A Coarse-Grained Biophysical Model of E. coli and Its Application to Perturbation of the rRNA Operon Copy Number
AA Coarse-Grained Biophysical Model of E.coli and ItsApplication to Perturbation of the rRNA Operon CopyNumber
Arbel D. Tadmor , Tsvi Tlusty
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, Israel
Abstract
We propose a biophysical model of
Escherichia coli that predicts growth rate and an effective cellular composition from aneffective, coarse-grained representation of its genome. We assume that
E. coli is in a state of balanced exponential steady-state growth, growing in a temporally and spatially constant environment, rich in resources. We apply this model to a seriesof past measurements, where the growth rate and rRNA-to-protein ratio have been measured for seven
E. coli strains withan rRNA operon copy number ranging from one to seven (the wild-type copy number). These experiments show thatgrowth rate markedly decreases for strains with fewer than six copies. Using the model, we were able to reproduce thesemeasurements. We show that the model that best fits these data suggests that the volume fraction of macromoleculesinside
E. coli is not fixed when the rRNA operon copy number is varied. Moreover, the model predicts that increasing thecopy number beyond seven results in a cytoplasm densely packed with ribosomes and proteins. Assuming that under suchovercrowded conditions prolonged diffusion times tend to weaken binding affinities, the model predicts that growth ratewill not increase substantially beyond the wild-type growth rate, as indicated by other experiments. Our model thereforesuggests that changing the rRNA operon copy number of wild-type
E. coli cells growing in a constant rich environment doesnot substantially increase their growth rate. Other observations regarding strains with an altered rRNA operon copynumber, such as nucleoid compaction and the rRNA operon feedback response, appear to be qualitatively consistent withthis model. In addition, we discuss possible design principles suggested by the model and propose further experiments totest its validity.
Introduction
The rRNA ( rrn ) operons of
E. coli have the important role ofdetermining ribosome synthesis in the cell (c.f. [1–6] for reviews).These operons are unique in the sense that a wild-type (WT)
E. coli cell carries seven copies of this operon per chromosome [7] (otherbacteria have copy numbers ranging between 1 and 15 [8]). Thiscopy number also appears to be selectively maintained.
S.typhimurium for example, from which
E. coli is thought to havediverged 120–160 million years ago [9], also has seven copies ofthis operon [10] and in evolution experiments of up to 10 generations no deviations from the WT copy number have beenobserved [11]. These findings raise the question of whatunderlying mechanisms, if any at all, fixed this copy number tobe seven and not six or eight. In other cases it has been shown thatthe WT genome configuration maximizes fitness [12–14]. Thus,can it be shown that this copy number maximizes fitness?In general, this question is hard to answer because the naturalenvironment of E. coli is expected to vary both spatially andtemporally [15,16], thereby invoking complex physiologicalresponses in the cell that are complicated to model. We thereforeconsider a much simpler scenario, where a resource-rich environment is spatially and temporally constant, and where thecell is at a state of balanced exponential steady-state growth [3],such that it has a well defined and reproducible growth rate andphysiological state [17,18]. In such an environment, cells that canoutcompete their rivals will takeover the population and thus fixtheir genotype. Thus we refer to fitness in the narrow sense thatprevious authors have used [11,14], i.e. it is the potential capacityof a cell in exponential growth to outcompete another strainpopulation wise. Therefore, for an exponentially growing cell in aconstant and rich environment, fitness would be, by definition, thegrowth rate of that cell.Experimental evidence indeed suggests that for exponentiallygrowing cells, cells with altered rRNA operon copy numbers havea lower growth rate. In a series of experiments, the Squires grouphas measured the growth rate and cell composition of seven strainsof
E. coli with rRNA operon copy numbers ranging from one toseven copies per chromosome [19]. All strains were grown in thesame nutrient rich environment and measurements were per-formed on cells in exponential phase. These experiments show thatcells with fewer than six rRNA operons have a considerable lowergrowth rate [19] (c.f. Figure 2A presented later in the text). Forexample, cells with five functional rRNA operons have a 21%ower growth rate than WT cells, while cells with only onefunctional rRNA operon have a 50% lower growth rate than WTcells [19]. In addition, a strain carrying extra rRNA operons on aplasmid exhibited a 22% reduction in growth rate relative to a WTcontrol strain with a plasmid expressing nonfunctional rRNA [20].To gain further insight into these findings, we sought toformulate a model of
E. coli that could predict phenotype, such asgrowth rate and cell composition, directly from DNA relatedparameters, such as the rRNA operon copy number, while keepingthe complexity of the model to a minimum. The model of
E. coli proposed here differs from existing models of
E. coli in severalrespects. Traditionally,
E. coli has been monitored in different orchanging environments [17,21–23], and existing models haveattempted to predict
E. coli’s response to such environmentalperturbations [12,23–26]. However, since disparate environmentsare expected to induce disparate genetic networks, we anticipatethat such a strong perturbation will be difficult to capture in asimple model that attempts to predict phenotype from DNArelated parameters (c.f. S2.3 in Text S1). Existing models of
E. coli tend to fall into two classes. One class includes very complexmodels, involving tens to hundreds of equations [12,24,25], whichdo not lend themselves to simple interpretation. The other classinvolves simple and elegant models of
E. coli that followed theCopenhagen school [22,23,26,27] (see [17] for review). Theseclassic models, however, do not relate genome to growth rate andcomposition, nor do they make reference to certain key physicalprocesses in the cell elucidated since. Included in our currentmodel are the relationships of genome to growth rate and cellcomposition, reflecting key physical processes now better under-stood, such as RNA polymerase (RNAp)-promoter interaction[3,28], RNAp autoregulation [29], ribosome-ribosome binding site(RBS) interaction [30–33], mRNA degradation [34–40], DNAreplication initiation [41–43] and macromolecular crowding (seebelow) [44–51]. In addition, we have attempted to find the middleground in terms of complexity by coarse-graining, for simplicity,certain features of the cell: in the spirit of previous works [28,52],the genome has been lumped into a small set of ‘‘gene classes’’ thatrepresent all transcription and translation within the cell for the given environment. Similarly, the cell composition was reduced toa small set of variables accounting for the macromolecule contentof the cell. The resulting type of model is referred to as a
Coarse-Grained Genetic Reactor (CGGR).Another point of difference with respect to existing models of
E.coli is that in this model we take into account possible globalbiophysical effects resulting from the high volume fraction ofmacromolecules in
E. coli , a state commonly termed ‘‘crowding’’[45]. Formulating such a biophysical model for
E. coli raises thebasic question: is the macromolecular volume fraction, W = V macro /V cell , inside E. coli constrained to be fixed or does it change forgenetically perturbed cells? We have explored both of thesepossibilities in what we refer to as the constrained ( W = const) and unconstrained ( W ? const) CGGR models.Using the CGGR modeling approach, we have modeled theseven strains engineered by the Squires group and have calculatedtheir growth rate and their effective cellular composition. We wereable to reproduce the experimental data within a model in whichmacromolecular volume fraction was allowed to change forgenetically perturbed cells. These findings, along with otherbiological considerations, seem to favor the unconstrained CGGRmodel (see Discussion). According to this model, increasing thechromosomal rRNA operon copy number beyond seven will over-crowd the cytoplasm with ribosomes and proteins. Under suchover-crowded conditions, we expect that binding affinities willweaken due to prolonged diffusion times. As a result, given thisassumption, we show that the growth rate of an exponentiallygrowing cell in a constant rich medium will not increasesubstantially beyond its WT growth rate when the rRNA operoncopy number is increased beyond seven. Although we have notshown that the maximum in growth rate is a global maximum,since we only perturbed one genetic parameter, this result suggeststhat—at least for the case of a cell undergoing balancedexponential steady-state growth in a constant and rich medi-um—basic kinetic and biophysical considerations may have animportant role in determining an optimal rRNA operon copynumber (see Discussion).Besides explaining the Squires data, the unconstrained CGGRmodel is qualitatively consistent with observations regardingnucleoid compaction in the inactivation strains and with the rrn feedback response originally observed by Nomura and coworkers(see Methods and Discussion). Thus, the CGGR model may offeran initial conceptual framework for thinking about E. coli as awhole system at least for the simplified environment considered.More complex genetic networks may subsequently be embeddedinto this model enabling one to analyze them in the larger, wholecell framework. Such a model may also help elucidate how
E. coli works on a global scale by making experimentally testablepredictions and suggesting experiments (see Discussion). We willalso consider possible insights into the ‘‘design principles’’ of
E. coli suggested by the CGGR model, such as intrinsic efficiency ofresource allocation and decoupling of DNA replication regulatorymechanisms from cell composition.
Methods
The Cell as a Coarse-Grained Genetic Reactor (CGGR)
Our goal is to formulate a model of
E. coli that predictsphenotype, such as growth rate and the cell composition, fromparameters directly related to the genome, while keepingcomplexity to a minimum. To reduce the complexity of thisproblem we coarse-grained both input parameters (the genome)and output parameters (the cell composition and growth rate). Thegenome (input) was lumped into four basic gene classes : RNA
Author Summary
A bacterium like
E. coli can be thought of as a self-replicating factory, where inventory synthesis, degrada-tion, and management is concerted according to a well-defined set of rules encoded in the organism’s genome.Since the organism’s survival depends on this set of rules,these rules were most likely optimized by evolution.Therefore, by writing down these rules, what could onelearn about
Escherichia coli?
We examined
E. coli growingin the simplest imaginable environment, one constant inspace and time and rich in resources, and attempted toidentify the rules that relate the genome to the cellcomposition and self-replication time. With more than4,400 genes, a full-scale model would be prohibitivelycomplicated, and therefore we ‘‘coarse-grained’’
E. coli bylumping together genes and proteins of similar function.We used this model to examine measurements of strainswith reduced copy number of ribosomal-RNA genes, andalso to show that increasing this copy number overcrowdsthe cell with ribosomes and proteins. As a result, thereappears to be an optimum copy number with respect tothe wild-type genome, in agreement with observation. Wehope that this model will improve and further challengeour understanding of bacterial physiology, also in morecomplicated environments. olymerase (RNAp), ribosomal protein (r-protein), stable RNAand bulk protein. The gene classes are represented by geneticparameters such as: genetic map locations, promoter strengths, RBSstrengths, mRNA half-lives and transcription and translationtimes, all of which can be, in principle, linked to the DNA. Geneticparameters have been determined based on empirical data for theWT growth rate (or very close to it) and represent all transcriptionand translation within the cell at that growth rate (see Results formore details).The cell composition (output) was reduced to the following fivemacromolecule classes: free functional RNAp, total RNAp, freefunctional ribosomes, total ribosomes and bulk protein. The bulkprotein class represents all other cell building and maintenanceproteins in the cell [53]. The concentration of these macromol-ecules, together with the growth rate constitutes six state variablesthat define the cell state . Table 1 gives an example of the observedWT cell state at 2.5 doub/h. In the Discussion we consider theapplicability of this choice of coarse-graining. The Feedback Mechanisms within a Coarse-GrainedModel of
E. coli
After coarse-graining the cell, one can map the various feedbackmechanisms that exist between these coarse-grained components,as illustrated in Figure 1A. Transcription of the various geneclasses by RNAp [29] is depicted on the left, and translation ofmRNA by ribosomes on the right. Ribosomes are shown to beassembled by combining rRNA with r-protein. r-proteins synthesisrate is regulated to match the rRNA synthesis rate [1] as indicatedby the black arrow in Figure 1A.RNAp naturally has positive feedback to all promoters, andribosomes have positive feedback to all RBSs. In the case ofRNAp, it has been shown [29] that the bb subunits, which limitthe production of RNAp (c.f. discussion in [17]), repress their owntranslation, and the functional, assembled RNAp holoenzymerepresses transcription of bb . While the details of the RNApautoregulation are still being elucidated, the latter finding suggeststhat the apparent fast response of the negative translationalautoregulation of the bb operon keeps the concentration of totalRNAp fixed, at least approximately (for a detailed discussion seeS2.1 in Text S1). The level of RNAp may also be modulated byguanosine 5 -diphosphate 3 -diphosphate (ppGpp) [3,17,28],however, since ppGpp levels were measured to be constant forstrains with increased or decreased number of rRNA operons[20,54], this modulation will not be relevant in this analysis (seediscussion below).Finally, there is the feedback arising from the translation-degradation coupling, indicated in Figure 1A by the dashed greenline connecting ribosomes to mRNA degradation. Ribosomesbound to the RBS of mRNA protect the mRNA from degradationby preventing RNase E – thought to be the primary endonuclease Table 1.
The cell state at m = 2.5 doub/h, 37 u C. Statevariable m (doub/h) n bulk n RNAp n RNAp,free n ribo n ribo,free Measuredvalue 2.5 5.76 ? n i ) was either directly measured or estimated from empiricaldata and is given in units of molec/cell. Measurement error is expected to bearound 15%, mostly due to culture-to-culture variation [17]. More details can befound in Table S2. Figure 1. The coarse-grained genetic reactor (CGGR) model of
E. coli . (A) A schematic diagram of the cell as a CGGR. This figuredepicts the various gene classes (double stranded DNA on the left),mRNA (single stranded RNA on the right) and their expressed products,stable RNA and the various positive ( R ) and negative ( x ) feedbacksbetween these components. The colored tips on the DNA and mRNArepresent the promoter binding sites and the ribosome binding sites(RBSs) respectively. The white boxes denote the process of assembly offunctional complexes from immature subunits. The light blue back-ground represents the finite volume of the cell in which reactions takeplace, which is determined by the DNA replication initiation system (seetext and S2.2 in Text S1). (B) The basic reactions taking place in the cell: K m , i , V max i are the Michaelis-Menten (MM) parameters for transcriptioninitiation of gene class i [3,28], followed by transcription at a rate of c,i . L m , i , U max i are the MM parameters of gene class i for translationinitiation (i.e. binding of a 30S ribosome subunit to a RBS) [30,31],followed by translation at a rate of l,i . Note that in this scheme, the30S-RBS binding affinity, L m,i , includes the 30S interaction with thesecondary structure of mRNA [30]. mRNA degradation is primarilyachieved via the endonuclease RNase E [35] and is assumed to followMM like kinetics ( J m , i , W max i ) [36,37]. In the scheme considered here,RNase E competes with the 30S subunits in binding to a vacant RBS[38,39], functionally inactivating it when it binds [38]. This results in acoupling between translation and degradation [34]. Delays due toassembly are incorporated separately. The MM parameters, togetherwith the time constants and map locations define genetic parameters for the various gene classes, which can be easily ‘‘tweaked’’ by base pairmutations (c.f. Table 2 and Table S1).3 n E. coli [35,55] – from binding to the 5 end of the mRNA andcleaving it (for recent reviews see [34,38,39]). RNase E isconsidered to be the partially or fully rate-determining step inthe mRNA degradation process [35,55–59]. Here we havemodeled this effect by allowing only mRNA’s with a vacantRBS to be cleaved [38,39,60]. This feedback manifests itself asdependence of mRNA half-life on the probability that the RBS isvacant, as suggested by observation [34,56,61] (see S2.7 in Text S1and discussion further below).We shall refer to the feedbacks depicted in Figure 1A as internalfeedbacks . If the macromolecular volume fraction is allowed tochange, then an additional internal feedback arises due to the factthat the binding affinities of RNAp and ribosomes to theircorresponding binding sites may change due to crowding effects.This kind of dependence on the crowding state of the cell offers anadditional feedback path not explicit in Figure 1A. We willelaborate on this point in the Results section. Also not explicit inFigure 1A is DNA replication that determines gene concentration.This issue will be further discussed below. The Feedback Response of the rRNA Operons
It has long been known that there is some form of feedbackcontrol on rRNA operons that responds to any artificial attempt tomanipulate ribosome synthesis [1–6,20,62], yet the source of thisfeedback has remained controversial. Since we will be consideringperturbations on the rRNA operon copy number, which affectribosome synthesis, it is pertinent to identify any additionaleffectors that apply a feedback within the system.Nomura and his coworkers noted that cells with increasednumber of rRNA operons did not exhibit a significant increase inrRNA transcription [62], i.e. the transcription per rRNA operondecreased by means of some feedback. Furthermore, the absenceof this feedback in cells overproducing nonfunctional rRNA, andthe observation of a feedback response in strains in whichribosome assembly was blocked, suggests that complete ribosomesare involved in the feedback response [62]. This became known asthe ‘‘ribosome feedback regulation model’’ (c.f. discussions in[1,3,5]). Direct effect of ribosomes on rRNA transcription couldnot, however, be observed in vitro [62], and it was suggested bythese authors that this regulation may be achieved indirectly [62].Further experiments indicated that the feedback depends ontranslating ribosomes (or translational capacity) rather than freeribosomes [1,63]. Later studies have demonstrated the feedbackresponse for various other perturbations that attempted toartificially manipulate ribosome synthesis rate, including: increas-ing rRNA operon copy number [20,64,65], decreasing rRNAoperon copy number [54], overexpressing rRNA from aninducible promoter [66], deleting the fis gene [20,67] (see below),muting the rpoA gene coding for the a subunit of RNAp [20,68]and more (c.f. [2,20]). Since many of these perturbations [20], aswell as perturbations in nutritional conditions [2,69], correlatedwith changes in the concentration of ppGpp and nucleosidetriphosphate (NTP), Gourse and his coworkers have proposed thatNTP and ppGpp are the feedback regulators [6,69]. In addition,these authors have suggested a model where translating ribosomesconsume or generate NTP and ppGpp and thus are able toachieve homeostasis of rRNA expression on a rapid time scale[6,69]. Yet these authors also point out that these effectors cannotexplain the feedback response specifically associated with changesin rRNA gene dosage [64] (the perturbations considered in thisstudy). In this case, it has been demonstrated that the smallmolecule ppGpp has no effect on rRNA synthesis rate both in thecase where rRNA gene dosage was increased [20] or decreased[54] since ppGpp concentration remains constant in these strains (also indicating that tRNA imbalance was not a problem in thosestrains). In addition, feedback inhibition due to increased rRNAgene dosage was of the same magnitude in both wild-type cells andstrains lacking ppGpp [70]. Similarly, the concentration of thesmall effector NTP was shown to be constant when decreasing orincreasing the rRNA gene dosage [64]. In a different study, NTPconcentration decreased by only a small amount (20%) whenrRNA gene dosage was increased [20], such that those authorsconcluded that the small change in NTP concentration appears tobe insufficient to account for the entire effect on transcriptioninitiation. Due to these findings, Gourse and coworkers concludedthat there may be additional mediators involved in feedbackcontrol of rRNA expression when altering the rRNA operon genedosage [2,20]. We show that internal feedbacks may account, atleast partially, for the feedback response, although an additionaleffector may still be involved. In the Discussion we analyze modelpredictions and compare them to observations regarding thiseffect. We will also discuss the predicted feedback response in thecontext of Nomura and coworkers’ feedback model and show thatthere appears to be no contradiction between the two. Additional Factors Affecting rRNA Expression
In addition to the small molecules mentioned above, rRNAtranscription is further modulated by transcription factors like Fis,HN-S and DskA, as well as the UP element [1,2,4–6,69,71],however there is currently no experimental evidence to suggestthat these factors are linked to the feedback response to alteredrRNA gene dosage. DskA, for example, a small molecule thatbinds to RNAp, is thought to amplify effects of small nucleotideeffectors such as ppGpp and NTP [4,6,72]. DskA concentration,however, was found to be unchanged with growth rate and growthphase and therefore it apparently does not confer a novel type ofregulation on rRNA synthesis [3,72] and is thus considered to be aco-regulator rather than a direct regulator [4]. Fis stimulatesrRNA transcription by helping recruit RNAp to the promoterthrough direct contact with the a subunit of RNAp, while the UPelement, a sequence upstream of the promoter, binds the a subunitof RNAp and can greatly stimulate rRNA transcription [2,4–6,71]. Although Fis levels change throughout the growth cycle[4,5], strains lacking Fis binding sites retain their regulatoryproperties [2,5,67] indicating that fis is not essential for regulationof rRNA transcription during steady-state growth [67], andperhaps just plays a role in control during nutritional shift-ups andonset of the stationary phase [5]. HN-S concentration changeswith the growth phase of an E. coli culture [2,69,71] and is thoughtto be associated with regulation related to stress [15], particularlyin stationary phase [4,69]. Since there is currently no directevidence that shows that any of these or other factors are asso-ciated with the feedback response to rRNA gene dosage perturba-tion, no such factors were included in the proposed models, yetfuture experiments may prove otherwise (c.f. Discussion).
Kinetic Equations
The biochemical reactions that make up the feedback networkillustrated in Figure 1A are approximated, for simplicity, byMichaelis-Menten type kinetics [3,28,30,36,37], as is illustrated inFigure 1B. These reactions include: stable RNA synthesis, bulkprotein synthesis and bulk mRNA decay. Since ribosomes and thebulk of proteins in
E. coli are stable on timescales of several genera-tions [35,73], their degradation can be neglected compared to thefast doubling time of the cell ( ,
30 min). We also do not need toexplicitly consider r-protein synthesis since ribosome synthesis islimited by rRNA [1]. Finally we note that the free RNAp in thesereactions may include RNAp bound nonspecifically to DNA and n rapid equilibrium with it [28] that may locate promoters by atype of 1-D sliding mechanism [74]. In the current model, allinactive RNAp was assumed to be associated with pause genes (c.f.[28] and e.g. Table S4) and thus inaccessible to promoters.However, it may be that some of these inactive RNAp moleculesare just bound nonspecifically to the DNA [28] perhaps serving asan additional reservoir of RNAp.Since the Squires strains were measured under steady-stateconditions, we consider next the steady-state equations implied byFigure 1B. The CGGR Steady-State Equations
The reactions in Figure 1B can be readily expressed as rateequations and analyzed at steady-state. Although the fullderivation is rather lengthy (see S2.5 in Text S1), the finalequations lend themselves to simple interpretation. The averagetranscription [3,75] and translation [30,31,76] initiation rates aregiven by the usual Michaelis-Menten relations V i ~ V max i z K m , i (cid:1) n RNAp , free , U i ~ U max i z L m , i (cid:1) n ribo , free ð Þ where n i denotes the concentration of species i , V max i and U max i are the maximum transcription and translation initiation rates ofthe i -th gene class respectively, and K m,i and L m,i are RNApholoenzyme and 30S ribosome subunit binding affinities of the i -thgene class to their corresponding binding sites respectively,measured in units of concentration (see Table 2 for notation listand units). Using this notation, the RNA transcript synthesis rateper unit volume is v i = d i V i (where d i is the gene concentration ofthe i -th gene class) and the number of translations per mRNA is u i ~ U i T fun = i . ln 2 , where T fun = i is the functional half-life of the i -thgene class mRNA. Therefore, the protein synthesis rate per unitvolume of gene class i is v i u i . In this notation, the five equations ofstate take the form: i ð Þ n RNAp & constii ð Þ n bulk ~ a v bulk u bulk iii ð Þ n ribo ~ a v rrn iv ð Þ n RNAp ~ n RNAp , free z t c , bulk v bulk z t c , r (cid:1) protein v r (cid:1) protein z t c , rrn v rrn (cid:2) (cid:3) z { e { at RNAp ð Þ n RNAp v ð Þ n ribo ~ n ribo , free z t l , bulk v bulk u bulk z t l , r (cid:1) protein v rrn (cid:2) (cid:3) z { e { at ribo ð Þ n ribo ð Þ where t c,i and t l,i are the times to transcribe and translate the i -thgene class respectively, and t i is the average assembly time forcomponent i (the boxes in Figure 1A). Equation (i) states that thetotal RNAp concentration is constant. This is due to ourassumption that the negative autoregulation of RNAp is ideal.This somewhat naı¨ve model for the autoregulation of RNAp canbe, in principle, replaced with a more sophisticated modeldescribing the steady-state response of the negative transcriptionaland translational autoregulation of RNAp, once the details of thismechanism are known. Equations (ii) and (iii) are the bulk proteinand ribosome synthesis equations respectively, assuming exponen-tial growth, i.e. dilution at a rate of a = m ln 2 , where m is the doubling rate. Note that v rrn is the total ribosome synthesis rate perunit volume. Finally, (iv) and (v) are conservation equations forRNAp and ribosomes within the cell. In Eq. (iv), these termsinclude (left to right): free RNAp, bound RNAp and immatureRNAp (a modified version of Eq. (iv) was first derived in [28]).Similar terms exist in the ribosome conservation equation (v). Thecontribution of RNAp to the conservation equations was neglectedsince it constitutes less than 2% of the total protein mass [17]. Notethat in the second term of (v), the number of bound ribosomes tothe r-protein class is determined by the time it takes to translate allr-proteins and the total rrn transcription rate , due to the matching ofr-protein synthesis rate and rRNA synthesis rate throughregulation at the r-protein mRNA level [1]. The ribosomeconservation equation (v) is equivalent to the previously derivedresult [3]: a = ( N ribo / P ) b r c p , where N ribo is the number ofribosomes per cell, P is the total number of amino acids inpeptide chains, b r is the fraction of actively translating (bound)ribosomes and c p is the peptide chain elongation rate.Explicit expressions for functional and chemical half-lives ofbulk protein, and their dependence on the concentration offree ribosomes, can also be derived from Figure 1B, takinginto account the negative autoregulation of RNase E (c.f. S2.5and Eq. S15 in Text S1). For example, one can show thatthe functional half-life of bulk protein mRNA is given by T fun = bulk ~ T fun , o = bulk z n ribo , free (cid:1) L m , bulk (cid:2) (cid:3) , where T fun , o = bulk is agenetic parameter denoting the functional half-life of bulk mRNAin the absence of ribosomes. Thus, mRNA half-life increases withthe probability that the RBS is occupied. This relation reflectstranslation-degradation coupling trends observed between mRNAdegradation and translation [34,39], further discussed in S2.7 ofText S1.To extract the cell composition from Eq. 2 we requirean expression for the gene concentrations, d i ( m ), of the variousgene classes. This expression is given by linking [43] theCooper-Helmstetter model of DNA replication [77,78] andDonachie’s observations regarding the constancy of the initiationvolume [41,79]: d i m ð Þ ~ V ini ln 2 X j { m i , j C m ð Þ where V ini is the initiation volume, defined as the ratio of the cellvolume at the time of replication initiation and the number oforigins per cell at that time, m i,j represents the map location of the j -th gene in the i -th gene class relative to the origin of replication(0 m i,j C is the C period, the time required toreplicate the chromosome (roughly 40 min). Recent observationsand modeling of the replication initiation mechanism in E. coli [41,42] suggest that the initiation volume is regulated to be fixed,and therefore it should be independent of genetic perturbationsthat do not target that regulation (Tadmor and Tlusty, inpreparation). See S2.2 in Text S1 for further details. Thus, wecan use Eq. 3 to predict the gene concentration for the geneticallyperturbed cells considered here.Equation set 2 provides us with five equations of state . We now testwhether these equations are consistent with observed WT cellstates.
Results
The CGGR Model Can Reproduce the WT Cell State
We wish to see whether given measured genetic parameters at aspecified growth rate, we can reproduce the cell state, namely thegrowth rate of the cell and its coarse-grained composition (Table 1). or the case of growth at 2.5 doub/h, all genetic parameters,except the Michaelis-Menten parameters for translation initiation( U max bulk and L m,bulk ) are based on (1) previous estimates derivedfrom empirical data for this growth rate [28], (2) global mRNAhalf-life measurements at 37 u C in LB broth [40], and (3) genelengths and map locations obtained from the sequenced genome of
E. coli . These genetic parameters are summarized in Table S1. U max bulk was set at several plausible values (above observed averagetranslation initiation rates [3,80,81] and below the maximum limitwhere ribosomes are close-packed), with the remaining parametersestimated to minimize the mean square error (MSE) with respectto the WT cell state (Table 1). Errors in estimation of the cell statewere no more than 6% of the observed WT cell state and withinexperimental error bounds of these measurements ( , L m,bulk is of same order of magnitude as theconcentration of free ribosome, n ribo,free , indicating that the RBSsare not saturated by free ribosomes, in agreement with perviousstudies [30–33]. Further details are given in S1.1.1 of Text S1. rRNA Operon Inactivation Experiments: The Squires Data In the series of experiments that we consider here, Asai et al.[19] have measured the growth rate and rRNA to total proteinratio of seven
E. coli strains, with rRNA operon copy numbersranging from one to seven per chromosome (Figure 2). Since allstrains were grown in a constant environment of Luria-Bertanibroth at 37 u C ( m = 2.0 doub/h for the WT strain), the CGGRmodel is applicable. We first reconstructed the WT geneticparameters and the relevant physical constants (C periods andelongation rates) for a growth rate of 2 doub/h (c.f. Table S5 andS1.2 in Text S1 for a detailed account). Next, by analyzing thepublished lineage of these strains (Table S6) we derived the genetic Table 2.
CGGR variables, parameters and constants.
GeneticParameters units V max i Maximum transcription initiation rate of the i-th gene class U max i Maximum translation initiation rate of the i-th gene class mRNA K m,i Binding affinity of RNAp holoenzyme to the i-th gene class promoter m m) L m,i Binding affinity of the 30S ribosome subunit to the i-th gene class mRNA RBS m m) t c,i Average time to transcribe the i-th gene class ( = L i /c i ) min t l,i Average time to translate the mRNA of the i-th gene class ( = L i /3c p ) min m i,j Map location of the j-th gene in the i-th gene class dimensionless T fun , o = i Functional half-life for the i-th gene class mRNA in the absence of ribosomes min V ini Initiation volume ( m m) Cellstatevariables n RNAp
Concentration of total RNAp m m) n RNAp,free
Concentration of free functional RNAp m m) n ribo Concentration of total ribosomes m m) n ribo,free Concentration of free functional (30S) ribosomes m m) n bulk Concentration of bulk protein m m) a Specific growth rate ( a = m ? ln(2), where m is the doubling rate) ParametersandconstantsfortheunconstrainedCGGR c ribo/load Production cost of one ribosome or load protein dimensionless n Minimum cell density m m) ParametersandconstantsfortheconstrainedCGGR M bulk Bulk protein cutoff m m) c max p Maximal elongation rate aa/min h Hill coefficient dimensionless
Otherparameters,variablesandconstants c p Peptide chain elongation rate aa/min c i RNA chain elongation rate of the i-th gene class bp/min C C period min d i Gene concentration of the i-th gene class m m) L i Length of the i-th gene class base pairs v k Volume of a macromolecule belonging to the k-th species ( m m) W Macromolecule volume fraction = V macro /V cell dimensionless Genetic parameters, cell state variables, and other variables and constants associated with the CGGR models. Gene classes labeled by index i include: rrn , r-protein, bulkprotein and any load genes. arameters for each specific strain (Table S7). In S1.3 of Text S1we explain which genetic parameters for the WT cells can becarried over to the inactivation strains and which parameterschange, and how. The WT cell state at 2 doub/h is given in TableS2 and the genetic parameters at 2 doub/h for the WT cell andthe inactivation strains are summarized in Table S5 and Table S7,respectively.Given the genetic parameters, we set to solve Eq. 2 for thedifferent strains. However, in order to solve for the CGGR cellstate, which consists of six state variables, we need an additionalrelation which apparently does not arise from kinetic consider-ations. A hint to the solution may lie in the fact that so far we haveneglected the function of the bulk protein and biophysicalconsiderations such as macromolecular crowding. Macromolecular Crowding and the Function of the BulkProtein
The in vivo milieu of
E. coli is extremely crowded withmacromolecules [45] with typical values of macromolecule volumefraction W = V macro /V cell of 0.3–0.4 [46]. Observations of WT E.coli in varying environments suggest that the macromolecular massdensity of the interior of the cell is more or less a constant [23]. Ifwe neglect the contribution of RNAp, mRNA and DNA ( ,
6% at2.5 doub/h [17]) this is roughly equivalent to stating that W : v ribo n ribo z v bulk n bulk z v load n load ~ const ð Þ where v i is the volume occupied by a particle belonging to the i -thspecies (c.f. Table S2), and with potential contribution from ‘‘loadgenes’’ that express products not utilized by the cell and pose apure burden, like antibiotic resistance for example. Equation 4,which balances bulk protein against ribosomes, leads to acontradiction: it appears from this model, that by geneticperturbations, e.g. by increasing the rRNA operon copy number,one could construct a cell composed almost entirely of ribosomeswith no bulk proteins to support it, or vice versa. To resolve thisdifficulty we need to take into account the fact that some of thebulk proteins are required to support ribosome synthesis.One possible resolution is to introduce a mechanism that wouldlimit protein and ribosome synthesis when bulk protein density isreduced. For example, one could assume that the peptide chainelongation rate, c p , is given by c p ~ c max p . z M bulk = n bulk ð Þ h h i ,where h is some Hill coefficient, c max p is the maximal elongationrate and M bulk is a cutoff. M bulk may depend on the environment,reflecting the dependence of c p on the environment [17]. Note that c p affects our system of equations through the translation times t l,i . Figure 2. Comparison of rRNA operon inactivation data of Asaiet al. [ ] to CGGR models and predictions for higher rRNAoperon copy numbers. (A) Growth rate as a function of rRNA operoncopy number per chromosome. The maximum standard error of growthrate measurements was 0.07 [19]. (B) rRNA to total protein ratio, wheretotal protein is given by total amino acids in the form of r-proteins andbulk proteins. Measurement error was not available for this data. In thecase of the constrained model, solutions were not obtainable above acopy number 11. (C) Ribosome efficiency, defined as e r = a ? P/N ribo [3,19](see text), was obtained by dividing the growth rate in (A) by the ratio ofrRNA to total protein in (B). All curves are normalized to WT values at2.0 doub/h. The legend to all figures is given in (A). The kink observed forcopy number 2 is due to strong expression of lacZ in this strain used forinactivation. Beyond the WT rRNA gene dosage, rRNA operons wereadded at the origin (also see S1.4 in Text S1). The rRNA chain elongationrate, c rrn , was assumed to be fixed in these simulations.7 his criterion along with Eq. 2 and Eq. 4 define the constrainedCGGR model.However, is the macromolecular volume fraction, W , reallyconstant? The phenomenological evidence indicating that W isroughly constant has been obtained for WT cells in differentenvironments, and not for a suboptimal mutant growing in a givenenvironment like the Squires strains. Indeed, it has been proposedthat W can vary by adjusting the level of cytoplasmic water tocounter changes in the external osmotic pressure [82]. Theseobservations suggest that W = const is apparently not a universallaw in E. coli .An alternative resolution, which does not hypothesize that W = const, could be to postulate a cost criterion , which states that theamount of ribosomes that the cell can produce is limited byresources, such as ATP, amino acids, etc., that are made availableby the bulk proteins. Assuming that bulk protein concentration, n bulk , is proportional to its demand, i.e. to total ribosomeconcentration, n ribo , and also to possible load protein concentra-tion, n load , the criterion takes the form: n bulk ~ n z c ribo n ribo z c load n load ð Þ where c i are the costs and n is some minimal density of the cell(e.g. housekeeping proteins, membrane building proteins etc.),assumed to be more or less constant. c ribo , for example, is definedas the number of bulk proteins per cell, N bulk , required to increasethe number ribosomes in the cell, N ribo , by one, given a fixedenvironment E (i.e. sugar level, temperature, etc.), a fixed cellvolume and a fixed number proteins, N j , expressed from all othergenes (akin to the definition of a chemical potential): c i ~ L N bulk L N i (cid:4) (cid:5) E , V cell , N j = i ð Þ In other words, to synthesize and support one additional ribosomeper cell, in a constant environment, cell volume etc., according tothis definition, would require an additional c ribo bulk proteins percell ( c ribo is dimensionless). An equivalent way to interpret Eq. 5 isto say that c ribo is the capacity of a ribosome to synthesize bulkproteins: one additional ribosome added to the cell will synthesize c ribo bulk proteins. Thus, at steady-state, cost and capacity aredifferent sides of the same coin. The costs, c i , depend on theenvironment since the cost of producing and maintaining aribosome in a rich environment is expected to be lower than thecost in a poor environment due to the availability of readymaderesources that otherwise the cell would need to produce on itsown. The hypothesized costs, c i , can therefore be thought of as effective environment-dependent genetic parameters and could, inprinciple, be estimated from knowledge of the genetic networksinvoked in a given growth environment. Note that Eq. 4 isactually a special case of Eq. 5 for certain negative costs. Eq. 5also crystallizes the difference between bulk proteins and loadproteins: the latter are a burden for the former. The cost criteriontogether with Eq. 2 define the unconstrained CGGR model. Thefinal equation set for both models is summarized in S2.6 ofText S1.From an experimental point of view it should be possible todiscern between the two hypotheses: one model (Eq. 5) predicts apositive slope for the n bulk vs. n ribo curve, whereas the other model(Eq. 4) predicts a negative slope. In the Discussion we suggest howthe cost criterion may naturally occur in the cell. Global Crowding Effects.
Since macromolecular volumefraction can change in the unconstrained CGGR model due to Eq. 5, it is essential to examine how crowding can affect the inputgenetic parameters. We considered two possible crowdingscenarios (c.f. S2.4 in Text S1). In the ‘‘transition state’’ scenarioit was assumed that holoenzyme-promoter and 30S-RBS bindingaffinity are transition state limited, that is, the probability that anassociation complex will decay to a product is small comparedwith the probability that it will dissociate back into the reactants[83]. Typically such reactions display an increase in efficiency ascrowding is initially increased and eventually decrease in efficiencysince in the limit of high fractional volume occupancy, allassociation reactions are expected to be diffusion limited andhence slowed down [44]. The forward rate of transition statereactions is predicted to display a bi-modal dependence on themacromolecular volume fraction W [44,51,83–85]. Such a bi-modal dependence has been observed experimentally in vitro [44,86]. Assuming binding affinities weaken in the limit of highvolume fraction we expect that in such a case the binding affinitywill display a maximum (Figure S8A). In the transition statebinding scenario we have further assumed the binding affinity hasbeen evolutionary tuned to be maximal at the WT value of W ( = 0.34 [46]), similar to the temperature optimum commonlyexhibited for RNAp/promoter complexes [75].A second, ‘‘diffusion limited’’ scenario, assumed that allreactions were diffusion limited, that is almost every associationcomplex will become a product [83]. For this scenario, bindingaffinities were assumed to decay exponentially (Figure S8B), as hasbeen observed in vitro for diffusion coefficients [47,51,87], andsuggested for the forward rate in diffusion limited reactions[44,51,83–85]. Thus, both models assumed that binding affinitiesdecay at high W ( . W , end of the mRNAappear to cancel out with the crowding effects on the bindingaffinity of RNase E to its own mRNA (c.f. S2.7 in Text S1 formore details). Comparison to the Experimental Data of Squires
With the CGGR models at hand, we now compute the cellstates for each of the seven strains used in the Squires rRNAoperon inactivation experiments. We will use this data to fit for theunknown environment dependent parameter in each of theCGGR models: c ribo for the unconstrained model and M bulk forthe constrained model. In the case of the unconstrained model, thepredicted rRNA to total protein ratio was more sensitive to c ribo than the predicted growth rate, with a best fit for the former at c ribo < bulk proteins per ribosome (for mean square errors referto Figure S1). For comparison, a 70S ribosome is about 70 bulkproteins in mass. Note that c ribo has a rather limited range ofvalues since 0 , c ribo , n bulk / n ribo .
101 via Eq. 5.For the constrained CGGR model, the minimum Hillcoefficient to yield a solution that did not diverge in growth ratefor copy numbers greater than 7, which contradicts observation(see Introduction), was h = 2 (see e.g. Figure S2 for a fit with h = 1).For h = 2, M bulk was chosen minimize the MSE with respect to thegrowth rate, which displayed a minimum, and the best fit was chieved for M bulk > ? molec/WT cell (for all MSEs c.f.Figure S1). Attempting to minimize the MSE with respect to therRNA to total protein ratio resulted in a slightly lower MSE(though still higher than the MSE for the unconstrained model fit),however solutions diverged in growth rate for copy numbersgreater than 7, again contradicting observation. Increasing the Hillcoefficient so as to penalize the peptide chain elongation rate, c p , for higher copy numbers did not remedy this and growth ratecontinued to diverge for copy numbers greater than 7 (c.f. FigureS3) rendering such solutions inapplicable. Finally, increasing theHill coefficient beyond 2 did not improve the overall MSE toeither the growth rate or to the rRNA to total protein ratio (FigureS1). Thus the fit for the constrained model presented hererepresents the best fit, over all parameter range, which does notcontradict observation.Figures 2A and 2B show the observed growth rate and rRNA tototal protein ratio plotted against the best fits of these models. Inboth cases, the fit to the observed data was reasonable, howeverthe model for which macromolecular volume fraction, W , was notconstrained gave an overall better fit indicating a preference forthat model. Further evidence in favor of this model and against theconstrained model will be considered in the Discussion. Thedeviation observed for the D Free RNAp and Free Ribosomes Self-Adjust to CounterChanges in Binding Affinities Due to Crowding
Whereas the macromolecular volume fraction W in theconstrained model is, by definition, constant, the unconstrainedCGGR model predicts that W increases with the number of rRNAoperons with consequences on binding affinities (Figure 3). Thisincrease in the macromolecular volume fraction is due to anincrease in both ribosome concentration and bulk proteinconcentration due to the relation imposed by the cost criterion(Eq. 5; also c.f. Figure S5). Quite surprisingly, the fit to the Squiresdata depends very little on the crowding scenario chosen. Thisresults from a self-adjusting homeostasis mechanism: it is the ratiosof free RNAp and free ribosomes with respect to theircorresponding binding affinities that govern the transcriptionand translation rates (Eq. 1). Hence, although the binding affinitieschange with W , the concentrations of free RNAp and freeribosomes counterchange to stabilize these ratios (see Figure S4and S1.6 in Text S1). The efficiency of the homeostaticmechanism diminishes as the degree of crowding is increasedabove , W (Figure 2A and 2B). Translation-Degradation Coupling
Due to translation-degradation coupling, bulk mRNA half-lifewas predicted to mildly increase with rRNA operon copy numberfor all models. In both crowding scenarios, bulk mRNA half-lifeincreased from about 0.8 of the WT half-life to about 1.2 of theWT half-life. The increase in mRNA half-life is caused by theincrease in the ratio of the RBS binding affinity and theconcentration of free ribosomes with rRNA operon copy number(Figure S4). This ratio reflects the probability that a RBS isoccupied, thereby protecting the mRNA from cleavage.
Beyond a Copy Number of Seven
Increasing the rRNA operon copy number beyond 7 (at maplocation 0), we found that both CGGR models exhibit a shallowoptimum plateau for of the growth rate in the range of 7–12 copies, with the maximum occurring at a copy number of 10–11for the diffusion limited scenario, and 11–12 for the transition statescenario. In the case of the unconstrained models, overcrowdingcontributed to the formation of this maximum (e.g. there is nomaximum in the unrealistic model where binding affinities areassumed to be independent of W ). A striking difference between themodels is in their predictions regarding the rRNA to total proteinratio. This ratio strongly diverges in the constrained model at highcopy numbers because ribosomes are formed at the expense ofbulk protein (see Discussion). A Simplified Model
For the data of the Squires strains, the unconstrained CGGRcan be approximated by a simplified three-state model involvingonly n ribo , n bulk , and m (c.f. S3 in Text S1): i ð Þ n ribo ~ g rrn = m ii ð Þ n bulk ~ g bulk n ribo = m iii ð Þ n bulk ~ n z c ribo n ribo ð Þ where g rrn and g bulk are effective genetic parameters that areestimated from the WT cell state. Equation (i) reflects ribosomesynthesis, (ii) reflects bulk protein synthesis and (iii) is the costcriterion. Interestingly, in a different context of WT cells measured Figure 3. Predicted effect of crowding on the rRNA promoterbinding affinity for two crowding scenarios.
In the transition statescenario, binding affinities initially strengthen as macromolecularcrowding is increased due to increased entropic forces, while in thediffusion limited scenario binding affinities weaken as macromolecularcrowding is increased due to increased diffusion times. In both cases,binding affinities weaken when macromolecular crowding is increasedbeyond the WT crowding state due to increased diffusion times (seealso Figures S8 and S2.4 in Text S1). Quantitatively, affinities can vary byup to a factor of 5 (transition state limited scenario) to 16 (diffusionlimited scenario), measured as the ratio of the maximum and minimumvalues of binding affinities in the range of rRNA operon copy numbersconsidered. Similar results are obtained for RBS binding affinities.Binding affinities were normalized to WT values. Insert: predictedmacromolecular volume fraction, W , as a function of rRNA operon copynumber per chromosome.A Coarse-Grained Model of E. coli and Applications9 n varying environments, a relation similar to Eq. (ii) has beenobserved [21]. Solving Eq. 7 for the growth rate we obtain m ~ c ribo g rrn n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z n g bulk g rrn c ribo r { ! ð Þ In the limit g rrn R ‘ , m g bulk / c ribo , suggesting that in the absenceof crowding effects, growth rate would be limited by theproduction cost of a ribosome. To fit to experiments where therRNA operon copy number is manipulated, we approximate that g rrn R g rrn ? copy /7. The best fit to the Squires data was obtainedfor c ribo . Ribosome Efficiency
Ribosome efficiency has been previously defined as e r ; b r c p = a P / N ribo [3,19] where c p is the peptide chain elongation rate and b r is the fraction of actively translating ribosomes. For wild-typecells, b r = 80%, and is independent of growth rate [17].Genetically perturbed cells may however respond differently[19]. For example, the simplified model predicts that e r = a P / N ribo = ln 2( g bulk L bulk + m L r-protein ), where L i is the length of gene class i and m is given by Eq. 8. Since c p is assumed to be fixed in theunconstrained/simplified models, ribosome efficiency is thereforeexpected to decrease purely due to kinetic considerations.Crowding effects tend to either increase or decrease ribosomeefficiency, depending on the scenario. In Figure 2C we plot theribosome efficiency for the various crowding scenarios in theunconstrained CGGR model, for the constrained CGGR modeland for the simplified three-state model. We see that the datapoints lie between the diffusion limited crowding scenario and thetransition state crowding scenario, which possibly indicates thatthe in vivo crowding scenario is somewhere between being diffusionlimited and transition state limited. Overall however, the diffusionlimited model was a better predictor of ribosome efficiency thanthe transition state model and its deviation from the observed datapoints was on the order of the maximum error for these points (themaximum deviation from experimental data points is , in vivo case (seeDiscussion). The solution for which binding affinities areindependent of crowding (the ‘‘no crowding’’ scenario) also fitsthe data due to the proposed homoeostasis mechanism for W , , E. coli . Finally, thesimplified model appears to adequately trace the observedribosome efficiency.
The Initiation Rate of a Single rRNA Operon
Figure 4 shows the initiation rate of a single rRNA operon, V rrn ,as a function of the rRNA operon copy number as predicted bythe unconstrained model (Eq. 1). The solid lines represent modelswhere the rRNA chain elongation rate was assumed to be constant(85 nuc/sec [17]; Table S5). Both unconstrained models exhibitan increase in rRNA expression per operon as copy number isdecreased from 19 copies per chromosome down to 3 copies perchromosome (in the case of the diffusion limited model) and 5 copies per chromosome (in the case of the transition state model).This trend is in agreement with the feedback response mechanism,especially for the diffusion limited model (see Discussion). It hasbeen shown that the rRNA chain elongation rate (but apparentlynot mRNA chain elongation rate) increases in inactivation strainsfrom ,
90 nuc/sec in a WT strain to ,
135 nuc/sec in a strainwith four inactivated rRNA operons [54], but remains constant instrains with increased rRNA gene dosage [65]. To check howthese finding affect our predictions, we also included a modelwhere rRNA chain elongation rate decreased linearly from160 nuc/sec for one functional rRNA operon per chromosome,to 85 nuc/sec for the WT strain (dashed lines in Figure 4). Indeed,the feedback response seems to be stronger for the inactivationstrains when assuming that rRNA chain elongation rate increasesas more operons are inactivated. Quantitatively, for the diffusionlimited model, rRNA expression from a single operon increasedfrom 0.6 of the WT expression for 19 chromosomal rRNAoperons to about 1.1 of the WT expression for 3 chromosomalrRNA operons. Finally, the ‘‘no crowding’’ scenario exhibited amilder feedback response due to departure from the homeostasisdiscussed earlier.
Discussion
The goal of the coarse-grain genetic reactor (CGGR) approachis to attempt to link global phenotypes, such as growth rate andcell composition, directly to genetic parameters, while keeping themodel as simple as possible by means of coarse-graining. Thepresent CGGR models assumed the simplest type of environment,namely a spatially and temporally constant environment that isunlimited in resources. The models attempt to explain a series of
Figure 4. Initiation rate of a single rRNA operon ( V rrn ) forvarious crowding scenarios in the unconstrained CGGR model. Solid lines are predictions for the case where the rRNA chain elongationrate is assumed to be fixed at its WT value of 85 nuc/sec [17], while thedashed lines take into account the observed effect of rRNA operon copynumber on rRNA elongation rates [54,65] (see text for further details). V rrn is given by Eq. 1. In each case considered, c ribo was obtained byfitting to the Squires data.10 xperiments performed by the Squires group [19] in which growthrate and cell composition have been measured for seven E. coli strains with varying rRNA operon copy numbers. The genome ofall seven strains has been coarse-grained, and their correspondingcell state was calculated based on the CGGR models.We considered two possible CGGR models, one in which themacromolecular volume fraction is constrained to be fixed, andone in which macromolecular volume fraction is unconstrained.We have seen that the unconstrained CGGR model appears togive an adequate fit to experimental data, while the fit for theconstrained CGGR model is rather poor (despite the latter havingan additional degree of freedom). Yet beyond the fit of theunconstrained model to the Squires data, this model also appearsto be consistent with additional observations regarding strains withaltered rRNA operon copy numbers. For example, the uncon-strained CGGR model predicts that growth rate decreases forhigher rrn copy numbers, as indicated by observation. Forcomparison, the best fit of the constrained CGGR model actuallypredicted that growth rate increases for rRNA operon copynumbers greater than 7, contradicting observation. In addition,both models predict that the concentration of ribosomes (andribosomes per cell) decreases with rRNA operon copy number(Figure S5), as was shown in measurements of an earlier set ofinactivation strains engineered by the same group, with rRNAoperon copy number ranging from three to seven [54]. Below wediscuss further evidence in support of the unconstrained CGGRmodel: observations regarding the nucleoid size in strains withaltered rRNA gene dosage appear to be consistent with thecrowding predictions of this model. Finally, the unconstrained(diffusion limited) CGGR model is in agreement with the trendassociated with the feedback response, and appears to be inqualitative agreement with measurements of this effect. Theproposed model is also consistent with the feedback modelproposed by Nomura and coworkers, as will be discussed furtherbelow. The constrained CGGR model, on the other hand, inaddition to yielding an inferior fit to the Squires data, is alsoproblematic from a biological standpoint. This model shouldpredict that ppGpp levels rise due to a shortage in an essentialfactor such as charged tRNAs [3,6,89]. However, ppGpp levelswere observed to be constant in similar rRNA inactivation strainswith up to four inactivations [54]. In addition, the constrainedmodel appears to be considerably more complicated than theunconstrained model in that it necessitates some kind ofhomeostasis mechanism for keeping the volume fraction fixed, towhich there is no experimental evidence as far as we know, whilethe unconstrained model does not necessitate any additionalbiological mechanisms (see below). In fact, evidence fromosmotically stressed cells indicates that the volume fraction ofmacromolecules can change quite considerably [82]. Indeed, theseexperiments indicate growth rate can be limited by crowding [82],just as predicted by the unconstrained CGGR model (see below).Since the macromolecular volume fraction in the unconstrainedCGGR model is not constant, we needed to consider crowdingeffects on association reactions such as transcription initiation andtranslation initiation. We investigated two possible crowdingscenarios: one in which all association reactions are diffusionlimited and one in which all association reactions are transitionstate limited and have been evolutionally tuned to be maximal atthe WT volume fraction. Both crowding scenarios give anadequate fit to the growth rate and rRNA to total protein ratiodata, thanks to the homeostasis mechanism involving free RNApand free ribosomes. However, the diffusion limited model seems togive a slightly better fit when considering the feedback responseand ribosome efficiency data, possibly indicating a preference for this model. Indeed, it has been proposed that the in vitro in vitro measured on rates are of the order of the diffusion limit [90]. Inaddition we have proposed a simplified version of the uncon-strained CGGR model, which is a three-variable model and isincluded since it is an analytically solvable reduction of the morecomplicated six state model. We have shown, however, that sincethe simplified model does not take into account the physical effectsof crowding, its predictions for strains with increased rRNAoperon gene dosage is unrealistic. Hence the full unconstrainedCGGR model is the biophysical model that we propose to berelevant for
E. coli growing in balanced exponential steady-stategrowth in a rich medium.
Nucleoid Compaction in the Inactivation Strains
Further support for the reduction of macromolecular volumefraction in the rRNA inactivation strains may perhaps be found influorescence images of the WT Squires strain vs. the D D rrn operonsentirely deleted from the genome (S. Quan and C. L Squires,personal communication). These results are consistent withcrowding effects [48] predicted by the unconstrained CGGRmodel, effects that are absent in the constrained CGGR model. The Feedback Response of the rRNA Operons
While both unconstrained CGGR models exhibited a decreasein the expression of a single rRNA operon as rRNA gene dosagewas increased, as is observed in the feedback response, in the caseof the inactivation strains, the diffusion limited model appeared tobe in better agreement with the feedback response than thetransition state model (Figure 4). In the former model, rRNAexpression from a single rRNA operon increased as rRNA operoncopy number was decreased from 19 copies per chromosome to 3copies per chromosome. The transition state model exhibited thisdependence only up to an rRNA operon copy number of 5. Theincrease in the rRNA operon expression is due to an increase inthe ratio of free RNAp concentration and the rRNA operonbinding affinity (Figure S4). It is interesting to note that in thediffusion limited scenario, it is actually the changes in bindingaffinities, and not free RNAp, which correct for the observed trendof the feedback response, as free RNAp concentration is actuallypredicted to increase when the number or rRNA operons perchromosome is increased (Figure S4B). Furthermore, we foundthat a model in which the rRNA chain elongation rate increaseswhen inactivating rRNA operons, as observed experimentally[54], exhibits a slightly stronger feedback response wheninactivating rRNA operons than a model that assumes that thisparameter is constant.Quantitatively, in the case of the diffusion limited scenario withvariable rRNA chain elongation rate, the rRNA expression from asingle operon increased from 0.6 of the WT expression for 19chromosomal rRNA operons to about 1.1 of the WT expressionfor 3 chromosomal rRNA operons. Although rRNA operonsynthesis rate was not measured for the inactivation strainsconsidered here, we can qualitatively compare these predictions toexperiments with other strains. Strains in which four rRNA perons were inactivated exhibited a 1.4 to 1.5 increase rRNAoperon expression relative to a WT background, where expressionwas measured as b -galactosidase activity from WT P1 promoterfragments fused to a lacZ reporter gene and normalized toexpression from a WT background [64] (we are not aware ofmeasurements for lower copy numbers). In a similar mannerrRNA expression was shown to decrease by a factor of 0.65 to 0.8with respect to the WT background in strains in which rRNA genedosage increased by using plasmids expressing rRNA (the plasmidcopy number was not specified) [64]. In a different study by thesame group, the initiation rate in strains with increased rRNAoperon copy number was obtained based on counting the numberof RNAp bound to rRNA operons using electron microscopy andmeasurement of the rRNA elongation rate, and yielded 0.66 of theWT initiation rate [65]. Although the predicted feedback responsefor the inactivated strains is somewhat weaker than the responseobserved experimentally, the overall trend appears to be inqualitative agreement with the feedback response, i.e. as the rRNAoperon copy number is increased, the transcription from a singlerRNA operon decreases. We note however that the geneticmakeup of the inactivated strains tested above differed from theinactivated strains of Asai et al. [91], especially in the respect thatin the former strains, each inactivated rRNA operon expressedantibiotic resistance, which may have had adverse effects on thecell. The fact that the elicited feedback response is not as strong asthe one observed experimentally in the inactivation strains mayalso possibly be a consequence of some of the simplifyingassumptions made in this model (e.g. ideal RNAp autoregulationor the somewhat naı¨ve crowding models assumed) or perhapsindicate the presence of an additional mediator (see below).The unconstrained CGGR model also predicts that bulkmRNA transcription would be affected by the change in rRNAgene dosage since in the current model bulk RNAp bindingaffinity has the same response to changes in macromolecularcrowding as the rRNA binding affinity. The effect may be,however, somewhat alleviated by the fact that bulk mRNA bindingaffinity is proposed to be about 3 times stronger than the P1 rRNApromoter (which is the major site for the feedback response [92]) atthis growth rate (Table S4), thus closer to saturation, and can evenbe ,
30 times stronger in poor medium (Table 2 in [28]), althoughis has also been suggested that RNAp promoters may require thesame or less RNAp than other RNA promoters for transcription[93]. Also, in principle rRNA and bulk promoters could responddifferently to crowding. When measured experimentally, mRNApromoters did in fact exhibit some reduction when the feedbackresponse was induced using increased rRNA gene dosage: whileexpression of a P1- lac
Z fusion decreased by 0.45 relative to acontrol with WT rRNA gene dosage, spc or lac UV5 promotersfused to lac
Z decreased by , Effect of Increased rRNA Operon Copy Number onGrowth Rate
Extrapolating to higher copy numbers suggests that the WTgrowth rate in a constant and rich environment is nearly maximal.In an experiment with increased rRNA gene dosage, whereppGpp concentration was shown to be constant, the growth rate ofa strain carrying extra rRNA operons on a plasmid indeeddecreased by 22% relative to a WT strain carrying a controlplasmid expressing nonfunctional rRNA [20], in agreement withthe trend predicted by the model. In another experiment withincreased rRNA gene dosage, growth rate decreased relative toWT cells containing a control plasmid, and rRNA to total proteinratio was more or less constant (thus appearing to favor theunconstrained CGGR model) although the authors argue thatthere may be tRNA imbalance in these strains [94]. In addition,the unconstrained CGGR model predicted that ribosome andbulk protein concentration increase with rRNA operon copynumber (Figure S5) thus leading to an increase in themacromolecular volume fraction (Figure 3, insert). This increaseis due to the cost criterion hypothesis (Eq. 5), which correlated theconcentration of bulk protein in the cell with the concentration ofribosomes.
The Optimum in Growth Rate
The biophysical origin of the predicted upper limit on growthrate with respect to the rRNA operon copy number, suggested bythe unconstrained CGGR model, is overcrowding of thecytoplasm with ribosomes and with bulk proteins supporting/synthesized by those additional ribosomes via the cost criterionrelation (Eq. 5). As rRNA operon copy number is increased, theconcentration of ribosomes and bulk protein increases (Figure S5)leading to an increase in macromolecular volume fraction in thecell (Figure 3, insert).
In vitro experiments suggest that in a crowdedenvironment diffusion times increase [47,51,87]. If in anovercrowded environment, when all reactions are thought to be iffusion limited [44,51,83–85], increased diffusion times causebinding affinities to weaken, then overcrowding will reduce theefficiency of transcription initiation and translation initiation(Figure 4 and Figure S4). This reduction in efficiency ultimatelycauses the growth rate to decrease at high rRNA operon copynumbers. In the scenario where binding affinities were assumed tobe independent of the level of crowding in the cell (the ‘nocrowding’ scenario in Figure 2A), growth rate continued toincrease as rRNA operon copy number increased, indicating thatthe reduction in growth rate in the transition state and diffusionlimited crowding scenarios was due to crowding effects. See alsoFigure S6 for a breakdown of the different contributions in theribosome synthesis equation, Eq. 2iii. Interestingly, a similarphenomenon may be occurring in osmotically stressed cells. It hasbeen shown experimentally that the growth rate of osmoticallystressed cells is correlated with the amount of cytoplasmic water inthose cells [82] leading those authors to propose that increaseddiffusion times of biopolymers due to crowding may be limitinggrowth rate. This conclusion appears to be in accord with ourfindings.The fact that the maximum in growth rate is so shallow maysuggest that in a natural environment for E. coli there areadditional constraints in the system. In nature,
E. coli is likely toexperience chronic starvation conditions like in water systems, aswell as fluctuating environments like in the host intestine [15,16].Indeed, it has been shown that
E. coli ’s growth rate displays a morepronounced dependence on the rRNA operon copy number in achanging environment compared to a constant one [15], and thata high rRNA operon copy number enables
E. coli and otherbacteria to adapt more quickly to changing environments[15,95,96].Finally we wish to point out that the optimum we have shown isonly with respect to rrn copy number perturbations of a WT
E. coli genome, and therefore may possibly not be a global one. A highergrowth rate could perhaps be attained when consideringperturbations of all genetic parameters.
Efficiency and Decoupling of the Replication InitiationModule
The unconstrained CGGR model suggests possible insights intothe design principles of
E. coli . The model introduces the conceptof a cost per gene class, akin to a chemical potential. In theabsence of load genes for example, the cost criterion basicallymeasures the number of bulk proteins needed to support thesynthesis of ribosomes (or vice versa). This criterion implies that thecell is efficient : bulk protein is utilized to its full potential and is notstored as inventory for later use. This is true even for geneticallyperturbed (i.e. suboptimal) cells. A similar notion of efficiency wassuggested by Ecker and Schaechter in the context of WT cellsgrowing in different environments [21]. How then is the costcriterion realized by the cell? Perhaps the cost criterion is realizedsimply by virtue of internal feedback. If, for example, the rRNAoperon copy number is slightly increased, resulting in a smallincrease in ribosome concentration, D n ribo , the transient deficit inbulk protein ( D n bulk ) will be compensated for, at steady-state, bythe extra ribosomes when D n bulk ( = c ribo D n ribo ) bulk proteins aresynthesized. n bulk therefore increases to the minimum concentrationneeded to sustain these excess ribosomes. Thus, the cost criterionobviates the need for a homeostatic mechanism for keeping W fixed. Nevertheless, direct experimental proof for the cost criterionis currently lacking.An additional engineering principle suggested by the CGGRmodels is related to the DNA replication mechanism. Replicationenters the model through the C period and the initiation volume (Eq. 3), both of which are regulated to be roughly constant [23,41]and thus in principle unaffected by genetic perturbations (Tadmorand Tlusty, in preparation). Since this implies that geneconcentrations do not depend strongly on growth rate (see FigureS7 and S2.2 in Text S1), this result suggests that the regulatorymechanism of replication initiation may be designed to bedecoupled from the cell state. Such a scheme may simplify thetask of engineering global regulation mechanisms such as the oneresponsible for rRNA regulation in different growth conditions orgrowth phases. Assumptions and Further Predictions
The CCGR models rely on many assumptions, the validity ofwhich should be questioned. One possibility is that the coarse-graining has discarded ‘‘hidden variables’’. Such variables mayinclude, for example, the structure of the nucleoid andtranscription factors associated with it (which can affect globaltranscription [97]), or the osmotic response of the cell [82]. Inaddition, strong genetic perturbations may lead to ribosomeinstability [98] and possibly induce a stress response with globaleffects. Other concerns may be possible additional factorsregulating rRNA synthesis alluded to earlier, the validity of theassumptions regarding the function of the bulk protein and theexistence of limiting resources even in a rich environment. In aresource limited environment for example, state variables relatedto the energy metabolism of the cell would probably come intoplay. Although, regarding limitation of resources, as was pointedout in the Introduction, it has been demonstrated experimentallythat the concentration of NTP is constant or changes by only asmall amount when altering the rRNA operon copy number[20,64], and ppGpp is also constant in these strains [20,54]. Thelatter observation suggests that the cell is not limited, for example,by the availability of amino acid, charged tRNAs or carbon[69,89] (see also [6]). Another concern may be that some portionof the inactive RNAp, which was assumed to be inaccessiblebecause of pausing, is actually nonspecifically bound to DNA [28]and might serve as an additional reservoir of RNAp fortranscription initiation. With all these difficulties in mind, theadvantage of the CGGR modeling approach is that it offers aninitial conceptual framework for thinking about
E. coli whilemaking quantitative predictions. Such tests can be useful inidentifying factors that have been left out in this round of coarse-graining and can be subsequently added. Examples of quantitativepredictions include: (i) non-constancy of the macromolecularvolume fraction in genetically perturbed cells (Figure 3, insert) (ii)state variables and their relations, e.g. the cost criterion (Figure S5)(iii) decay of binding affinities at high volume fractions (Figure 3and Figure S8; raising the more general question of the nature ofcrowding effects on equilibrium constants) (iv) increase in bulkmRNA half-life with rRNA operon copy number. Yet another testto this model may be to increase rRNA gene dosage beyond theWT gene dosage, where the differences between the CGGRmodels is much more pronounced [28] (Figure 2B). Although thefocus here was on altering the rRNA operon copy number, othergenetic perturbations can be considered, like adding non-nativeproteins that only serve as a load on the cell. In such a case, in vivo diffusion times are expected to be increased due to increasedcrowding. Green fluorescent protein (GFP) diffusion coefficient didin fact appear to decrease in
E. coli cells overexpressing GFP,however GFP dimerization may have contributed to this effect, asnoted by Elowitz et al. [99]. Finally, the proposed model maysuggest testable predictions for the effect of genetic noise onprotein expression and growth rate. upporting Information Text S1
Complete Supporting Information
Table S1
Genetic parameters for
E. coli growing at 1 and2.5 doub/h, 37 u C. Table S2
Cell state and additional parameters for variousgrowth conditions.
Table S3
Reconstruction of the WT cell state for 1 and2.5 doub/h, 37 u C. Table S4
Transcription related parameters for 2 doub/h, 37 u C. Table S5
Genetic parameters for
E. coli growing at 2 doub/h,37 u C. Table S6
Lineage of the rrn inactivation strains.
Table S7
Genetic parameters for the rrn inactivation strains at2 doub/h, 37 u C. Figure S1
Mean square errors with respect to the Squires data.(A) Unconstrained CGGR MSE. Square root of the mean squareerror (MSE) as a function of c ribo in estimation of the growth rateand the rRNA to total protein ratio measured by Asai et al. [19].This graph was computed as follows: for a given n , optimal L m,bulk and c ribo that minimize the square error between anestimated WT cell state and the observed WT cell state wereobtained (see S1.1.1 in Text S1). Next, for those optimal L m,bulk and c ribo values, the growth rate curve and the rRNA/total proteincurve were calculated for the various rrn inactivation strains (c.f.S1.3 in Text S1) and the MSEs were calculated between these twocurves and the data points, yielding two errors for a given n (orequivalently c ribo ). Next, n is increased and the process isrepeated. The minimum MSE for the rRNA to total protein ratio(which displayed more sensitivity to c ribo than the growth rate) wasobtained for c ribo = n = 2.8 ? molec/WT cell). Circlesmark the cost for which W would be fixed in an unconstrainedCGGR model (i.e. when c i = v i /v bulk , which is equivalent to theconstrained CGGR model with h = 0 ). (B) Constrained CGGRMSE. Square root of the MSE in estimation of the growth rate andthe rRNA to total protein ratio as a function of M bulk and the Hillcoefficient h , for a model where W is assumed to be fixed, and c p ~ c max p . z M bulk = n bulk ð Þ h h i . This graph was computed asfollows: for a given c max p and h , optimal L m,bulk and M bulk thatminimize the square error between the estimated WT cell stateand the observed WT cell state were obtained. Note that thissquare error included the error between the estimated WT c p andthe observed WT value of c p at 2 doub/h (20 aa/sec). The error inprediction of the WT cell state was on the order of a few percent(data not shown). Next, for those optimal L m,bulk and M bulk values,the growth rate curve and the rRNA/total protein curve werecalculated for the various rrn inactivation strains and the MSE wascalculated between these curves and the data points. Next, c max p isincreased and the process repeated. The minimum Hill coefficientto yield a solution that did not diverge in growth rate for high rrn copy numbers was h = 2 (see e.g. Figure S2 for fit with h = 1). For h = 2, M bulk was chosen to minimize the growth rate error yielding: M bulk = 7.4 ? molec/WT cell ( c max p ~
73 aa = sec ).Solutions that minimized the rRNA/total protein MSE (corre-sponding to the minimum possible value for c max p , i.e. >
21 aa/sec)diverged in growth rate for copy numbers greater than 7 (seeFigure S3). In addition, the MSE did not improve for higher Hillcoefficients, as shown. Note that the minimization procedure in (A)and (B) are equivalent if we map M bulk « c ribo , c max p < n . (C)Simplified 3-state model MSE. Square root of the MSE inestimation of the growth rate and the rRNA to total protein ratioas a function of c ribo for the simplified model. Stars indicateminima. Circles indicate the same as in (A). The minima almostcoincide and were obtained for c ribo . U max bulk was set to 80 ini/min and therRNA chain elongation rate, c rrn , was assumed to be constant. Figure S2
Fit for the constrained CGGR model with Hillcoefficient h = 1. Comparison of the constrained CGGR modelwith Hill coefficient h = 1 to (A) growth rate measurements and (B)rRNA to total protein ratio measurements of Asai et al [19]. M bulk was chosen such that the product of growth rate error and rRNAto total protein error was minimal, yielding M bulk = 5.7 ? molec/WT cell ( c max p ~
45 aa = sec ). For MSE see Figure S1B. Notethat for h = 1, growth rate diverges with copy number. rRNAchain elongation rate, c rrn , was assumed to be constant in thissimulation. Figure S3
Fit for the constrained CGGR model with higher Hillcoefficients. Comparison of the constrained CGGR model withHill coefficients of 2, 4, 6, 8, and 10 to (A) growth ratemeasurements and (B) rRNA to total protein ratio measurementsof Asai et al [19]. We show the h = 2 case for both c max p ~
73 aa = sec ( M bulk = 7.4 ? molec/WT cell; as in Figure 2) and c max p ~
21 aa = sec . For all other cases, c max p was set to 21 aa/secand corresponds to the minimum possible value for M bulk. , a valuethat according to Figure S1B minimizes the MSE for the rRNA tototal protein ratio. This figure demonstrates that all solutions with c max p ~
21 aa = sec diverge in growth rate for rrn copy numbersgreater that 7. Higher Hill coefficients ( .
10) appear to benumerically unstable or insolvable for high copy numbers. Legendto both figures is given in (A).
Figure S4
Free RNAp and free ribosomes with respect tocorresponding binding affinities for various crowding scenarios. (A)Model prediction for n RNA p,free , n RNA p,free /K m,i and W for thetransition state limited and no crowding scenarios as a function ofthe rrn operon copy number. In the no crowding scenario the plotsfor n RNAp ,free and n RNA p,free /K m,i coincide. (B) Same as (A) but forthe diffusion limited scenario. (C) Model prediction for n ribo,free and n ribo,free /L m,i for the transition state limited and no crowdingscenarios as a function of the rrn operon copy number. (D) Same as(C) but for the diffusion limited scenario. All curves are normalizedto WT values at copy number 7. Note that in the diffusion limitedscenario, when rRNA operons are inactivated, free RNApconcentration actually decreases. The reasons for this are thatfirst, although the rRNA operons are inactivated, they continue tobe partly transcribed (c.f. S1.3 in Text S1). Second, as rRNAoperons are inactivated, growth rate is reduced (Figure 2A), whichtends to slightly increase gene concentrations via Eq. 3 (c.f. FigureS7B). Finally, there is the contribution of increased transcriptioninitiation. When rRNA operons are increased beyond seven copiesper chromosome, free RNAp concentration increases mainlybecause transcription initiation is reduced due to diminished inding affinities. See main text and S1.6 in Text S1 for furtherexplanations. Figure S5
Predictions for bulk protein and ribosome concen-trations as a function of the rrn operon copy number. (A) Totalconcentration of ribosomes (ribosomes per unit volume) in theconstrained and unconstrained CGGR models as a function of the rrn operon copy number. (B) Concentration of bulk protein(proteins per unit volume) in the constrained and unconstrainedCGGR models as a function of the rrn operon copy number. Solidlines are for fixed rrn chain elongation rate, c rrn = const, anddashed lines are for c rrn ? const, as described in the main text. Allcurves are normalized to WT cell state values (at copynumber = 7). Figure S6
Breakdown of the ribosome synthesis equation tocomponents for the diffusion limited scenario. (A) Variables inunits of concentration. d rrn - rrn gene concentration (total rRNAoperon copy number per unit volume); V rrn - rrn initiation rate peroperon (init/min/operon); n ribo - ribosome concentration (ribo-somes per unit volume), and m - growth rate. These parameters aretied by Eq. 2iii: a = d rrn ? V rrn /n ribo . (B) Variables in units of molec/cell. D rrn - rrn gene dosage (total rRNA operon copy number percell); N ribo - number of ribosomes per cell. These parameters aretied by Eq. 2iii: a = D rrn ? V rrn /N ribo . This simulation is for thediffusion limited scenario assuming that the rRNA chainelongation rate, c rrn , is variable, as described in the main text.All curves are normalized to WT cell state values (at copynumber = 7). Figure S7
Gene dosage and gene concentration as a function ofgrowth rate. (A) Gene dosage and (B) gene concentration for the rrn gene class and bulk gene class. C and D periods wereinterpolated based on data from table 2 of [17] as a second orderpolynomial in m . For this simulation we assumed that 66 evenlydistributed bulk genes are expressed (c.f. map locations in TableS1). The initiation volume, V ini , was assumed to be fixed[41,43,100]. See also main text and S2.2 in Text S1 for furtherexplanations. Figure S8
Dependence of binding affinities on the volumefraction W for the various crowding scenarios. (A) Normalizedinverse equilibrium constants, K m and L m (in units of 1/M),for the RNAp holoenzyme (radius 5.57 nm) and the 30S ribosomesubunit (radius 6.92 nm), respectively, in the transition statelimited model. The water molecule radius was taken to be0.138 nm [101] and the radius of the background crowding agentwas taken to be 3.06 nm [46]. (B) Normalized K m and L m forthe diffusion limited model (curves overlap). All curves werenormalized to values at the WT volume fraction of W = 0.34. SeeS2.4 in Text S1 for more details. Acknowledgments
We thank Uri Alon, Rob Phillips, Carlos Bustamante, Shalev Itzkovitz,Irene A. Chen, Michael L. Shuler, Evgeni V. Nikolaev, Gideon Schreiber,Maarten H. de Smit, Moselio Schaechter, Hans Bremer, Tomer Kalisky,Alon Zaslaver, Erez Dekel, David Wu, Elisha Moses, Roy Bar-Ziv, andOfer Vitells for their insightful comments. We are especially grateful for theassistance of Catherine L. Squires and Selwyn Quan.
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Washington, D.C.: ASM Press. pp 1458–1496.90. de Smit MH, Duin Jv (2003) Ribosomes on standby: a prelude to translational(re)initiation. In: Lapointe J, Brakier-Gingras L, eds. Translation mechanisms.1 edition ed. GeorgetownTX: Landes Bioscience. pp 298–321.91. Asai T, Zaporojets D, Squires C, Squires CL (1999) An Escherichia coli strainwith all chromosomal rRNA operons inactivated: Complete exchange of rRNAgenes between bacteria. Proc Natl Acad Sci U S A 96: 1971–1976.92. Gourse RL, Deboer HA, Nomura M (1986) DNA Determinants of Ribosomal-Rna Synthesis in Escherichia-Coli - Growth-Rate Dependent Regulation,Feedback Inhibition, Upstream Activation, Antitermination. Cell 44: 197–205.93. Barker MM, Gaal T, Gourse RL (2001) Mechanism of regulation oftranscription initiation by ppGpp. II. Models for positive control based onproperties of RNAP mutants and competition for RNAP. J Mol Biol 305:689–702.94. Baracchini E, Bremer H (1991) Control of Ribosomal-Rna Synthesis inEscherichia-Coli at Increased Rrn Gene Dosage - Role of GuanosineTetraphosphate and Ribosome Feedback. J Biol Chem 266: 11753–11760.95. Stevenson BS, Schmidt TM (2004) Life history implications of rRNA gene copynumber in Escherichia coli. Applied and Environmental Microbiology 70:6670–6677. 96. Klappenbach JA, Dunbar JM, Schmidt TM (2000) RRNA operon copynumber reflects ecological strategies of bacteria. Appl Environ Microbiol 66:1328–1333.97. Ishihama A (1999) Modulation of the nucleoid, the transcription apparatus,and the translation machinery in bacteria for stationary phase survival. GenesCells 4: 135–143.98. Dong HJ, Nilsson L, Kurland CG (1995) Gratuitous Overexpression of Genesin Escherichia-Coli Leads to Growth-Inhibition and Ribosome Destruction.J Bacteriol 177: 1497–1504.99. Elowitz MB, Surette MG, Wolf PE, Stock JB, Leibler S (1999) Protein mobilityin the cytoplasm of Escherichia coli. 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Phillies GDJ (1986) Universal Scaling Equation for Self-Diffusion byMacromolecules in Solution. Macromolecules 19: 2367–2376.88. Jain C, Belasco JG (1995) Rnase-E Autoregulates Its Synthesis by Controllingthe Degradation Rate of Its Own Messenger-Rna in Escherichia-Coli -Unusual Sensitivity of the Rne Transcript to Rnase-E Activity. Genes Dev 9:84–96.89. Cashel M, Gentry DR, Hernandez VJ, Vinella D (1996) The StringentResponse. In: Neidhardt FC, Curtiss RIII, Ingraham JL, Lin ECC, Low KB etal., eds. Escherichia coli and Salmonella typhimurium: cellular and molecularbiology. 2nd ed. Washington, D.C.: ASM Press. pp 1458–1496.90. de Smit MH, Duin Jv (2003) Ribosomes on standby: a prelude to translational(re)initiation. In: Lapointe J, Brakier-Gingras L, eds. Translation mechanisms.1 edition ed. GeorgetownTX: Landes Bioscience. pp 298–321.91. Asai T, Zaporojets D, Squires C, Squires CL (1999) An Escherichia coli strainwith all chromosomal rRNA operons inactivated: Complete exchange of rRNAgenes between bacteria. Proc Natl Acad Sci U S A 96: 1971–1976.92. Gourse RL, Deboer HA, Nomura M (1986) DNA Determinants of Ribosomal-Rna Synthesis in Escherichia-Coli - Growth-Rate Dependent Regulation,Feedback Inhibition, Upstream Activation, Antitermination. Cell 44: 197–205.93. Barker MM, Gaal T, Gourse RL (2001) Mechanism of regulation oftranscription initiation by ppGpp. II. Models for positive control based onproperties of RNAP mutants and competition for RNAP. J Mol Biol 305:689–702.94. Baracchini E, Bremer H (1991) Control of Ribosomal-Rna Synthesis inEscherichia-Coli at Increased Rrn Gene Dosage - Role of GuanosineTetraphosphate and Ribosome Feedback. J Biol Chem 266: 11753–11760.95. Stevenson BS, Schmidt TM (2004) Life history implications of rRNA gene copynumber in Escherichia coli. Applied and Environmental Microbiology 70:6670–6677. 96. Klappenbach JA, Dunbar JM, Schmidt TM (2000) RRNA operon copynumber reflects ecological strategies of bacteria. Appl Environ Microbiol 66:1328–1333.97. Ishihama A (1999) Modulation of the nucleoid, the transcription apparatus,and the translation machinery in bacteria for stationary phase survival. GenesCells 4: 135–143.98. Dong HJ, Nilsson L, Kurland CG (1995) Gratuitous Overexpression of Genesin Escherichia-Coli Leads to Growth-Inhibition and Ribosome Destruction.J Bacteriol 177: 1497–1504.99. Elowitz MB, Surette MG, Wolf PE, Stock JB, Leibler S (1999) Protein mobilityin the cytoplasm of Escherichia coli. J Bacteriol 181: 197–203.100. Donachie WD (1968) Relationship between Cell Size and Time of Initiation ofDNA Replication. Nature 219: 1077.101. Berg OG (1990) The Influence of Macromolecular Crowding on Thermody-namic Activity - Solubility and Dimerization Constants for Spherical andDumbbell-Shaped Molecules in a Hard-Sphere Mixture. Biopolymers 30:1027–1037.102. Neidhardt FC, Ingraham JL, Schaechter M (1990) Physiology of the BacterialCell: A Molecular Approach: Sinauer Associates Inc.103. Grunberg-Manago M (1996) Regulation of the expression of aminoacyl-tRNAsynthetases and translation factors. In: Neidhardt FC, Curtiss RIII,Ingraham JL, Lin ECC, Low KB et al., eds. Escherichia coli and Salmonellatyphimurium: cellular and molecular biology. 2nd ed. Washington, D.C.: ASMPress. pp 1432–1457.104. Forchham J, Lindahl L (1971) Growth Rate of Polypeptide Chains as aFunction of Cell Growth Rate in a Mutant of Escherichia-Coli 15. J Mol Biol55: 563–&.105. Zhang X, Dennis P, Ehrenberg M, Bremer H (2002) Kinetic properties of rrnpromoters in Escherichia coli. Biochimie 84: 981–996.106. Ellwood M, Nomura M (1980) Deletion of a Ribosomal Ribonucleic-AcidOperon in Escherichia-Coli. J Bacteriol 143: 1077–1080.107. Brosius J (1984) Plasmid Vectors for the Selection of Promoters. Gene 27:151–160.108. Close TJ, Rodriguez RL (1982) Construction and Characterization of theChloramphenicol-Resistance Gene Cartridge - a New Approach to theTranscriptional Mapping of Extrachromosomal Elements. Gene 20: 305–316.