (s) ds)−N−α2,γY7(

∫ t

0


3 (s) ds)


ZN,γ4 (t) = ZN,γ

4 (0) +N−2Y6(

∫ t

0


3 (s) ds)−N−2Y18(∫ t

0

κ18NγZN,γ

4 (s) ds)

ZN,γ5 (t) = ZN,γ

5 (0) +N−2Y5(

∫ t

0


3 (s) ds)−N−2Y16(∫ t

0

κ16NγZN,γ

5 (s) ds)

ZN,γ6 (t) = ZN,γ

6 (0) + Y7(

∫ t

0


3 (s) ds) + Y8(

∫ t

0

κ8Nγ−2ZN,γ

7 (s) ds)

+Y12(

∫ t

0


9 (s) ds) + Y15(

∫ t

0

κ15Nγ−1ZN,γ

4 (s)ZN,γ7 (s) ds)

−Y9(∫ t

0


2 (s)ZN,γ6 (s) ds)

−Y10(∫ t

0


6 (s)ZN,γ8 (s) ds)− Y17(

∫ t

0

κ17Nγ−2ZN,γ

6 (s) ds)

ZN,γ7 (t) = ZN,γ

7 (0) + Y9(

∫ t

0


2 (s)ZN,γ6 (s) ds)

−Y8(∫ t

0

κ8Nγ−2ZN,γ

7 (s) ds)

−Y15(∫ t

0

κ15Nγ−1ZN,γ

4 (s)ZN,γ7 (s) ds)


ZN,γ8 (t) = ZN,γ

8 (0) +N−α8,γY1(

∫ t

0

κ1Nγ ds) +N−α8,γY12(

∫ t

0


9 (s) ds)

−N−α8,γY10(

∫ t

0


6 (s)ZN,γ8 (s) ds)

−N−α8,γY11(

∫ t

0


8 (s) ds)

ZN,γ9 (t) = ZN,γ

9 (0) +N−α8,γY10(

∫ t

0


6 (s)ZN,γ8 (s) ds)

−N−α8,γY12(

∫ t

0


9 (s) ds)

ZN,γ2,3 = ZN,γ

2,3 (0) +N−α2,γY4(

∫ t

0


1 (s) ds)

+N−α2,γY8(

∫ t

0

κ8Nγ−2ZN,γ

7 (s) ds)

−N−α2,γY9(

∫ t

0


2 (s)ZN,γ6 (s) ds)


γ = 0

α1 α2 α3 α4 α5 α6 α7 α8 α9

1 0 0 2 2 0 0 0 0

For γ = 0, {ZN,02 , ZN,0

3 , ZN,08 } converge to the solution of

Z02(t) = Z0

2(0) + Y3(

∫ t

0

κ3Z03(s) ds) + Y4(

∫ t

0

κ4Z01(0) ds)

−Y2(∫ t

0

κ2Z02(s) ds)

Z03(t) = Z0

3(0) + Y2(

∫ t

0

κ2Z02(s) ds)− Y3(

∫ t

0

κ3Z03(s) ds)

Z08(t) = Z0

8(0) + Y1(

∫ t

0

κ1 ds)


0 20 40 60 80 1000

2

4

6

8

Time in seconds

X2 in

num

bers

Full system

X2:σ32

0 20 40 60 80 1000

2

4

6

8

Time in seconds

X2 in

num

bers

Reduced system

X2:σ32

0 20 40 60 80 1000

5

10

15

Time in seconds

X3 in

num

bers

Full system

X3:Eσ32

0 20 40 60 80 1000

5

10

15

Time in secondsX

3 in n

umbe

rs

Reduced system

X3:Eσ32


0 20 40 60 80 1000

100

200

300

400

500

Time in seconds

X8 in

num

bers

Full system

X8:Rec−Prot

0 20 40 60 80 1000

100

200

300

400

500

Time in seconds

X8 in

num

bers

Reduced system

X8:Rec−Prot


γ = 1

α1 α2 α3 α4 α5 α6 α7 α8 α9

0 0 0 2 2 0 0 1 1

For γ = 1, {ZN,12,3 , Z

N,16 , ZN,1

7 , ZN,18 } converges to the solution of

Z12,3(t) = Z1

2,3(0) + Y4(

∫ t

0

κ4Z11(0) ds)

Z16(t) = Z1

6(0) + Y7(

∫ t

0

κ7Z1

3(s) ds) + Y12(

∫ t

0

κ12Z19(0) ds)

+Y15(

∫ t

0

κ15Z14(0)Z1

7(s) ds)− Y10(∫ t

0

κ10Z16(s)Z1

8(s) ds)

Z17(t) = Z1

7(0)− Y15(∫ t

0

κ15Z14(0)Z1

7(s) ds)

Z18(t) = Z1

8(0) +

∫ t

0

κ1 ds

Z1

3(t) =κ2Z

12,3(s)

κ2 + κ3


0 200 400 600 800 10000

20

40

60

80

Time in seconds

X2+

X3 in

num

bers

Full system

X2+X

3:σ32+Eσ32

0 200 400 600 800 10000

20

40

60

80

Time in seconds

X2+

X3 in

num

bers

Reduced system

X2+X

3:σ32+Eσ32

0 200 400 600 800 10000

10

20

30

40

50

60

Time in seconds

X6 in

num

bers

Full system

X6:JKE−complex

0 200 400 600 800 10000

10

20

30

40

50

60

Time in seconds

X6 in

num

bers

Reduced system

X6:JKE−complex


0 200 400 600 800 10004

5

6

7

8

9

10

Time in seconds

X7 in

num

bers

Full system

X7:J−σ32 complex

0 200 400 600 800 10004

5

6

7

8

9

10

Time in seconds

X7 in

num

bers

Reduced system

X7:J−σ32 complex

0 200 400 600 800 10000

1000

2000

3000

4000

5000

Time in seconds

X8 in

num

bers

Full system

X8:Rec−Prot

0 200 400 600 800 10000

1000

2000

3000

4000

5000

Time in seconds

X8 in

num

bers

Reduced system

X8:Rec−Prot


γ = 2

α1 α2 α3 α4 α5 α6 α7 α8 α9

0 1 1 2 2 0 0 2 2

For γ = 2, {ZN,21 , ZN,2

2,3 , ZN,24 , ZN,2

5 , ZN,28 , ZN,2

9 } converges to the solution of

Z21(t) = Z2

1(0) + Y13(

∫ t

0

κ13 ds)− Y14(∫ t

0

κ14Z21(s) ds)

Z22,3(t) = Z2

2,3(0) +

∫ t

0

[κ4Z

21(s)− κ9Z2

2(s)Z2

6(s))]ds

Z24(t) = Z2

4(0) +

∫ t

0

(κ6Z23(s)− κ18Z2

4(s)) ds

Z25(t) = Z2

5(0) +

∫ t

0

(κ5Z23(s)− κ16Z2

5(s)) ds

Z28(t) = Z2

8(0) +

∫ t

0

(κ1 − κ7Z23(s)− κ11Z2

8(s)) ds

Z29(t) = Z2

9(0) +

∫ t

0

κ7Z23(s) ds


Z13(t) =

κ3Z12,3(s)

κ2 + κ3Z1

3(t) =κ2Z

12,3(s)

κ2 + κ3

Z2

6(s) =κ7Z

23(s) + κ12Z

29(s)

κ10Z28(s)

Z2

7(t) =κ9Z

22(t)Z6(t)

κ15Z24(t)


0 2000 4000 6000 8000 100009

9.5

10

10.5

11

Time in seconds

X1 in

num

bers

Full system

X1:σ32 mRNA

0 2000 4000 6000 8000 100009

9.5

10

10.5

11

Time in seconds

X1 in

num

bers

Reduced system

X1:σ32 mRNA

0 2000 4000 6000 8000 100000

200

400

600

Time in seconds

X2+

X3 in

num

bers

Full system

X2+X

3:σ32+Eσ32

0 2000 4000 6000 8000 100000

200

400

600

Time in secondsX

2+X

3 in n

umbe

rs Reduced system

X2+X

3:σ32+Eσ32


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

Time in seconds

X2 in

num

bers

Full system

X2:σ32

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

20

40

60

80

100

Time in seconds

X2 in

num

bers

Full system

X2:σ32

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

Time in seconds

X3 in

num

bers

Full system

X3:Eσ32

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

100

200

300

400

500

600

Time in seconds

X3 in

num

bers

Full system

X3:Eσ32


0 2000 4000 6000 8000 100000

5000

10000

Time in seconds

X4 in

num

bers

Full system

X4:FtsH

0 2000 4000 6000 8000 100000

5000

10000

Time in seconds

X4 in

num

bers

Reduced system

X4:FtsH

0 2000 4000 6000 8000 100000

5000

10000

15000

Time in seconds

X5 in

num

bers

Full system

X5:GroEL

0 2000 4000 6000 8000 100000

5000

10000

15000

Time in secondsX

5 in n

umbe

rs

Reduced system

X5:GroEL


0 2000 4000 6000 8000 100000

2

4

6

8

10

Time in seconds

X6 in

num

bers

Full system

X6:JKE−complex

0 2000 4000 6000 8000 100000

2

4

6

8

10

Time in seconds

X6 in

num

bers

Reduced system

X6:JKE−complex

0 2000 4000 6000 8000 100000

2

4

6

8

Time in seconds

X7 in

num

bers

Full system

X7:J−σ32 complex

0 2000 4000 6000 8000 100000

2

4

6

8

Time in secondsX

7 in n

umbe

rs

Reduced system

X7:J−σ32 complex


0 2000 4000 6000 8000 100000

5000

10000

15000

Time in seconds

X8 in

num

bers

Full system

X8:Rec−Prot

0 2000 4000 6000 8000 100000

5000

10000

15000

Time in seconds

X8 in

num

bers

Reduced system

X8:Rec−Prot

0 2000 4000 6000 8000 100000

5000

10000

Time in seconds

X9 in

num

bers

Full system

X9:Rec−Prot−J

0 2000 4000 6000 8000 100000

5000

10000

Time in secondsX

9 in n

umbe

rs

Reduced system

X9:Rec−Prot−J


Things I haven’t told youKang and Kurtz (2013), Kang, Kurtz, and Popovic (2014)

In general, additional balance conditions are needed.

A systematic approach to averaging fast components.

How to derive appropriate diffusion/Langevin approximations.

Things I think would be usefulHow to automate the analysis.

Criteria for “optimal” selection of the scaling exponents.

How to systematically incorporate biological constraints.


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Hye-Won Kang and Thomas G. Kurtz. Separation of time-scales and model reduction for stochastic reactionnetworks. Ann. Appl. Probab., 23(2):529–583, 2013. ISSN 1050-5164. doi: 10.1214/12-AAP841. URL http://dx.doi.org/10.1214/12-AAP841.

Hye-Won Kang, Thomas G. Kurtz, and Lea Popovic. Central limit theorems and diffusion approximations formultiscale Markov chain models. Ann. Appl. Probab., 24(2):721–759, 2014. ISSN 1050-5164. doi: 10.1214/13-AAP934. URL http://dx.doi.org/10.1214/13-AAP934.

Thomas G. Kurtz. Strong approximation theorems for density dependent Markov chains. Stochastic ProcessesAppl., 6(3):223–240, 1977/78.

R. Srivastava, M. S. Peterson, and W. E. Bentley. Stochastic kinetic analysis of escherichia coli stress circuitusing sigma(32)-targeted antisense. Biotechnol. Bioeng., 75:120–129, 2001.

R. Srivastava, L. You, J. Summers, and J. Yin. Stochastic vs. deterministic modeling of intracellular viralkinetics. J. Theoret. Biol., 218(3):309–321, 2002. ISSN 0022-5193.

N. G. van Kampen. Stochastic processes in physics and chemistry. North-Holland Publishing Co., Amsterdam,1981. ISBN 0-444-86200-5. Lecture Notes in Mathematics, 888.

http://link.aip.org/link/?JCP/115/1716/1

http://link.aip.org/link/?JCP/115/1716/1

http://dx.doi.org/10.1186/1752-0509-6-143

http://dx.doi.org/10.1186/1752-0509-6-143

http://dx.doi.org/10.1214/12-AAP841




Abstract

Stochastic Models for Chemical Reactionskurtz/Lectures/BASEL17M.pdf · Stochastic Models for Chemical Reactions Specifying Markov chains Specifying inﬁnitesimal behavior Chemical

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