Journal of Theoretical Biology 229 (2004) 501–521

ARTICLE IN PRESS

*Correspond

Science, The U

153-8505, Japan

E-mail addr

0022-5193/$ - se

doi:10.1016/j.jtb

Multivariate analysis of noise in genetic regulatory networks

Ryota Tomiokaa,b,*, Hidenori Kimurab,c, Tetsuya J. Kobayashib, Kazuyuki Aiharab,d,e

aDepartment of Mathematical Engineering and Information Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo,

Bunkyo-ku, Tokyo 113-8656, JapanbDepartment of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, 7-3-1 Hongo,

Bunkyo-ku, Tokyo 113-8656, JapancBio-Mimetic Control Research Center, The Institute of Physical and Chemical Research, 2271-130 Anagahora, Shimoshidami,

Moriyama-ku Nagoya, 463-0003 Japand Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, JapaneERATO Aihara Complexity Modelling Project, JST, 45-18 Oyama, Shibuya-ku, Tokyo 151-0065, Japan

Received 17 October 2003; received in revised form 19 April 2004; accepted 29 April 2004

Available online 19 June 2004

Abstract

Stochasticity is an intrinsic property of genetic regulatory networks due to the low copy numbers of the major molecular species,

such as, DNA, mRNA, and regulatory proteins. Therefore, investigation of the mechanisms that reduce the stochastic noise is

essential in understanding the reproducible behaviors of real organisms and is also a key to design synthetic genetic regulatory

networks that can reliably work. We use an analytical and systematic method, the linear noise approximation of the chemical master

equation along with the decoupling of a stoichiometric matrix. In the analysis of fluctuations of multiple molecular species, the

covariance is an important measure of noise. However, usually the representation of a covariance matrix in the natural coordinate

system, i.e. the copy numbers of the molecular species, is intractably complicated because reactions change copy numbers of more

than one molecular species simultaneously. Decoupling of a stoichiometric matrix, which is a transformation of variables,

significantly simplifies the representation of a covariance matrix and elucidates the mechanisms behind the observed fluctuations in

the copy numbers. We apply our method to three types of fundamental genetic regulatory networks, that is, a single-gene

autoregulatory network, a two-gene autoregulatory network, and a mutually repressive network. We have found that there are

multiple noise components differently originating. Each noise component produces fluctuation in the characteristic direction. The

resulting fluctuations in the copy numbers of the molecular species are the sum of these fluctuations. In the examples, the limitation

of the negative feedback in noise reduction and the trade-off of fluctuations in multiple molecular species are clearly explained. The

analytical representations show the full parameter dependence. Additionally, the validity of our method is tested by stochastic

simulations.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Stochastic gene expression; Noise reduction; Linear noise approximation; Lyapunov equation; Decoupling of a stoichiometric matrix

1. Introduction

Control of intracellular noise is crucial for livingorganisms. Biochemical reactions are intrinsically noisy(Ozbudak et al., 2002; Elowitz et al., 2002; Blake et al.,2003) due to the low copy numbers of the molecularspecies, such as DNA, mRNA, and regulatory proteins.

ing author. Aihara Laboratory, Institute of Industrial

niversity of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo

. Tel.: +81-3-5452-6693; fax: +81-3-5452-6694.

ess: ryotat@sat.t.u-tokyo.ac.jp (R. Tomioka).

e front matter r 2004 Elsevier Ltd. All rights reserved.

i.2004.04.034

However, most of the cellular events are ordered andreproducible despite the underlying randomness in theirbuilding blocks (Rao et al., 2002). One importantunsolved problem is the explanation of how robustnessto this randomness in real organisms is achieved.Understanding such mechanisms not only gives deepinsight into the design principles of real organisms, butalso is a crucial key for engineering reliable syntheticgenetic regulatory networks for biotechnological andtherapeutic applications.One strategy that can be used to reduce the noise is to

increase the copy numbers of the molecular species

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521502

enough so that the fluctuations become insignificantcompared to the whole number of the molecular species.However, such strategies are energetically inefficient.The strategies that real organisms have developed seemto be both efficient and robust. A recent study of theembryo-to-embryo variability of morphogen proteinprofiles in Drosophila (Houchmandzadeh et al., 2002)has revealed that, the Hb boundary position is preciselyregulated at the downstream of the Bicoid gradient,which has high embryo-to-embryo variability. Interest-ingly, the Hb protein profile displays high variabilityexcept at the center of embryo length, which is the exactposition of the boundary. This result suggests that (i) inreal organisms, not all fluctuations are fatal, (ii)organisms can reduce the fluctuations in significantmolecular species at the cost of increasing fluctuations inthe other molecular species that are less significant, andconversely (iii) when designing a synthetic geneticregulatory network, one might increase fluctuations insome of the molecular species just to reduce thefluctuations in the others. Thus, it is necessary toanalyse the fluctuations of multiple molecular species.However, it is insufficient to analyse the fluctuation ofeach molecular species independently, because usuallychemical reactions require two or more molecularspecies, thus the correlation between the fluctuationscannot be ignored. Therefore, we need to evaluate notonly variance, but also covariance as statistical measuresof noise in genetic networks.Various methods for evaluating noise in genetic

regulatory networks have been proposed recently. Theyare roughly categorized into two groups. Monte-Carlosimulations of the chemical master equation (CME)based on the Gillespie method (Gillespie, 1977) are oneof the most frequently used numerical methods (Arkinet al., 1998; Elowitz and Leibler, 2000; Blake et al.,2003). They can fully reproduce the probabilistic anddiscrete nature of biochemical reactions. However, theyrequire huge computational time in order to obtain areliable estimation of the distribution, especially inmultivariate problems, and give us little intuition intothose mechanisms behind specific observations whichdetermine phenotypic noisy behaviors.On the other hand, analytical methods using the

probability generating functions (Berg, 1978; Peccoudand Ycart, 1995; Thattai and van Oudenaarden, 2001;Swain et al., 2002), the Langevin equation (Ozbudaket al., 2002; Simpson et al., 2003), or the Fokker-Planckequation (Hasty et al., 2000; Kepler and Elston, 2001)have also been applied to genetic regulatory networks.Although they clearly show the fundamental mechan-isms that determine noise, they may become intractablydifficult for nonlinear or multivariate problems.In this paper, we apply an analytical and multivariate

method called the linear noise approximation (LNA)(van Kampen, 1992) of the CME, to evaluate the

fluctuations around deterministic stable equilibriumstates. It is rigorously derived from the CME and canbe systematically applied to arbitrary N-dimensionalproblems.In addition, we propose the decoupling of a stoichio-

metric matrix in advance to the LNA to facilitateanalytical studies. When a given stoichiometric matrix isdecouplable, there exist a transformation of variables,which (i) guarantees that each of the reaction channelschanges only one variable for a single firing, (ii)diagonalizes the diffusion matrix, and (iii) simplifiesthe representation of the covariance matrix into a linearcombination of N terms. These terms represent noisecomponents that originate from different noise sources.Moreover, these noise components have characteristicmagnitudes and directions (the definitions of thedecomposition of noise and the decoupling of astoichiometric matrix are given in Appendices A andB). Decoupling of a stoichiometric matrix enables us tounderstand the mechanisms behind mere observations,such as, how noise can be decomposed into its origins, inwhich direction the major noise component is, fluctua-tions in which molecular species can be reduced withoutaffecting others, and which fluctuations cannot be.We apply this method to investigate the multivariate

stochasticity in three fundamental genetic regularity net-works. That is, a single-gene autoregulatory network, atwo-gene autoregulatory network, and a mutually repres-sive network. First, we consider a single-gene autoregula-tory network in which dimers formed in solution bind tothe DNA to repress the transcription. We show thatautoregulations can reduce one of the two noise compo-nents, namely, the protein birth and death noise, but not theother noise component, namely, the monomer–dimer

fluctuation noise. This is clearly shown in analyticalrepresentation of the covariance matrix. Additionally, weshow that these two noise components produce fluctua-tions in the characteristic directions. Second, in order toinvestigate the mechanisms that can change direction ofnoise, we consider a two-gene autoregulatory network. Weshow that the direction of noise can be drastically changedby changing the balance between the positive regulationand the negative regulation. Furthermore, we demonstratethis by a biologically plausible network. Finally, we applyour method to a mutually repressive network to demon-strate that our method is applicable to systems withmultiple equilibrium states.This paper is organized as follows. In Section 2, we

model the stochastic dynamics of a genetic regulatorynetwork with the CME and show the derivation of theLNA (van Kampen, 1992). In Section 3, we show themain results of applying this method to three types ofnetworks. In Section 4, we discuss the advantages of ourmethod, the connection with the previous studies, and thebiological implications of our results. Additionally,Appendix A shows the definition of the decomposition

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 503

of noise and the exact form of the decomposed 2� 2covariance matrix; Appendix B shows the definition ofthe decoupling of a stoichiometric matrix and thenecessary and sufficient condition for a given stoichio-metric matrix to be decouplable; Appendix C shows theproof of stability and uniqueness conditions; the para-meter values used in this paper are listed in Appendix D;the comparison of the analytical results and stochasticsimulations are shown in Appendix E.

2. Theory

2.1. Linear noise approximation

A genetic regulatory network can be modeled as achemically reacting system consisting of N molecularspecies fS1;y;SNg and M reaction channelsfR1;y;RMg inside a region with a fixed volume O andat a constant temperature. We specify the dynamical stateof the system by the copy numbers of molecular speciesX ¼ ðX1;X2;y;XN Þ

TAZþN ; where T denotes transposi-tion. The stoichiometric matrix is defined as A ¼faijgAZN�M where aij is the change in the copy numberof Si produced by a single firing of the reaction channel Rj :The propensity function WðXÞ ¼ ðW1ðXÞ;y;WMðXÞÞT isdefined as follows (Gillespie, 2000): WjðXÞDt denotes theprobability that the reaction channel Rj fires within thenext infinitesimal time interval Dt; given that the system isin a state X ; for j ¼ 1;y;M :When there are sufficiently large copy numbers of

molecular species, a propensity function can be rewrittenas WðXÞCOwðX=OÞ where wðxÞ is a function of theconcentration x � X=O: Therefore, in the thermodynamiclimit O-N; we obtain the well known macroscopicreaction rate equation (RRE) by taking the concentrationx as the state variable and omitting the O1=2 orderfluctuation in x; as follows:

dx

dt¼ AwðxÞ: ð1Þ

In mesoscopic systems with large but finite copynumbers, such as, genetic regulatory networks, X shouldbe regarded as random variable X: The CME (vanKampen, 1992) is widely used to describe the timeevolution of the probability density function (PDF)PðX ; tÞ � ProbðXðtÞ ¼ XÞ; i.e. the probability that thesystem is in a state X at time t; under a certain initialdistribution PðX ; 0Þ:

dPðX ; tÞdt

¼XMj¼1

½WjðX ajÞPðX aj; tÞ

WjðXÞPðX ; tÞ�; ð2Þ

where aj ¼ ða1j ;y; aNjÞT is the stoichiometric coefficient

of the reaction channel Rj :

We do not aim to solve the CME directly. Rather, wecharacterize the solution by the moments of X: We canobtain the following equation representing the timeevolution of the first-order moment /XS by multiplyingboth sides of Eq. (2) by X and taking summation over allvariables X1;y;XN (van Kampen, 1992):

d/XSdt

¼ /AWðXÞS � /FðXÞS: ð3Þ

Similarly, we can obtain the equation for the second-order moment /XXTS by multiplying both sides ofEq. (2) by XXT and taking the summation (van Kampen,1992):

d/XXTSdt

¼ /FðXÞXTSþ/XFTðXÞSþ/DðXÞS; ð4Þ

where the diffusion matrix (van Kampen, 1992) DðXÞ isdefined as follows:

DðXÞ � fdijðXÞgij ; dijðXÞ �XMk¼1

aikajkWkðXÞ

ði; j ¼ 1;y;NÞ: ð5Þ

In general, the time evolutions of the moments aredescribed by linear but infinite dimensional ODEs. Thetime evolution of the m-th order moment depends on thed þ m 1-th order moment, where d denotes the dimen-sion of the highest term of X inWðXÞ: Accordingly, whendX2; the time evolutions of any moments depend onhigher moments that depend in turn on much highermoments and so on. Eqs. (3) and (4) become closed simul-taneous ODE only when WðXÞ is a linear function of X :Now, we consider the case when the PDF PðX; tÞ is

distributed around the deterministic solution X ¼ O/ðtÞtightly enough so that we can approximate WðXÞ bylinearizing it around X ¼ O/ðtÞ; where /ðtÞ is thesolution of the following deterministic equation, whichis equivalent to the macroscopic RRE (1) in thethermodynamic limit O-N:

dO/ðtÞdt

¼ FðO/ðtÞÞ: ð6Þ

The state vector X is decomposed as X ¼ O/ðtÞ þ E;using the random variable E to denote the deviationfrom the deterministic term O/ðtÞ: Therefore, Eq. (3)can be rewritten as follows:

d/O/ðtÞ þ ESdt

¼ /FðO/ðtÞÞ þ Kð/ðtÞÞEþ OðjEj2ÞS;

ð7Þ

where Kð/Þ is the Jacobian matrix of the deterministicsystem (Eq. (6)):

Kð/Þ �@F ðOxÞ@Ox

����x¼/

C@AwðxÞ

@x

����x¼/

ðO-NÞ

!:

Similarly rewriting Eq. (4) and truncating the OðjEj2Þterms in the Taylor series of the both equations yield the

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521504

following equations representing the time evolutions ofl � /ES and S � /EETS/ES/EST:

dl

dt¼ Kð/ðtÞÞl; ð8Þ

dSdt

¼Kð/ðtÞÞSþ SKTð/ðtÞÞ þ DðO/ðtÞÞ

þ@DðOxÞ@Ox

����x¼/ðtÞ

l: ð9Þ

Now, we have the closed simultaneous equationsrepresenting the time evolution of the mean l and thecovariance matrix S of the random fluctuations aroundthe deterministic solution.Here, we make the following two assumptions:

1. Along the deterministic solution /ðtÞ; Kð/ðtÞÞ isstable, in other words, the real parts of all theeigenvalues of Kð/ðtÞÞ are negative for all t:

2. The solution of Eq. (6) under a certain initialcondition /ð0Þ ¼ /0 is attracted to a stable equili-brium point /eq:

The first assumption is necessary to justify the lineariza-tion we have performed above. The second assumptionis necessary to evaluate the fluctuations at deterministicstable equilibrium points as we show next.It is well known that Eqs. (8) and (9) have a unique

stable equilibrium point under the two assumptionsshown above. Therefore, it is straightforward to showthat there exists a stable distribution around /eq withmean l ¼ 0 and covariance matrix S that satisfies thefollowing equation called the Lyapunov equation:

Kð/eqÞSþ SKTð/eqÞ þ DðO/eqÞ ¼ 0: ð10Þ

This equation can be solved for arbitraryN; because it canalways be transformed into linear NðN þ 1Þ=2-dimen-sional simultaneous equations. In addition, Eq. (10) can bedecomposed into N2 equations. The solution of Eq. (10) isrewritten as a linear combination of the solutions of theseequations. In the special case when the stoichiometricmatrix A is decouplable, this decomposition is significantlysimplified into a linear combination of no more than N

terms rather thanN2 terms. These terms can be consideredas representations of all the noise components thatoriginate from different noise sources (see Appendices Aand B for detail).In summary, one can obtain the covariance matrix of

the stable distribution around a deterministic stableequilibrium point by taking the following steps:

1. Find two matrices, the stoichiometric matrix A andthe propensity function WðXÞ of the system to beconsidered.

2. Find a stable equilibrium point / ¼ /eq of Eq. (6).3. Calculate two matrices Kð/eqÞ and DðO/eqÞ:4. Solve the Lyapunov equation (10).

Note that the derivation shown above, though simple,gives the same result as the original derivation (vanKampen, 1992) for the first- and the second-ordercumulants of a stationary distribution, and even forthose cumulants of the transient distribution under theassumptions EBOðO1=2Þ and O-N to eliminate thelast term in Eq. (9).

2.2. Measure of noise

We use the covariance matrix S itself as the measureof noise unless otherwise explicitly noted. Normalizedcovariance matrix S0 is partly used, which is equivalentto the coefficient of variation (CV). The normalizedcovariance is defined as S0 ¼ SSST; where matrix S ¼diagð1=ðOfeq

1 Þ; 1=ðOfeq2 Þ;y; 1=ðOfeq

N ÞÞ denotes the nor-malization by the mean values, i.e. the deterministicequilibrium point.Furthermore, we use the following characteristic

values of the noise:

Zmax �ffiffiffiffiffiffiffiffiffilmax

p: the maximum noise component;

vmax : the maximum noise direction;

where lmax and vmax denote the largest eigenvalue of S0

and its eigenvector, respectively.

2.3. Representing noise

Here, we make a note on the terms specially used inthis paper to describe the two-dimensional distributions.We use the term in-phase direction to denote the

direction in which two random variables increase ordecrease in phase with each other. Similarly, we use theterm anti-phase direction to denote the direction inwhich two random variables increase or decrease in anti-phase with each other.We use noise ellipsoids, as shown in Figs. 1(b) and (d).

The noise ellipsoids show the shapes of the GaussianPDFs by their 1s equiprobability curves.Furthermore, we use parameter space plots, as shown

in Figs. 1(c) and 2(c). These figures show the noisecharacteristics for various combinations of two para-meters chosen as x- and y-axis. Solid curves representthe contour curves of the maximum noise componentZmax; which we call the equi-noise curves. Each shortarrow represents the maximum noise direction vmax:

3. Results

3.1. Single-gene autoregulatory network

Let us consider a model of an autoregulatory gene, i.e.a gene that represses its own transcription. The noise

ARTICLE IN PRESS

0 50 100 150 200 250 300

50

100

150

200

250

dim

ers

monomers

Kd =2.4Kd =5.6

Kd =13Kd =32

Kd =75

Kd =1.8⋅ 102

Kd =4.2⋅ 102

Kd =103

0.108460.10846

0.16270.1627

0.244040.24404

0.36606

0.36606

0.5491

0.5491

0.823640.82364

1.23551.2355

1.85322.7798

10-3

10-2

10-1

100

101

102

transcription initiation rate (1/s)

nega

tive

feed

back

str

engt

h

0.8 0.9 1 1.1 1.2

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

normalized monomers

norm

aliz

ed d

imer

s

ρ=0ρ=3.7ρ=37ρ=367

(b)(a)

(d)(c)

gene

kdKd-1

φ

Promoter

kr

monomers(X1-2X2)

n×

2×

kd

α(x2)g[n]

mRNA

dimers(X2)

Fig. 1. (a) A reaction model of a single-gene autoregulatory network. This model consists of two molecular species, the monomer and dimer forms of

the expressed protein, and four reactions, namely, protein synthesis, dimerization, dissociation, and degradation. The dimers formed in solution

repress the transcription of the gene. (b) Noise reduction by autoregulation. The noise ellipsoids show the shapes of the Gaussian distributions in the

normalized coordinate system. The negative feedback strength r is increased from r ¼ 0 (no repression) to r ¼ 367 (strong repression). The directionof the protein birth and death noise (chained line) and the monomer–dimer fluctuation noise (dotted line) are shown. (c) The parameter space plot with

the transcription initiation rate a and the negative feedback strength r as x- and y-axis, respectively. Equi-noise (Zmax) curves (solid curves) are

shown. Each short arrow represents the maximum noise direction (vmax) at the corresponding parameter value. (d) Control of the direction of noise

by changing the dimer dissociation constant Kd : The crosses show the equilibrium points. The noise ellipsoids show the 1s equiprobability curves ofthe probability distributions around the equilibrium points. The direction of the protein birth and death noise (chained line) and the monomer–dimer

fluctuation noise (dotted line) are shown.

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 505

reduction by autoregulation has been extensively studiedboth theoretically (Thattai and van Oudenaarden, 2001;Bundschuh et al., 2003; Simpson et al., 2003; Kobayashiand Aihara, 2003) and experimentally (Becskei andSerrano, 2000). However, since the expressed proteinsoften form dimers in solution and bind to the operatorregion as dimers (Ptashne, 1987), multivariate analysis isnecessary.This model consists of four reactions schematically

illustrated in Fig. 1(a), namely, protein synthesis,dimerization, dissociation, and degradation. Thesereactions are represented as follows:

| ���!aðx2Þg½n�nP ðn ¼ 0; 1; 2;yÞ;

2P "kd K1

d

kd

P2;

P!kr |;

where | denotes the absence of effective molecularspecies; P and P2 denote the monomer and the dimer

forms of the expressed proteins, respectively. Thestate variable X ¼ ðX1;X2Þ denotes the copy numbersof the total proteins and the dimers, respectively.This coordinate system is chosen so that the stoichio-metric matrix A is decoupled, i.e. each of the reactionchannels changes only one state variable for a singlefiring as shown below (see Appendix B). For proteinsynthesis, we consider prokaryotic translation as afast process through which proteins are released intothe cytoplasm in sharp bursts according to a geometricdistribution (Berg, 1978; McAdams and Arkin,1997; Thattai and van Oudenaarden, 2001).The transcription initiation rate aðx2Þ is a monotonicallydecreasing function of x2 � X2=O; the concentrationof the dimers, because we assume that protein–DNAbinding occurs in the dimer forms of the proteins.Therefore, the protein synthesis is modeled as aset of reactions that produces n proteins with rateaðx2Þg½n� for n ¼ 0; 1; 2;y; where g½n� ¼ Bn=ðB þ1Þnþ1 ðn ¼ 0; 1; 2;yÞ denotes the PDF of the geometricdistribution with average B (the average burst size).

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521506

The stoichiometric matrix A and the propensityfunction WðXÞ for this network are as follows:

A ¼n 1 0 0

0 0 1 1

� ;

WðXÞ ¼

OaðX2=OÞg½n�

krX1

O1kdððX1 2X2ÞðX1 2X2 1ÞÞ=Kd

kdX2

0BBB@

1CCCA:

ð11Þ

The deterministic equation (6) is calculated as follows:

df1dt

¼ aðf2ÞB krf1; ð12Þ

df2dt

¼ kdfðfmðf1;f2Þðfmðf1;f2Þ O1ÞÞ=Kd f2g;

ð13Þ

where fmðf1;f2Þ � f1 2f2 is the concentration of themonomers.Furthermore, the matrices Kð/Þ and DðO/Þ are

calculated as follows:

Kð/Þ ¼

kr a0ðf2ÞB

kdð2fm O1Þ=Kd kdf1þ 2ð2fm O1Þ=Kdg

� ;

DðO/Þ ¼ O

kraðf2ÞB

krð2B þ 1Þ þ f1

n o0

0 kdfðfmðfm O1ÞÞ=Kd þ f2g

0@

1A;

ð14Þ

where a0ðf2Þ ¼ da=dx2jf2 : Here, summation over all n

is taken and the relationsP

N

n¼0 ng½n� ¼ B andPN

n¼0 n2g½n� ¼ 2B2 þ B are used.

It can be shown that Eqs. (12) and (13) have theunique equilibrium point /eq ¼ ðfeq

1 ;feq2 ÞT that is stable

(see Appendix C.1). Therefore, we do not consider thewhole function aðx2Þ; rather we consider two parametersa � aðfeq

2 Þ and r � a0ðfeq2 ÞB=kr:We call r the negative

feedback strength, because here we consider only theautoregulatory network with a0ðfeq

2 Þo0:Because DðO/Þ is a diagonal matrix, the solution of

the Lyapunov equation (10) is decomposed into twonoise components that originate from different noisesources (see Appendix A for detail). When the dimer-ization and the dissociation are much faster thanthe degradation ðk11=k22 ¼ kr=ðkdgÞ51Þ; the solution(Eq. (A.6)) is further reduced as follows:

SCmt

1þ B

1þ rZ1 Z

Z Z2

� �

þ mdg1r2k111þ rZ

rk111þ rZ

rk111þ rZ

1

2664

3775; ð15Þ

where mt � Ofeq1 is the copy number of the total

proteins at the equilibrium point, md � Ofeq2 the copy

number of the dimers at the equilibrium point, r �k12=k11 ¼ a0ðfeq

2 ÞB=kr the negative feedback strength,Z � k21=k22 ¼ ð2feq

m O1Þ=ðKd þ 2ð2feqm O1ÞÞ; g �

k22=kd ¼ 1þ 2ð2feqm O1Þ=Kd ; k11 � k11=ðk11 þ k22Þ

¼ kr=ðkr þ kdgÞ:The first term of Eq. (15) represents the noise

component generated by the protein synthesis anddegradation, namely, the protein birth and death noise.The scalar part mtð1þ BÞ=ð1þ rZÞ and the eigenvectorð1; ZÞT of the matrix part ð1; ZÞTð1; ZÞ represent themagnitude and the direction of the fluctuation producedby this noise component, respectively. The second termof Eq. (15) represents the noise component generated bythe monomer–dimer fluctuation, namely, the monomer–

dimer fluctuation noise. The magnitude is mdg1; which isindependent of the negative feedback strength r; andsimilarly, the direction of this noise component isrepresented by the 2� 2 matrix multiplied from theright. The ð2; 2Þ component of this matrix is independentof the negative feedback strength r and is dominant forsmall r: Therefore, the direction of this second noisecomponent is approximately in the direction thatchanges the number of the dimers but holds the numberof total proteins constant, which is denoted by ð0; 1ÞT:Fig. 1(b) shows the noise ellipsoids with four different

values of the negative feedback strength r: All otherparameter values are listed in Appendix D. Here, thenegative feedback effectively represses the protein birth

and death noise in the in-phase direction. However, themonomer–dimer fluctuation noise in the anti-phase

direction is not affected. This is clearly explained abovein the analytical expression (Eq. (15)). The directions ofthe protein birth and death noise ð1; ZÞT (chained lines)and the directions of the monomer–dimer fluctuation

noise ð0; 1ÞT (dotted lines) are shown. For visualization,we transformed the two vectors in the ðX1;X2Þ

T spaceinto the normalized coordinate system of the copynumbers of the monomers and the dimers.Fig. 1(c) is the parameter space plot with the

transcription initiation rate a and the negative feedbackstrength r as x- and y-axis, respectively. Here, inaddition to the equi-noise curves, the direction of noiseis shown as a short arrow for each pair of parameters.Increasing the negative feedback strength r reduces themaximum noise component Zmax when r is below acertain value. However, there is a critical negativefeedback strength above which Zmax cannot further bereduced. Moreover, the maximum noise direction vmax

changes from the protein birth and death noise (arrows

ARTICLE IN PRESS

gene bgene a

Protein A (X1)

φ

kr1

φ

Promoter a Promoter b

n×

α(x2)g[n]

'n×mRNA mRNA

Protein B (X2)

β(x1)g[n']

kr2

(a)

(b)

(c)

0.5 1 1.5

0.5

1

1.5

normalized protein A

norm

aliz

ed p

rote

in B

ρα=0.92ρα=1.8ρα=2.8ρα=3.7

ρβ=0.92ρβ =1.8ρβ=2.8ρβ=3.7

0.20

6990.

2069

90.20699

0.206990.20699

0.24838

0.24838

0.24

8380.

24838

0.298060.29806

0.29

8060.

2980

6

0.357670.35767

0.35

767

0.42921

0.42

921

10-2

100

10 -2

10-1

100

10-1

positive regulatory strength

nega

tive

regu

lato

ry s

tren

gth

Fig. 2. (a) A reaction model of a two-gene autoregulatory network.

This model consists of two molecular species, protein speciesA andB;and four reactions, namely, syntheses and degradations of A and B:The proteinA represses the transcription of the gene b and the protein

B activates the transcription of the gene a. (b) Control of the direction

of noise by genetic regulation. Noise ellipsoids are shown for different

values of the positive regulatory strength ra (broken lines) or thenegative regulatory strength rb (solid lines). The chained lines and thedotted lines show the directions denoted by the two vectors ð1;rbÞ

T

and ðra; 1ÞT; respectively. (c) The parameter space plot with the

positive regulatory strength ra and the negative regulatory strength rbas x- and y-axis, respectively. Equi-noise (Zmax) curves (solid curves)

are shown. Each short arrow represents the maximum noise direction

(vmax) at the corresponding parameter value.

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 507

pointing the upper right) to the monomer–dimer fluctua-

tion noise (arrows pointing the upper left) at the criticalvalue of r: Therefore, the autoregulation is effective inreducing the noise generated by protein synthesis anddegradation, but not the noise generated by monomer–dimer fluctuation.Fig. 1(d) shows the noise ellipsoids for different values

of the dimer dissociation constant Kd with the number oftotal proteins kept constant. The equilibrium point movesfrom low-monomers state to high-monomers state as Kd

increases. The direction of the protein birth and death noise

(chained line) changes from the dimer (upward) directionto the monomer (rightward) direction, but the direction ofthe monomer–dimer fluctuation noise (dotted line) isunchanged. When the fluctuation of the monomers islarge, the fluctuation of the dimers is small, and vice versa.Therefore, there exists a trade-off between the fluctuationin monomers and that in dimers.

3.2. Two-gene autoregulatory network

Next, we present a simple example that the directionof noise in a genetic network can be controlled bychanging the network parameters.Let us consider a model of an autoregulatory network

of two genes. This network, illustrated in Fig. 2(a),consists of two protein species A and B; and twogenes gene a and gene b coding the proteins, respectively.Here, we assume that proteinA represses the transcrip-tion of gene b, while protein B activates the transcrip-tion of gene a. For simplicity, we omit dimerizationand regard protein monomer molecules to be respon-sible for transcription regulation. Thus, this modelconsists of two variables X1 and X2; namely, the copynumbers of protein species A and B; respectively,and four reactions, namely, syntheses and degradationsof A and B: These reactions can be represented asfollows:

| ����!aðx2Þg1½n�nA ðn ¼ 0; 1; 2;yÞ;

A!kr1 |;

| ����!bðx1Þg2½n0�n0B ðn0 ¼ 0; 1; 2;yÞ;

B!kr2 |;

The stoichiometric matrix A and the propensityfunction WðXÞ are as follows:

A ¼n 1 0 0

0 0 n0 1

� ;

WðXÞ ¼

OaðX2=OÞg1½n�

kr1X1

ObðX1=OÞg2½n0�

kr2X2

0BBB@

1CCCA: ð16Þ

Accordingly, the deterministic equation (6) is calcu-lated as follows:

df1dt

¼ aðf2ÞB1 kr1f1; ð17Þ

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521508

df2dt

¼ bðf1ÞB2 kr2f2; ð18Þ

where B1 and B2 denote the average numbers of proteinmolecules synthesized from mRNAs (the average burstsize) for the two genes, respectively.Furthermore, the matrices Kð/Þ and DðO/Þ are

calculated as follows:

Kð/Þ ¼kr1 a0ðf2ÞB1

b0ðf1ÞB2 kr2

� ;

DðO/Þ ¼ O

kr1aðf2ÞB1

kr1ð2B1 þ 1Þ þ f1

� �0

0 kr2bðf1ÞB2

kr2ð2B2 þ 1Þ þ f2

� �0B@

1CA;

ð19Þ

where a0ðf2Þ � da=dx2jx2¼f2 and b0ðf1Þ � db=dx1jx1¼f1 :It can be shown that Eqs. (17) and (18) have the

unique equilibrium point /eq ¼ ðfeq1 ;f

eq2 ÞT that is stable,

because protein A regulates protein B negatively, andprotein B regulates protein A positively (see AppendixC.2). Therefore, similarly to the single-gene case, weuse four parameters a � aðfeq

2 Þ; ra � a0ðfeq2 ÞB1=kr1; b �

bðfeq1 Þ; and rb � b0ðfeq

1 ÞB2=kr2: We call ra and rbthe positive and the negative regulatory strength,respectively.Because DðO/Þ is a diagonal matrix, the solution of

the Lyapunov equation (10) is decomposed into twonoise components, namely, the protein A noise and theprotein B noise, which are generated by the synthesesand degradations of these two protein species. For thecase when the degradation rate constants of the twogenes are equal kr1 ¼ kr2; the solution (Eq. (A.6)) isfurther reduced as follows:

S ¼1

2m1ð1þ B1Þ

1 0

0 0

� þ

1

1þ rarb

1 rbrb r2b

!" #

þ1

2m2ð1þ B2Þ

0 0

0 1

� þ

1

1þ rarb

r2a rara 1

� " #;

ð20Þ

where m1 � Ofeq1 and m2 � Ofeq

2 : Here, note thatð1;rbÞ

T or ðra; 1ÞT becomes approximately the eigen-

vector of each term when one of the regulation is muchstronger than the other, i.e., rbbra or rabrb;respectively, and the product of them is small rarb51:Fig. 2(b) shows the noise ellipsoids for different values

of the positive ra or the negative rb regulation. All otherparameter values are listed in Appendix D. Thedistribution is spherically symmetric without the regula-tions. The positive regulation makes the distributioncorrelated to the in-phase direction. On the other hand,the negative regulation makes the distribution correlatedto the anti-phase direction. These results demonstratethat the direction of the noise in this network can be

controlled in the opposite way by changing the strengthof positive or negative regulation.Fig. 2(c) is the parameter space plot that shows this

change more clearly. When the positive and the negativeregulatory strengths are equal (ra ¼ rb; on the diagonalline), the distribution is symmetric, in other words, hasno correlation. However, when ra > rb; the distributionbecomes correlated to the in-phase direction and viceversa. When ra and rb are small, the change in thedirection of noise is smooth. However, when they arelarge, the change across the diagonal line of Fig. 2(c) isabrupt. This implies that a small change in parametervalues in this region can drastically change thephenotypic noise characteristics.To explain these observations, we set the transcription

initiation rates a ¼ b and the average burst sizes B1 ¼B2; according to the parameters used for the abovecalculations. Now, Eq. (20) is reduced as follows:

1

2mð1þ BÞ

1þ1þ r2a1þ rarb

ra rb1þ rarb

ra rb1þ rarb

1þ1þ r2b1þ rarb

0BBB@

1CCCA; ð21Þ

where m � m1 ¼ m2; and B � B1 ¼ B2:The change in the direction of noise across the

diagonal line is explained by the fact that the sign ofthe covariance is determined by the difference of ra fromrb: In addition, two terms, ð1þ r2aÞ=ð1þ rarbÞ and ð1þr2bÞ=ð1þ rarbÞ in Eq. (21) explain the abrupt change inthe direction of noise. These two terms can be rewrittenas ð1þ arÞ=ð1þ aÞ where a ¼ rarb and r ¼ ra=rb (orr ¼ rb=ra for the latter term) are the product and theratio of the two parameters. r denotes the deviationfrom the diagonal line. The coefficient a=ð1þ aÞ denotesthe sensitivity of that change in the ð1; 1Þ or ð2; 2Þelement of Eq. (21) produced by the change in r; thiscoefficient becomes large as a increases. Therefore, inthe region of large a; the direction of the maximumeigenvector of the covariance matrix (Eq. (21)) changesto the rightward direction ðra > rbÞ or to the upwarddirection ðraorbÞ as soon as the parameters deviatefrom the diagonal line.

3.3. A biologically plausible model of the two-gene

autoregulatory network

We consider the biologically plausible model of thetwo-gene autoregulatory network schematically illu-strated in Fig. 3(a). It incorporates the Escherichia coli

lacI gene and the Ptrc-2 promoter as the repressiveregulatory pair, and the bacteriophage l cI gene and thePRM promoter with mutant operator OR3

as theactivatory regulatory pair, in which CI cannot repressits own transcription (Ptashne, 1987). Here, the func-tions aðx2Þ and bðx1Þ in Eq. (16) are modeled by Hill

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 509

functions as follows:

aðx2Þ ¼a0ðx2=KaÞ

ha

1þ ðx2=KaÞha; ð22Þ

bðx1Þ ¼b0

1þ ð x11þð½IPTG�=KI Þ

hI=KbÞ

hb; ð23Þ

where, x1 � X1=O and x2 � X2=O denote the concentra-tions of LacI and CI proteins, respectively. Here,isopropyl-b-thiogalactopyranoside (IPTG), which inac-

cIlacIPRM Ptrc-2

IPTG (inducer)

(a)

(b)

(c)

0 100 200 300 400 5000

100

200

300

400

500

LacI

CI

16µM25µM40µM

63µM

100µM

159µM

252µM

M400µ

φ 2-nullclines

φ1-nullcline

0.5 1 1.50.6

0.8

1

1.2

1.4

normalized LacI

norm

aliz

ed C

I

16 µM

25 µM

40 µM

63 µM

100 µM

159 µM

252 µM

400 µM

Fig. 3. (a) A biologically plausible two-gene autoregulatory network.

The lacI-Ptrc-2 and the cI-PRM gene-promoter yield negative and

positive regulations, respectively. IPTG is the inducer that inactivates

the repression of Ptrc-2 promoter by lacI. (b) Control of the shape of

the probability distribution by IPTG induction. The IPTG concentra-

tion is increased from 16 to 400 mM: The nullclines of Eqs. (17) and(18) with regulatory functions denoted by Eqs. (22) and (23) are

shown. The noise ellipsoids are shown at each equilibrium point for

each IPTG concentration. The results of stochastic simulations are also

shown (see noise ellipsoids in broken lines). (c) Control of the direction

of noise by IPTG induction. The noise ellipsoids in the normalized

coordinate system are shown from ½IPTG� ¼ 16 mM to ½IPTG� ¼400 mM:

tivates LacI repressor, is incorporated in order tocontrol the repression of the Ptrc-2 promoter by LacI(Gardner et al., 2000).Fig. 3(b) shows the noise ellipsoids for different

concentrations of IPTG at deterministic equilibriumpoints. The equilibrium point moves from a low to ahigh copy number state as the IPTG concentrationincreases. Meanwhile, the direction of the noise aroundthe equilibrium point changes from the in-phase direc-

tion to the anti-phase direction. This is more clearlyshown in Fig. 3(c). This can be explained as follows. Thegradient of the positive regulatory function aðx2Þ at theequilibrium point is large at low IPTG concentrationsand becomes small as IPTG increases, while the gradientof the negative regulatory function bðx1Þ is not so muchaffected. Therefore, ra > rb holds at low IPTG concen-trations and rb > ra holds at high IPTG concentrations.We note that the analytical results are in good

correspondence with the stochastic simulation of thedetailed model (see the noise ellipsoids in broken lines ofFig. 3(b), see Appendix E for detail).

3.4. Mutually repressive switch

As the last model, we consider the mutually repressivetwo-gene network schematically illustrated in Fig. 4(a).This kind of networks have been extensively studied,and shown both theoretically (Wolf and Eeckman, 1998;Cherry and Adler, 2000; Gardner et al., 2000) andexperimentally (Gardner et al., 2000; Ozbudak et al.,2004) that it can be mono-stable or bi-stable withhysteresis depending on parameter values that can beexternally controlled. In other words, it can function asa genetic toggle switch. Here, we aim to shed light on thestochastic nature of the bifurcation phenomenon in thegenetic switch, which has been conventionally analyseddeterministically. In addition, we test the validity ofapplying our analytical method to each equilibriumpoint independently in multistable systems.In this model, the E. coli lacI gene and the Ptrc-2

promoter pair, and the bacteriophage l cI gene and thePL promoter pair are incorporated to produce mutualrepression. IPTG is also incorporated to control therepression of the Ptrc-2 promoter by LacI.We model mutual repression by setting both aðx2Þ and

bðx1Þ in Eq. (16) as monotonically decreasing functionsas follows (Gardner et al., 2000):

aðx2Þ ¼a0

1þ ðx2=KaÞha; ð24Þ

bðx1Þ ¼b0

1þ ð x11þð½IPTG�=KI ÞhI

=KbÞhb; ð25Þ

where, x1 � X1=O and x2 � X2=O denote the concentra-tions of LacI and CI proteins, respectively.

ARTICLE IN PRESS

cIlacIPL Ptrc-2

IPTG (inducer)

(a)

0 200 400 600

0

100

200

300

400

500

LacI

CI

20µM18µM10µM

0µM

φ2-nullclines

φ1-nullcline

(b)

Fig. 4. (a) A two-gene mutually repressive switch (Gardner et al.,

2000). The lacI-Ptrc-2 and the cI-PL gene-promoter pairs yield mutual

repression. IPTG is the inducer that inactivates the repression of Ptrc-2

promoter by lacI. (b) Noise in IPTG-induced genetic switching.

The IPTG concentration is increased from a bistable state

(½IPTG� ¼ 0 mM) to just before (½IPTG� ¼ 20 mM) the deterministicbifurcation (½IPTG� ¼ 20:184 mM). The nullclines of Eqs. (17) and (18)with regulatory functions denoted by Eqs. (24) and (25) are shown.

The noise ellipsoids are shown at each stable equilibrium point for

each IPTG concentration. Crosses and circles represent stable

equilibrium points and saddle points, respectively.

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521510

Fig. 4(b) shows the noise ellipsoids for differentconcentrations of IPTG at deterministic stable equili-brium points. As the negative regulation of Ptrc-2 byLacI is relaxed by the IPTG induction, the lower-rightequilibrium point with high LacI and low CI disappearsthrough saddle–node bifurcation, however the upper-left equilibrium point with low LacI and high CIremains stable. Applying the LNA to each stableequilibrium point gives us further information. We cansee that the noise around the lower-right equilibriumpoint grows rapidly as the IPTG concentration in-creases. The noise ellipsoids cover the saddle pointbefore the deterministic bifurcation occurs. On the otherhand, noise around the upper-left equilibrium point isalmost unchanged during the IPTG induction.We note that the analytical results are practically in

good correspondence with the stochastic simulations ofthe detailed model. The comparisons are shown inAppendix E.

3.5. Summary of results

In this section, we have applied the LNA of the CMEalong with the decoupling of a stoichiometric matrix (see

the discussions in the next section and Appendix B) tothree fundamental types of genetic regulatory networks,that is, a single-gene autoregulatory network, a two-gene autoregulatory network, and a mutually repressivenetwork.In the single-gene autoregulatory network, we have

found that there exist two differently originating noisecomponents, namely, the protein birth and death noise

and the monomer–dimer fluctuation noise. Each of themhas its own characteristic magnitude and direction. Theautoregulation is effective in reducing the former noisecomponent but not the latter. Additionally, we havefound that the dimer dissociation constant Kd deter-mines not only the distribution of proteins betweenmonomers and dimers but also the distribution of noisebetween the fluctuations of the monomers and dimers.This result has been clearly shown as the change in thedirection of noise from the dimer direction to themonomer direction.Next, we focused on the mechanism that changes the

direction of noise. We investigated a two-gene auto-regulatory network. We have shown that the directionof noise can be controlled drastically by changing thebalance between the positive regulation and the negativeregulation. Furthermore, with a biologically plausibletwo-gene autoregulatory network consisting of the lacI

gene and the cI gene, we have shown that the directionof the noise around the equilibrium point changes fromthe in-phase direction to the anti-phase direction by theIPTG induction.Finally, we have applied our method to a two-gene

toggle-switch network. We have evaluated the stochasticfluctuations around the two equilibrium points in thegenetic switch under the IPTG induction. We havefound that while deterministically one of the two stableequilibrium points disappears through the saddle–nodebifurcation, the noise around the disappearing equili-brium point grows rapidly. The noise ellipsoid coversthe saddle point before the deterministic bifurcationoccurs. On the other hand, the noise around theremaining equilibrium point is almost unchanged duringthe IPTG induction.

4. Discussion and conclusion

4.1. LNA in the analysis of noise in genetic regulatory

networks

In spite of the intrinsic nature of noise in geneticregulatory networks, the diversities of the copy numbersand the time scales among the molecular species havebeen keeping both numerical and analytical analyses ofthe noise difficult. In this paper, we have used the LNAof the CME for the noise analysis, which was originallyproposed in (van Kampen, 1992). It is an analytical

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 511

method that can evaluate stochastic fluctuations arounddeterministic stable equilibrium points. It is rigorouslyderived from the CME and can be systematicallyapplied to arbitrary N-dimensional problems. One needsonly to (i) find the stoichiometric matrix A and thepropensity function WðXÞ; (ii) find a stable equilibriumpoint of Eq. (6), (iii) calculate two matrices K and D

there, and (iv) solve the Lyapunov equation (10). Here,we emphasize that no manipulation of the CME isnecessary when we utilize this method.We remark that the LNA has two different impacts.

The first point is that it enables us to make a quick andaccurate estimation of the PDF in arbitrary N-dimen-sional problems. This has been pointed out andemphasized in the previous study (Elf et al., 2003; Elfand Ehrenberg, 2003) that applied the LNA to near-critical phenomena in biochemically reacting systems.This factor is important for the lack of an efficientnumerical method that is generally and systematicallyapplicable to arbitrary N-dimensional problems. How-ever, mere numerical calculation gives little insight intothe mechanisms behind the evaluated fluctuation andhardly improve our understanding of intracellularphenomena. Additionally, it should be noted that thecumulant evolution equation can improve the accuracyof the calculation with almost the same computationalcost (Kobayashi and Aihara, 2003).The second and the most important point is that it

enables us to obtain the analytical representation of acovariance matrix, which preserves the full parameterdependence. This analytical representation is usuallyintractably complicated. Therefore, we have proposedthe decoupling of a stoichiometric matrix in advance tothe LNA.The decoupling of a stoichiometric matrix A is a

transformation of variables that guarantees that each ofthe reaction channels changes only one variable for asingle firing. If such a transformation of variables exist,i.e., A is decouplable, the diffusion matrix D (defined asEq. (5)) is diagonalized, and the representation of thecovariance matrix is decomposed into a linear combina-tion of N different noise components as follows:

S ¼XN

i¼1

niiMði;iÞ; ð26Þ

where nii � diið/eqÞ=ð2kiið/

eqÞÞ represents the magni-tude of the i-th noise component and M ði;iÞ; which is apositive semidefinite matrix, represents the characteristicdispersion produced by the i-th noise component.Therefore, we call the maximum eigenvector of M ði;iÞ

the direction of the i-th noise component (see AppendixA for detail).The decoupling of a stoichiometric matrix signifi-

cantly simplifies the representation of a covariancematrix into a set of N distinct noise components, rather

than a single N � N covariance matrix. Therefore, itextends the power of the LNA from the quickevaluation of fluctuation to an understanding of themechanisms behind such fluctuation.

4.2. Diagonalization of K or D

In the previous study (Elf and Ehrenberg, 2003) thediagonalization of K was used with the LNA.Both the diagonalization of K and D are transforma-

tions of variables. Because the diagonalization of K

elucidates the time scale of the dynamics, it has greatimportance in the deterministic dynamical systemsanalysis. On the other hand, the diagonalization of D

has two remarkable points, which are especiallyimportant in the stochastic analysis.The first point is that it removes the second-order

correlations of those probabilistic jumps in the copynumbers which are produced by randomly occurringreactions. This is explained as follows. The diffusionmatrix D can be defined as the second order jumpmoment (van Kampen, 1992) as follows:

DðXðtÞÞ

� limDt-0

/ðXiðt þ DtÞ XiðtÞÞðXjðt þ DtÞ XjðtÞÞSDt

� ij

¼ limDt-0

PMk¼1 aikajkWkðXðtÞÞDt þ OðDt2Þ

Dt

( )ij

¼XMk¼1

aikajkWkðXðtÞÞ

( )ij

;

where Xiðt þ DtÞ ði ¼ 1;y;NÞ is the random variablethat denotes the copy number of the molecular species Si

at time t þ Dt conditional on the copy number XiðtÞ attime t: Here, the second-order central jump moment, i.e.the covariance, can be similarly defined and is equal toD because the product /Xiðt þ DtÞ XiðtÞS/Xjðt þDtÞ XjðtÞS is OðDt2Þ: Thus, the diagonalization of D

removes the second-order correlations of the jumps.The second point is the distinction between the

stoichiometry represented by A and the kineticsrepresented by WðXÞ: If there exists a transformationof variables that decouples a stoichiometric matrix A;i.e. A is decouplable, this transformation diagonalizes D;as mentioned in the previous section. Therefore, thediagonalization of D is possible independently of thekinetics WðXÞ when A is decouplable. Although,because of the positive definiteness of D; a transforma-tion of variables that diagonalizes D also exists evenwhen A is not decouplable, it depends not only on thestoichiometry A but also the kinetics WðXÞ: Therefore,we do not consider such case. We note that theimportance of this distinction has been phenomenolo-gically recognized as the burst size effect (Thattai and

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521512

van Oudenaarden, 2001; Ozbudak et al., 2002) and hasbeen clearly pointed out in (Elf and Ehrenberg, 2003).

4.3. The decomposition of noise into intrinsic and

extrinsic noise components

The decomposition of noise into intrinsic andextrinsic noise components (Elowitz et al., 2002; Swainet al., 2002; Paulsson, 2004) can be considered as thespecial case of the decomposition of noise proposed inthis paper (see Appendix A for detail). However, themotivations seems to be greatly different.The decomposition proposed in Paulsson (2004) is

motivated by the experimental study (Elowitz et al.,2002) where the question was how much of the observednoise arises from the network under consideration(intrinsic noise) and how much is caused by thefluctuation of the molecular species outside of it(extrinsic noise). Therefore, only the action of theextrinsic variable to the intrinsic variable was consideredand the action in the opposite direction was omitted(Paulsson, 2004). Additionally, the covariance of thetwo variables were omitted and only the variance of theintrinsic variable was used as the measure of noise.On the other hand, the decomposition (Eq. (26))

proposed in this paper aims to simplify the solution ofthe Lyapunov equation (10). In general, there is noreason to consider some molecular species as moresignificant than the others without any a priori knowl-edge. Thus, we did not concentrate on specific pathwaysbut considered all the molecular species and reactionchannels as equally significant. Therefore, M ði;iÞ of eachdecomposed noise component reflects the overall con-nections of the network.Moreover, in Section 3, we have shown that the

direction of noise is important even in understanding themechanisms behind the fluctuation of each molecularspecies independently. Therefore, the decomposition ofnoise (Eq. (26)) is a key to gain deep insight into thesemechanisms.

4.4. Biological implications

4.4.1. Protein birth and death noise

The first term of Eq. (15), namely, the protein birth

and death noise, can be regarded as the natural extensionof (Thattai and van Oudenaarden, 2001) to a multi-variate system. The scalar part is proportional to thetotal number of proteins mt: The coefficient of thisproportionality is 1þ B in the absence of autoregula-tion; it decreases as the negative feedback strength rincreases. Furthermore, the 2� 2 matrix ð1; ZÞð1; ZÞT

multiplied to the scalar part represents how this noise isdistributed between the fluctuations of the monomersand dimers.

4.4.2. Reducing noise by changing the direction of noise

The reduction of noise by homodimer formation(Bundschuh et al., 2003) and heterodimer formation(Morishita and Aihara, 2004) previously analysedemploy essentially the same mechanism with the changein the direction of noise (Fig. 1(d)) in the single-geneautoregulatory network of this paper. The mechanism,in our words, changes the direction of noise from one ofthe molecular species, e.g. monomers, to the others, e.g.dimers. Indeed, the direction of the protein birth anddeath noise ð1; ZÞT is determined by the dimer dissocia-tion constant Kd ; which was shown to change greatly theeffectiveness of reducing noise in (Bundschuh et al.,2003). In other words, those mechanisms change thedistribution of noise between molecular species tightlycoupled in the stoichiometry. Our decoupling approachhas worked well by taking an appropriate coordinatesystem that decouples the stoichiometric matrix anddecomposes the noise into the protein birth and death

noise and the monomer–dimer fluctuation noise.

4.4.3. Noise in synthetic genetic regulatory networks

The multivariate analysis enables us to estimatequantitatively the stochastic fluctuation around equili-brium points in multi-stable networks. It gives us a clearcriterion to design a feasible genetic switch (Cherry andAdler, 2000; Kobayashi et al., 2003) that can operatereliably even in low copy numbers. This is importantbecause a synthetic network that require high copynumbers of the molecular species in order to operatereliably might disrupt the homeostasis of the host cellinto which it is transformed.

4.5. Future directions

Recent advance in measurement technologies andaccumulation in experimental data are increasing theneed of multivariate analysis. However, mere numericalcalculation techniques lack the ability to improve ourunderstandings of the mechanisms or the structuresbehind specific observations. We have proposed the useof the LNA along with the systematic decoupling of astoichiometric matrix. The decoupling of a stoichio-metric matrix is clearly suited for the analysis of themultivariate stochasticity in a large-scale biochemicalsystem because it simplifies the solution of the Lyapu-nov equation and facilitates analytical treatment. Thismethod can be used to understand the fundamentalstructures that generate, enhance, or reduce the noiseand to reveal not only how organisms can fight againstthe noise but also how they can utilize it. Furthermore,the trade-off between the fluctuations of multiplemolecular species may be the key to understand theevolutionary advantages that have led real networksinside the organisms to such structures. These theore-tical predictions could be confirmed by carefully

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 513

designed experiments based on these predictions. Theunderstanding of the mechanisms is also essential in suchexperiments. An analytical method, such as the one pre-sented here, can provide a quick and accurate predictionover a range of parameters. They might accelerate theprocess from modelings to experiments, thus providemore hypotheses with opportunity to be tested.

Acknowledgements

The authors thank Dr. Hiroyuki Okano, Mr.Yoshihiro Morishita, and Dr. Yoh Iwasa for theirhelpful discussions and comments. The authors alsoacknowledge the valuable comments of the anonymousreviewers that greatly improved this paper. This study ispartially supported by Grant-in-Aid No.12208004 andby the Superrobust Computation Project in 21stCentury COE Program on Information Science andTechnology Strategic Core from the Ministry ofEducation, Culture, Sports, Science, and Technology,the Japanese Government.

Appendix A. The solution of the Lyapunov equation and

the decomposition of noise

In this section, we show that the solution of theLyapunov equation (10) can be decomposed into N � N

equations that do not include D; and the solution of theoriginal Lyapunov equation (10) is written as a linearcombination of the solutions of these equations. Inaddition, when D is diagonal, the solution is furthersimplified into a linear combination of N terms. Then,the decomposition of noise into its origins and thedirections and magnitudes of noise are clearly defined.We consider the following Lyapunov equation:

KSþ SKT þ D ¼ 0; ðA:1Þ

where K is a N � N stable matrix and D is a positivesemidefinite matrix of the same size.

Remark A.1. The solution of Eq. (A.1) is unique andcan be written explicitly as follows (Gaji!c and Qureshi,1995):

S ¼Z

N

0

eKtDeKTt dt: ðA:2Þ

However, this representation is not suited for ananalytical study. Therefore, we write the solution ofEq. (A.1) as follows:

S ¼XN

i¼1

XN

j¼1

nijMði; jÞ; ðA:3Þ

where nij � dij=ððkii þ kjjÞÞ; and M ði; jÞ; which is asymmetric matrix, is the solution of the followingequations:

KM ði; jÞ þ M ði; jÞKT ðkii þ kjjÞ1

2ðeie

Tj þ eje

Ti Þ

� ¼ 0 ði; j ¼ 1;y;NÞ; ðA:4Þ

where fe1;y; eNg is the standard basis in RN : This iseasily proven by taking a summation of the left-handside of Eq. (A.4) weighted by nij for i; j ¼ 1;y;N: Here,note that M ði; jÞ ¼ M ð j;iÞ holds.Let NðN þ 1Þ=2-dimensional vectors *r and *d be *r ¼

VðSÞ � ðs11;y;s1N ;s22;y;s2N ;y; sNNÞTARNðNþ1Þ=2

and *d � ðd11; 2d12;y; 2d1N ; d22; 2d23;y; 2d2N ;y;dN1;N1; 2dN1;N ; dNNÞ

TARNðNþ1Þ=2: The followingNðN þ 1Þ=2 simultaneous equation that is equivalentto Eq. (A.1) does uniquely exist:

*K *r ¼ *d: ðA:5Þ

Moreover the coefficients *K can be algorithmicallyobtained (Gaji!c and Qureshi, 1995).

Remark A.2. M ði; jÞ ð jXiÞ is equivalent to the p-thcolumn vector of ðkii þ kjjÞ *K1:

M ði; jÞ ¼ V1ðfðkii þ kjjÞ *K1gp-th column vectorÞ;

where p ¼ ð2N þ 2 iÞði 1Þ=2þ ð j i þ 1Þ:

The proof is straightforward because the only non-zero component in the right-hand side of Eq. (A.5)equivalent to Eq. (A.4) is the p-th component and isequal to ðkii þ kjjÞ:When D is diagonal, the solution (Eq. (A.3)) can be

written in a more meaningful manner:

S ¼XN

i¼1

niiMði;iÞ;

where nii ð¼ dii=ð2kiiÞÞ is the solution of the followingscalar Lyapunov equation:

kiinii þ niikii þ dii ¼ 0; ði ¼ 1;y;NÞ:

Therefore, it is natural to call nii the magnitude of the i-thnoise component. On the other hand,M ði;iÞ; which is nowa positive semidefinite matrix, represents the character-istic dispersion produced by the i-th noise component.Therefore, we call the maximum eigenvector of M ði;iÞ thedirection of the i-th noise component. Here, note that thedirection of noise is not a ‘‘directed’’ notion, i.e., v andvrepresent the same direction of noise.The magnitude of noise nii ði ¼ 1;y;NÞ reflects only

the characteristics of a small set of reactions. Forexample, for protein synthesis and degradation, dii ¼2krð1þ BÞm and kii ¼ kr give nii ¼ ð1þ BÞm; where kr;B; and m are the degradation rate constant, the averageprotein synthesis from mRNA, and the number ofproteins at the deterministic equilibrium point. On the

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521514

other hand, Mði;iÞ ði ¼ 1;y;NÞ is independent of D andreflects the global connections of the network, repre-sented by K ; e.g. binding to the other proteins or geneticregulation, as shown in the main text.The exact form of decomposed covariance matrix S

for N ¼ 2; with which we have dealt in the main text, isas follows:

S ¼ n11k11

k11 þ k22

1 0

0 0

� þ1

Dk222 k21k22

k21k22 k221

!( )

þn22k22

k11 þ k22

0 0

0 1

� þ1

Dk212 k12k11

k12k11 k211

!( );

ðA:6Þ

where D � k11k22 k12k21: Here, note that ðk22;k21ÞT

and ðk12; k11ÞT are the eigenvectors of the second and

the fourth terms, respectively.Furthermore, we show two special cases. The first

case is when the time-scales of the two components arefar apart (k11; k125k21; k22), Mð1;1Þ and M ð2;2Þ can bereduced as follows:

M ð1;1Þ ¼k11

k11 þ k22

1

Dk222 k21k22

k21k22 k221

!;

M ð2;2Þ ¼0 0

0 1

� :

Here, the maximum (and the only) eigenvectors ofMð1;1Þ

and M ð2;2Þ is ðk22;k21ÞT and ð0; 1ÞT; respectively. This

approximation has been used for the single-geneautoregulatory network in the main text.The second case is when the interaction is unidirec-

tional, k12 ¼ 0; which is the same assumption as thatused in (Paulsson, 2004). This gives the equivalent resultas the previous study:

S ¼

n11 n11k21

k11 þ k22

n11k21

k11 þ k22n22 þ n11

k21

k22

� 2k22

k11 þ k22

0BBB@

1CCCA:

Appendix B. Decoupling of a stoichiometric matrix

First, we define the decoupled stoichiometric matrix.

Definition B.1. A stoichiometric matrix A is said to bedecoupled if each column vector of A has only one non-zero component.

When A is decoupled, each of the reaction chan-nels changes only one variable for a single firing.

Therefore, each reaction channel corresponds to a statevariable.

Definition B.2. A stoichiometric matrix A is said to bedecouplable if there exist a regular matrix TAZN�N suchthat TA is decoupled.

If A is decouplable, T is obtained by performing theGaussian elimination on A (Lancaster and Tismenetsky,1985).As mentioned in the main text, the next fact is

important.

Remark B.1. The sufficient condition for the diffusionmatrix D to be diagonal is that the stoichiometric matrixA is decoupled.

The proof is straightforward from the definition(Eq. (5)). Additionally, we do not consider the un-necessary case, because though it is possible thatPM

k¼1 aikajkWkðXÞ ¼ 0 ð8iajÞ holds for a specificWðXÞ; any perturbation in the rate constants or thestate variable X can break this equality.As a summary, given an arbitrary stoichiometric

matrix A; if A is decouplable, there exist a transformationof variables that makes the diffusion matrix D diagonal,and the covariance matrix a linear combination ofN terms, as shown in the previous section. Theoverall picture of our theory is schematically shown inFig. 5.Next, we present a necessary and sufficient condition

for A to be decouplable. Let us define a subset ofreactions R0ðDR ¼ fR1;y;RMgÞ of size M0 ðpMÞand an N � M0 matrix A0: R0 is defined as themaximum subset of reactions whose stoichiometricmatrix A0 consists of pairwise linearly independentcolumn vectors. Therefore, the original stoichio-metric matrix A can be rewritten as A ¼ ðA0;A1Þwhere A0AZN�M0 and A1AZN�ðMM0Þ: Here,for each column vector a1 of A1; there exist a rationalnumber s and a column vector a0 of A0 thatsatisfies a1 ¼ sa0; e.g. when s ¼ 1 the reaction corre-sponding to a1 is the reverse reaction of that corre-sponding to a0:Generally, the whole set of reactions R can be

partitioned into equivalence classes under a equivalencerelation of pairwise linear dependence. R0 isthe set of class representatives of these equivalenceclasses. If R0 is linearly independent, each equivalenceclass of reaction channels can be made to correspondto a state variable. This is formally expressed asfollows.

Theorem B.1. A is decouplable if and only if A0 has full

column rank.

ARTICLE IN PRESS

Fig. 5. The overall picture of the decoupling of a stoichiometric

matrix, the diagonalization of a diffusion matrix, and the decomposi-

tion of noise.

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 515

Proof. Sufficiency: Because A0 has full column rank,M0pN: Therefore, we can choose T as

T ¼SðAT0A0Þ

1AT0

AT0>

!;

where S ¼ diagðs1;y; sM0ÞAZM0�M0 is an appropriate

diagonal matrix that makes T an integer matrix, andA0>AZN�ðNM0Þ is a basis of the orthogonal comple-ment space of A0:

Necessity: Any decoupled stoichiometric matrix *A ¼TA can be rewritten as

*A ¼ ð *A0; *A1Þ;

where

*A0 ¼S

0

� AZN�M0 ;

S ¼ diagðs1;y; sM0ÞAZM0�M0 ; and *A1 is similarly de-

fined as A1: Therefore, A ¼ T1ð *A0; *A1Þ: Clearly, thefirst M0 columns of A have full column rank and foreach column vector a1 of T1 *A1; there exist a columnvector a0 of T1 *A0 that is pairwise linearly dependentto a1: &

For example, the following chemical system isdecouplable:

ðB:1Þ

The decoupled stoichiometric matrix *A ¼ TA is ob-tained as follows:

ðB:2Þ

The correspondence between the equivalence classes ofreaction channels and the molecular species is welldescribed by a bipartite graph (Temkin et al., 1996) of adecoupled stoichiometric matrix. The bipartite graphs ofthe original and the decoupled stoichiometric matricesare shown in Fig. 6.The following chemical system is not decouplable.

whose Gaussian elimination ends up in the followingform:

where the reverse reactions R2; R4; and R6 wereomitted.

Appendix C. Proof of the stability conditions

C.1. Single-gene autoregulatory network

We prove that the equilibrium point of Eqs. (12) and(13) is unique and the Jacobian Kð/Þ (Eq. (14)) is stablefor all / ¼ ðf1;f2Þ

T: Here, we assume that there aremore than 0.5 molecules of monomers as an average,i.e., 2Ofm 1 > 0: This is satisfied in all cases in thispaper. In addition, we use a0ðf2Þo0 for all f2; since weare considering an autoregulatory gene.

ARTICLE IN PRESS

R1

R2

R3

R4

R5

R1

R2

R3

R4

R5

A

B

C

D

A

(A+B)-(C+D)

(A+B)-D

A+B

U V U V

(B.1) (B.2)

Fig. 6. The bipartite graph of the original (left, see Eq. (B.1)) and

decoupled (right, see Eq. (B.2)) stoichiometric matrices. A bipartite

graph (U,V,E) of a stoichiometric matrix A is defined as follows

(Temkin et al., 1996): U ¼ fujgj¼1;y;M is the set of vertices

corresponding to reaction channels. V ¼ fvigi¼1;y;N is the set of

vertices corresponding to molecular species. E ¼ feijg is the set ofedges. eij denotes the edge between uj and vi: Here, eij exists if and only

if aija0: Note that the stoichiometric matrix that a bipartite graphrepresents is not unique in this definition, which is sufficient for the

current purpose.

Table 1

Parameter values used for the single-gene autoregulatory network

Parameters Values

Cell volume O ¼ 109 ð¼ 1:66� 1015ðlitersÞ �6:02� 1023Þ

Transcription initiation rate a ¼ 0:011 ðs1ÞNegative feedback strength r ¼ 0Average protein synthesis

per mRNA

B ¼ 11 (proteins/mRNA)

Dimer dissociation constant Kd ¼ 20 (nM)Dimer dissociation rate constant kd ¼ 1 ðs1ÞProtein degradation rate constant kr ¼ 0:0003 ðs1)

Table 2

Parameter values for the two-gene autoregulatory network

Parameters Values

Cell volume O ¼ 109 ð¼ 1:66� 1015ðlitersÞ �6:02� 1023Þ

Transcription initiation rate a ¼ b ¼ 0:011 ðs1ÞPositive regulatory strength ra ¼ 0Negative regulatory strength rb ¼ 0Average protein synthesis

per mRNA

B1 ¼ B2 ¼ 11 ðproteins=mRNAÞ

Protein degradation rate constant kr1 ¼ kr2 ¼ 0:0003 ðs1Þ

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521516

Uniqueness: The equilibrium condition ofEqs. (12)=(13)=0 can be rewritten as follows:

fmðaðf2ÞB=kr;f2Þðfmðaðf2ÞB=kr;f2Þ O1Þ

¼ Kdf2; ðC:1Þ

where fmðf1;f2Þ ¼ f1 2f2 is the concentration of themonomers. By differentiating the left-hand side ofEq. (C.1) with respect to f2; we obtain

ð2fmðaðf2ÞB=kr;f2Þ O1Þða0ðf2ÞB=kr 2Þo0:

Thus, the solution of Eq. (C.1) is unique.Stability: The trace and the determinant of the matrix

Kð/Þ satisfy the following inequalities:

tr½Kð/Þ� ¼ fkr þ kd ð1þ 2ð2fm O1Þ=KdÞgo0;

det½Kð/Þ� ¼ kdkrf1þ ð2 a0ðf2ÞB=krÞð2fm O1Þ=Kdg

> 0:

The eigenvalues of Kð/Þ are the solutions of thefollowing characteristic equation:

l2 tr½Kð/Þ�lþ det½Kð/Þ� ¼ 0:

Thus, both of the eigenvalues of Kð/Þ have negative realparts for all / ¼ ðf1;f2Þ

T:

C.2. Two-gene autoregulatory network

We prove that the equilibrium point of Eqs. (17) and(18) is unique and the Jacobian Kð/Þ (Eq. (19)) is stablefor all / ¼ ðf1;f2Þ

T: Here, we use a0ðf2Þb0ðf1Þo0 for all

/ ¼ ðf1;f2ÞT; considering that the protein A represses

the gene b coding protein B; and the protein B activatesthe gene a coding protein A:

Uniqueness: The equilibrium condition ofEqs. (17)=(18)=0 can be rewritten as follows:

aðbðf1ÞB2=kr2ÞB1=kr1 ¼ f1: ðC:2Þ

By differentiating the left-hand side of Eq. (C.2) withrespect to f1; we obtain

a0ðbðf1ÞB2=kr2Þb0ðf1ÞB1B2=ðkr1kr2Þo0:

Thus, the solution of Eq. (C.2) is unique.Stability: The trace and the determinant of the matrix

Kð/Þ satisfy the following inequalities:

tr½Kð/Þ� ¼ ðkr1 þ kr2Þo0;

det½Kð/Þ� ¼ kr1kr2 a0ðf2Þb0ðf1ÞB1B2 > 0:

Thus, both of the eigenvalues of Kð/Þ have negative realparts for all / ¼ ðf1;f2Þ

T:

Appendix D. Parameter values

Tables 1–3 show the parameter values we used for thesingle-gene autoregulatory network (in Section 3.1), thetwo-gene autoregulatory network (in Section 3.2), andthe biologically plausible two-gene autoregulatory andmutually repressive networks (in Sections 3.3 and 3.4).All the parameter values were held constant at thesevalues unless explicitly noted. We fixed the cell volumeat 1:66� 1015 (liters) to be consistent with the E. coli

cell volume B1015 (liters) (Alberts et al., 2002), and tomake mathematical treatment easy. That is, this makes1 molecule=cell correspond to the 1 nM concentration.The transcription initiation rate a; the average burst size

ARTICLE IN PRESS

Table 3

Parameter values for the biologically plausible two-gene autoregula-

tory network and the two-gene mutually repressive network

Parameters Values

Cell volume O ¼ 109 ð¼ 1:66�1015ðlitersÞ � 6:02� 1023Þ

Maximum transcription

initiation rate

a0 ¼ b0 ¼ 0:011 ðs1Þ

DNA–regulatory protein binding

constant

Ka ¼ Kb ¼ 50 (nM)

DNA–regulatory protein binding

Hill coefficient

ha ¼ hb ¼ 2

Average protein synthesis per

mRNA

B1 ¼ B2 ¼ 11 (proteins/mRNA)

Protein degradation rate

constant

kr1 ¼ kr2 ¼ 0:0003 ðs1)

IPTG–LacI binding constant KI ¼ 29:618 ðmMÞIPTG–LacI binding Hill

coefficient

hI ¼ 1

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 517

B; and the dimer dissociation constant Kd were obtainedfrom (Shea and Ackers, 1985), which were valuesmeasured for the bacteriophage l cI gene and the CIprotein. The dimerization rate kd is obtained from Arkinet al. (1998). The protein degradation rate kr wasadjusted to yield a half-life roughly equal to the E. coli

cell cycle (40 min).

Table 4

Parameter values for the full models of biologically plausible two-gene

autoregulatory network and two-gene mutually repressive network

Parameters Values

Cell volume 1:66� 1015ðlitersÞk1 1 ðnM1s1Þk2 2500 ðs1Þk3 1 ðnM1s1Þk4 1 ðs1Þk5 0:011 ðs1Þk6 1:1 ðs1Þk7 0:1 ðs1Þk8 0:1 ðmM1s1Þk9 2:962 ðs1Þk10 0:0003 ðs1Þk11 0:0003 ðs1Þk12 1 ðnM1s1Þk13 2500 ðs1Þk14 1 ðnM1s1Þk15 1 ðs1Þk16 0:011 ðs1Þk17 1:1 ðs1Þk18 0:1 ðs1Þk19 0:0003 ðs1Þ

Appendix E. Comparison of the analytical results and the

stochastic simulations

We compare the results in Sections 3.3 and 3.4 withdetailed stochastic simulations composed of 12 variablesand 19 reactions. We use the next reaction method(Gibson and Bruck, 2000), which is an efficient andexact stochastic simulation algorithm based on a Monte-Carlo method well known as the Gillespie method(Gillespie, 1977).For the two-gene autoregulatory network in Section

3.3, we model that the transcription of the LacI mRNAinitiates only from the lacI gene whose promoter PRM isbound by two copies of CI, which is denoted by DLac2;as follows:

DLac2!k5DLac2 þmLac;

where k5 denotes the open complex formation rateconstant. For the two-gene mutually repressive networkin Section 3.4, we model that the transcription of theLacI mRNA initiates only from the lacI gene whosepromoter PL is not bound, which is denoted by DLac0;as follows:

DLac0!k5DLac0 þmLac:

All the other reactions are the same for the twonetworks as follows:

DLac0 þ CI"k1

k2

DLac1;

DLac1 þ CI"k3

k4

DLac2;

mLac!k6mLacþ Lac;

mLac!k7 |;

Lacþ IPTG"k8

k9

Lac¼ IPTG;

Lac!k10 |;

Lac¼ IPTG!k11IPTG;

DCI0 þ Lac"k12

k13

DCI1;

DCI1 þ Lac"k14

k15

DCI2;

DCI0!k16DCI0þmCI;

mCI!k17mCIþ CI;

mCI!k18 |;

CI!k19 |;

where DLaci ði ¼ 0; 1; 2Þ denotes the lacI gene whosepromoter PRM or PL bound by i copies of CI; mLac,LacI, and LacI=IPTG denote the LacI mRNA, theLacI protein, and the LacI protein bound by IPTG,respectively. DCIi; mCI, and CI are similarly used. Allthe values of the rate constants ki are listed in Table 4.

ARTICLE IN PRESS

LacI

CI

[IPTG]=16µM

0 100 200 300 400 5000

100

200

300

400

500

-1

-0.5

0

0.5

1

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

-4

CI=50.4

LacI

PD

F

[IPTG]=16µM

0 100 200 300 4000

0.5

1

1.5x 10

-4

LacI=203.2

CI

PD

F

[IPTG]=16µM

(a)

(b)

(c)

Fig. 7. Stochastic simulation of the two-gene autoregulatory network

in Section 3.3 with ½IPTG� ¼ 16 mM: (a) The two-dimensional

histogram of the simulation result and the LNA (the noise ellipsoid

in broken line). The histogram is shown in a grayscale, corresponding

to the logarithmic probability density normalized by the 1s probabilitydensity log10ðP � 2p

ffiffiffiffiffiffiffiffiffiffiffijSE je

pÞ: (b) The cross section of the probability

density function with CI fixed at the deterministic equilibrium point.

(c) The cross section of the probability density function with LacI fixed

at the deterministic equilibrium point.

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521518

E.1. Autoregulatory network

The comparisons between the LNA and the sto-chastic simulation are shown for two parameters:½IPTG� ¼ 16 mM in Fig. 7 and ½IPTG� ¼ 252 mM in

LacI

CI

[IPTG]=252µM

0 200 400 6000

100

200

300

400

500

-1

-0.5

0

0.5

1

(a)

0 200 400 600 8000

1

2

3

4

5x 10

-5

CI=242.8

LacI

PD

F

[IPTG]=252µM

(b)

(c)0 100 200 300 400 500 600

0

1

2

3

4

5x 10

-5

LacI=386.9

CI

PD

F

[IPTG]=252µM

Fig. 8. Stochastic simulation of the two-gene autoregulatory network

in Section 3.3 with ½IPTG� ¼ 252 mM: (a)–(c) are similar to those inFig. 7.

ARTICLE IN PRESS

LacI

CI

[IPTG]=10µM

0 200 400 600

0

100

200

300

400

500

600

-1

-0.5

0

0.5

1

0 100 200 300 400 500 600 7000

0.5

1

1.5

2x 10

-4

CI=12.0

LacI

PD

F

[IPTG]=10µM

0 50 100 150 2000

0.5

1

1.5

2

2.5

3x 10

-4

LacI=381.2

CI

PD

F

[IPTG]=10µM

(a)

(b)

(c)

Fig. 9. Stochastic simulation of the two-gene mutually repressive

network in Section 3.4 with ½IPTG� ¼ 10 mM: The initial values for thesample paths are set at the lower-right equilibrium point. (a)–(c) are

similar to those in Fig. 7.

LacI

CI

[IPTG]=20µM

0 200 400 600

0

100

200

300

400

500

600

-1

-0.5

0

0.5

1

0 100 200 300 400 500 600 7000

1

2

3

4

5x 10

-5

CI=27.5

LacI

PD

F

[IPTG]=20µM

0 50 100 150 2000

1

2

3

4

5

6x 10

-5

LacI=309.6

CI

PD

F

[IPTG]=20µM

(a)

(b)

(c)

Fig. 10. Stochastic simulation of the two-gene mutually repressive

network in Section 3.4 with ½IPTG� ¼ 20 mM: The initial values for thesample paths are set at the lower-right equilibrium point. (a)–(c) are

similar to those in Fig. 7.

R. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521 519

Fig. 8. The initial values of all the sample paths are setat the deterministic equilibrium point for each para-meter; the number of the sample paths is 4000 and

the length of each path is 20000s. Here, the axis,LacI denotes the sum of the IPTG bound and unboundLacIs. Figs. 7(a) and 8(a) show the two-dimensional

ARTICLE IN PRESSR. Tomioka et al. / Journal of Theoretical Biology 229 (2004) 501–521520

histograms in a grayscale corresponding to the normal-ized logarithmic probability density log10ðP � 2p

ffiffiffiffiffiffiffiffiffiffiffijSE je

pÞ;

i.e. the logarithm of the probability density (P) normal-ized by the 1s probability density (e1=2=ð2p

ffiffiffiffiffiffiffiffijSE j

pÞ)

where jSE j is the determinant of the covariance matrixSE calculated from the estimated probability distribu-tion. Thus, the correspondence between the zero cross-ing regions and a 1s equi-probability curve is a goodmeasure of the accuracy of an analytical result.Figs. 7(b) and 8(b) and Figs. 7(c) and 8(c) showthe cross sections of the PDFs at the deterministicequilibrium points with fixed CI and fixed LacI,respectively.Both Figs. 7 and 8 show that the analytical results are

in good agreement with the simulations. Note that theanalytical representation of the 2 variable model(Eq. (21)) and applying the LNA directly to the full 12variable model (result not shown) yield almost the sameresults.

E.2. Mutually repressive network

The comparisons between the LNA and the stochasticsimulation are shown for two parameters: ½IPTG� ¼10 mM in Fig. 9 and ½IPTG� ¼ 20 mM in Fig. 10. Theinitial values of all the sample paths are set at thedeterministic lower-right equilibrium point for eachparameter; the number of the sample paths is 16,000and the length of each path is 40000s. Figs. 9(a) and10(a), 9(b) and 10(b), and 9(c) and 10(c) show the two-dimensional histograms, the CI fixed cross sections, andthe LacI fixed cross sections, similarly to those in theautoregulatory network. The cross sections are shown atthe lower-right equilibrium points.Figs. 9(a), (b) and 10(a), (b) show the good agree-

ment of the analytical results to the simulations. Thetrue PDFs are far from Gaussian when the copynumbers are extremely small (see Figs. 9(c) and 10(c)).However, the analytical results approximate the truePDFs fairly well even in these cases. Similarly to theautoregulatory network, the analytical representa-tion (Eq. (21)) and applying the LNA directly to thefull 12 variable model (result not shown) yield almostthe same results.

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