PHYSICAL REVIEW E 92, 012120 (2015)

Approximate probability distributions of the master equation

Philipp Thomas*

School of Mathematics and School of Biological Sciences, University of Edinburgh, Edinburgh EH8 9YL, United Kingdom

Ramon Grima†

School of Biological Sciences, University of Edinburgh, Edinburgh EH8 9YL, United Kingdom(Received 13 November 2014; revised manuscript received 3 April 2015; published 13 July 2015)

Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations canrarely be obtained. We here derive an analytical approximation of the time-dependent probability distributionof the master equation using orthogonal polynomials. The solution is given in two alternative formulations:a series with continuous and a series with discrete support, both of which can be systematically truncated.While both approximations satisfy the system size expansion of the master equation, the continuous distributionapproximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast,the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shownto lead to particularly simple analytical expressions for the probability distributions of molecule numbers inmetabolic reactions and gene expression systems.

DOI: 10.1103/PhysRevE.92.012120 PACS number(s): 02.50.Ey, 87.18.Tt, 05.40.−a, 87.23.−n

I. INTRODUCTION

Master equations are commonly used to describe fluctua-tions of particulate systems. In most instances, however, thenumber of reachable states is so large that their combinatorialcomplexity prevents one from obtaining analytical solutions tothese equations. Explicit solutions are known only for certainclasses of linear birth-death processes [1], under detailedbalance conditions [2], or for particularly simple examplesin stationary conditions [3]. Considerable effort has beenundertaken to approximate the solution of the master equationunder more general conditions, including time dependence andconditions lacking detailed balance [4,5].

A common technique addressing this issue was given by vanKampen in terms of the system size expansion (SSE) [6]. Themethod assumes the existence of a specific parameter, termedthe system size, for which the master equation approaches adeterministic limit as its value is taken to infinity. The leading-order term of this expansion describes small fluctuations aboutthis limit in terms of a Gaussian probability density, calledthe linear noise approximation (LNA). This approximationhas been widely applied in biochemical kinetics [7] butalso in the theory of polymer assembly [8], epidemics [9],economics [10], and machine learning [11]. The benefitof the LNA is that it yields generic expressions for theprobability density. Its deficiency lies in the fact that, strictlyspeaking, it is valid only in the limit of infinite system size.Hence one generally suspects that its predictions becomeinaccurate when one studies fluctuations that are not too smallcompared to the mean and therefore implying non-Gaussianstatistics.

Higher-order terms in the SSE have been employed tocalculate non-Gaussian corrections to the LNA for the first fewmoments [12–14]; alternative methods are based on momentclosure [15]. It is, however, the case that the knowledge of

*philipp.thomas@ed.ac.uk†ramon.grima@ed.ac.uk

a limited number of moments does not allow us to uniquelydetermine the underlying distribution functions. Reconstruc-tion of the probability distribution therefore requires additionalapproximations such as the maximum entropy principle [16]or the truncated moment generating function [17], whichgenerally yield different results. While the accuracy ofthese repeated approximations remains unknown, analyticalexpressions for the probability density can rarely be obtainedor might not even exist [18]. A systematic investigationof the distributions implied by the higher-order terms inthe SSE, without resorting to moments, is therefore stillmissing.

We here analytically derive, for the first time, as far aswe know, a closed-form series expansion of the probabilitydistribution underlying the master equation. We proceed byoutlining the expansion of the master equation in Sec. Iand briefly review the solution of the leading-order termsgiven by the LNA in Sec. II. While commonly the SSEis truncated at this point, we show that the higher-orderterms can be obtained using an asymptotic expansion of thecontinuous probability density. The resulting series is givenin terms of orthogonal polynomials and can be truncatedsystematically to any desired order in the inverse systemsize. Analytical expressions are given for the expansioncoefficients.

Thereby we establish two alternative formulations of thisexpansion: a continuous and a discrete one, both satisfying theexpansion of the master equation. We show that for linearbirth-death processes, the continuous approximation oftenfails to converge reasonably fast. In contrast, the discreteapproximation introduced in Sec. III accurately convergesto the true distribution with increasing truncation order. InSec. IV, we show that for nonlinear birth-death processes,renormalization is required for achieving rapid convergenceof the series. Our analysis is motivated by the use ofsimple examples throughout. Using a common model of geneexpression, we conclude in Sec. VI that the new method allowsus to predict the full time dependence of the molecule numberdistribution.

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PHILIPP THOMAS AND RAMON GRIMA PHYSICAL REVIEW E 92, 012120 (2015)

II. SYSTEM SIZE EXPANSION

As a starting point, we focus on the master equationformulation of biochemical kinetics. We therefore consider aset of R chemical reactions involving a single species confinedin a well-mixed volume �. Note that for chemical systems thesystem size coincides with the reaction volume. We denote bySr the net change in the molecule numbers in the rth reactionand by γr (n,�) the probability per unit time for this reactionto occur. The probability of finding n molecules in the volume� at time t , denoted by P (n,t), then obeys the master equation

dP (n,t)

dt=

R∑r=1

(E−Sr − 1)γr (n,�)P (n,t), (1)

where E−Sr is the step operator defined as E−Sr g(n) =g(n − Sr ) for any function g(n) of the molecule numbers [6].Note that throughout the article, deterministic initial conditionsare assumed. The system size expansion now proceeds byseparating the instantaneous concentration into a deterministicpart, given by the solution of the rate equations [X], and afluctuating part ε,

n

�= [X] + �−1/2ε, (2)

which is van Kampen’s ansatz. The expansion of the masterequation can be summarized in three steps:

(i) Using Eq. (2) one expands the step operator

E−Sr γr (n,�)P (n,t) = γr (n − Sr,�)P (n − Sr,t)

= e−�−1/2Sr∂ε γr (�[X] + �1/2ε,�)

×P (�[X] + �1/2ε,t), (3)

where ∂ε denotes ∂∂ε

.(ii) Next, the probability for the molecule numbers is cast

into a probability density �(ε,t) for the fluctuations using vanKampen’s ansatz,

�(ε,t) = �1/2P (�[X] + �1/2ε,t), (4)

which is essentially a change of variables. Note that this stepimplicitly assumes a continuous approximation �(ε,t) of theprobability distribution as thought in the original derivation ofvan Kampen [6].

(iii) It remains to expand the propensity about the deter-ministic limit

γr (�[X] + �1/2ε,�) =∞∑

k=0

�−k/2 εk

k!

∂kγr (�[X],�)

∂[X]k. (5)

Note that γr (�[X],�) is just the propensity evaluated at themacroscopic concentration and hence it must depend explicitlyon �. We assume that the propensity possesses a power seriesin the inverse volume

γr (�[X],�) = �

∞∑s=0

�−sf (s)r ([X]). (6)

For mass-action kinetics, for instance, the propensity is

given by γr (n,�) = �1−�r kr�r !( n�r

), where �r is the reac-

tion order of the rth reaction. Using the Taylor expansionof the binomial coefficient, we have f (0)

r ([X]) = kr [X]�r ,

f (s)r ([X]) = kr [X]�r−sS�r ,�r−s , and f (s)

r = 0 for s � �r , whereS denotes the Stirling numbers of the first kind. Note alsothat effective propensities being deduced from mass actionkinetics have an expansion similar to Eq. (6). The Michaelis-Menten propensity γr (n,�) = �kr

nn+K�

[19], for instance, has

f (0)r ([X]) = kr

[X][X]+K

and f (s)r ([X]) = 0 for s > 0.

Substituting now Eqs. (3)–(6) into Eq. (1) and rearrangingthe result in powers of �−1/2, we find(

∂

∂t− �1/2 d[X]

dt

∂

∂ε

)�(ε,t)

=[−�1/2

R∑r=1

Srf(0)r ([X])

∂

∂ε+

N∑k=0

�−k/2Lk

]�(ε,t)

+O[�−(N+1)/2]. (7)

Equating terms to order �1/2 yields the macroscopic rateequation

d[X]

dt=

R∑r=1

Srf(0)r ([X]). (8)

The higher-order terms in the expansion of the master equationcan be written out explicitly,

Lk =�k/2�∑s=0

k−2(s−1)∑p=1

Dk−p−2(s−1)p,s

p![k − p − 2(s − 1)]!(−∂ε)pεk−p−2(s−1),

(9)

where �·� denotes the ceiling value and the coefficients aregiven by

Dqp,s =

R∑r=1

(Sr )p∂q

[X]f(s)r ([X]), (10)

which depend explicitly on the solution of the rate equation (8).Note that in the following the abbreviation Dq

p = Dq

p,0 is used.

III. EXPANSION OF THE CONTINUOUSPROBABILITY DENSITY

We here study the time-dependent solution of the partialdifferential equation approximation of the master equation,Eq. (7). We therefore expand the probability density of Eq. (4),

�(ε,t) =N∑

j=0

�−j/2πj (ε,t) + O[�−(N+1)/2], (11)

which also allows the expansion of the time-dependentmoments to be deduced in closed-form. Finally, we recoverthe stationary solution as a particular case.

A. Linear noise approximation

Substituting Eq. (11) into Eq. (7) and equating terms toorder O(�0), we find(

∂

∂t− L0

)π0 = 0, (12)

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APPROXIMATE PROBABILITY DISTRIBUTIONS OF THE . . . PHYSICAL REVIEW E 92, 012120 (2015)

where L0 = −∂εJ ε + 12∂2

εD02 is a Fokker-Planck operator

with linear coefficients and J = D11 is the Jacobian of the

rate equation. The probability density of fluctuations aboutthe macroscopic concentration, described by ε, is given by acentered Gaussian,

π0(ε,t) = 1√2πσ 2(t)

exp

[− ε2

2σ 2(t)

], (13)

which acquires time dependence via its variance σ 2(t). Thelatter satisfies

∂σ 2

∂t= 2J (t)σ 2 + D0

2(t), (14)

which is the familiar LNA result [6]. In the following we willdrop the time dependence of the coefficients for convenienceof notation.

B. Higher-order terms

Substituting Eq. (11) into Eq. (7), rearranging the remainingterms, and equating terms to order �−j/2, we find(

∂

∂t− L0

)πj (ε,t) = L1πj−1 + · · · + Ljπ0

=j∑

k=1

Lkπj−k(ε,t). (15)

This system of partial differential equations can be solvedusing the eigenfunction approach. We consider(

∂

∂t− L0

)m = λmm, (16)

which is solved by λm = −mJ and m = ψm(ε,t)π0(ε,t) with

ψm(ε,t) = π−10 (−∂ε)mπ0 = 1

σmHm

(ε

σ

). (17)

The functions Hm denote the Hermite orthogonal polynomialswhich are given explicitly in Appendix A. To verify the solu-tion of the eigenvalue problem, we set m+1 = (−∂ε)m andobserve that (∂t − L0)m+1 = −Jm+1 − ∂ε(∂t − L0)m.Using this in Eq. (16), we obtain λm+1 = (−J + λm) fromwhich the result follows because λ0 = 0 and 0 = π0.

Using the completeness of the eigenfunctions, we canwrite πj (ε,t) = ∑∞

m=0 a(j )m (t)ψm(ε,t)π0(ε,t). We verify in

Appendix B that the j th order term in the expansion involvesonly the first Nj = 3j eigenfunctions. The continuous SSEapproximation is consequently given by the asymptotic ex-pansion

�(ε,t) = π0(ε,t)

⎡⎣1 +

N∑j=1

�−j/2Nj∑

m=1

a(j )m (t)ψm(ε,t)

⎤⎦

+O[�−(N+1)/2], (18)

for which the coefficients can be determinedusing the orthogonality of the functions ψm, i.e.,σ 2n

n!

∫dε ψn(ε,t)ψm(ε,t)π0(ε,t) = δm,n.

C. The equation for the expansion coefficients

The coefficients a(j )n are now determined by inserting

the expansion of πj into Eq. (15), multiplying the resultby σ 2n

n!

∫dε ψn(ε,t), and performing the integration. Using

Eq. (16), the left-hand side of Eq. (15) becomes

σ 2n

n!

∑m

∫dε ψn(ε,t)

(∂

∂t− L0

)a(j )

m ψm(ε,t)π0(ε,t)

=(

∂

∂t− nJ

)a(j )

n . (19)

The calculation of terms in the summation on the right-handside of Eq. (15) is greatly simplified by defining the integral

Iαβmn = σ 2n

n!α!β!

∫dε ψn(ε,t)(−∂ε)αεβψm(ε,t)π0(ε,t), (20)

which yields

σ 2n

n!

∫dεψn(ε,t)Lkψm(ε,t)π0(ε,t)

=�k/2�∑s=0

k−2(s−1)∑p=1

Dk−p−2(s−1)p,s Ip,k−p−2(s−1)

mn . (21)

Using Eqs. (19) and (21) in Eq. (15), we find that thecoefficients satisfy the following set of ordinary differentialequations:(

∂

∂t− nJ

)a(j )

n

=j∑

k=1

Nj−k∑m=0

a(j−k)m

�k/2�∑s=0

k−2(s−1)∑p=1

Dk−p−2(s−1)p,s Ip,k−p−2(s−1)

mn ,

(22)

where we have assumed a(j )n = 0 for n > Nj . Explicitly, the

nonzero integrals are given by

Iαβmn =σβ−α+n−m

α!

min(n−α,m)∑s=0

(m

s

)

× [β + α + 2s − (m + n) − 1]!!

[β + α + 2s − (m + n)]!(n − α − s)!, (23)

and zero for odd (α + β) − (m + n). Here (2k − 1)!! = (2k)!2kk!

is the double factorial. Along with Eq. (23), in Appendix B weverify the following two important properties of the asymptoticseries solution given deterministic initial conditions: (i) Wehave Nj = 3j and hence Eq. (22) indeed yields a finite numberof equations and (ii) a

(j )n vanishes for all times when (n + j )

is odd.Finally, we note that Dq

p,s and Ipqmn are generally time

dependent because they are functions of the solution of therate equation and the LNA variance. Explicit expressions forthe approximate probability density can now be evaluated toany desired order.

D. Moments of the distribution

The solution for the probability density enables one toderive closed-form expressions for the moments. These are

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PHILIPP THOMAS AND RAMON GRIMA PHYSICAL REVIEW E 92, 012120 (2015)

obtained by multiplying Eq. (18) by∫

dε εβ and performingthe integration using Eq. (B4) of Appendix B. We find

〈εβ〉 =N∑

j=0

�−j/2�β/2�∑k=0

β!

2kk!σ 2ka

(j )β−2k + O[�−(N+1)/2], (24)

where a(j )0 = δ0,j and �·� denotes the floor value. In par-

ticular, it follows that the mean and variance are givenby 〈ε〉 = ∑N

j=1 �−j/2a(j )1 + O[�−(N+1)/2] and 〈ε2〉 = σ 2 +

2∑N

j=1 �−j/2a(j )2 + O[�−(N+1)/2].

It is now evident that the coefficients of the expansionare intricately related to the system size expansion of thedistribution moments. Naturally, one may seek to invert thisrelation. Indeed, as we show in Appendix C, given theexpansion for a finite set moments, the coefficients in Eq. (18)can be uniquely determined. In particular, to construct theprobability density to order �−j/2 one requires the expansionof the first 3j moments to the same order. Thus the problemof moments provides an equivalent route of systematicallyconstructing solutions to the master equation.

E. Solution in stationary conditions

Of particular interest is the expansion of the probabilitydensity under stationary conditions. Implicitly, we assume herethat the rate equation, Eq. (8), has a single asymptoticallystable fixed point, and hence the LNA variance is given byσ 2 = D0

2/(−2J ). Setting the time derivative on the left-handside of Eq. (22) to zero, we find that the coefficients of Eq. (18)can be expressed in terms of lower-order ones,

a(j )n = − 1

nJ

j∑k=1

�k/2�∑s=0

k−2(s−1)∑p=1

× Dk−p−2(s−1)p,s

3(j−k)∑m=0

a(j−k)m Ip,k−p−2(s−1)

mn . (25)

For example, truncating after terms of order �−1, we obtain

�(ε) =π0(ε) + �−1/2[a

(1)1 ψ1(ε) + a

(1)3 ψ3(ε)

]π0(ε)

+ �−1[a

(2)2 ψ2(ε) + a

(2)4 ψ4(ε) + a

(2)6 ψ6(ε)

]π0(ε)

+ O(�−3/2). (26)

The nonzero coefficients to order �−1/2 are given by

a(1)1 = −σ 2D2

1

2J − D01,1

J ,

(27)

a(1)3 = −σ 4D2

1

6J − σ 2D12

6J − D03

18J ,

while those to order �−1 are

a(2)2 = − a

(1)1

(D01,1

2J + D12

4J + 3σ 2D21

4J

)− a

(1)3

3D21

2J

− D02,1

4J − σ 2D11,1

2J − σ 2D22

8J − σ 4D31

4J ,

a(2)4 = − a

(1)1

( D03

24J + σ 2D12

8J + σ 4D21

8J

)− D0

4

96J − σ 2D13

24J

− σ 4D22

16J − σ 6D31

24J − a(1)3

(D01,1

4J + 3D12

8J + 7σ 2D21

8J

),

a(2)6 =1

2

(a

(1)3

)2. (28)

The accuracy of this distribution approximation is studiedthrough an example in the following.

F. The continuous approximation fails under lowmolecule number conditions

We now study the SSE solution for a linear birth-deathprocess, i.e., its propensities depend at most linearly on themolecular populations. Specifically, we consider the synthesisand decay of a molecular species X,

∅k0�k1

X. (29)

The master equation is constructed using S1 = +1, γ1 = �k0,S2 = −1, γ2 = k1n, and R = 2 in Eq. (1). The exact stationarysolution of the master equation is a Poisson distribution withmean �[X], where [X] = k0/k1. The coefficients in Eq. (10)are then given by

Dmn = δm,0k0 + (−1)nk1(δm,0[X] + δm,1), (30)

and Dmn,s = 0 for s > 0. The leading-order corrections to the

LNA given by Eqs. (26)–(28) lead to very compact expressionsfor the expansion coefficients and are given by

a(1)3 = [X]

6, a

(2)4 = [X]

24, a

(2)6 = [X]2

72(31)

and a(1)1 = a

(2)2 = 0.

Though the continuous approximation is expected toperform well at large values of �, we are particularly interestedin its performance when the value of � is decreased. Since theexpansion is carried out at constant average concentration,low values of � typically imply low numbers of moleculesand non-Gaussian distributions. In Fig. 1(a) we show that forparameters yielding half a molecule on average, the continuousapproximation obtained in this section, given by Eq. (18)together with Eqs. (25) and (30), is unsatisfactory since ashigher orders are taken into account, one observes largeoscillations in the tails of the distribution. In the followingsection we show that the disagreement arises due to theassumption that the support of the distribution is continuousrather than discrete as implied by the master equation.

IV. DISCRETE APPROXIMATION OF THEPROBABILITY DISTRIBUTION

The aim of this paragraph is to establish a discreteformulation of the distribution approximations. To clarify thisissue, we note that the exact characteristic function G(k,t) =∑∞

n=0 eiknP (n,t) is a 2π -periodic function and hence can beinverted as follows:

P (n,t) =∫ π

−π

dk

2πe−iknG(k,t). (32)

We now associate our continuous approximation,Eq. (18), with this characteristic function, i.e.,G(k,t) = ∫ ∞

−∞ dε eik�([X]+�−1/2ε)�(ε,t). Substituting this

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APPROXIMATE PROBABILITY DISTRIBUTIONS OF THE . . . PHYSICAL REVIEW E 92, 012120 (2015)

(a)

(b)

FIG. 1. (Color online) Linear birth-death process. We considerthe reaction system (29) in stationary conditions. (a) We compare theexact Poisson distribution (gray) to the continuous SSE approxima-tion [Eq. (18) together with Eqs. (25) and (30)] truncated after the�0 (LNA, blue line), �−1 (green), and �−3 terms (red) for parametervalues k0 = 0.5, k1 = 1, and � = 1 giving half a molecule on average.We observe that the continuous approximation becomes increasinglynegative and tends to oscillations with increasing truncation order.(b) In contrast, the discrete approximation shows no oscillations, andthe overall agreement with the exact Poisson distribution (gray bars)improves with increasing truncation order.

together with Eq. (11) into Eq. (32) one establishes aconnection formula between these discrete and continuousapproximations via the convolution

P (n,t) =N∑

j=0

�−j/2∫ ∞

−∞dε K(n − �[X] − �1/2ε)πj (ε,t)

+ O(�−(N+1)/2), (33)

with kernel

K(s) =∫ π

−π

dk

2πe−iks = sin(πs)

πs.

The convolution can be used to define the derivatives of thediscrete probability via

∂nP (n,t) =∫ ∞

−∞dε K(n − �[X] − �1/2ε)(�−1/2∂ε)�(ε,t),

(34)

and hence it satisfies E−Sj P (n,t) = ∫ ∞−∞ dε K(n − �[X] −

�1/2ε)e−�−1/2∂εSj �(ε,t), as well as γj (n,�)P (n,t) =∫ ∞−∞ dε K(n − �[X] − �1/2ε)γj (�[X] + �1/2ε,�)�(ε,t)

for analytic γj . It then follows from the fact that P (n,t)and �1/2�(�−1/2(n − �[X]),t) have the same characteristicfunction expansion that (i) both approximations possess thesame asymptotic expansion of their moments and that (ii)they satisfy the same expansion of the master equation.

For example, to leading order �0, Eq. (33) replaces theconventional continuous LNA estimate, π0 given by Eq. (13),with a discrete approximation,

P0(n,t) = 1

2

e− y2

2�2

√2π�

[erf

(iy + π�2

√2�

)− erf

(iy − π�2

√2�

)],

(35)

where y = n − �[X], �2 = �σ 2 is the LNA’s estimate forthe variance of molecule numbers, and erf is the error functiondefined by erf(x) = 2√

π

∫ x

0 e−t2dt .

Associating the �−j/2 term of Eq. (18) with πj in Eq. (33),higher-order approximations can now be obtained from

P (n,t) = P0(n,t) +N∑

j=1

�−j/23j∑

m=1

a(j )m (−�1/2∂n)mP0(n,t)

+O(�−(N+1)/2). (36)

The above follows from the definition of the eigenfunctions,Eq. (17), and using the derivative property of the convolutiongiven by Eq. (34). Note that the coefficients in this equation areexactly the same as given in Eq. (18) and hence are determinedby Eq. (22). One can verify two limiting cases: (i) as � → 0and �[X] being integer valued, then P0(n) = K(n − �[X]) =δn,�[X] is just the Kronecker δ as required for deterministicinitial conditions; (ii) as � → ∞ with y/� constant, theprobability distribution P0 reduces to the density π0 given byEq. (13) and hence it follows that in this limit the continuousand discrete series give the same results.

A. The discrete approximation performs well for linearbirth-death processes

For the linear birth-death process in the previous section,in Fig. 1(b) we show that the discrete approximation givenby Eq. (36) with Eq. (31) is in good agreement with the truedistribution when truncated after the terms of order �−1 andshows no oscillations. This agreement is remarkable giventhe compact form of the solution given by Eq. (31) and (36).The approximation is almost indistinguishable from the exactresult when the series is truncated after the �−3 terms usingEqs. (25) and (30) in Eq. (36). We hence conclude that thediscrete series approximates better the underlying distributionof the master equation than the continuous approximation.

B. The discrete approximation fails for nonlinearbirth-death processes

Next, we turn our attention to the analysis of nonlinear birth-death processes, i.e., a process whose propensities dependnonlinearly on the number of molecules. A particular feature

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PHILIPP THOMAS AND RAMON GRIMA PHYSICAL REVIEW E 92, 012120 (2015)

of such processes is that the LNA estimates for mean andvariances are generally no longer exact but agree with those ofthe true distribution only in the limit of large system size [13].

As an example, we here consider a simple metabolicreaction confined in a small subcellular compartment ofvolume � with substrate input,

∅h0−→ S, (37a)

S + Eh1�h2

Ch3−→ E. (37b)

The reactions describe the input of substrate moleculesS and their catalytic conversion by enzyme species E viathe enzyme-substrate complex C. The SSE of the averageconcentrations correcting the macroscopic rate equations havebeen extensively studied [12]. Since our theory applies to asingle species only, we here consider a reduced model in whichreaction (37b) is modelled via an effective propensity: Thisgives S1 = +1, γ1 = �k0, and S2 = −1, γ2 = �k1

nn+�K

. Thissimplification is valid when the enzyme-substrate associationis in rapid equilibrium, which holds when [ET ] � K andh3 � h2, where [ET ] is the total enzyme concentration [19].

The parameters in the reduced model are related to those in thedeveloped model by k0 = h1, k1 = h3[ET ], and K = h2/h1.This reduced master equation is solved exactly by a negativebinomial distribution [20].

The system size coefficients are obtained from Eq. (10) andare given by

Dmn = δm,0k0 + (−1)nk1

∂m

∂[X]m[X]

K + [X], (38)

and Dmn,s = 0 for s > 0. In Fig. 2(a) and 2(b), we consider two

parameter sets corresponding to low and moderate numbersof substrate molecules, respectively. We observe that incontrast to the linear case, the discrete approximation of thenonlinear birth-death process tends to oscillate with increasingtruncation order. This issue is addressed in the followingsection.

V. RENORMALIZATION OF NONLINEARBIRTH-DEATH PROCESSES

Van Kampen’s ansatz, Eq. (2), bears the particularly simpleinterpretation that for linear birth-death processes ε denotes thefluctuations about the average given by the solution of the rate

(a) (b)

(c) (d)

FIG. 2. (Color online) Nonlinear birth-death process. A metabolic reaction with Michaelis-Menten kinetics, scheme (37), is studied usingthe reduced model described in Sec. IV B. The exact stationary distribution is a negative binomial (shown in gray). (a) The discrete SSEapproximation given by Eq. (36) with Eq. (25) and (30) is shown in the low-molecule-number regime (k0/k1 = 0.25, 1 molecule on average)when truncated after the �0 (blue), �−3/2 (green), and �−4 terms (red dots). We observe that the expansion tends to oscillations and negativevalues of probability as the truncation order is increased. (b) Similar oscillations are observed for moderate molecule numbers (k0/k1 = 0.9, 27molecules on average) for the discrete series truncated after the �0 (blue), �−3/2 (green), and �−3 terms (red lines). In (c) and (d) we show theapproximations corresponding to the same parameters used in (a) and (b), respectively, but obtained using the renormalization procedure givenby Eq. (41) with Eq. (42) as described in the main text. The renormalized approximations avoid oscillations and are in excellent agreement withthe true probability distributions (gray bars). We note that for the cases (b) and (d) the continuous and discrete approximations give essentiallythe same result. The remaining parameters are given by � = 10 and K = 0.1.

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APPROXIMATE PROBABILITY DISTRIBUTIONS OF THE . . . PHYSICAL REVIEW E 92, 012120 (2015)

equation [X]. As noted in the previous example, for nonlinearbirth-death processes these estimates are only approximate.Their asymptotic series expansions will therefore requireadditional terms that compensate for the deviations of theLNA from the true concentration mean and variance. It wouldtherefore be desirable to find an approximation for nonlinearprocesses that yields more accurate mean and variance thanthe LNA, for instance, by rewriting van Kampen’s ansatz as

n

�= [X] + �−1/2〈ε〉︸ ︷︷ ︸

mean

+ �−1/2ε︸ ︷︷ ︸fluctuations

. (39)

Here ε = ε − 〈ε〉 denotes a centered variable that quantifiesthe fluctuations about the true average which is a prioriunknown. These estimates, however, can be approximatedusing the SSE beforehand, and the asymptotic expansion of thedistributions then can be performed about these new estimates.This idea is called renormalization and makes use of the factthat the terms correcting mean and variances can be summedexactly. As we show in the following the resummation allowsus to better control the convergence by effectively reducingthe number of terms in the summation while at the same timeit retains the accuracy of the expansion.

The system size expansion of the moments, Eq. (24),yields the following estimates for mean and variance of thefluctuations:

〈ε〉 =N∑

j=0

�−j/2a(j )1 + O(�−(N+1)/2), (40a)

σ 2 = σ 2 +N∑

j=1

�−j/2σ 2(j ) + O(�−(N+1)/2), (40b)

respectively, where σ 2(j ) = 2[a(j )

2 − Bj,2({χ !a(χ)1 }j−1

χ=1)/j !] andBj,n are the partial Bell polynomials [21].

The renormalization procedure amounts to replacing y withy = (n − �[X] − �1/2〈ε〉) and �2 with �2 = �σ 2 in Eq. (35)and associating a new Gaussian P0(n) with these estimates.The renormalized expansion is then given by

P (n,t) = P0(n,t) +N∑

j=1

�−j/23j∑

m=1

a(j )m (−�1/2∂n)mP0(n,t)

+O(�−(N+1)/2), (41)

where the renormalized coefficients can be calculated from thebare ones using

a(j )m =

j∑k=0

3k∑n=0

a(k)n κ

(j−k)m−n (42)

and

κ(n)j = 1

n!

�j/2�∑m=0

(−1)(j+m)n−m∑

k=j−2m

(n

k

)

×Bk,j−2m

({χ !a(χ)

1

})Bn−k,m

({χ !

2σ 2

(χ)

}), (43)

where, again, Bk,n({xχ }) denote the partial Bell polynomi-als [21]. The result is verified at the end of this section.

Note that the renormalized series has generally less nonzerocoefficients since by construction a

(j )1 = a

(j )2 = 0. Note that

for linear birth-death processes, mean and variance are exactto order �0 (LNA), and hence for this case expansion (36)coincides with Eq. (41).

For example, truncating after the �−1 terms, from Eq. (40)it follows that 〈ε〉 = �−1/2a

(1)1 + O(�−3/2) and σ 2 = σ 2 +

�−1[2a(2)2 − (a(1)

1 )2] + O(�−3/2). Using Eq. (42) the renor-malized coefficients can be expressed in terms of the bareones,

a(1)1 = 0, a

(1)3 = a

(1)3 , (44a)

a(2)2 = 0, a

(2)4 = a

(2)4 − a

(1)1 a

(1)3 , a

(2)6 = a

(2)6 . (44b)

This result can be used, for instance, to renormalizethe stationary solution using the bare coefficients given inSec. III E, Eqs. (27) and (28). The nonzero renormalizedcoefficients evaluate to

a(1)3 = −σ 4D2

1

6J + σ 2D12

6J + D03

18J , (45a)

a(2)4 = − D0

4

96J − σ 2D13

24J − σ 4D22

16J − σ 6D31

24J

− a(1)3

(3D1

2

8J + 3σ 2D21

4J

),

a(2)6 = 1

2

[a

(1)3

]2. (45b)

Note that for linear birth-death processes Dmn,s = 0 for s > 0

and m > 1, and hence the above equations reduce to Eqs. (27)and (28).

A. The renormalized approximation performs well fornonlinear birth-death processes

For the metabolic reaction (37), mean and variance canbe obtained to be 〈ε〉 = �−1/2ς + O(�−2) and σ 2 = σ 2 +�−1ς (ς + 1) + O(�−2), where ς = [X]/K is the reducedsubstrate concentration and σ 2 = Kς (ς + 1). Substitutingnow Eq. (38) into Eqs. (45), we obtain the expansioncoefficients

a(1)3 = σ 2

6(2ς + 1), (46a)

a(2)4 = σ 2

24[6ς (ς + 1) + 1], a

(2)6 = 1

2

(a

(1)3

)2, (46b)

which determine the renormalized series expansion to order�−1. Using Eq. (25), (38), and (42) we can give the next orderterms to order �−3/2 analytically,

a(3)3 = a

(1)3

K, a

(3)5 = a

(1)3

20[12ς (ς + 1) + 1],

a(3)7 = a

(1)3 a

(2)4 , a

(3)9 = 1

6(a(1)

3 )3. (46c)

In Figs. 2(c) and 2(d) we compare the renormalized ap-proximation given by Eq. (41) with the respective bareapproximations in Figs. 2(a) and 2(b). We observe that the

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PHILIPP THOMAS AND RAMON GRIMA PHYSICAL REVIEW E 92, 012120 (2015)

renormalization technique avoids oscillations and even thesimple analytical approximation given by Eqs. (46) is inreasonable agreement with the exact result. We note that theasymptotic approximations shown in Figs. 2(c) and 2(d) arealmost indistinguishable for higher truncation orders.

B. Proof of the renormalization formula

The renormalized coefficients can in principle be obtainedby matching the expansions given by Eq. (36) and (41) viatheir characteristic functions. For convenience we considerthe characteristic function of the series (18)

G(k) = G0(k)

⎡⎣1 +

∞∑j=1

�−j/23j∑

n=1

a(j )n (ik)n

⎤⎦, (47)

with G0(k) = e−(kσ )2/2 being the characteristic function solu-tion of the LNA π0(ε) and we have omitted the explicit timedependence to ease the notation. We are now looking for adifferent expansion with corrected estimates for the mean andvariance.

G(k) = G0(k)

⎡⎣1 +

∞∑j=1

�−j/23j∑

n=1

a(j )n (ik)n

⎤⎦, (48)

Note that G0(k) = eik〈ε〉e−(kσ )2/2 is the characteristic functionfor a Gaussian random variable with mean 〈ε〉 and variance σ 2

given by Eqs. (40).Equating now Eq. (47) and (48), we find

1 +∞∑

j=1

�−j/23j∑

n=1

a(j )n (ik)n

= G0(k)

G0(k)

⎡⎣1 +

∞∑j=1

�−j/23j∑

n=1

a(j )n (ik)n

⎤⎦. (49)

Expanding the prefactor in the above equation in powers of k

and then in �, we have

G0(k)

G0(k)=

∞∑j=0

(ik)j κj =∞∑

n=0

�−n/22n∑

j=0

(ik)j κ (n)j , (50)

from which Eq. (42) follows, which expresses the newcoefficients a

(j )n in terms of the bare ones a

(j )n . It remains to

derive an explicit expression for the κ(n)j . The expansion in

powers of (ik) yields

κj =�j/2�∑m=0

(−1)(j+m) 〈ε〉j−2m

(j − 2m)!

(σ 2−σ 2

2

)m

m!. (51)

We now expand the first term in inverse powers of � using thepartial Bell polynomials

1

(j − 2m)!

( ∞∑n=1

�−n/2a(n)1

)j−2m

=∞∑

n=1

�−n/2

n!

n∑k=0

δj−2m,kBn,k

({χ !a(χ)

1

}), (52)

and similarly for the second term,

1

m!

(1

2

∞∑n=1

�−n/2σ 2(n)

)m

=∞∑

n=0

�−n/2

n!

n∑k=0

δm,kBn,k

({χ !

2σ 2

(χ)

}). (53)

Using the above expansions in Eq. (51) and rearranging inpowers of �−1/2, Eq. (43) for the coefficients κ

(n)j follows.

Finally, one associates with the centered variable ε =ε − 〈ε〉 a Gaussian π0(ε) with variance σ 2. It thenfollows from inverting Eq. (48) that �(ε) = π0(ε) +∑N

j=1 �−j/2 ∑3j

n=1 a(j )n ψn(ε)π0(ε) + O[�−(N+1)/2]. Associat-

ing now the �−j/2 term of this equation with πj in Eq. (33),the discrete series for P (n,t) given by Eq. (41) follows.

VI. APPLICATION

The models studied so far have been useful to develop themethod. It remains to be demonstrated, however, that it remainsaccurate in cases where analytical solution is not feasible, as,for instance, for out-of-steady-state and nondetailed balancesystems. We here consider the synthesis of a protein P whichis degraded through an enzyme

∅h0−→ M

h1−→ ∅, Mh2−→ M + P, (54a)

P + Eh3�h4

Ch5−→ E, (54b)

where M denotes the transcript, E the enzyme, and C thecomplex species as studied in Ref. [22]. Since our theoryapplies only to a single species, we consider the limiting casein which the protein dynamics represents the slowest timescale of the system. It has been shown [23] that when speciesM is degraded much faster than the protein P , the proteinsynthesis (54a) reduces to the transition S1 = +z, γ1 = �k0

in which z is a random variable following the geometricdistribution ϕ(z) = 1

1+b( b

1+b)z

with average b, which is calledthe burst approximation. Similarly to the metabolic reactionstudied in Sec. IV, the enzymatic degradation process (54b)can be reduced to S2 = −1, γ2 = �k1

n�K+n

with a nonlineardependence on the protein number n. The master equationdescribing the protein number is then given by

d

dtP (n) = �

∞∑z=0

(E−z − 1)k0ϕ(z)P (n)

+�(E+1 − 1)k1n

�K + nP (n). (55)

The relation between the parameters in the reduced and thedeveloped model are given by k0 = h0h2/h1, b = h2/h1,k1 = h5[ET ], K = h5/h3, where [ET ] denotes the totalenzyme concentration. This description involves countablymany reactions: one for the degradation of the protein andone for each value of z. Therefore, the reactions cannot obeydetailed balance in steady state. The system size coefficientsnow follow from Eq. (10) and are given by

Dmn = δm,0k0〈zn〉ϕ + (−1)nk1

∂m

∂[X]m[X]

K + [X], (56)

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APPROXIMATE PROBABILITY DISTRIBUTIONS OF THE . . . PHYSICAL REVIEW E 92, 012120 (2015)

and Dmn,s = 0 for s > 0, where 〈zn〉ϕ = ∑∞

z=0 znϕ(z) =1

1+bLi−n( b

1+b) denotes the average over the geometric dis-

tribution in terms of the polylogarithm function [24]. Thedeterministic equation is given by

d[X]

dt= k0b − k1[X]

K + [X], (57)

which follows from the expression for D01. Using the Jacobian

J = D11 and D0

2 in Eq. (14), we find that the LNA varianceobeys

∂σ 2

∂t= − 2k1K

([X] + K)2σ 2 + k0b(1 + 2b) + k1[X]

K + [X]. (58)

The ODEs given by Eq. (57) and (58) are integrated numer-ically and the solution is used in Eq. (35) from which theleading-order approximation follows. Higher-order approxi-mations are now be obtained by using Eq. (56) in (22) whichgovern the time evolution of the coefficients a

(j )m (t) and using

the result in Eqs. (41) and (42). We assume deterministic initialconditions with zero proteins, meaning a

(j )m (0) = δm,0δj,0.

In Fig. 3(a) we compare the time evolution obtained bythe leading-order approximation P0 and Eq. (41) truncatedafter the �−3 term. The latter distributions are in excellentagreement with the distributions sampled using the stochasticsimulation algorithm [25]. In particular, unlike the leadingorder approximation, these describe well mode, skewness, andtails of the distribution. We note that also the mean and varianceof these distribution approximations are in excellent agreementas verified in inset of Fig. 3(a).

Despite the overall good agreement, in Fig. 3(b) weshow that there are discrepancies at very short times where,again, the distribution approximations tend to oscillations.Motivated by this numerical observation, we speculate that thisbehavior of the expansion is due a temporal boundary layer ascommonly observed in singular perturbation expansions [26].Theoretically, the layer must be located at times of the sameorder as the expansion parameter, i.e., t = (�K)−1/2 min ≈13 s, coinciding with the simulation in Fig. 3(b). This suggeststhat our approach only describes the outer solution. Furtheranalysis would be required to investigate also the inner solutionwhich is beyond the scope of this article.

VII. DISCUSSION

We have here presented an approximate solution method forthe probability distribution of the master equation. The solutionis given in terms of an asymptotic series expansion that canbe truncated systematically to any desired order in the inversesystem size. For biochemical systems with large numbers ofmolecules, we have derived a continuous series approximationthat extends van Kampen’s LNA to higher orders in the SSE.In low-molecule-number conditions, we have found that thiscontinuous approximation becomes inaccurate. Instead, inmost practical situations, the prescribed discrete distributionapproximations incorporating higher-order terms in the SSEbetter capture the underlying solution of the master equation.While the terms to order �−1 have been given explicitly, wefound that for the examples studied here up to the �−3 or �−4

terms had to be taken into account to accurately characterize

(a)

(b)

FIG. 3. (Color online) Predicting transient distributions of geneexpression. The dynamics of protein synthesis with enzymatic degra-dation, scheme (54), is studied using the burst approximation (55).(a) We compare the time dependence of the renormalized discreteapproximations to exact stochastic simulations at times 1, 2, and14 min. The overall shape (mode, skewness, distribution tails) ofthe simulated distributions (bars) is in excellent agreement with theseries approximation when truncated after the �−3 terms (solid lines)but not when only �0 are taken into account (dashed lines). Thisagreement is also observed for the first two moments shown in theinset: While the �−3 approximation (blue solid line) agrees withthe moment dynamics of the simulated distributions (dots) of thereduced model (55), the �0 approximation underestimates the mean(gray solid line) and variance by 25%. The area within one standarddeviation of the mean obtained from simulations is shown in blue,and the boundary obtained from the approximations are shown asdashed lines (�0 gray, �−3 blue). (b) Despite the good agreementshown in (a) we found that at very short times (12 s, blue solid line)the series truncated after the �−3 terms tends to oscillations whichquickly disappear for later times (24 s, green; 48 s, red solid line).See the main text for discussion. Parameters are k0� = 15 min−1,k1� = 100 min−1, K� = 20, � = 100, and b = 5. Histograms wereobtained from 10 000 stochastic simulations.

these non-Gaussian distributions. We note, however, that theasymptotic expansion cannot generally guarantee the positivityof the probability law. These undulations are particularlypronounced in the short-time behavior of the expansion studiedin Sec. VI, which our theory does not describe.

Previous means of solving the master equation have eitherbeen numerical in nature [27] or have focused on the inverseproblem, i.e., reconstruction of the probability density from themoments. While a numerical solution for the master equationof a single species is rather straightforward, we expect our

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PHILIPP THOMAS AND RAMON GRIMA PHYSICAL REVIEW E 92, 012120 (2015)

procedure to become computationally advantageous whengeneralized to the multivariate case where numerical solutionis usually prohibitive because of combinatorial explosion.

Methods based on moments typically require approx-imations such as moment closure [16] and also requirethe prior assumption of the first few moments containingall information on the probability distribution. Conversely,using the system size expansion, we have here obtained theprobability distribution directly from the master equationwithout the need to resort to moments. This method enjoysthe particular advantage over previous ones whereby the firstfew terms of this expansion can be written down explicitlyas a function of the rate constants and for any number ofreactions. For small models we have demonstrated that theprocedure leads to particularly simple expressions for thenon-Gaussian distributions. This development could proveparticularly valuable for parameter estimation of biochemicalreactions in living cells.

ACKNOWLEDGMENTS

We thank Claudia Cianci and David Schnoerr for carefulreading of the manuscript.

APPENDIX A: USEFUL PROPERTIES OF THEHERMITE POLYNOMIALS

We here briefly review some properties of the Hermiteorthogonal polynomials. The polynomials can be defined interms of the derivatives of a centered Gaussian π0 with varianceσ 2,

Hn

(ε

σ

)= π−1

0 (ε)(−σ∂ε)nπ0(ε). (A1)

An explicit formula is

Hn

(ε

σ

)=

�n/2�∑k=0

(n

2k

)(−1)k

(ε

σ

)n−2k

(2k − 1)!! . (A2)

These functions are orthogonal1n!

∫ ∞−∞ dε Hm( ε

σ)Hn( ε

σ) π0(ε) = δnm, with respect to the

Gaussian measure π0. The derivative satisfies

(σ∂ε)mHn

(ε

σ

)= n!

(n − m)!Hn−m

(ε

σ

). (A3)

Since these polynomials are complete, every function f (ε)in L2(R,π0) (not necessarily positive) can be expanded asf (ε) = ∑∞

n=0 bnHn( εσ

)π0(ε), where the coefficients are givenby bn = 1

n!

∫dε Hn( ε

σ)f (ε). We note because H0( ε

σ) = 1 and

π0 is normalized, we must have b0 = 1 if∫

dε f (ε) = 1.

APPENDIX B: EXPLICIT DERIVATION OF EQ. (23) ANDTHE PROPERTIES OF THE EXPANSION COEFFICIENTS

Changing variables ε = xσ and letting Iαβmn =

σα−β+m−nIαβmn, the integral (20) can be written

Iαβmn = 1

n!α!β!

∫dxHn(x)(−∂x)αxβHm(x)π0(x), (B1)

where π0(x) is a centered Gaussian with unit variance. Usingpartial integration, property (A3), and the relation

Hα(x)Hβ(x) = α!β!min(α,β)∑

s=0

Hα+β−2s(x)

s!(α − s)!(β − s)!, (B2)

given in Ref. [28], one obtains

Iαβmn = 1

α!β!

min(n−α,m)∑s=0

(m

s

)∫dx xβHm+n−α−2s(x)π0(x)

(n − α − s)!.

(B3)

The remaining integral can now be evaluated as the momentsof the unit Gaussian which yield∫

dx xbHa(x)π0(x) = b!

(b − a)!

∫dx xb−aπ0(x)

= b!

(b − a)!(b − a − 1)!! (B4)

for even (b − a) � 0 and zero otherwise. Explicitly, the matrixelements are given by

Iαβmn = 1

α!

min(n−α,m)∑s=0

(m

s

)

× (β + α + 2s − (m + n) − 1)!!

[β + α + 2s − (m + n)]!(n − α − s)!, (B5)

for even (α + β) − (m + n) and zero otherwise. Note that theabove quantity is strictly positive. Note also that the argumentof the double factorial is taken to be positive and hencethe summation is nonzero only if α + β + 2 min(n − α,m) �m + n and hence for even β = 2k we have n = m + α ± 2l,

while for odd β = (2k + 1) we have n = m + α ± (2l + 1),with l = 0, . . . ,k.

The integral formula can be used to verify two importantproperties of the solution of Eq. (22) given deterministic initialconditions: (i) We have Nj = 3j and hence Eq. (22) indeedyields a finite number of equations. (ii) The coefficients a

(j )n

for which (n + j ) is odd vanish at all times.To verify property (i), let Nj be the index of the highest

eigenfunction required to order �−j/2. Using Eq. (22) one canshow that a

(j )Nj

∼ a(j−1)Nj−1

Ip,3−p

Nj−1,Njfor p ∈ {1,2,3}. By virtue of

the properties given after Eq. (23), we find Nj = Nj−1 + 3.Since for deterministic initial conditions we have N0 = 0, itfollows that Nj = 3j .

Finally, we verify property (ii). To the summation inEq. (22) there contribute only terms for which Ip,k−p−2(s−1)

mn

is nonzero. Hence, by the condition given after Eq. (23),k − (m + n) is an even number. Considering the equation fora

(j )n for which n + j is even, it follows that in the summation on

the right-hand side of Eq. (22) there appear only coefficients forwhich m + (j − k) is even. Conversely, for n + j being odd,m + (j − k) must be odd. Hence the pairs of equations for a

(j )n

for which (j + n) is even or odd are mutually uncoupled. Fordeterministic initial conditions, only terms with j + n evendiffer from zero initially from which the result follows.

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APPROXIMATE PROBABILITY DISTRIBUTIONS OF THE . . . PHYSICAL REVIEW E 92, 012120 (2015)

APPENDIX C: SOLUTION TO THE PROBLEM OFMOMENTS USING THE SYSTEM SIZE EXPANSION

Having obtained the moment expansion in terms of thecoefficients a

(j )n , it would be desirable to invert this relation

and the coefficients in terms of the expansion of the moments.This can be derived using the completeness of the Hermitepolynomials and writing the probability density as �(ε) =∑∞

n=0 bnHn( εσ

)π0(ε), where the bn = 1n!

∫dεHn( ε

σ)�(ε) can

be expressed in terms of the moments using Eq. (A2), asfollows:

bn = 1

n!

�n/2�∑k=0

(n

2k

)(−1)k

〈εn−2k〉σn−2k

(2k − 1)!!. (C1)

Assuming now that the moments can be expanded in a seriesin powers of �, i.e.,

〈εβ〉 =N∑

j=0

�−j/2[εβ]j + O(�−(N+1)/2), (C2)

the bn can be matched to the coefficients an in Eq. (18)using σnbn = ∑N

j=0 �−j/2a(j )n + O[�−(N+1)/2], from which

one obtains

a(j )n = 1

n!

�n/2�∑k=0

(n

2k

)(−σ 2)k(2k − 1)!![εn−2k]j , (C3)

with [ε0]j = δj,0. The above formula relates the expansion ofthe moments to the expansion of distribution functions. It isnow evident that the system size expansion of the distributioncan be constructed from the system size expansion for a finiteset of moments.

Specifically, to order �−1/2 the nonzero coefficients evalu-ate to

a(1)1 = [ε]1, a

(1)3 = 1

3!

([ε3]3

1 − 3σ 2[ε]1), (C4)

while the coefficients to order �−1 are given by

a(2)2 = 1

2[ε2]2, a

(2)4 = 1

4!([ε4]2 − 6σ 2[ε2]2),

a(2)6 = 1

6!(45σ 4[ε2]2 − 15σ 2[ε4]2 + [ε6]2). (C5)

A different series is obtained using the Edgeworth expansionwhich, instead of using the system size expansion of themoments, Eq. (C2), proceeds by scaling the cumulants bya size parameter.

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Economics (Cambridge University Press, Cambridge, 2001).[11] T. Heskes, J. Phys. A: Math. Gen. 27, 5145 (1994).[12] R. Grima, Phys. Rev. Lett. 102, 218103 (2009); P. Thomas,

A. V. Straube, and R. Grima, J. Chem. Phys. 133, 195101(2010).

[13] R. Grima, P. Thomas, and A. V. Straube, J. Chem. Phys. 135,084103 (2011).

[14] C. Cianci, F. Di Patti, D. Fanelli, and L. Barletti, Eur. Phys. J.Special Topics 212, 5 (2012); P. Thomas, C. Fleck, R. Grima,and N. Popovic, J. Phys. A: Math. Theor. 47, 455007 (2014).

[15] S. Engblom, Appl. Math. Comput. 180, 498 (2006); R. Grima,J. Chem. Phys. 136, 154105 (2012); A. Ale, P. Kirk, and M. P.Stumpf, ibid. 138, 174101 (2013).

[16] V. Sotiropoulos and Y. N. Kaznessis, Chem. Eng. Sci. 66, 268(2011); P. Smadbeck and Y. N. Kaznessis, Proc. Natl. Acad. Sci.USA 110, 14261 (2013); A. Andreychenko, L. Mikeev, and V.Wolf, ACM Trans. Model. Comput. Simul. 25, 12 (2015).

[17] C. Cianci, F. Di Patti, and D. Fanelli, Europhys. Lett. 96, 50011(2011).

[18] T. M. Cover and J. A. Thomas, Elements of Information Theory,2nd ed. (John Wiley & Sons, Hoboken, NJ, 2012).

[19] P. Thomas, A. V. Straube, and R. Grima, J. Chem. Phys. 135,181103 (2011); K. R. Sanft, D. T. Gillespie, and L. R. Petzold,IET Syst. Biol. 5, 58 (2011).

[20] J. Paulsson, O. G. Berg, and M. Ehrenberg, Proc. Natl. Acad.Sci. U.S.A. 97, 7148 (2000).

[21] L. Comtet, Advanced Combinatorics, 1st ed. (Springer,Netherlands, 1974); The partial Bell polynomials are definedas Bn,k({xχ }) = ∑′ n!

j1!..jn−k+1! ( x11! )j1 ..( xn−k+1

(n−k+1)! )jn−k+1 , where the

summation∑′ is such that j1 + . . . + jn−k+1 = k and j1 +

2j2 + . . . + (n − k + 1)jn−k+1 = n and {xχ } denotes the se-quence (x1,x2,x3, . . .). These are available in Mathematica viathe function BellY .

[22] P. Thomas, H. Matuschek, and R. Grima, in IEEE InternationalConference on Bioinformatics Biomed (BIBM), 2012 (IEEE,Piscataway, NJ, 2012), pp. 1–5.

[23] V. Shahrezaei and P. S. Swain, Proc. Natl. Acad. Sci. USA 105,17256 (2008).

[24] The polylogarithm is defined by Lis(x) = ∑∞k=1

xk

ks and im-plemeted in Mathematica by PolyLog [x,s].

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[25] D. T. Gillespie, J. Phys. Chem. 81, 2340 (1977); ex-act realizations of the process described by Eq. (55) areobtained using the Gillespie’s algorithm with the incre-ments of the synthesis reaction being replaced by indepen-dently and identically geometrically distributed integers ofmean b.

[26] F. Verhulst, Methods and Applications of Singular Perturbations(Springer, New York, 2006).

[27] B. Munsky and M. Khammash, J. Chem. Phys. 124, 044104(2006).

[28] N. N. Lebedev, Special Functions and Their Applications(Dover, New York, 1972).

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