Division of Mycobacterial Research, National Institute for Medical Research, London NW7 1AA, UK
Correspondence
Robert A. Cox
rcox{at}nimr.mrc.ac.uk
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ABSTRACT |
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INTRODUCTION |
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Knowledge of mycobacterial physiology has been limited by the slow growth of the major pathogens in the laboratory and other technical problems such as cell aggregation. The availability of the sequence of the genome of M. tuberculosis (Cole et al., 1998) and limited data for the macromolecular composition of its very close relative Mycobacterium bovis bacille CalmetteGuérin (BCG) (Winder & Rooney, 1970
) provide the basis for gaining insight into the growth of the tubercle bacillus and other mycobacteria through a comparative approach. In this study a single quantitative framework was sought to relate genomic and macromolecular properties to the rates of protein synthesis of M. bovis BCG and two other representative bacterial species, namely Streptomyces coelicolor A3(2) and Escherichia coli B/r. The three species range in their maximum specific growth rates (µmax) from 0·029 h1 (M. bovis BCG) to 1·73 h1 (E. coli).
S. coelicolor A3(2), which is Gram-positive, soil-dwelling and filamentous, is an aerial-mycelium-producing actinomycete (Shahab et al., 1996); the genome, which is linear rather than circular, is almost twice the size of that of M. tuberculosis. E. coli belongs to a group of organisms, Enterobacteriaceae, that are Gram-negative, rod-shaped and capable of ultra-fast growth (Neidhardt, 1996
). E. coli was defined as an ultra-fast grower (Cox, 2003
) because, when it is growing at its maximum rate the generation time, tD, is less than the time, C, needed to replicate the genome (C>tD), leading to new-born cells possessing more than one genome equivalent. In contrast, M. tuberculosis (a slow-grower) and S. coelicolor (a fast-grower) are thought to replicate their genomes once only during the cell division cycle.
The concept of a virtual or schematic cell was developed in this study to provide an instantaneous view of macromolecular synthesis, particularly protein synthesis, carried out by cells growing at their maximum rate. The quantitative approach not only provides a succinct way of summarizing a wide range of data but also provides models amenable to mathematical manipulation. This integrative approach was used to calculate the specific growth rate of hypothetical E. coli cells having one or two rRNA (rrn) operons per genome as models for mycobacteria and to explore factors limiting the maximum specific growth rate of the three species studied.
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THEORETICAL ANALYSES |
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The concept of the minimal cell.
As the supply of nutrients become scarce E. coli cells become smaller (Bremer & Dennis, 1996) and eventually attain a minimal size and a characteristic macromolecular composition (Jacobsen, 1974
, cited by Ingraham et al., 1983
). It is proposed that minimal cells are a general feature of bacterial growth. By using the superscript
, I indicate the value of the corresponding variable (
, etc.) for the minimal cell.
Significance of properties of cells growing at their maximum rate.
The maximum specific growth rate, µmax, is attained when cells are amply supplied with the most favourable nutrients. An asterisk (mp(av)*, etc.) is used to denote a property of a cell growing at its maximum rate. A cell's capacity for growth is defined by the properties of minimal cells and of cells growing at their maximum rate.
The concept of a virtual schematic cell.
The formulation is based on the notions of an average gene, an average mRNA and an average protein (see Table 1) and on a few key equations derived from data for macromolecular compositions of cells growing at different rates. The model cell provides an instantaneous view of macromolecular synthesis, particularly protein synthesis, carried out by an average cell growing at its maximum rate.
Quantitative aspects of cell processes
DNA synthesis.
Except for ultra-fast growth, tD is equal to the sum of periods relating to DNA synthesis [see equation (1)] where B is the period between cell division and the start of DNA replication and D is the period following the completion of DNA synthesis and cell division:
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The rate of elongation of DNA per replication fork (DNA, bp per fork s1), the size of the genome (lg, bp) and C (h) are, by definition related by equation (2):
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Transcription/translation coupling.
Coupling between the processes of bacterial transcription and translation has long been accepted (Stent, 1964; Byrne et al., 1964
). This coupling was vividly shown by electron microscopy (Miller et al., 1970
) because ribosomes were found to be attached to nascent transcripts, demonstrating that translation accompanies transcription. The same technique revealed that, at any instant, the chromosome of an individual E. coli cell is largely transcriptionally inactive and that few, if any, free polyribosomes are found in the cytoplasm. Although the chromosome is largely transcriptionally inactive, a high proportion of its genes are expressed during the lifetime of the cell, as shown by proteome (Tonella et al., 1988
) and transcriptome analysis (Bernstein et al., 2002
). Transcription/translation coupling is thought to require that the rates of mRNA elongation (
mRNA, nucleotides h1) and peptide chain elongation (
aa, amino acid residues h1) are co-ordinated (Bremer & Dennis, 1996
); see equation (4), where the factor 3 reflects the number of nucleotides per codon:
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If the amount of mRNA is represented by nm·nuc(av) nucleotides, and nR(av) is the number of ribosomes and R the fraction of ribosomes actively synthesizing proteins, then the ratio nm·nuc(av)/
R·nR(av)=
nucleotides per ribosome reflects the organization of an average polyribosome. The minimum value of
reflects the size of the footprint of an initiation form of RNA polymerase (RNAP) bound to a promoter, which is approximately 80 bp (Krummel & Chamberlin, 1989
). Accordingly, it is assumed that there is a minimum of 80 nucleotides of mRNA per ribosome. Neglecting secondary structure, this segment corresponds to a length of approximately 27 nm (270 Å), which is comparable to the diameter of a ribosome of 25 nm (Noller & Nomura, 1996
). The maximum value of
corresponds to the length of the region protected by ribosomes plus a region smaller than the number of nucleotides that form a binding site for a degradosome. An arbitrary value of 100 nucleotides (34 nm) was assumed for protected mRNA; that is, unprotected stretches of mRNA of 20 nucleotides or more were considered to be capable of binding degradosomes and, hence, to be very rapidly degraded. Kinetic studies summarized by Bremer & Dennis (1996)
led to estimates for
of 5580 nucleotides of mRNA per ribosome depending on the growth rate of E. coli (see Table 3
).
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Protein synthesis.
The rate-limiting step in protein synthesis is the interaction of a ternary complex of aminoacyl-tRNA, EF-Tu and GTP with the A site of the ribosome. The concentrations of aminoacyl-tRNA and EF-Tu are conveniently expressed as the number of copies of aminoacyl-tRNA per ribosome (naat/R) and the number of copies of EF-Tu per ribosome (nEF/R). It is assumed that both naat/R and nEF/R have characteristic values for a particular species during the growth cycle, irrespective of the specific growth rate (Cox, 2003).
The key equations for describing protein synthesis during exponential growth concern the specific protein synthesis rate, p(av) (fg protein synthesized per cell h1), which is defined by equation (8a):
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Previously, it was shown (Cox, 2003) for E. coli that
p(av) or
aa(av) is proportional to the third power of the RNA concentration [see, for example, equation (12)] for values of (mRNA(av)/mdc(av))
b/a.
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Rate of energy consumption.
Cell proliferation requires the uptake and consumption of energy. Approximately half of the energy is used for the synthesis of macromolecules, of which 90 % or more is used for protein synthesis (Ingraham et al., 1983). The formation of each peptide bond needs the consumption of 4·2 high-energy phosphate bonds. Thus, the rate of energy consumption,
E ATP equivalents h1, is estimated in terms of ATP equivalents by equation (13):
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RESULTS AND DISCUSSION |
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The macromolecular properties of E. coli B/r grown at 37 °C, in five different media (media AE), S. coelicolor A3(2) grown at different specific growth rates (µ=0·024 h1 to µ=0·300 h1) at 30 °C and M. bovis BCG (µ=0·029 h1) grown at 37 °C are presented respectively in Table 3 (based on the review of Bremer & Dennis, 1996
), Table 4
(based on Shahab et al., 1996
) and Table 5
(based on Winder & Rooney, 1970
; Cox, 2003
).
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The average time, trrn, for the transcription of an rrn operon was found to be 1·2 min by direct measurement, although the relation rRNA=6
aa [see equation (4)] suggests that
rRNA should be dependent on growth rate so that trrn would be expected to decrease from 1·4 min in medium A to 0·8 min in medium E. Irrespective of the specific growth rate, the complement, nrrn(av), of rrn operons was estimated to be in excess of the minimum number,
, needed to achieve the observed rate of rrn transcription. In other words, the specific rRNA synthesis rate could have been achieved by using fewer rrn operons more intensively. For discussion, see equations (6b) and (6c) of the Theoretical Analyses section and equations (6A) and (7A) of the Appendix. As shown in Table 3
, at high specific growth rates (µ=1·73 h1)
rRNA was found to be suboptimal (
rRNA<6
aa residues h1), values of iRNAP/ilim were found to range from 0·06 (µ=0·42 h1) to 0·90 (µ=1·73 h1), and
[see equation (6c)] was found to range from 0·9 operons (µ=0·42 h1) to 22·0 operons (µ=1·73 h1). The fraction
, which measures the relative activity of each rrn operon, was found to range from 0·07 (µ=0·42 h1) to 0·61 (µ=1·73 h1). Thus, E. coli growing at its maximum rate (µ=1·73 h1) has a complement of rrn operons in excess of its needs for pre-rRNA synthesis. It was estimated (see Appendix) that the maximum specific growth rates attainable for hypothetical strains with one, two, three and four rrn operons per genome were, respectively, 0·42 h1, 1·00 h1, 1·40 h1 and 1·70 h1. These conclusions are in accord with the specific growth rates observed on the progressive inactivation of rrn operons present in the E. coli genome (Condon et al., 1993
, 1995
). It was found that the specific growth rate was maintained by increasing the activities (both
rRNA and iRNAP/ilim) of the remaining operons when the number of functional operons per genome was reduced from seven to four. A reduction from seven to three operons led to a slower growth rate and to smaller cells, in agreement with the estimated value for three rrn operons per genome mentioned above. The unique property of ultra-fast growers is that, beyond the transition point from fast to ultra-fast growth (tD=C*+D*) the number of genome equivalents per cell, g(av), increases to two or more, with a concomitant increase in the number, nrrn(av), of rrn operons per cell [see equation (14)]:
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The rate of ribosome synthesis needed to support the maximum specific growth rate of 1·73 h1 could not be achieved without increasing nrrn(av) from 12·4 to a minimum of 22 operons through polyploidy.
Both the dry cell mass and volume of E. coli were found to increase 4·5-fold as µ increased from 0·42 h1 to 1·73 h1 (see Table 3). Empirically, linear plots were obtained by plotting µ2 against mdc(av) or v(av) (see Fig. 1
), which are summarized by equations (15a) and (15b):
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By definition, the slope, q=4·2x103, in equation (15a) is equal to , which is a measure of growthdirected metabolic activity of the cell. The relationship between dry cell mass and parameters such as the number of rrn operons per cell (nrrn(av)) and the peptide chain elongation rate (
aa) was made explicit [see equation (16)] by equating the righthand sides of equations (7) and (15a) and rearranging:
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The RNA concentration (mRNA(av)/mdc(av)) reflects the ribosome concentration needed to achieve a particular value of µ. The relation between µ and the product of (mRNA(av)/mdc(av)) and aa is illustrated in Fig. 2
and is described by equation (10b), where
=1x104, in accord with the theoretical value. The relations between µ and (mRNA(av)/mdc(av)), and between µ and mRNA(av)/v(av), which are illustrated in Fig. 3
are described by equations (17a) and (17b):
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The macromolecular compositions of S. coelicolor A3(2), grown at different specific growth rates are summarized in Table 4. The average numbers of genomes per cell (ng(av)) are provisional and more precise values may lead to an adjustment of dry cell mass (mdc(av)), protein content (mp(av)), etc., but would not affect ratios such as mRNA(av)/mdc(av). Equations (6a) and (6b) were used to analyse features of rRNA synthesis. In cells grown optimally (µmax=0·3 h1) the rate of rRNA synthesis was found to be slow in comparison with the high rate of DNA synthesis (see Table 4
). Briefly, the time taken to synthesize one copy of prerRNA was calculated to be 4·8 min, corresponding to
rRNA*/3600 s h1=19 nucleotides s1. In contrast with E. coli, the rate of initiation of prerRNA synthesis was found to be high (iRNAP/ilim
1), indicating that, when growing optimally, S. coelicolor A3(2) has little spare capacity for prerRNA synthesis.
The correlations between µ2 and mdc(av) (or v(av)), which are summarized by equations (19a) and (19b), are presented in Fig. 1.
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It was estimated that =320 fg and mdc(av)*=430 fg (µ=0·3 h1), corresponding to
=1·0 fl and v(av)*=1·3 fl. Comparison of the slopes of equations (15a) and (19a) reveals that the growthdirected metabolic activity per fg dry cell mass of S. coelicolor A3(2) is less than onesixth of the activity of E. coli B/r.
The relation between µ and the product of RNA concentration (mRNA(av)/mdv(av)) and the peptide chain elongation rate (aa) is illustrated in Fig. 2
and is described by equation (10b), where
=1x104, in accord with the theoretical values for E. coli. This result is to be expected if constants such as the sizes of the rRNA species, the fraction of ribosomes actively synthesizing protein, etc., are very similar for both species. The concentration of RNA was found to vary by a little more than twofold over a 12·5fold increase in µ (see Fig. 2
), according to equations (20a) and (20b):
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Values of RNA concentrations ( ) for limiting cells were calculated (intercept/slope) to be 0·061 fg RNA per fg dry cell mass and 21 fg per fl cell volume, respectively, similar to values deduced for limiting cells of E. coli [see discussion following equations (17a) and (17b)]. Within experimental error (see Fig. 4
), the protein synthesis rate was found to be proportional to the third power of the RNA concentration [see equations (21a) and (21b)]:
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Properties of cells of S. coelicolor A3(2) and E. coli B/r synthesizing protein at similar rates (51 and 42 fg h1, respectively) were compared. It was found that cells of S. coelicolor A3(2) (µ=0·30 h1) had a higher number (22 000 ribosomes) and a higher concentration (18 200 ribosomes fl1) of ribosomes than cells of E. coli (µ=0·42 h1), which had 6800 ribosomes (11 000 ribosomes fl1). Application of equation (9) reveals that the peptide chain elongation rate aa/3600 s hh is lower for S. coelicolor A3(2) than for E. coli B/r; namely 3·17 and 14·0 amino acid residues per ribosome s1 respectively. These values of
aa are thought to reflect both the different temperatures of growth (30 and 37 °C, respectively) and the rate of interaction between the ternary complex formed between aminoacyltRNA, EFTu and GTP with the Asite of the ribosome. It is proposed that lower concentrations of aminoacyltRNA and of EFTu relative to ribosomes (Cox, 2003
) would diminish both the rate of ternary complex formation and the rate of peptide bond formation (
aa). Similarly, the 13fold differences in the numerical values of
[see equation (12a)] found for E. coli B/r [equation (18a)] and S. coelicolor A3(2) [equation (21a)] are attributable to differences in both the temperature of growth and the number of copies per ribosome of aminoacyltRNA and EFTu. A rough estimate, which was made on the basis of the assumption that the kinetic constants for peptide bond formation are similar for both species, suggests that the numbers of copies of aminoacyltRNA per ribosome and EFTu per ribosome present in S. coelicolor A3(2) are onethird of the numbers of copies per ribosome present in E. coli.
The transcriptional and translational properties of S. coelicolor A3(2) growing (µ=0·30 h1) near to its maximum rate (µ0·35 h1) are summarized in Fig. 5(b)
as a virtual schematic cell.
The analysis of the data for S. coelicolor A3(2) growing optimally leads to provisional values for DNA, trrn, iRNAP, the number of copies of EFTu per ribosome and the number of aminoacyltRNA molecules per ribosome. Each of these parameters may be measured by experiment.
Genomic and macromolecular properties of M. bovis BCG
The genomes of M. tuberculosis and E. coli are very similar in their sizes, the numbers of ORFs per genome and the sizes of their average proteins (see Table 1). The average macromolecular properties of cells grown at 37 °C (tD=24h, µ=0·029) are summarized in Table 5
; data for M. bovis BCG are presented in Figs 3
5. The average cell volume was estimated as 0·96±0·06 fl (or µm3); see Table 5
. The concentration of RNA (mRNA(av)/mdc(av)=0·042 fg RNA per fg dry cell mass) was found to be close to the minimum value inferred for viable cells (see Fig. 3
).
Equations derived for E. coli were applied to M. bovis BCG. For example the value of the RNA concentration (mRNA(av)/mdc(av)) derived by means of equation (17a) for µ=0·029 h1 was found to be 0·064 fg RNA per fg dry cell mass compared with the observed value (see Table 5) of 0·042 fg RNA per fg dry cell mass; substitution of
p(av) for M. bovis BCG in equation (18a) yielded a value of mRNA(av)/mdc(av)=0·041 fg RNA per fg dry cell mass. In contrast, calculations based on equations (20a) and (21a) derived for S. coelicolor yielded estimates close to twice the observed value: namely 0·092 and 0·096 fg RNA per fg dry cell mass respectively. In this respect E. coli is a better model for M. bovis BCG than is S. coelicolor. Equation (10b), which was found to apply to both E. coli and S. coelicolor (see Fig. 2
), was found to apply also to M. bovis BCG. Substitution of mRNA(av)/mdc(av)=0·042 fg RNA per fg dry cell mass and
aa=2 amino acids incorporated into protein per ribosome s1 into the equation led to a value of µ=0·030 h1 compared with the observed value of 0·029 h1.
The activity of rrn operons is described by rRNA and iRNAP. A value of
rRNA/3600 s h1=12 residues s1 was made on the basis of the assumption that
rRNA=6
aa residues h1. There is no reported value for
rRNA of M. bovis BCG. However, a very close relative of M. tuberculosis was found (Harshey & Ramakrishnan, 1977
) to synthesize prerRNA in 7·6 min, corresponding to
rRNA/3600 s h1=12·2 nucleotides s1 for prerRNA. An average value of iRNAP=12 RNAP complexes per rrn operon was calculated from
R(av)/nrrn(av) for
rRNA/3600 s h1=12·2 nucleotides s1. The rate of initiation of transcription of rrn operons of M. bovis BCG is inferred to be close to 12 % of the theoretical maximum (iRNAP/ilim=0·12). The transcriptional and translational properties of M. bovis BCG are summarized in Fig. 5(c)
. The metabolic activity of the average cell is encapsulated by the parameters C*=10·3 h, trrn*·60 min h1=7·6 min, tap*·60 min h1=2·8 min, and
E/3600 s h1=0·64x105 ATP equivalents s1. The low rates of macromolecular synthesis require a low rate of nutrient uptake and generation of energy, which may be a valuable asset for growth within macrophages.
The observation that equations (17a) and (18a) obtained for E. coli appear valid for M. bovis BCG suggests that equation (7) relating µ to nrrn(av) may apply not only to E. coli but also to M. bovis BCG and other mycobacteria. Upper limits for the maximum specific growth rates of mycobacteria were equated with the values of µmax calculated for E. coli possessing either one or two rrn operons per genome, namely 0·69 h1 and 1·00 h1 respectively. Mycobacteria grow more slowly than these estimated values.
Mycobacterium marinum, which is a very close relative of M. tuberculosis as judged by the relatedness of their 16S rRNA gene sequences, fatty acid profile analysis and DNADNA hybridization (Tønjum et al., 1998), has a single rrn operon per genome (C. HelgueraRepetto, R. A. Cox & J. A. GonzalezyMerchand, unpublished work). The maximum specific growth rate of M. marinum was found to be 0·173 h1 (tD=4 h) at 30 °C (Clark & Shepard, 1963
), which compares with 0·69 h1 calculated for hypothetical cells of E. coli. Mycobacterium chelonae, which also possesses a single rrn operon per genome (Domenech et al., 1994
), was found to grow at 30 °C with a µmax of 0·13 h1 (tD=5·4 h) (M. C. Nuñez & M. J. Garcia, unpublished work). Mycobacterium smegmatis, which is known to have two rrn operons per genome (Bercovier et al., 1986
), has a µmax of 0·28 h1 (tD=2·5 h) at 37 °C (GonzalezyMerchand et al., 1999
), compared with tD=1·00 h calculated for hypothetical cells of E. coli.
The tubercle bacillus has a single rrn operon per genome, which was shown to be under growth rate control when transferred to M. smegmatis (Verma et al., 1999). When growing at its maximum rate (see above) M. bovis BCG was found to use about 12 % of the potential activity of its rrn operon. Thus, it is inferred that the slow growth of M. tuberculosis, M. bovis BCG and other members of the complex reflects cell metabolism and not the possession of a single rrn operon per genome. Members of the M. tuberculosis complex have genomes which have gained a large number (56 or more) of insertion sequences through horizontal transfer (Brosch et al., 2000
, 2002
), which may have attenuated their capacities for growthdirected cell metabolism, leading to slow growth.
Concluding remarks
The framework devised to explore the growth of bacteria was applied to three species with maximum growth rates ranging from µmax=0·029 h1 to µmax=1·73 h1 (see Fig. 5). The examination of the properties of E. coli indicates that the requirements for ultrafast growth include: (i) highly efficient mechanisms for the uptake and metabolism of a nutrient source of carbon and energy; (ii) a capacity for the cell to become polyploid and thereby increase the average number of rrn operons per cell; (iii) a capacity for rapid rRNA synthesis (trrn*·60 min h1=1·2 min); (iv) a capacity for rapid protein synthesis for example, an average protein of 330 amino acid residues can be synthesized in 15 s; this rate is achieved through high concentrations of ribosomes, aminoacyltRNA and EFTu.
S. coelicolor A3(2) growing optimally was found to synthesize macromolecules at contrasting rates. Specifically, DNA synthesis was estimated to proceed at a rate similar to that observed for E. coli (µ=1·73 h1). In contrast, the predicted rates for rRNA synthesis and for synthesis of an average protein are relatively slow, namely, trrn*·60 min h1=4·8 min and tap*·60 mins h1=1·74 min. However, each rrn operon appears to be fully engaged in prerRNA synthesis.
Equations derived to describe the growth of E. coli were found to describe the growth of M. bovis BCG, in contrast with equations for S. coelicolor A3(2). Using my equations relating the number of rrn operons per genome to maximum specific growth rate, it is expected that M. tuberculosis with one rrn operon should be capable of growing much faster than it actually does. Therefore, it is suggested that the high number of insertion sequences in this species attenuates growth to much lower values. M. tuberculosis has the ability to persist in the form of a longterm asymptomatic infection (for review see Stewart et al., 2003) and knowledge of the properties of minimal cells of this pathogen is relevant to this phenomenon.
Finally, the framework described in the Theoretical Analyses section is based on the availability of a minimum set of data which includes: (i) the size of genome; (ii) the number of rrn operons per genome; (iii) the DNA : RNA : protein ratios obtained at several growth rates and, preferably, a knowledge of the number of copies of aminoacyltRNA and EFTu per ribosome. These data provide a range of insights into cell physiology.
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APPENDIX |
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However, nR(av)/naa(av) is proportional to mRNA(av)/mdc(av), leading to equation (3A) [equation (10b) of the Theoretical Analyses section], where =
R·
rRNA·(Mr[aa(av)]/Mr[nuc(av)])/(
m·lrRNA); mp(av)=
m·mdc(av); naa(av)=mp(av)/(maa(av)/NA) and nR(av)=
rRNA·[mRNA(av)/lrRNA·Mr[nuc(av)]/NA)], where Mr[aa(av)] is the average mass of an amino acid residue, NA is Avogadro's constant,
rRNA is the fraction of RNA that is rRNA, lrRNA is the number of nucleotides per ribosome, and Mr[nuc(av)] is the average mass of a nucleotide.
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The specific ribosome synthesis rate is defined by equation (4A):
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Equation (7A) was used in the following way. First, appropriate values were assigned to the product iRNAP·rRNA and to naa(av). A value, µx, was then assigned to µ allowing the specific protein synthesis rate (
=mp(av)·µx) to be calculated. The ratio mRNA(av)/mdc(av) was then obtained by means of equation (12) of the Theoretical Analyses section, allowing
aa to be evaluated by means of equation (3A). Then, the only unknown quantity is xrrn(av). Provisionally, a value of 6
aa residues h1 was assigned to
rRNA, allowing iRNAP to be calculated.
The validity of equations (6A) and (7A) was established in the following way. The effects of reducing the number of rrn perons per genome on µ were calculated for E. coli B/r grown in medium E. Wildtype E. coli was found to have he following properties (see Table 3): µ=1·73 h1, nrrn(av)*=35·9 operons, iRNAP*=68 copies/operon,
rRNA/3600 s h1=85 residues s1. Please note that the reported value of
rRNA is less than 6
aa, possibly owing to feedback control of rRNA synthesis. An rrn operon functioning at its maximum rate is characterized by iRNAP=75 copies per rrn operon and
rRNA/3600 s h1=126 residues s1 [see equation (6c) of the Theoretical Analyses section]. The minimum value of nrrn(av) required for µ=1·73 h1 was calculated on the basis of the assumption that reducing nrrn(av) would lead to the remaining operons functioning at full capacity; that is iRNAP=75 copies per rrn operon,
rRNA/3600 s h1=6
aa/3600 s h1=126 residues s1, naa(av)=8·5x109 amino acids,
F=7·25x103, µ=1·73. The value of nrrn(av)=19·7 operons, which was obtained by means of equation (7A), is equivalent to four rrn operons per genome [nrrn(av)=(4/7)·35·9=20·5 operons]. A similar calculation revealed that a value of µ=1·5 h1 (tD=0·47 h) required a minimum value of nrrn(av)=15·5 operons, corresponding to three rrn operons per genome. These estimates are in accord with the results obtained by progressively reducing the number of functional rrn operons present in the E. coli genome (Condon et al., 1993
, 1995
).
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ACKNOWLEDGEMENTS |
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Received 12 June 2003;
revised 2 December 2003;
accepted 20 January 2004.
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