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Growth-rate-dependent dynamics of a bacterial genetic oscillator

Matteo Osella1, 2, ∗ and Marco Cosentino Lagomarsino1,2, 3

1Genomic Physics Group, UMR7238 CNRS “Microorganism Genomics”2University Pierre et Marie Curie, 15, rue de l’Ecole de Medecine, Paris, France

3Dipartimento di Fisica, University of Torino, Via P. Giuria 1, Torino, Italy(Dated: November 8, 2018)

Gene networks exhibiting oscillatory dynamics are widespread in biology. The minimal regulatorydesigns giving rise to oscillations have been implemented synthetically and studied by mathematicalmodeling. However, most of the available analyses generally neglect the coupling of regulatorycircuits with the cellular “chassis” in which the circuits are embedded. For example, the intracellularmacromolecular composition of fast-growing bacteria changes with growth rate. As a consequence,important parameters of gene expression, such as ribosome concentration or cell volume, are growth-rate dependent, ultimately coupling the dynamics of genetic circuits with cell physiology. This workaddresses the effects of growth rate on the dynamics of a paradigmatic example of genetic oscillator,the repressilator. Making use of empirical growth-rate dependences of parameters in bacteria, weshow that the repressilator dynamics can switch between oscillations and convergence to a fixed pointdepending on the cellular state of growth, and thus on the nutrients it is fed. The physical supportof the circuit (type of plasmid or gene positions on the chromosome) also plays an important role indetermining the oscillation stability and the growth-rate dependence of period and amplitude. Thisanalysis has potential application in the field of synthetic biology, and suggests that the couplingbetween endogenous genetic oscillators and cell physiology can have substantial consequences fortheir functionality.

PACS numbers: 87.17.Aa,87.18.Vf,87.16.Yc,82.40.Bj

I. INTRODUCTION

Oscillatory behavior is widespread and fundamentallyimportant in biological systems, from circadian clocks tocell-cycle control [1]. At the level of a single cell, oscil-lations can be sustained by regulatory circuits, the ba-sic components of which are nucleic acids, proteins, andbiochemical interactions [2, 3]. However, naturally oc-curring regulatory systems are complex. Thus, our pathto rationalize their behavior has to pass through simpli-fications and basic underlying principles [4]. The studyof minimal circuits, by mathematical modeling and ex-perimental synthetic engineering, allows a quantitativeapproach to the study of gene expression [5–8].

This simplifying approach is not only useful for study-ing the extant regulatory circuits. Understanding the ba-sic principles and dynamical properties of regulatory net-works makes it possible to design and produce syntheticcircuits that can perform specific functions in a pre-dictable manner [9–12], with potentially relevant futurebiotechnological and biomedical applications [11, 13]. Aparadigmatic example of a synthetic genetic oscillator,and one of the earliest in vivo realizations [14], is the so-called “repressilator”, a three-gene cyclic circuit whereeach gene protein-product represses the synthesis of itssuccessor (Fig. 1A).

From a physics standpoint it is desirable to build min-imal, but effective and testable models, with qualita-

∗correspondence to: matteo.osella@upmc.fr

tive and possibly quantitative predictive power for ex-periments. To this end, dealing with controlled, well-characterized, and isolated systems is necessary. Unfor-tunately, every genetic circuit, endogenous or syntheti-cally implemented in a living cell, cannot be truly con-sidered isolated from cellular processes. These processesare strongly affected by the physiological state of thecell [15], and thus by the environmental conditions andtype/availability of nutrients. For example, the macro-molecular composition of bacterial cells in “steady ex-ponential” growth (i.e. constantly dividing at the samerate [50]) changes substantially with growth rate (the rateof cell proliferation) [16–18]. Importantly, in this case,growth rate appears to encapsulate most of the physio-logical changes. In other words, cells growing on differ-ent nutrients but at similar growth rates are in manyways equivalent, considering important global cell pa-rameters such as concentrations of ribosomes and RNApolymerases (the molecular machines effecting transla-tion and transcription respectively). Such growth-ratedependent parameters affect gene expression, coupling itsdynamics with the cell state, with relevant consequencesfor genetic circuit functioning [15, 19, 20].The emergent quantitative “laws” of bacterial physiol-

ogy, linking cell composition and growth rate, are remi-niscent of those of thermodynamics, and were the subjectof recent advances using physical modeling [17, 18]. Pre-viously, they were in part captured by early studies inthe 1950’s and 60’s [16]. However, to date, they are notcompletely characterized. For E. coli, the best studiedbacterial species, the growth rate dependences of variouscellular parameters were evaluated phenomenologicallyin a study by Klumpp and coworkers [15], compiling re-

2

sults from multiple experiments. Leveraging on these em-pirical data, they analyzed the growth-rate dependenceof the steady state protein concentration of a constitu-tively expressed (i.e. unregulated) gene and few othersimple genetic circuits, such as the self-regulator and thetoggle switch. Their modeling strategy will be brieflyreviewed in section IIA. Beyond steady state, knowingthe empirical growth-rate dependence of gene expressionparameters allows in principle to revisit the dynamicalproperties of genetic circuits, introducing the physiolog-ical cell state as a new player.

This work addresses the growth-rate dependence of abiological oscillator dynamics, and considers the effectof cell state on a repressilator, integrated on either aplasmid or the E. coli chromosome. From the physi-cal modeling viewpoint, this entails understanding themodels and phenomenological laws that relate the cir-cuit parameters to the cell physiological state. Incorpo-rating these features in otherwise fairly standard mod-els leads to predictions that are experimentally relevant.After reviewing the conditions for oscillations and therepressilator dynamics at fixed-growth rate (and extend-ing some known results), we will show how the physiol-ogy of the cell can alter qualitatively and quantitativelythe dynamical properties of the system. The main resultis that the conditions for stable oscillations, as well astheir amplitude and period are growth-rate dependent.This implies that a genetic oscillator can display distinctdynamical behaviors in different environments and nu-trient conditions. Additionally, the circuit support, e.g.type of plasmid or chromosomal position of the genes,also contributes to its dynamics, through gene dosage.Specifically, the range of growth rates in which oscilla-tions are observable will vary with support, in a mannerthat could be exploited both biologically and technolog-ically. Our predictions, although based on a simplifiedmodel, are experimentally testable in a straightforwardway on synthetically realized repressillator circuits, andmore in general set a framework for the more complexcase of endogenous oscillators in bacteria, prominentlythe cell cycle and circadian clocks [21].

The paper is organized as follows. Section II reviewsthe modeling strategy adopted to include the growth rateas a variable in gene expression (Subsection IIA) andexplains how this strategy can be applied to a mathe-matical description of the repressilator (Subsection II B).Appendix A reviews the Cooper-Helmstetter model [22],an empirical model of DNA replication in fast-growingbacteria that is an essential part of our approach, and jus-tifies in more detail some of the model assumptions pre-sented in Section II. Section III contains the quantitativeanalysis of the circuit, and in particular of the role playedby the cellular state of growth on the dynamics. Morespecifically, Subsection III A analyzes the symmetric re-pressilator, showing the possibility of a growth-mediatedswitch between oscillations and convergence to a stablefixed point. Subsection III B focuses on the asymmetriccase, with particular emphasis on the role of gene chro-

mosomal position on the circuit behavior. Finally, thelast section discusses the implications of the results fromboth a biological and a physical modeling standpoint.

II. MODEL

A. Growth-rate dependence of gene expression

This section reviews the approach of Klumpp andcoworkers [15], and the necessary assumptions in orderto extend it to the analysis of circuit dynamics. Thismodeling strategy constitutes the basis for our descrip-tion of the repressilator, introduced in section II B.In absence of regulation, the dynamics of messenger

RNA (mRNA) levels and protein concentrations, denotedwith m and p respectively, is described by two equations

m =g αm

V− βmm

p = αpm− βpp , (1)

where αm and αp are the transcription and translationrates respectively, βm and βp the degradation rates ofmRNAs and proteins, and V the cell volume. The pa-rameter g is the gene copy number. If the gene support isa plasmid, g represents the (mean) plasmid copy number.For genes integrated on the chromosome, the copy num-ber can vary because of DNA replication, especially atfast growth rates, when multiple copies of the genome arereplicated at the same time. This phenomenon wherebygene dosage is modified by DNA replication is describedby the Cooper-Helmstetter model [22], and reviewed inAppendix A1. Note that for fast E. coli growth, g in-creases with decreasing distance ℓ from the replicationorigin (illustrated in Fig. 1B), since the cell engages over-lapping rounds of DNA replication in order to allow fastgrowth [23].In bacteria, proteins are typically stable, with a life-

time longer than the cell cycle, while mRNAs have alifetime of just a few minutes [24]. Therefore, the lossof protein is mainly due to dilution through growthand cell division, so that an effective degradation rateβp = µ ln2 (where µ is the growth rate) can be safelyused in most cases. On the other hand, the fast time-scale of mRNA dynamics allows a quasi-equilibrium ap-proximation. Thus, Eq. 1 can be reduced to

p =g αmαp

βmV− βpp. (2)

Rescaling time with the dilution rate βp, all the growth-rate dependence can be factorized in a single term F (µ),

p =g αmαp

βmβpV− p = p∗F (µ)− p . (3)

Normalizing the growth-rate dependence F (µ) suchthat F (µ) = 1 for µ = 1 db/hr (i.e. doublings per

3

hour), the parameter p∗ represents the steady-state con-centration of the constitutive (unregulated) gene at µ =1 db/hr. Thus, it is a measure of the degree of basal ex-pression. Klumpp and coworkers [15], refer to this quan-tity as the “promoter strength”. However, this terminol-ogy might be slightly misleading, as the term actually in-cludes non-transcriptional parameters, such as the trans-lation efficiency or the gene copy number at µ = 1 db/hr.Thus, we will refer to it as “basal expression” level in thefollowing.

In principle, all the cellular parameters in F (µ) maydisplay a growth-rate dependence. The volume V isknown experimentally to change with growth rate [16,25], with faster-growing bacteria being larger thanslower-growing ones. The protein degradation rate βp isa direct consequence of dilution due to cell growth and di-vision, at least for stable proteins, and therefore, as previ-ously discussed, it is a linear function of the growth rate.By contrast, the empirical growth-rate independence ofthe mRNA degradation rate βm has a less obvious in-terpretation [15]. In order to sustain fast growth and alarge cell mass, the cellular abundance of the transcrip-tional and translational machinery is expected to increaseat fast growth, and indeed this is what can be observedexperimentally [16]. However, the rates of transcriptionand translation (αm and αp) of a gene only reflect theavailability of “free” (active but not already engaged inany process) RNA polymerases and ribosomes. This frac-tion can be quite hard to estimate [26], requiring severalempirical measurements, and is in general not merelyproportional to the total abundance. In fact, while thecellular ribosome content increases strongly with growthrate, the translation rate is apparently constant in differ-ent nutrient conditions [15]. The gene dosage g is prob-ably the best-characterized parameter, since it is cap-tured by the Cooper-Helmstetter model [22]. Its increasewith growth rate is a consequence of the coupling be-tween DNA replication and growth. To sum up, whileit is possible to try to rationalize at least some of thegrowth-rate dependences of the basic parameters, thereis in general no a priori quantitative expectation (andsometimes not even a basic intuition) for them, and thusfor F (µ). Hence, F (µ) can be fully defined only throughthe empirical knowledge of the growth-rate dependencesof the different parameters.

For E. coli, the numerical values of these parameterswere collected for five growth rates µ between 0.6 db/hrand 2.5 db/hr [15]. These values are reported in thesupplementary material of Klumpp et al. (Cell 2009, ref-erence [15]), more specifically in Table S1. The growth-rate dependences of the various parameters are plottedin Figure 1 of the same paper. As a combination of theseparameters, F (µ) decreases in a weakly nonlinear fash-ion in the available range of growth rates (interpolationwill be used in the following when needed). The growth-rate dependence of F (µ), and thus of the concentrationof the protein product of a constitutive gene, is reportedin Figure 2D and in Figure 3 of the same reference [15].

terminus of

replication (Ter) ℓ=1

origin of

replication (Ori) ℓ=0 gene

positio

n ℓ

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tance

from

Ori

A

B

FIG. 1: (Color online) A. Scheme of the repressilator.

Each node represents the expression level of one gene, linksrepresent repressive regulatory interactions, as described byEq. (4). B. Scheme of the E. coli chromosome, illus-trating the coordinate ℓ, used for the positions where a genecan be inserted.

The empirical growth-rate dependences that defineF (µ) are based on experimental measurements of theaverage cellular properties in a growing cell population.Thus, they are in principle also dependent on the age dis-tribution (where “age” stands for stage of the cell cycle)of the population. However, the differences between aver-ages calculated over a cell cycle and over a cell populationwith the age distribution determined by the exponentialgrowth are not quantitatively significant, as discussed indetail in Appendix A. Therefore, the available empiricalmeasurements of all the key parameters, which are basedon population averages, will be used throughout this pa-per as an approximation of their cell-cycle averages.

Moreover, we will not consider explicitly the cell-cycledynamics of cellular parameters, which, for sufficientlylong time scales, can be averaged out using for exampleeffective gene dosage or effective protein dilution rate. Asa consequence, Eq. 3 can accurately describe the dynam-ics of protein concentration on time scales longer thanthe generation time [27], as it will always be the case inthe following.

4

B. The repressilator

The repressilator is a genetic network consisting ofthree genes, each of which encodes for a “transcriptionfactor” (i.e. a protein able to control the transcriptionlevel of one or a set of target genes by binding to theirupstream genomic region) that represses the expressionof its target in a cycle of regulations (Fig. 1A). The effectof this negative regulation can be modeled phenomeno-logically as a multiplicative factor rescaling the targetproduction rate (the positive term on the right-hand sideof Eq. 3) with a non-linear function of the repressor con-centration (called “Hill function”) [5, 6]:

R(x/k) =1

1 + (x/k)n, (4)

where the Hill coefficient n defines the degree of co-operativity (determining the steepness of R), while thedissociation constant k specifies the repressor concentra-tion at which the production rate is half of its consti-tutive value. With this mathematical representation oftranscriptional regulation, the repressilator circuit canbe described by three equations, one for each gene, basedon the gene expression model discussed in the previoussection:

x1 = x∗

1F1(µ)R(x2/k1)− x1

x2 = x∗

2F2(µ)R(x3/k2)− x2

x3 = x∗

3F3(µ)R(x1/k3)− x3 , (5)

where the terms x∗

i are the basal expression levels. Theyrepresent the steady-state concentration of protein i inabsence of repression at µ = 1 db/hr. Therefore, the“basal expression level”, as defined here for a negativelyregulated gene, is the maximal expression level that canbe achieved in a fixed growth condition (µ = 1 db/hr) inabsence of regulation [51].

The growth rate functions Fi(µ) can be gene-specificdue to the dependence of gene dosage gi on the chromo-somal gene position.

Assuming similar dissociation constants for the threepromoters k1 ≃ k2 ≃ k3 = k, the protein concentrationscan be rescaled with k

x1 = x∗

1F1(µ)R(x2)− x1

x2 = x∗

2F2(µ)R(x3)− x2

x3 = x∗

3F3(µ)R(x1)− x3. (6)

With these notations, the basal expression levels x∗

i

have dimensionless units, as they are protein concentra-tions rescaled with the dissociation constant k.

III. RESULTS

A. Growth rate affects qualitatively and

quantitatively the dynamics of a symmetric

repressilator

We start the analysis from the simplified case of asymmetric repressilator, i.e. the three genes have ap-proximately the same production/degradation rates formRNA and proteins, as well as the same growth-ratedependence of parameters (this is attained for examplewhen all genes are integrated on a single plasmid or at asimilar distance from the origin of replication on a chro-mosome). This is the most studied case, and was ap-proached with different mathematical descriptions [4, 28–30]. The symmetric approximation was originally pro-posed to explain the behavior of the synthetic realizationin vivo [14].In the symmetric case, the functions encoding the

growth-rate dependence F (µ) and the basal expressionlevels x∗ are exactly the same for each of the three genes,simplifying Eq. 6 to

xi = x∗F (µ)R(xi+1)− xi , (7)

where i ∈ [1, 3] and x4 ≡ x1.

1. At fixed growth rate, the dynamics is determined bybasal expression and cooperativity

Before addressing the effects of the growth-rate depen-dence of parameters, we characterize the circuit dynamicsat fixed growth conditions, for simplicity at µ = 1 db/hrwhere F (µ) = 1. In general, the symmetric repressilatorcan display stable oscillations, arising through a Hopfbifurcation [28]. More specifically, Eqs. 7 have an oscil-latory solution if the condition for cooperativity n > 2is satisfied, as can be shown in a straightforward wayconsidering the symmetry of the system [31].For a given steepness of the repression function satisfy-

ing the condition n > 2, the stability of the limit cycle issolely determined by the basal expression x∗. The valuesof x∗ that ensure stability of the oscillatory state can becalculated using linear stability analysis and the resultingstability diagram is shown in Fig. 2. Essentially, an in-crease of either cooperativity or basal expression can helpthe stabilization of oscillations. This result is common toall the different repressilator descriptions proposed in theliterature [4].Furthermore, the two parameters n and x∗ determine

the amplitude and period of oscillation (Fig. 3). The ap-proximately linear and logarithmic dependences of ampli-tude and period respectively that emerge from numericalintegration of Eqs. 7 can be rationalized by the follow-ing rough but conceptually simple argument. Each genein the repressilator tends to oscillate between two statescorresponding to its maximally repressed state and its

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FIG. 2: (Color online) Stability diagram of the symmetric repressilator. The left panel shows the parameter space givingrise to oscillations. The basal expression x∗ is dimensionless, measured as protein concentration at µ = 1 db/hr in absenceof repression in units of the dissociation constant k. High repression cooperativity and high basal expression levels (“strongpromoters”) can ensure stable oscillation. The right panel plots are illustrative examples of the dynamics in correspondencewith two sets of parameter values ({n = 4, x∗ = 1.8} and {n = 4, x∗ = 5}), showing the time evolution of the three proteinconcentrations x1, x2, x3 (in units of k). The red continuous curve describes the dynamics of x1, the blue dashed curve x2, andthe orange dot-dashed curve x3.

maximally activated state. In a simplified situation of astep repression function (n ≫ 1) that switches the tar-get gene on and off after its equilibration, the proteinconcentration would go from xi ≃ 0 (fully repressed) toxi ≃ x∗ (fully activated), thus leading to a linear depen-dence on basal expression of the oscillation amplitude.On the other hand, the oscillation period depends on thetimescales of gene activation and deactivation. For ex-ample, the time τ required to go from the fully activatedstate xi(t) ≃ x∗ to xi(t) ≃ 1, when, assuming a step-likerepression function, it releases the repression of its tar-get, is given by x∗e−τ = 1 → τ = ln(x∗). This expressionsuggests that the basal expression contributes logarithmi-cally to the deactivation timescale, which is compatiblewith the dependence of the period on x∗ measured bynumerical integration shown in Fig. 3B.

2. Increasing the growth rate can destabilize oscillations

Let us now consider a specific experimental realizationof the repressillator, which uses a set of genetic compo-nents with some intrinsic parameters. Given the basalexpression level and the cooperativity of repression, de-fined by the specific properties of the actual genes andpromoters implementing the repressilator, the growth-rate dependence of parameters defines a vertical path inthe stability diagram of Fig. 2, since the “effective” basalexpression F (µ) x∗ decreases nonlinearly with growthrate. This path can cross the border of stability of thelimit cycle, defining a maximum growth rate at whichstable oscillations can be sustained. More precisely, thebasal expression level x∗ defines the position of the pathat µ = 1 db/hr, while the function F (µ) is related toits length. In fact, given the experimentally accessiblegrowth rates, F (µ) identifies the span of effective basal

expressions that can be explored changing the growthrate, and thus the upper and lower bound of the path.Therefore, both factors contribute to establish the rangeof growth rates in which oscillations are expected exper-imentally.

Remarkably, the circuit can show qualitatively differ-ent dynamics depending on growth rate, and hence onnutrient conditions. Fig. 4 shows this for the case of co-operativity n = 3. Sustained oscillations can be observedat slow growth, while in fast-growth conditions the dy-namics can converge to a stable fixed point. The parame-ter range where oscillations are stable depends on wherethe repressilator is integrated through the gene dosagefactor g appearing in F (µ) (see Eq. 3).

If the repressilator is integrated on a plasmid, as theoriginal in vivo experimental realization [14], the genedosage g is simply given by the plasmid copy num-ber, which has a plasmid-specific growth-rate dependencethat can be quite strong [15, 32]. Fig. 4 compares theparameter regions corresponding to convergence to sta-ble oscillation or to a fixed point for a repressilator in-tegrated on the two plasmids R1 and pBR322 for whichthe copy number has been measured in different growthmedia [15].

The dosage of a chromosomal gene, instead, is deter-mined by its genomic position as set by the Cooper-Helmstetter model (see Appendix A and Fig. 1B). There-fore, for a repressilator integrated on the chromosome,the normalized circuit distance ℓ from the replication ori-gin (Ori) defines the growth-rate dependence of the dy-namics (where ℓ = 0 represents Ori and ℓ = 1 the repli-cation terminus, Ter, Fig 1B). It should be noticed thatthe three genes composing the repressilator are assumedhere to be inserted approximately in a single chromoso-mal location (or equivalently in different replichores, theoppositely replicated chromosome halves, but at the same

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basal expression x* [units of k]

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FIG. 3: (Color online) Amplitude and period of oscilla-

tion of a symmetric repressilator. The plots show oscil-lation amplitude (A) and period (B) as a function of the basalexpression level for three different values of cooperativity n.Both quantities are plotted in rescaled (dimensionless) units.The amplitude (concentration) is rescaled by the dissociationconstant k, while the period has units of protein degradationrate βp. For each value of n, Eqs. 7 were integrated numer-ically with values of x∗ in the range 0.1 − 40 and step-size0.1. For each numerical solution, the oscillation amplitudeand period were evaluated after convergence to a stable limitcycle.

distance ℓ from Ori, see Fig. 1B). The effects of the imbal-ance in gene dosage generated by different gene locationson the chromosome are explored in section III B 2.

3. Period and amplitude of oscillation are growth ratedependent

Fig. 4 shows that increasing the basal expression x∗

(which we recall also includes parameters not based ontranscription) and the steepness of the repression func-tion n can make the limit cycle solution stable in a widerange of conditions. However, even if a repressilator isdesigned to exhibit oscillations in the experimentally ac-cessible conditions, the growth rate is still expected toinfluence the oscillation amplitude and period in a mea-surable way (i.e. by a factor that can exceed five, Fig. 5).Indeed, the effective basal expression F (µ) x∗ decreaseswith increasing growth rate, because of the functionalform of F (µ), leading to a reduced amplitude and periodof oscillation. Integrating Eqs. 7 numerically for parame-ter values corresponding to different growth rates allows

plasm

idR1

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Oscillations

ba

sa

l e

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n x

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ℓ=0.5

ℓ=0

FIG. 4: (Color online) The repressilator has different

dynamics in different growth conditions. At fixed re-pression cooperativity (n = 3 in this plot), the repressilatorcan show stable oscillation or convergence to a steady statedepending on the cell growth rate µ and the basal expres-sion x∗ (in units of k). The circuit physical support (plasmid,chromosome) defines the range of growth rates at which alimit cycle can be observed. The cases of circuit integrationon plasmids (R1 and pBR322) and on chromosome near theorigin of replication (ℓ = 0) and between the origin and theterminus (ℓ = 0.5) are represented in the plot.

to predict the oscillation period and amplitude for a chro-mosomally integrated repressilator or for a plasmid im-plementation, if the scaling of the plasmid copy numberwith growth rate is known, The presence of nonlinearitiesin the system makes the dependence on growth rate ofthese two variables nontrivial, as represented in Fig. 5.

B. The effect of intrinsic and position-induced

asymmetry on the repressilator dynamics

Differences in intrinsic properties of the genes compos-ing the circuit, such as affinity for RNA polymerase orribosome binding, or gene dosage imbalances due to thespecific gene location on the chromosome, lead to unequalparameter values in the equations for the dynamics of xi.This situation is generally referred to as “asymmetric”.A certain degree of asymmetry is expected for genericsynthetic realizations of the repressilator. However, withthe exception of a few studies [29, 33], its consequences onthe dynamics have not been fully characterized theoreti-cally [4]. In the modeling framework adopted here, intrin-sic gene properties are summarized by the basal expres-sion level. Additionally, using the Cooper-Helmstettermodel (see Appendix A) it is possible to account for theposition-dependent scaling of gene dosage with growthrate.The two possible contributions to repressilator asym-

metry, intrinsic gene properties and gene dosage, andtheir effects on the dynamics can be analyzed separately.We first address the dynamics at fixed growth rate of a re-pressilator composed of genes with different basal expres-

7

FIG. 5: (Color online) The oscillation amplitude and pe-

riod of a symmetric repressilator vary significantly

with growth rate. The plots show the scaling with growthrate of oscillation amplitude (concentration in units of k) andperiod (time in units of protein degradation rate) for a “strongpromoter” (basal expression x∗ = 70 in k units) and cooper-ativity n = 3. The three curves correspond to integrationof the repressilator on plasmid R1, plasmid pBR322, and tochromosomal integration of the three genes near the origin ofreplication (ℓ = 0). A repressilator located near the replica-tion terminus (ℓ = 1, not shown) follows a growth-rate depen-dence of amplitude and period similar to the case of plasmidpBR322.

sion levels, and subsequently explore the consequences ofposition-induced asymmetry at different growth rates fora repressilator made of genes with equal intrinsic prop-erties, but different chromosomal location.

1. Effects of asymmetry at fixed growth rate

We consider the simplified situation in which only oneof the genes of the repressilator differs from the othersin its basal expression by a factor A, but the three genesshare the same growth rate dependence F (µ). Thus, thesingle additional parameter A introduced in the modelmeasures the level of circuit asymmetry. The system ofequations describing the dynamics of a repressilator de-signed this way is

x1 = x∗F (µ)R(x2)− x1

x2 = x∗F (µ)R(x3)− x2

x3 = A x∗F (µ)R(x1)− x3 . (8)

At fixed growth rate (for simplicity we take the caseof growth rate µ = 1db/hr, where F (µ) = 1), linear sta-bility analysis can be applied to study the fixed pointstability. As in the symmetric case described above, aHopf bifurcation stands between the system convergenceto a stable fixed point and the oscillatory solution. Foreach repression cooperativity level n, the stability dia-gram can be drawn as a function of the basal expressionx∗ and the asymmetry level A, as shown in Fig. 6 forn = 3. This diagram essentially shows that a high degreeof asymmetry destabilizes the oscillations. Therefore, ifthe goal is to engineer a stable oscillator, a roughly sym-metric design is generally preferable.However, the minimum of the boundary curve be-

tween the two asymptotic dynamical behaviors (contin-uous purple curve in Fig. 6) does not correspond to thesymmetric case (log10(A) = 0). This result indicates thata symmetric system showing damped oscillations (as itis generally the case for parameter values just below theboundary in Fig. 6) can be pushed toward a stable oscil-lation regime by slightly increasing the basal expressionlevel of just one gene, if the resulting asymmetry is nottoo strong.Moreover, the presence of a gene with a different basal

expression breaks the symmetry in the dynamics of pro-tein concentrations, making the oscillation amplitude (orthe stable fixed point) gene-specific, as shown in Fig. 6for two parameter sets. Thus, a possible test of the effec-tive symmetry of experimental repressilator realizationswould entail measuring the oscillation amplitude of twofluorescently-tagged protein products of genes in the cir-cuit. More specifically, the level of asymmetry, intro-duced by the presence of a gene with different intrinsicproperties, affects the oscillation period and amplitude ina predictable way and with a gene-specific signature onthe oscillation amplitude (Fig. 7). Therefore, the globaldynamical properties of the repressilator can be tunedsimply by changing the parameters relative to a singlegene.

2. The chromosomal position of genes affects the circuitdynamical properties at varying growth rates

To isolate the effect of genomic position of genes onthe dynamics of a chromosomally integrated repressila-tor, we consider the case of a circuit that is symmetricin terms of intrinsic gene properties, but where a geneis placed at a varying distance from the replication ori-gin. This situation is illustrated in Fig. 8A, where thegenes with protein product concentrations x1 and x2 areplaced in contiguous positions, thus at approximately thesame distance ℓ12 from Ori, while the gene correspondingto x3 is in position ℓ3. In modeling terms, the equiva-lence of intrinsic gene properties corresponds to the as-sumption of identical effective basal expression xiFi(µ)at extremely slow growth, where the growth-dependenceof gene dosage gi for each gene i is irrelevant, since gi ≃ 1

8

-1 0 0.5 1 1.5 20

5

10

15

20

ba

sal e

xpre

ssio

n x

*

[units o

f k]

asymmetry level log10(A)

Stable fixed point

Oscillations

-0.5

150 160 170 180 190 2000

5

10

15

20

10 20 30 40 50 60 70 800

1

2

3

4

time [units of βp]

time [units of βp]

x1, x

2, x

3

[units o

f k]

x1, x

2, x

3

[units o

f k]

FIG. 6: (Color online) Stability diagram of the asymmetric repressilator The parameter space is divided in the regionscorresponding to convergence of the dynamics to a limit cycle and to a stable fixed point. The two parameters considered arethe basal expression x∗ (in units of k) and the level of asymmetry A (the factor by which the basal expression of one of thegenes differs from the two others). The cooperativity is fixed to n = 3 but the qualitative shape of the phase diagram holdsfor cooperativity values n > 2. While a high degree of asymmetry reduces the robustness of the oscillatory state, a slightasymmetry can stabilize the damped oscillations showed by the symmetric system. The right panel shows an example of thedynamics of the three protein concentrations for two sets of parameter values ({x∗ = 12, A = 10} and {x∗ = 4.5, A = 10}).The red continuous curve describes the dynamics of x1, the blue dashed curve refers to x2, and the orange dot-dashed to x3.

1

pe

riod o

f oscill

ation

[units o

f β

p]

-1 0 2 34

6

8

10

12

asymmetry level log10(A) asymmetry level log10(A)

asymmetry level log10(A) asymmetry level log10(A)

1-1 0 2 3

1-1 0 2 3 1-1 0 2 3

x*=100x*=75

x*=50

x*=50

x*=100

x*=75

x*=75x*=75

x*=50

x*=100x*=100

x*=50

am

plit

ud

e o

f x2

oscill

ation

[un

its o

f k]

am

plit

ud

e o

f x1

oscill

ation

[un

its o

f k]

am

plit

ud

e o

f x3

oscill

ation

[un

its o

f k]

80

0

20

40

60

80

0

20

40

60

1000

800

600

400

200

0

FIG. 7: (Color online) Influence of asymmetry on pe-

riod and amplitude of oscillation. The figure shows thenonlinear effects of the asymmetry induced by the presenceof gene x3 with a basal expression level that differs by a fac-tor A from the other genes’ basal expression x∗. The period(upper left plot) and the gene specific oscillation amplitudes(other plots) are shown as a function of the asymmetry levellog10(A) for three basal expression values that ensure oscilla-tions in the whole range of asymmetry explored. The curvesare obtained by measuring period and amplitude of numericalsolutions of Eq. 8 for values of log10(A) spaced by 0.01 in therange {−1, 3}.

for every genomic position. In other words, the followingrelation is satisfied: x∗ F (µ ≃ 0) = x∗

3 F3(µ ≃ 0), wherex∗ and F (µ) are the basal expression level and growth-dependence function of genes 1 and 2, and F3(µ) differsfrom F (µ) only because of the position-dependent scaling

of gene dosage with growth rate.It is straightforward to verify that the dynamics of a

repressilator satisfying the above conditions is describedby Eq. 8 with A = 2µC∆ℓ, where ∆ℓ = ℓ12 − ℓ3 and Cis the time required for DNA replication (C ≃ 40 minfor fast growth [34]). The factor A is simply the ra-tio between the gene copy numbers, g3/g1(2) as givenby the Cooper- Helmstetter model (Appendix A), andencodes the growth-rate dependent level of asymmetryinduced by gene position in an otherwise symmetricalcircuit. Note that the sign of the relative position ∆ℓhas relevance. For example, the configuration with twogenes near the replication origin and the third near theterminus (∆ℓ = −1) leads to a quantitatively differentdynamics with respect to the opposite configuration with∆ℓ = 1. The repressilator dynamics can be analyzed fordifferent values of ∆ℓ to explore the effects of gene posi-tion.Fig. 8B shows a stability diagram obtained moving one

gene along the chromosome while the other two genes arenear the replication origin (ℓ12 = 0), for different levels ofbasal expression x∗. Analogously, the stability diagramin Fig. 8C is obtained keeping ℓ12 = 0.5 and varying thethird gene position ℓ3. Interestingly, this analysis showsthat at fixed growth rate the repressilator can converge toa limit cycle or to oscillations depending on where genesare integreted on the chromosome, highlighting the im-portance of including the chromosomal gene coordinatesas variables in genetic circuit models. The positional ef-fect is even more evident if two gene positions are varied(data not shown).As discussed in the previous section, the oscillation

stability can be reinforced by increasing the basal ex-pression of one gene. A simple way to increase the basalexpression of one gene for a chromosomally integrated

9

Ori

ℓ12=

0.5

ℓ3

Ori

X1

ℓ

ℓ3

Δℓ=ℓ12 - ℓ3

X2

X3

Ter

E

Ter

ℓ3

ℓ12=0

Ter

relative position Δℓ

gro

wth

ra

te μ

[d

b/h

r]0.5

1.0

1.5

2.0

2.5

0.2 0.4

x*=6

x*=4

x*=8

x*=10

-0.4 -0.2 0

-1 -0.8

no

rma

lize

do

scill

atio

n a

mp

litu

de

μ=0.6 db/hr

X1

X2

X3

relative position Δℓ

relative position Δℓ

μ=2.5 db/hr X3

X1

X2

no

rma

lize

do

scill

atio

n

am

plit

ud

e

no

rma

lize

do

scill

atio

n a

mp

litu

de

gro

wth

ra

te μ

[d

b/h

r]

relative position Δℓ 0

0.5

1.0

1.5

2.0

2.5

-0.6 -0.4 -0.2

x*=4

x*=6

x*=8

x*=10

no

rma

lize

do

scill

atio

n

am

plit

ud

e

A B D

0.20.40.60.8

1.21.0

-1 -0.8-0.6 -0.4-0.2 0relative position Δℓ

μ=0.6 db/hr

X1

X2

X3

0.60.81.01.2

1.61.4

0.60.81.01.2

1.61.4

0.20.40.60.8

1.21.0

-1 -0.8-0.6 -0.4-0.2 0relative position Δℓ

μ=2.5 db/hrX3

X1X2

-0.4 -0.2 0 0.2 0.4

-0.4 -0.2 0 0.2 0.4

C

Oscillations

Stable fixed

point

Stable fixed

point

Oscillations

FIG. 8: (Color online) Effect of gene-dosage induced asymmetry on oscillation stability (A) Scheme of the repressilatorchromosomal configuration analyzed here. Two genes are placed at approximately the same distance ℓ12 from the origin ofreplication (possibly on opposite arms), while the third is in a different position ℓ3. This genomic configuration induces a growth-rate dependent copy-number imbalance between genes, as a consequence of DNA replication. The difference ∆ℓ = ℓ12 − ℓ3measures the effective asymmetry level (∆ℓ = 0 corresponds to the symmetric case) since the ratio between the gene dosages(the factor A in Eq. 8) is 2µC∆ℓ, where C is the DNA replication time. (B,C) The region of oscillation stability is plotted as afunction of growth rate µ and relative position ∆ℓ for different values of the basal expression x∗. The genomic configurationsanalyzed, and thus the values of ∆ℓ, are obtained moving one gene along the chromosome while keeping the others near thereplication origin (ℓ12 = 0) in plot B, or in a mid position (ℓ12 = 0.5) in plot C (see the corresponding schemes on the left).The plots show how gene positions influence the range of growth rates in which stable oscillations take place. (D,E) Effect ofthe relative position ∆ℓ on the oscillation amplitude, normalized with its value in the symmetric case, of the three proteins isplotted for two different growth rates (the basal expression is x∗ = 75 in units of k). The sets of positions considered in D andE are those shown in plot B and C respectively. The change in oscillation amplitude that can be obtained by placing the samegenes in different configurations depends on the growth conditions.

circuit is moving its position toward Ori, since this in-creases its average gene copy number. Indeed, gene po-sition has been used to modulate the oscillation featuresin an experimental synthetic implementation of anothergenetic circuit [35]. However, as we show for our case,the effect of the displacement of a gene is growth-ratedependent. In fact, while moving a gene from Ter to Oriallows an increment of its gene dosage of a factor 4 for aµ = 3 db/hr, this factor decreases with doubling time upto 1 for slow growth. The consequence of this observa-tion on the repressilator dynamics is evident looking atFig. 8B and C. The same change in configuration (changein ∆ℓ) has different effects depending on the growth rate.The modeling framework adopted here gives a quantita-tive prediction of the gain in oscillation stability that canbe achieved by moving the genes along the chromosomedepending on the experimental growth conditions, giving

an experimental guideline for the best gene insertion sitesin order to obtain the desired dynamics. Finally, Fig. 8Dand E show the oscillation amplitude dependence on therelative gene positions for the configurations describedin Fig. 8B and C respectively. This dependence changessignificantly at different growth rates, showing that theeffect of gene position is strongly influenced by the cellu-lar environment, and thus by growth rate. This featureis also relevant for synthetic biology, since identical ex-periments carried out with different nutrient levels couldin principle lead to different results.

IV. DISCUSSION

To sum up, this work addresses the dynamics of aparadigmatic bacterial oscillatory gene circuit, the re-

10

pressilator, using for the first time a modeling frame-work that accounts for relevant physiological parame-ters, through their growth-rate dependence [15]. Fromthe modeling viewpoint, this framework entails assum-ing that the parameters of the dynamical system arein fact dependent on a hidden “super-parameter”, thegrowth rate, which encapsulates cell physiology. Addi-tional models or experiments are required in order toobtain the dependence of each relevant parameter fromthe super-parameter. For the case of E. coli, all thisinformation is available. The parameters change theirvalue following the hidden variable, which makes the phe-nomenology of the dynamical system with respect to thestandard parameters less informative. Specifically, igno-rance of the role of the super-parameter and its behaviorprevents from obtaining the physically relevant phase di-agram.

A growth-rate induced “dynamical switch”. Our resultsshow that the dynamics is dependent on the growth ratein a fashion that is both qualitative and quantitative.Specifically, a symmetric repressilator will lose its oscil-latory state with increasing growth rate unless its basalexpression is sufficiently high, and this phenomenology isexpected to be observable in a wide range of experimen-tally accessible conditions. Indeed, the growth-mediatedswitch between different dynamics shown in Fig. 4 shouldbe simple to observe experimentally, given the typicalvalues of the parameters involved. The average proteincopy numbers per cell span different orders of magnitude,from 10−1 to 104 [24], while the values for the dissocia-tion constants have been reported to range between afew molecules [36] and a few thousands [37]. With thesenumbers, the basal expression values (i.e. protein con-centration at µ = 1 in units of k) analyzed here, suchas x∗ ∈ (0, 20) in the example in Fig. 4, are well in thephysiological range. This suggests that a repressilatordynamics characterized by loss of the oscillatory behav-ior at a critical growth rate should not be too difficult toobserve in the laboratory, and that changes in both os-cillation period and amplitude should also be quite likelyto be measurable experimentally. [52]

Dependence of the dynamics on the physical support.

Furthermore, the growth-rate dependent behavior of thecircuit is affected by the support it is embedded in, aplasmid or a chromosome, and for a chromosome on thedetailed coordinates of the three promoters. Fig. 4 and 5suggest that the stability of oscillatory behavior is moresensitive to variations in growth rate for a repressilatorintegrated on plasmids than on the chromosome, as plas-mid copy number can be strongly growth-rate dependent(as well as variable from cell to cell). However, integra-tion on a high copy-number plasmid can naturally in-crease the protein concentration (hence the “promoterstrength” parameter defined by Klumpp and coworkers),thus leading to more robust oscillations. Therefore, ifthe goal is engineering a stable synthetic oscillator, thereis probably a trade-off between the advantage of an in-creased basal expression typical of a high copy number

plasmid, and the unavoidable plasmid-specific growth-rate dependence (and cell-to-cell variability) of the copynumber [38].

On the other hand, for chromosomally integrated re-pressilators, the dynamics depends nontrivially on geneposition. Recently, it has been shown that the spatialordering of a set of “important” genes along the chro-mosome is strongly conserved between different bacte-rial species and largely corresponds to their expressionpattern during growth [39], pointing to a functional rolefor gene chromosomal position. The example of a puta-tive chromosome-integrated repressilator analyzed heresuggests that the dynamics of genetic networks, in fast-growing bacteria, should be influenced by the genomicposition of its components. For example, for the repres-silator case (Fig. 8), a variation in the relative positionof genes involved in a regulatory circuit can have dif-ferent consequences depending on the growth rate. Inthis perspective, the analysis of the phenotypic conse-quences of chromosomal rearrangements, such as largeinversions [40], should be revisited taking into accountthe growth conditions. More generally, it is temptingto speculate that the evolutionary pressure for keepinga certain gene order with respect to genome replicationmay be partially due to natural selection of specific net-work dynamics defined by the combination of gene posi-tions and cell growth state.

Role of noise. Some relevant considerations can bemade about the possible role of noise in the results givenhere within a purely deterministic framework. In gen-eral, noise can strongly affect the dynamics of a repres-silator. For example, in the in vivo realization of therepressilator [14] only about 40% of the cells displayedoscillations, with high cell-to-cell variability in oscilla-tion period and amplitude. Several studies have analyzedthe possible impact of noise, focusing on the stochasticitythat arises from the discrete nature of the molecular play-ers and from the inherent randomness of their interac-tions (together referred to as intrinsic noise) [14, 29, 41].The main result is that intrinsic noise can both play aconstructive role in oscillation robustness and a destruc-tive one. The constructive phenomenon can enlarge theparameter space of oscillations through a resonance ef-fect [41]. The destructive one causes strong cell-to-cellvariability in oscillation amplitude and period [14].

While intrinsic noise can be a relevant factor and couldpartially explain the experimentally observed variability,the dominant source of noise might be due to fluctua-tions in global cellular parameters (extrinsic noise), suchas ribosome or polymerases concentration. This has beenshown to be the case in E. coli, for relatively high expres-sion (more than approximately 10 proteins per cell) [24].Since oscillations in the repressilator generally requirestrong promoters (high basal expressions), the variabilityin the circuit dynamics is expected to be highly sensitiveto the extrinsic noise level, which adds up, for plasmids,to the aforementioned cell-to-cell copy-number variation.These considerations point to an interest in considering

11

the stochastic aspects of the circuit. However, in order toextend the mean-field model introduced here to analyzethe growth rate dependence of the cell-to-cell variabil-ity of the repressilator dynamics, it would be necessaryto know how the extrinsic noise scales with growth rate.Unfortunately, there are no experimental data concern-ing this scaling, making the extension of this work to thestochastic case premature.

Nevertheless, the possibility of a switch between dif-ferent dynamical regimes in response to the physiologi-cal cell state opens interesting considerations about therobustness of this oscillatory genetic circuit. Fluctua-tions in physiological parameters such as the growth ratefall in the broadly-defined category of extrinsic noise. Ithas been suggested that fluctuations in the growth rate,mainly through its influence on protein dilution, can ac-count for a considerable part of the measured extrinsicnoise [27, 42]. Our analysis suggests that fluctuations incell parameters linked to growth can introduce cell-to-cellvariability in the circuit dynamics (convergence to oscil-lation or to a stable fixed point) as well as in oscillationperiod and amplitude. Therefore, the reasons behind thelack of robustness of the repressilator realized in vivo [14]should be also searched in the variability of physiologicalparameters rather than focusing exclusively on intrinsicnoise effects.

Biological outlook. Finally, we believe these findingscould be relevant from both a systems biology and a syn-thetic biology perspective. There is a long list of endoge-nous oscillators in bacteria [21] that are interesting for theformer discipline, and need to be understood within theframework adopted here. The most important examplesare circadian clocks and the cell cycle itself. We previ-ously studied the dynamics of the DnaA[53] oscillatorycircuit, which is determinant in this last process [23]. Inthis case, the timescale of oscillations matches (by defini-tion) the cell cycle time, thus the approximations definedin section IIA are not valid. More complex models arerequired, and there is no commonly accepted theoreticalframework to describe this case. However, it is interestingto note that in the simple modeling framework adoptedhere, the period of oscillation decreases with growth ratewithout the need of specific additional regulation. In ab-sence of overlapping replication rounds, this is exactlythe kind of behavior desired for an oscillator regulatingthe triggering of DNA replication, such as the DnaA cir-cuit: a shortening of the initiation time is required whenthe cell volume grows faster to synchronize DNA repli-cation and volume doubling (the situation becomes morecomplex at fast growth [23]).

In contrast, circadian clocks need to be resilient tochanges in the cell doubling time, and thus in the growthrate, in order to keep a steady 24-hour period in vari-able environmental conditions, and thus can not mea-sure time using the cell cycle. The consequent decou-pling between the cell cycle and the circadian rhythmhas indeed been verified in cyanobacteria [43–45]. Ourresults suggest that the dynamics of a genetic oscilla-

tor is naturally strongly connected to the cellular growthrate. Therefore, specific regulatory mechanisms are re-quired in order to compensate for these effects and rendera circadian oscillator insulated from the growth state. Al-though circadian clocks appear to be primarily based onpost-translational circuits in bacteria [21], the proteinsinvolved are the result of a gene expression process, andthus in principle coupled with growth rate [15]. It wouldbe interesting to evaluate experimentally if the promotersregulating these proteins are more buffered as a functionof growth rate compared to others. Quantitative modelstaking into account the parameter dependence on growthrate, such as the one presented here, may be importantto pose the question of the growth-rate robustness of thecircadian cycle. For example, the circuit architecturesand the type of regulations selected by evolution to com-pose circadian oscillators might be, at least in part, con-strained by the implementation of the observed growth-rate independence.Finally, from a synthetic biology standpoint, changing

the conditions in which the cells are grown alters quanti-tatively the characteristics of the repressilator dynamicsin a predictable manner. This offers the possibility ofexternal control of the circuit behavior by simply oper-ating on macroscopic variable related to physiology suchas the type of nutrient supply or the temperature. Thisway, the engineering and control of the dynamics canbe performed by tuning environmental conditions in amodel-guided way, rather then by modifying the geneticcomponents, which can be technically complex.

Acknowledgements

We acknowledge useful discussions with Vittore F. Sco-lari, Mina Zarei and Bianca Sclavi. We thank MatthewAA Grant and Carla Bosia for critical reading of themanuscript. This work was supported by the the Inter-national Human Frontier Science Program Organization(Grant RGY0069/2009-C).

Appendix A: Population averages vs cell-cycle

averages

A common assumption in genetic circuit modeling isthat the contribution of the cell-cycle dynamics can beneglected, at least when one is interested in time scaleslonger than the doubling time, which is often the casegiven the typical high protein stability. This approxi-mation allows the use of effective parameters obtainedaveraging over the cell cycle. The contribution of proteindilution due to growth and division is incorporated asan effective degradation rate [27]. However, as discussedin the text, the growth-rate dependences of gene expres-sion parameters are derived from experimental observa-tions of average cellular properties in a population [16],which are also affected by the cell age distribution. A

12

quantitative estimate of the age-structure effects on mea-surements performed on an exponentially growing pop-ulation is especially important when their growth-ratedependence is in analysis. In fact, the fraction of cellsfound at a certain cell-cycle stage is itself a function ofthe growth rate.This Appendix discusses the quantitative difference be-

tween population and cell-cycle averages for two quanti-ties that are well characterized, the gene dosage due toDNA replication and the cellular volume. We will showthat in these two cases population averages can be usedin dynamic models of genetic circuits without introduc-ing significant errors. The assumption that this resultcan be generalized to other quantities justifies, althoughnot rigorously, the use of available experimentally esti-mated population averages in dynamic models for thosequantities whose time dependence (or even cell-cycle av-erages) are not known [15]. This is the case for ribosomeor polymerase concentrations at different growth rates,which are crucial to determine the growth-rate depen-dence of transcription and translation rates in Eq. 3.Thus, in our case, given the small quantitative dif-

ference between the two type of averages, the empiricalgrowth rate dependences (based on population averages)have been used in the analysis (and thus for all the pa-rameters quoted in the main text). The alternative useof cell-cycle averages is possible only for the gene dosageand volume, and does not alter significantly the results.Indeed, as reported in [27], for a constitutive gene theaverage protein concentrations calculated with respect toan age-structured population or with respect to the cellcycle differ by only a few percent.

1. Cooper-Helmstetter model and gene dosage

DNA replication in fast-growing bacteria such asE. coli typically starts from a single replication ori-gin (Ori) and proceeds bidirectionally along the circu-lar chromosome until it reaches the replication terminus(Ter). The Cooper-Helmstetter model [22] establishesthe relation between growth rate and replication timingsuch that DNA copies are produced on time for each new-born cell. The model is based on the empirical observa-tion that the time necessary for chromosome replication(called “C period”) and the time period between com-pletion of chromosome replication and the following celldivision (D period) are approximately constant (at leastfor fast-dividing cells [34], with doubling times less than1hr). Since at time C +D the cell divides, a time lag Xbefore initiation is necessary to make the total replica-tion time X +C +D an integer multiple of the doublingtime τ , “synchronizing” DNA replication and cell divi-sion. Thus, the following relation has to be satisfied

X + C +D = (n+ 1)τ , (A1)

where n = Int[C+Dτ

] is the integer number of times that τdivides C+D. Starting from this relation, it can be easily

FIG. 9: (Color online) Cell cycle and population average

of gene dosage. The gene dosage g averaged over the cellcycle (dashed blue lines) and averaged over a population inbalanced exponential growth (continuous red lines) are shownas a function of the growth rate for different chromosomal po-sitions ℓ (normalized distance from Ori). In the physiologicalrange of growth rates, the two quantities do not present asubstantial quantitative difference.

shown that the number of origins present at initiation isexactly 2n [16, 22]. More generally, we can consider agene at a chromosomal position defined by its normalizeddistance ℓ from Ori, i.e. ℓ = 0 represents a gene in Oriand ℓ = 1 in Ter. The copy number of this gene, g,changes during the cell cycle following

g(t) =

{

2n′

if 0 < t < (n′ + 1)τ − (C(1 − ℓ) +D)

2n′+1 if (n′ + 1)τ − (C(1 − ℓ) +D) < t < τ

,

where n′ = Int[C(1−ℓ)+D

τ]. Therefore, the gene dosage

averaged over the cell cycle (which could be measuredfollowing a single cell lineage and averaging over time),is given by

〈g(t)〉cell cycle =1

τ

∫ τ

0

g(t)dt =

= 2n′

[1− n′ + µ(C(1 − ℓ) +D)] . (A2)

On the other hand, the population age structure mustbe taken into account when evaluating the average genedosage in a cell population. For ideal “balanced expo-nential” growth with rate µ this distribution is givenby [27, 46]

a(t, µ) = 2 ln2 µ 2−µt. (A3)

Thus, the population average is

〈g(t)〉population =

∫ τ

0

a(t, µ)g(t)dt =

= 2µ[C(1−ℓ)+D] . (A4)

This is the expression typically used to evaluate the genedosage [15, 16]. As shown in Fig. 9, the difference be-

13

tween cell-cycle averages (dashed blue lines) and popula-tion averages (red continuous lines) in the physiologicalrange of growth rates is negligible for all gene positions.

2. Cell volume growth

Similar considerations can be carried out in the caseof the average cell volume. The functional form of thevolume increase in time during a cell cycle has longbeen debated [47], with two prevailing hypotheses of lin-ear growth (constant rate) or exponential growth (size-dependent rate), although more complex dependenceshave been proposed [48]. Recent experiments strongly

suggest an exponential growth [49], and we assumed thisfunctional form (note that the same reasoning could beapplied to linear growth straightforwardly, so this choicehas no consequences on any of the results). With a vol-ume growth of the form V (t) = V02

µt, the mean volumeover a cell cycle is 〈V (t)〉cell cycle = V0/ln2, while the in-tegration over the population leads to 〈V (t)〉population =2 ln2 V0. All the volume growth-rate dependence is hid-den in V0, and experimental results indicate that this de-pendence is approximately exponential [25]. For all thesituations considered in our study, we verified that thedifferent numerical factors introduced by averaging overthe cell cycle or over the population do not affect signif-icantly the growth-rate dependence of the mean volume.

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[50] This state is conventionally referred to as “steady” be-cause its maintenance requires that a nutrient sink (thecells) is constantly replenished with a source.

[51] Note that the term “basal expression” is sometimes foundin the literature with a different meaning, i.e. the low butdetectable level of expression that a negatively regulatedgene maintains in conditions of strong repression (alsocalled “leakage” level of expression).

[52] Note however that some of the dissociation constant val-ues reported in the literature were measured in vitro andthus could be underestimated with respect to in vivo val-ues, where non-specific binding plays an important role.

[53] DnaA is a protein responsible for the initiation of DNAreplication in several bacteria. Essentially, if present in asufficiently high concentration, it can promote the melt-ing of DNA strands at the replication origin.