Modeling operon dynamics: the tryptophan and lactose ... · C. R. Biologies 327 (2004) 211–224...

C. R. Biologies 327 (2004) 211–224

Biological modelling / Biomodélisation

Modeling operon dynamics:the tryptophan and lactose operons as paradigms

Michael C. Mackey a,∗, Moisés Santillán b, Necmettin Yildirim c

a Departments of Physiology, Physics and Mathematics, Centre for Nonlinear Dynamics, McGill University,3655 Drummond Street, Montreal, Quebec, Canada H3G 1Y6

b Esc. Sup. de Física y Matemáticas, Instituto Politécnico Nacional, 07738, México D.F., Mexicoc Atatürk Üniversitesi, Bilgisayar Bilimleri Uygulama ve Araştırma Merkezi, 25240 Erzurum, Turkey

Received 15 April 2003; accepted after revision 26 November 2003

Presented by Pierre Auger

Abstract

Understanding the regulation of gene control networks and their ensuing dynamics will be a critical component in theunderstanding of the mountain of genomic data being currently collected. This paper reviews recent mathematical modelingwork on the tryptophan and lactose operons which are, respectively, the classical paradigms for repressible and inducibleoperons. To cite this article: M.C. Mackey et al., C. R. Biologies 327 (2004). 2004 Published by Elsevier SAS on behalf of Académie des sciences.

Résumé

Modélisation de la dynamique de l’opéron : les opérons tryptophane et lactose comme paradigmes. L’étude dela régulation des réseaux de contrôles des gènes et des dynamiques qui en découlent sera une composante critique de lacompréhension de la masse de données génomiques collectées. Cet article fait le bilan des récents travaux de modélisationmathématique sur les opérons tryptophanes et lactoses, qui sont respectivement les paradigmes classiques pour les opéronsrépressibles et inductibles. Pour citer cet article : M.C. Mackey et al., C. R. Biologies 327 (2004). 2004 Published by Elsevier SAS on behalf of Académie des sciences.

Keywords: gene regulatory networks; nonlinear dynamics; computational molecular biology

Mots-clés : circuits de régulation des gènes ; dynamique non linéaire ; biologie moléculaire informatique

1. Introduction

Molecular biology as a field would probably notexist in its present form were it not for the impact

* Corresponding author.E-mail address: [email protected] (M.C. Mackey).

1631-0691/$ – see front matter 2004 Published by Elsevier SAS on behdoi:10.1016/j.crvi.2003.11.009

of the beautiful little book What is Life? [1] writtenby Ernst Schrödinger (one of the fathers of quantummechanics). What is Life? was the written account ofa series of lectures given by Schrödinger at the DublinInstitute for Advanced Studies in 1943. What is Life?was partially inspired by the work of the physicist MaxDelbrück and was probably instrumental in recruiting

alf of Académie des sciences.

http://

212 M.C. Mackey et al. / C. R. Biologies 327 (2004) 211–224

a whole generation of theoretical physicists like WalterGilbert, Leo Szilard, Seymor Benzer, and FrancisCrick away from physics per se and into the excitingnew area of ‘molecular biology’ [2]. The results arehistory, but it has not ended there. For example,the current director of the Whitehead Center forGenomic Research at the MIT is Eric Lander, whoseD. Phil. from Oxford was not in molecular biology butrather in mathematics (algebraic coding theory)! Themathematical and physical sciences continue to play alarge role in the whole area of molecular biology andsome of the potential areas of impact are detailed inthe March 2001 issue of Chaos [3].

The operon concept [4], introduced by Jacob andcoworkers in 1960 [5], has had a profound andlasting effect on the biological sciences. Not longafter the operon concept was developed, Goodwin[6] gave the first mathematical analysis of operondynamics. Griffith then put forward a more completeanalysis of simple repressible (negative feedback [7])and inducible (positive feedback [8]) gene controlnetworks, and Tyson and Othmer [9] have summarizedthese results. Extensions considering the stability ofinducible operons were published by Selgrade [10,11] and Ji-Fa [12], but none of these treatmentsconsidered the role of the DNA transcription andmRNA translation delays though Tyson and Othmerpointed out that both should be considered.

In this paper we present recent mathematical mod-eling work on operon dynamics. In Section 2 we treatthe (repressible) tryptophan operon, and in Section 3the (inducible) lactose operon is considered. In bothcases we have made every effort to construct biolog-ically realistic mathematical models and to make ac-curate parameter estimation from the biological lit-erature. In Section 4, we conclude the paper withsome general comments related to our philosophicalapproach to the modeling of these systems.

2. The repressible tryptophan operon

After Jacob et al. introduced the operon concept[5], the development of more refined experimentaltechniques and the acquisition of more data made itclear that in addition to repression there are manyregulatory mechanisms involved in the control ofoperons. These mechanisms vary from operon to

operon. One of the most intensively studied of thesesystems from an experimental point of view is thetryptophan operon. An excellent review of this systemand its regulatory mechanisms is to be found in [13].

2.1. The Bliss model

One of the first mathematical models of the tryp-tophan operon was introduced by Bliss et al. in 1982[14]. The independent variables of this model are thetrp mRNA concentration (M), the concentration of an-thranilate synthase (E) which, according to Bliss etal., of all the enzymes formed with the polypeptidesof the tryptophan operon is the most important froma regulatory point of view, and the tryptophan concen-tration (T ). The equations governing the dynamic evo-lution of these three quantities are

(1)dM

dt= KmOR(t − τm) − K1M

(2)dE

dt= KpM(t − τp) − K2E

and

(3)dT

dt= KtEI(T ) − G(T ) − KT

In Eq. (1), Km is the intrinsic rate of transcriptioninitiation of an operon that is not repressed, R(T )is the probability that an operon is not repressed attime t , τm is the time delay between initiation oftranscription and initiation of translation, O is the totaloperon concentration, and K1 is a positive constantaccounting for mRNA depletion due to dilution bygrowth and enzymatic degradation.

Bliss et al. assumed that R can be modeled by a Hillfunction

R(T ) = Kmr

Kmr + T mwith m = 4. In Eq. (2), Kp denotes the rate oftranslation initiation per mRNA molecule, τp is thetime delay between initiation of translation and theappearance of functional enzyme, and K2 is a positiveconstant accounting for enzyme depletion due todilution by growth and enzymatic degradation. Finally,in Eq. (3), Kt is the tryptophan production rateper enzyme, I (T ) is the fraction of enzyme notinhibited by the end product, G(T ) is the tryptophan

M.C. Mackey et al. / C. R. Biologies 327 (2004) 211–224 213

consumption rate, and K is the growth rate of thebacterial culture.

This model clearly considers two different regula-tory mechanisms, repression and enzymatic feedbackinhibition by tryptophan. R(T ) accounts for repres-sion, while I (T ) stands for feedback inhibition. Blisset al. also assert that I (T ) can be modeled by a Hillfunction

I (T ) = Kni

Kni + T nwith n = 2, and that the demand for tryptophan obeysa Michaelis–Menten-like equation

G(T ) = Gmax TT + Kg

The authors made a careful estimation of the para-meters in this model based on the available experimen-tal data. They solved the model equations numericallyand also performed an analytical stability analysis ofthe steady states. They were able to reproduce the re-sults of derepression experiments with cultures of wildand mutant E. coli strains reported in [15]. From theseexperiments, the tryptophan operon loses stability inmutant E. coli cell cultures that have a partial loss ofof feedback inhibition. The previously stable steadystate is replaced by an oscillatory production of tryp-tophan [15]. This mutation is modeled by increasingKi to ten times its wild type value. However, to ourknowledge, these experiments have never been suc-cessfully repeated. Moreover, more recent experimen-tal evidence has demonstrated that the function R(T ),which Bliss et al. used to model repression, is in dis-agreement with the experimental facts about the in-teraction between the trp operon and repressor mole-cules. The issue of oscillations in tryptophan controlwas revisited by Xiu et al. [16] in their investigation ofthe role of growth and dilution rates on stability.

2.2. The Sinha model

In 1988, Sinha [17] introduced a different modelfor the tryptophan operon regulatory system in whichthe DNA-repressor interaction is modelled in a moredetailed way. The independent variables of the Sinhamodel are the same as in the Bliss et al. model, i.e.the trp mRNA concentration (M), the anthranilate syn-thase enzyme concentration (E), and the tryptophan

concentration (T ). The dynamic equations for thesevariables are

(4)dM

dt= Km KoKd + KoT

KoKd + KoT + nRT O − K1M

(5)dE

dt= KpM − K2E

and

(6)dT

dt= KtE − Gmax − KT

The first term in the right-hand side of Eq. (4) is themRNA production rate, which is assumed to be pro-portional to the concentration of unrepressed oper-ons, with a proportionality constant Km. The concen-tration of unrepressed operons is calculated by tak-ing into account the fact that active repressor mole-cules reversibly bind operons to block transcriptioninitiation, and assuming that this reaction takes placesufficiently rapidly that it is in a quasi-steady state.From this, the concentration of unrepressed operonsis OKo/(Ko +RA), where Ko is the dissociation con-stant of the repressor-operon reaction, O is the totaloperon concentration, and RA is the concentration ofactive repressor molecules.

When produced by the trpR operon, repressormolecules are unable to repress trp operons. For thisrepression to take place, they need to be activatedby two tryptophan molecules which sequentially bindnon-cooperatively in two independent places. Fromthis, and assuming that this reaction takes place in aquasi-steady state, Sinha obtained the concentration ofactive repressor given by RA = nT R/(T +Kd), wheren = 2, R is the total repressor concentration, and Kdis the repressor activation dissociation constant.

Compared with the Bliss et al. [14] model, theSinha model considers the DNA-repressor and repres-sor-tryptophan interactions in a more detailed way.It also ignores the fact that anthranilate synthase isfeedback inhibited by tryptophan as well as ignoringthe time delays inherent to the system. These are bothimportant features of the tryptophan operon regulatorysystem.

It is also important to note that Sinha assumes aconstant tryptophan consumption rate Gmax. Observ-ing this, Sen and Liu [18] modified the Sinha modelto study the case of a non-linear tryptophan consump-


tion rate given by a Michaelis–Menten-like functionG(T ) = GmaxT/(T + Kg). Sinha, as well as Sen andLiu, studied the stability of their models for variousvalues of parameters Ko and K2 and found the cor-responding stability regions in the (Ko,K2) parame-ter space. However, they neither compare their modelswith experimental data nor do they discuss whether thevalues for Ko and K2 where stability is lost are phys-

iologically attainable in wild type or mutant bacterialstrains.

2.3. A more detailed model

Recently Santillán and Mackey [19] introduced amore detailed mathematical model of the tryptophanoperon regulatory system, which is shown schemati-cally in Fig. 1. In this model, all of the known reg-

Fig. 1. Schematic representation of the tryptophan operon regulatory system. See the text for a detailed description.


ulatory mechanisms of the tryptophan operon (repres-sion, transcriptional attenuation, and enzyme feedbackinhibition) are taken into account. The system inher-ent time delays due to transcription and translationare also included, as well as the mRNA dynamics.The model considers four independent variables: Theconcentration of trp operons with operators free tobe bound by mRNA polymerase molecules (OF), theconcentration of trp mRNA molecules with free TrpE-related ribosome binding sites (MF), the concentrationof the enzyme anthranilate synthase (E), and the tryp-tophan concentration (T ).

Although five different polypeptides are synthe-sized during the expression of the trp operon and theycombine to form the enzymes that catalyze the reac-tion pathway which synthesizes tryptophan from cho-rismic acid, this model concentrates on anthranilatesynthase. The reason for this focus is that this enzymeis the most important from a regulatory point of viewsince it is subject to feedback inhibition by tryptophanand is the first to play its catalytic role in the reactionpathway [13].

Anthranilate synthase is a complex of two TrpEand two TrpD polypeptides. From this, Santillánand Mackey assume that the anthranilate synthaseproduction rate is one half that of the TrpE polypeptideand look only at the TrpE-related ribosome bindingsites of the mRNA chain. The equations governing thedynamics of these variables are:

dOFdt

= KrKr + RA(T )

{µO − kpP

[OF(t)

(7)− OF(t − τp)e−µτp]} − µOF

dMFdt

= kpPOF(t − τm)e−µτm[1 − A(T )]

− kρρ[MF(t) − MF(t − τρ)e−µτρ

]

(8)− (kdD + µ)MF(t)

(9)dE

dt= 1

2kρρMF(t − τe)e−µτe − (γ + µ)E(t)

and

(10)dT

dt= KEA(E,T ) − G(T ) + F(T ,Text) − µT

All the parameters in this model were estimatedfrom reported experimental results, and the details ofthe parameter estimation process are in [19], which

also contains an extended description of the modelderivation.

The factor

Kr

Kr + RA(T )in Eq. (7) is the fraction of unrepressed operons.This follows from the fact that when active repres-sor reversibly binds free operons, the experimentallyreported reaction rates allow one to make a quasi-steady-state assumption. Kr is the dissociation con-stant of this reaction, and RA(T ) is the active re-pressor concentration. The experimentally reportedrates of the repressor activation reaction also supporta quasi-steady-state assumption. Repressor moleculesactivate when two tryptophan molecules sequentiallybind an inactive repressor in two independent sitesnon-cooperatively. From this

RA(T ) = RT (t)T (t) + Kt

where R is the total repressor concentration and Ktis the dissociation constant of the repressor activationconstant.

µ is the growth rate of the bacterial culture. Theterm µO stands for the operon production rate due toDNA replication, which was assumed to be such thatit keeps the total operon concentration at a constantvalue O , balancing dilution by growth.

The term

−kpP[OF(t) − OF(t − τp) exp(−µτp)

]

accounts for the rate of free operons binding bymRNA polymerase molecules and later freed when thepolymerases have traveled for a distance along theoperon in a time τp. kp is the reaction rate of theDNA-polymerase binding reaction, assumed to be ir-reversible, and P is the mRNA polymerase concen-tration. The exponential factor exp(−µτp) takes intoaccount the dilution of OF due to exponential growthduring the time τp.

In Eq. (8), τm is the time it takes for a polymeraseto produce a mRNA chain long enough to have anavailable TrpE-related ribosome binding site. A(T )is the probability for transcription to be prematurelyterminated due to transcriptional attenuation. Santillánand Mackey [19] take A(T ) = b[1− exp(−T/c)]. The


term

−kρρ[MF(t) − MF(t − τρ) exp(−µτρ)

]

in Eq. (8) accounts for the rate of mRNAs beingbound by ribosomes and liberated after they travel fora time τρ along the mRNA chain. kρ is the mRNA-ribosome binding reaction rate and ρ is the ribosomalconcentration.

The first term on the right-hand side of Eq. (9), isthe anthranilate synthase production rate. The factor1/2 is incorporated because, as mentioned earlier, theenzyme production rate is assumed to be one halfthat of TrpE polypeptide. The TrpE production rateequals the rate of ribosome binding free TrpE-relatedribosome binding sites a time τe ago. τe is the time ittakes for a ribosome to synthesize and release a TrpEpolypeptide.

The tryptophan production rate, the first term onthe right-hand side of Eq. (10), is assumed to be pro-portional to the concentration of active (uninhibited)enzymes (EA) with a proportionality constant K . An-thranilate synthase is inhibited when two tryptophanmolecules bind the TrpE subunits in a sequential co-operative reaction with a Hill coefficient of nH � 1.2.The enzyme feedback inhibition reaction rate con-stants support a quasi-steady state assumption, fromwhich

EA(E,T ) = E(T ) KnHi

T nH + KnHiKi is the dissociation constant of this reaction. Thetryptophan consumption rate constant is modeled asa Michaelis–Menten-type function

G(T ) = Gmax TT + Kg

Finally, E. coli can incorporate tryptophan from theenvironment. This can be managed by different trypto-phan permeases. The rate of tryptophan uptake is mod-eled by Santillán and Mackey as

F(T ,Text) = d Texte + Text[1 + T/f ]

Text is the external tryptophan concentration.All of the model parameters were estimated from

the available experimental data. The details are givenin Reference [19] supplementary web material. InTable 1 the value of all these parameters is shown.

Table 1The tryptophan model parameters as estimated in [19]

R � 0.8 µM O � 3.32 × 10−3 µMP � 2.6 µM k−r � 1.2 min−1ρ � 2.9 µM kr � 460 µM−1 min−1τp � 0.1 min k−i � 720 min−1τm � 0.1 min ki � 176 µM−1 min−1τρ � 0.05 min k−t � 2.1 × 104 min−1τe � 0.66 min kt � 348 µM−1 min−1γ � 0 min−1 kp � 3.9 µM−1 min−1kdD � 0.6 min−1 kρ � 6.9 µM−1 min−1nH � 1.2 µ � 1.0 × 10−2 min−1b � 0.85 c � 4.0 × 10−2 µMKg � 0.2 µM g � 25 µM min−1e � 0.9 µM d � 23.5 µM min−1f � 380 µM K � 126.4 min−1

The model was solved numerically, and the resultscompared with derepression experiments performedby Yanofsky and Horn [20] on wild-type bacterial cul-tures as well as trpL29 and trpL75 mutant strains ofE. coli. In these experiments, a bacterial culture is firstgrown in a medium with a high tryptophan concentra-tion during a period of time long enough for the trpoperon to reach a steady state. Then the bacteria arewashed and shifted to a medium without tryptophan,and the response of the anthranilate synthase activityis measured as a function of time after the nutritionalshift.

The trpL29 mutant strain has a mutation A toG at bp 29 in the leader region of the trp operon.This change replaces the leader peptide start codonby GUG, and decreases operon expression in cellsgrowing in the presence or absence of tryptophan.This mutation is simulated by Santillán and Mackeyby decreasing the rate constant kp to 0.004 times itsnormal value.

The strain trpL75 of E. coli has a mutation of G toA at bp 75 in the leader region of the trp operon. Thischange decreases the stability of the transcription an-titerminator structure, and increases transcription ter-mination at the attenuator. Consequently, it decreasesoperon expression in cells growing in the presence orabsence of tryptophan. This mutation was simulatedby increasing the value of parameter b, which deter-mines the probability of transcriptional attenuation athigh tryptophan levels. The normal value of this para-meter is b � 0.85, and to simulate the mutation it wasincreased to b � 0.9996.


Fig. 2. Experimentaly measured points [20] and simulated curves of anthranilate synthase activity vs. time after a nutritional shift (minimal +tryptophan medium to minimal medium), with (A) wild, (B) trpL29 mutant, and (C) trpL75 mutant strain cultures of E. coli. Two different setsof experimental results (crosses and pluses) for the wild-type strain are presented in A. One set of experimental points corresponding to thetrpL29 mutant strain is shown in B with circles. Another set of experimental points for the trpL75 mutant strain is shown in C with asterisks.The corresponding simulations are shown with solid lines in all of the three graphics. The wild-type experimental points and simulation arerepeated in B and C for comparison.

The experiments of Yanofsky and Horn were sim-ulated numerically. In all of the three cases (wild typeand two mutants), the model results show a reasonablequalitative agreement with the experiments, given thesimplifying assumptions inherent to the model. The re-sults of these simulations are shown in Fig. 2.

3. The inducible lac operon

3.1. Previous models

The lac operon is the classical bacterial exampleof an inducible system which encodes the genes for

the transport of external lactose into the cell and itsconversion to glucose and galactose. Many experimen-tal studies have examined the control of the genesinvolved in lactose metabolism in E. coli. One ofthe most recent mathematical modeling studies of thelac operon was carried out by Wong and cowork-ers [21]. Although many of the relevant biologicaldetails were taken into account in their model, theyfailed to treat the transcriptional and translational de-lays. Conversely, Mahaffy and Savev [22] modeledlac operon dynamics and included transcriptional andtranslational delays, but ignored the conversion of in-ternal lactose to allolactose by β-galactoside and thespontaneous production rate of mRNA together with


the individual degradation terms for the proteins. Inthis section we give a brief summary of the work on arecent model developed by Yildirim et al. [23] on thelac operon including much of the relevant biologicaldetail considered by Wong and coworkers [21] as wellas the transcriptional and translational delays consid-ered by Mahaffy and Savev [22]. A more completemathematical analysis of the model presented here canbe found in [23] and [24].

3.2. The mechanism

The lac operon consists of a promoter/operator re-gion and three larger structural genes lacZ, lacY, andlacA together with a preceding regulatory operon re-sponsible for producing a repressor (R) protein. Tounderstand the mechanism under this control networkeasily, it is helpful to refer to Fig. 3. In the absenceof glucose available for cellular metabolism, but inthe presence of external lactose (Le), lactose is trans-ported into the cell by a permease (P ). Intracellularlactose (L) is then broken down into allolactose (A)first and then glucose and galactose by the enzyme

β-galactosidase (B). The allolactose feeds back tobind with the lactose repressor and enable the tran-scription process to proceed. In the absence of allolac-tose (A) the repressor R binds to the operator regionand prevents the RNA polymerase from transcribingthe structural genes. However, if allolactose is present,a complex is formed between allolactose and the re-pressor that makes binding of the repressor to the op-erator region impossible. In that case, the RNA poly-merase bound to the promoter is able to initiate tran-scription of the structural genes to produce mRNA.When a mRNA chain, long enough for a ribosometo bind it, has been produced, the process of transla-tion is initiated. The lacZ gene encodes for the mRNAresponsible for the production of β-galactosidase (B)and transcription of the lacY gene produces mRNA ul-timately responsible for the production of a membranepermease (P ). The mRNA produced by transcriptionof the lacA gene encodes for the production of thio-galactoside transacetylase which is thought to not playa role in the regulation of the lac operon [25] and willnot be further considered here.

Fig. 3. Schematic representation of the lactose operon regulatory system. See the text for a detailed description.


There is another control mechanism known ascatabolite repression. The presence of glucose in theexternal medium inhibits the production of cAMP inthe bacterium. cAMP is necessary for transcriptioninitiation to take place efficiently. In the presentwork we ignore these regulatory mechanism sincewe compare with experimental in which there is noglucose in the culture medium and thus, cAMP isalways produced at a high rate.

3.3. A new mathematical model

The lac operon regulatory pathway is fairly com-plex: it involves three repressor binding sites (opera-tors), and two CAP binding sites. Moreover, one re-pressor can bind two different operators simultane-ously [26]. A detailed model of all these regulatoryelements is beyond the scope of this work. Nonethe-less, according to Yagil and Yagil [27], the fraction offree operators as a function of allolactose (A) can bemodeled via a Hill-type function of the form:

(11)f (A) = 1 + K1An

K + K1Anwhere n, K and K1 are parameters to be estimatedfrom experimental data. The dynamics of mRNAproduction are given by Eq. (12), which is derivedas follows. First, note that the production of mRNAfrom DNA via transcription is not an instantaneousprocess but requires a period of time τM before wehave a mRNA long enough to be bound by a ribosome.The rate of change of the ribosome binding sites’concentration (M) is a balance between a productionterm αMf and a loss term (γM + µ)M . The argumentof f in the production term is e−µτMAτM , whereAτM ≡ A(t − τM), to account for the time τM requiredto produce the ribosome binding site. The factor e−µτMaccounts for the growth dependent allolactose dilutionduring the transcriptional period. The loss term inEq. (12) is made up of an mRNA degradation term(γMM) and an effective loss due to dilution (µM).

(12)dM

dt= αM 1 + K1(e

−µτMAτM)n

K + K1(e−µτMAτM)n− (γM + µ)M

The dynamics of the concentration (B) of β-ga-lactosidase are described by Eq. (13) in which τBis the time required for β-galactosidase productionthrough mRNA translation. We assume that the rate

of production of B is proportional to the concentrationof M a time τB ago times a dilution factor due to cellgrowth during the translation process, (αB e−µτBMτB).The loss rate of B is given by (γB + µ)B .

(13)dB

dt= αB e−µτBMτB − (γB + µ)B

For the allolactose concentration (A) dynamics, thefirst term in Eq. (14) gives the β-galactosidase medi-ated gain in allolactose from the conversion of lactose.The second term accounts for allolactose loss via con-version to glucose and galactose (again mediated byβ-galactosidase). Since they are enzymatic reactions,both of these expressions are taken to be in Michaelis–Menten form. The last term takes into account thedegradation and dilution of allolactose.

(14)

dA

dt= αAB L

KL + L − βABA

KA + A − (γA + µ)A

The lactose concentration (L) dynamics are morecomplicated and given by Eq. (15). The first termin Eq. (15) accounts for the increase of intracellularlactose through the permease facilitated transport ofexternal lactose (Le). The second term deals withintracellular lactose loss to the extracellular fluidbecause of the reversible nature of the permeasemediated transport [28–31]. The third term accountsfor the conversion of lactose to allolactose as well asthe hydrolysis of lactose to glucose and galactose viaβ-galactosidase (B). The fourth term is the decrease ininternal lactose concentration due to degradation anddilution.dL

dt= αLP Le

KLe + Le− βL1P

L

KL1 + L(15)− αAB L

KL + L − (γL + µ)LThe permease concentration (P ) dynamics are

given by Eq. (16). The first term reflects the as-sumption that permease production is directly propor-tional to the mRNA concentration a time τP in thepast, where τP is the time it takes to translate a per-mease polypeptide. The exponential factor e−µτP ac-counts for dilution of mRNA concentration due to cellgrowth. The second term accounts for the degradationand dilution of permease.

(16)dP

dt= αP e−µτPMτP − (γP + µ)P


Table 2The estimated parameters for the lac operon model, taken from [23,24]

n 2 µmax 3.47 × 10−2 min−1γM 0.411 min−1 γB 8.33 × 10−4 min−1γA 0.52 min

−1 K 1000αM 9.97 × 10−4 mM min−1 τB 2.0 minαA 1.76 × 104 min−1 KL1 1.81 mMαB 1.66 × 10−2 min−1 KA 1.95 mMβA 2.15 × 104 min−1 τM 0.10 minKL 9.7 × 10−4 M γL 0.0 min−1γP 0.65 min−1 αL 2880 min−1αP 10.0 min−1 τP 0.83 minβL1 2.65 × 103 min−1 KLe 0.26 mMK1 2.52 × 10−2 (µM)−2

3.4. Analysis of the model and comparison withexperimental data

We have carried out an extensive search of theexisting literature for data that would allow us toestimate the model parameters in Eqs. (12)–(16) whichare summarized in Table 2, and the details of how wearrived at these parameter values can be found in [24].

The model as formulated in Eqs. (12)–(16) canhave one, two, or three steady states depending onthe values of the parameters (µ,Le). The details ofhow these steady states are determined are containedin [23] and [24]. The results of these considerations arepresented in Fig. 4. There we show in (Le, Ā) spacethe region where a non-negative steady state can exist.Note in particular that for a range of Le values theremay be coexisting two stable and one unstable steadystate values of the intracellular allolactose levels, Āand, consequently, of ( �M, �B, Ā, �L, �P ).

In other words, analytical and numerical stud-ies of the model predict that for physiologically re-alistic values of external lactose and the bacterialgrowth rate, a regime exists where there may be abistable steady state behavior, and that this corre-sponds to a cusp bifurcation in the model dynam-ics. This means that, as the external lactose con-centration Le is increased, the bacterial culture mustswitch from a low-internal-allolactose-concentrationuninduced steady state to a high-internal-allolactose-concentration induced steady state. For very low Levalues, only the uninduced state exists. Conversely, forvery high Le concentrations, only the induced state isavailable for the system. All these facts are in agree-

Fig. 4. The region in the (Le, Ā) space where a non-negative steadystate can exist as a function of external lactose levels Le for themodel when all parameters are held at the estimated values inTable 2 and when µ̄ = 2.26 × 10−2 min−1. The shaded area showsthe region where a steady state is not defined, while the solid lineis the locus of (Le, Ā) values satisfying the steady state. The insetbox shows that at large values of Le there is still a separation ofthe line for the steady state from the region where steady states arenot defined. Notice that for these values of the parameters there is arange of Le values for which there are three coexisting steady-statevalues of allolactose A. The point marked with a * corresponds tothe minimum lactose level for induction at this growth rate, andpredicts that induction should occur for a lactose concentration ofabout 62.0 µM.

ment with the experimentally-observed dynamic be-havior of E. coli cultures. On the other hand, the modelalso predicts a range of Le values in which both theuninduced and the induced stable steady states coex-ist. If this bistability behavior is real, an hysteresisphenomenon must be observed in inducing and unin-ducing the lac operon of a given bacterial culture bychanging the external lactose concentration. This pre-diction is qualitatively confirmed by the observationsin [32] and [33].

Although a full analysis of the stability propertiesof the model is not possible due to its complexity,we have found that the basic properties contained in areduced version of this model [23] (which consideredonly the dynamics of M , B , and A) are apparentlyretained in this much more complicated model as far aswe are able to ascertain analytically and numerically.

Given the parameter values determined in Table 2,we numerically solved the model equations to com-pare the predicted behavior with three distinct exper-imental data sets. The first data set is from [34]. In


Fig. 5. β-galactosidase activity vs. time when Le = 8.0×10−2 mM.The experimental datasets were taken from [34] for E. coli ML30(◦) and from [35] for E. coli 294 (�). The model simulation (solidline) was obtained using the parameters of Table 2 with a growthrate µ̄ = 2.26 × 10−2 min−1. For details see, [24].

there, the activity of β-galactosidase in an E. coli cul-ture (strain ML 30) was measured after the bacteriawere switched from a glucose-rich non-lactose to anon-glucose lactose-rich medium. The second data setis from [35]. In this paper, Pestka et al. studied spe-cific inhibition of translation of single mRNA mole-cules and gave data for the specific activity of β-ga-lactosidase versus time for E. coli 294 in the presenceof IPTG. These two data sets and the model simula-tion determined using Matlab’s dde23 [36] routineare shown in Fig. 5.

For this simulation, initial values for the variablewere chosen close to the steady state values whenLe = 8.0 × 10−2 mM. (With this value of Le there is asingle uninduced steady state.) To compare these twosets of experimental data with the model simulationpredictions, the data were scaled so the steady-statevalues of measured β-galactosidase activities andthose produced by the simulation were equal. As seenin Fig. 5, there is relatively good agreement betweenboth sets of experimental data and the model predictedtemporal approach of β-galactosidase activity to itssteady-state value.

In the experiments of Pestka et al. [35], the bacterialculture medium has no glucose all of the time. Thismeans that catabolite repression is not a factor in theregulation of the lac operon and thus, the comparisonwith our simulations is straightforward. This is not

the case with the experiments of Knorre [34] since,in such case, the bacterial were originally growing ina medium rich in glucose. Notice however that thetime course of both experimental data is quite similar.This does not necessarily justifies the comparisonof such experimental data with our model results,but highlights the necessity of including cataboliterepression in further mathematical models of the lacoperon in order to test its influence on the systemdynamic behavior.

As a third test of the model, a data set from Good-win [37] was used. In this paper, the dynamic behaviorof β-galactosidase was studied in chemostat culturesof E. coli synchronized with respect to cell divisionby periodic phosphate feeding at a period equal to thebacterial doubling time. Experimentally, oscillationsin β-galactosidase concentration were observed witha period equal to the feeding period. Since in these ex-periments, there is no glucose present in the environ-ment, we can attempt to simulate them with our model.

To mimic the periodic phosphate feeding in oursimulation, we assumed that the bacterial growth ratevaries as a function of time in manner given by

(17)µ(t) = µ̄ − α mod(t, T )Here, µ̄ is the maximal growth rate for the bacteria,T is the period of the feeding and α is a positiveparameter with dimension min−2. mod(t, T ) is afunction that gives the remainder on division of tby T . Selection of this type of function is motivated bythe observation that growth rates decrease as nutrientlevels fall and sharply increase after the addition ofnutrient.

The maximal value of µ(t) corresponds the timesthat phosphate was added. The minimal value of thisfunction represents the minimal amount of nutrientleft in the vessel. Assuming no nutrient is left �minutes before the addition of phosphate and letting� → 0, α can be estimated if the doubling time (T )is known: α � µ̄/T . In Fig. 6, we compare the dataon β-galactosidase activity from the forced culture asa function of time with the model predictions for afeeding period of T = 100 min which is the populationdoubling time. Again, there is a reasonable goodagreement between the model predicted curve and thisdataset.


Fig. 6. Oscillation in β-galactosidase activity in response to periodicphosphate feeding with period T = 100 min, which is the culturedoubling time, and µ̄ = 2.26 × 10−2 min−1. The experimentaldata (∗) together with the model simulation (solid line) using theparameters of Table 2 are presented. The experimental data are takenfrom [37]. In the numerical simulation, periodic phosphate feedingwas imitated by choosing a periodic function given by Eq. (17).The simulation was calculated by numerically solving the systemof delay differential equations given by Eqs. (12)–(16). For details,see [24].

4. Discussion and conclusions

Given the virtual flood of information that isemerging from the current initiatives in molecularbiology, we feel that mathematical modeling of thedynamics of molecular control systems will assume anessential role in the coming years. The reasons for thisare threefold:

1. Mathematical modeling offers a concise summaryof many biological facts and insights in an easilymanipulable language.

2. The consequences of analyzing these models canreveal potentially new dynamical behaviors thatcan be then searched for experimentally.

3. Failures of realistic mathematical models to ex-plain experimentally observed behavior oftenpoint to the existence of unknown biological de-tails and/or pathways, and thereby can act as aguide to experimentalists in their search for bio-logical reality.

These mathematical models will, however, only be ofuse to biologists if they are made biologically realis-

tic, and if every effort is made to identify the valuesof parameters appearing in the models from publisheddata or from experiments explicitly designed to mea-sure these parameters.

The paucity of dynamic data (as opposed to steady-state data) with which we were able to compare thepredictions of our models for the tryptophan and lac-tose operons highlights a serious problem in the in-teraction between experimentalists and mathematicalmodelers. Namely, we feel that there must be a closeand cooperative relationship between these two groupsif maximal use is to be made of the experimental datathat is being so laboriously collected and if the mathe-matical models constructed to explain these data.

The last area that we want to touch on is the na-ture of the mathematical models that are constructed.Those that we have discussed in this paper are all writ-ten with the explicit assumption that one is dealingwith large numbers cells (and hence of large numbersof molecules) so that the law of large numbers is oper-ative. However, the situation is quite different if one isinterested in the dynamics of small numbers (or single)prokaryotic or eukaryotic cells, for then the numbersof molecules are small. Adequate means to analyti-cally treat such problems do not exist in a satisfactoryform as of now [38], and one is often reduced to mostlynumerical studies [39–41]. The situation is analogousto examining the interactions between small numbersof interacting particles (where the laws of mechan-ics or electrodynamics hold), and then deriving fromthese formulations the behavior of large numbers ofidentical units as is done (not completely successfullyeven at this point) in statistical mechanics. We viewthis connection between the ‘micro’ and ‘macro’ lev-els as one of the major mathematical challenges facingthose interested in the understanding of gene controlnetworks.

Acknowledgements

We thank Dr Photini Pitsikas for her initial sugges-tion of these problems, Prof. Claire Cupples for heradvice in parameter estimation section and her generalcomments, and Mr. Daisuke Horike for many helpfuldiscussions. This work was supported by CGEPI–IPNand COFAA–IPN (México), TUBITAK (Turkey), MI-TACS (Canada), the Natural Sciences and Engineer-


ing Research Council (NSERC grant OGP-0036920,Canada), the ‘Alexander von Humboldt Stiftung’, the‘Fonds pour la formation de chercheurs et l’aide à larecherche’ (FCAR grant 98ER1057, Québec), and theLeverhulme Trust (UK).

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http://www.radford.edu/~thompson/webddes/

Modeling operon dynamics: the tryptophan and lactose operons as paradigmsIntroductionThe repressible tryptophan operonThe Bliss modelThe Sinha modelA more detailed model

The inducible lac operonPrevious modelsThe mechanismA new mathematical modelAnalysis of the model and comparison with experimental data

Discussion and conclusionsAcknowledgementsReferences

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