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Quantitative relationships for specific growth rates and macromolecular compositions of Mycobacterium tuberculosis, Streptomyces coelicolor A3(2) and Escherichia coli B/r: an integrative theoretical approach

Division of Mycobacterial Research, National Institute for Medical Research, London NW7 1AA, UK

Correspondence
Robert A. Cox
rcox{at}nimr.mrc.ac.uk

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORETICAL ANALYSES RESULTS AND DISCUSSION APPENDIX REFERENCES

 
Further understanding of the physiological states of Mycobacterium tuberculosis and other mycobacteria was sought through comparisons with the genomic properties and macromolecular compositions of Streptomyces coelicolor A3(2), grown at 30 °C, and Escherichia coli B/r, grown at 37 °C. A frame of reference was established based on quantitative relationships observed between specific growth rates (µ) of cells and their macromolecular compositions. The concept of a schematic cell based on transcription/translation coupling, average genes and average proteins was developed to provide an instantaneous view of macromolecular synthesis carried out by cells growing at their maximum rate. It was inferred that the ultra-fast growth of E. coli results from its ability to increase the average number of rRNA (rrn) operons per cell through polyploidy, thereby increasing its capacity for ribosome synthesis. The maximum growth rate of E. coli was deduced to be limited by the rate of uptake and consumption of nutrients providing energy. Three characteristic properties of S. coelicolor A3(2) growing optimally (µ=0·30 h^–1) were identified. First, the rate of DNA replication was found to approach the rate reported for E. coli (µ=1·73 h^–1); secondly, all rrn operons were calculated to be fully engaged in precursor-rRNA synthesis; thirdly, compared with E. coli, protein synthesis was found to depend on higher concentrations of ribosomes and lower concentrations of aminoacyl-tRNA and EF-Tu. An equation was derived for E. coli B/r relating µ to the number of rrn operons per genome. Values of µ=0·69 h^–1 and µ=1·00 h^–1 were obtained respectively for cells with one or two rrn operons per genome. Using the author's equation relating the number of rrn operons per genome to maximum 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 rate to still lower values.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORETICAL ANALYSES RESULTS AND DISCUSSION APPENDIX REFERENCES

 
Mycobacteria are rod-shaped, Gram-positive, acid-fast, aerobic and non-motile (Wayne & Kubica, 1986). Pathogenic mycobacteria such as Mycobacterium leprae and Mycobacterium tuberculosis have notable properties including slow growth [generation times of 12 days (Shepard, 1960) and 1 day (Wayne, 1994) respectively], an ability to survive within host cells (Armstrong & D'Arcy-Hart, 1971; Ferrari et al., 1999) and an ability to persist in a dormant state (Wayne & Hayes, 1996). Other mycobacteria are pathogenic to man or animals (Wayne & Kubica, 1986; Woods & Washington, 1987). Traditionally, mycobacteria are divided into slow-growers and fast-growers according to the time taken for colonies to appear on a solid medium (Wayne & Kubica, 1986). In general slow-growers were found to have a single rRNA (rrn) operon whereas fast-growers were found to have two rrn operons per genome (Domenech et al., 1994).

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 Calmette–Guérin (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 (µ_max) from 0·029 h^–1 (M. bovis BCG) to 1·73 h^–1 (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, t_D, is less than the time, C, needed to replicate the genome (C>t_D), 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.

	THEORETICAL ANALYSES

TOP ABSTRACT INTRODUCTION THEORETICAL ANALYSES RESULTS AND DISCUSSION APPENDIX REFERENCES

 
Bacterial strains used for the theoretical analyses.
Literature data of Escherichia coli B/r, Streptomyces coelicolor A3(2), Mycobacterium tuberculosis H37Rv and Mycobacterium bovis BCG provided the basis for these theoretical analyses. The genomic properties of these strains are summarized in Table 1.

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, µ_max, is attained when cells are amply supplied with the most favourable nutrients. An asterisk (m_p(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, t_D 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:

No biochemical significance has been attributed to B, which diminishes as µ increases so that B0 (Helmstetter, 1996). The D period is thought to be required to allow chromosome segregation and septum formation to take place.

The rate of elongation of DNA per replication fork (_DNA, bp per fork s^–1), the size of the genome (l_g, bp) and C (h) are, by definition related by equation (2):

The number of genome equivalents per average cell (n_g(av)) is given by equation (3) (Helmstetter & Cooper, 1968):

When B0, (C+D)/t_D1 and D/t_D0·25 then n_g1·6 genome equivalents per average cell, the maximum value for a fast-growing cell.

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 h^–1) and peptide chain elongation (_aa, amino acid residues h^–1) are co-ordinated (Bremer & Dennis, 1996); see equation (4), where the factor 3 reflects the number of nucleotides per codon:

Implicit in equation (4) is the notion that the rates of initiation of transcription and translation are co-ordinated. The process of translation is thought to protect mRNA from degradation by RNases (degradosomes) (Grunberg-Manago, 1999; Régnier & Arraiano, 2000; Leroy et al., 2002; Khodursky & Bernstein, 2003).

If the amount of mRNA is represented by n_m·nuc(av) nucleotides, and n_R(av) is the number of ribosomes and _R the fraction of ribosomes actively synthesizing proteins, then the ratio n_m·nuc(av)/_R·n_R(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 55–80 nucleotides of mRNA per ribosome depending on the growth rate of E. coli (see Table 3).

The specific ribosome synthesis rate, _R(av)(number of ribosomes synthesized per cell h^–1), is defined by equation (6a):

leading to equation (6b) because, by definition, _R(av) ribosomes h^–1and _rRNA(av) nucleotides h^–1/l_rrn are equal.

The term i_RNAP is the average number of RNAP holoenzymes transcribing an rrn operon, _rRNA(nucleotides) is the elongation rate of nascent precursor-rRNA (pre-rRNA) and l_rrn is the length (nucleotides) of a complete transcript of an rrn operon. _rRNA is thought to be approximately equal to 2_mRNA or 6_aa(Bremer & Dennis, 1996), possibly reflecting feedback control of rRNA synthesis. The ratio _R(av)/n_rrn(av), which describes the rate of transcription (i_RNAP·_rRNA) of a single rrn operon [see equation (6b)], forms the basis of a method for assessing the relationship between µ and n_rrn(av)[see Appendix]. The limiting value, i_lim, of i_RNAP is observed when the rate of initiation is equal to the rate of promoter clearance; then a loading of one RNAP per 80 nucleotides (the footprint of the RNAP initiating complex) would be expected; that is i_lim is equal to l_rrn/80, or 70–75 RNAPs per rrn operon. The ratio i_RNAP/i_lim provides a measure of the transcriptional activity of each rrn operon. The minimum number, , required to meet the observed specific ribosome synthesis rate is given by equation (6c); _rRNA is equated with 6 _aa residues h^–1.

Equation (7) was derived by relating the specific growth rate, µ_x, to the average number, x_rrn(av), of rrn operons per average cell; _F is a constant dependent on the growth medium (see Appendix).

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 (n_aat/R) and the number of copies of EF-Tu per ribosome (n_EF/R). It is assumed that both n_aat/R and n_EF/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 h^–1), which is defined by equation (8a):

or the equivalent equation (8b), where _aa(av) is the number of amino acids incorporated into protein per cell h^–1:

Alternatively, _aa(av) may be related to the number (n_R(av)) of ribosomes per cell and the peptide chain elongation rate _aa amino acids incorporated per ribosome h^–1 (Bremer & Dennis, 1996); see equation (9), where _R is the fraction of ribosomes actively synthesizing protein:

Equation (10a) may be derived by equating the right-hand sides of equations (8b) and (9) and rearranging to make µ the subject (Bremer & Dennis, 1996):

The term n_R(av)/n_aa(av) is an expression of RNA concentration. Equation (10b) is an alternative form of equation (10a) where =1x10^–4 for E. coli (see Appendix):

Although the terms that define are species specific, variations in the values of these terms between species are unlikely to be large and so is expected to remain close to the numerical value of 1x10^–4. It was reported for E. coli (Cox, 2003) that specific growth rates were related to RNA concentration by equation (11), where a (the slope) and b (the intercept on the µ axis) are constants.

The ratio b/a is equal to the RNA concentration of minimal cells (); by definition ; . Equation (11) is related to equation (10b) through and _aa*.

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 (m_RNA(av)/m_dc(av))b/a.

The slope is considered to be proportional to the product of kinetic constants, the number of copies of aminoacyl-tRNA per ribosome and the number of copies of EF-Tu per ribosome. The parameters determining are also thought to be reflected in _aa.

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 h^–1, is estimated in terms of ATP equivalents by equation (13):

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORETICAL ANALYSES RESULTS AND DISCUSSION APPENDIX REFERENCES

 
The genomic properties of E. coli B/r, S. coelicolor A3(2) and M. bovis BCG (see Table 1) reveal that in each case the average ORF is close to 1000 bp in length, encoding an average protein of approximately 330 amino acid residues. It is noted that the size of the genome of S. coelicolor is almost twice the size of the two other species. The sizes of the rRNA components and the RNA (rrn) operons are very similar.

The macromolecular properties of E. coli B/r grown at 37 °C, in five different media (media A–E), S. coelicolor A3(2) grown at different specific growth rates (µ=0·024 h^–1 to µ=0·300 h^–1) at 30 °C and M. bovis BCG (µ=0·029 h^–1) 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).

The average time, t_rrn, 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 t_rrn 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, n_rrn(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 h^–1) _rRNA was found to be suboptimal (_rRNA<6_aa residues h^–1), values of i_RNAP/i_lim were found to range from 0·06 (µ=0·42 h^–1) to 0·90 (µ=1·73 h^–1), and [see equation (6c)] was found to range from 0·9 operons (µ=0·42 h^–1) to 22·0 operons (µ=1·73 h^–1). The fraction , which measures the relative activity of each rrn operon, was found to range from 0·07 (µ=0·42 h^–1) to 0·61 (µ=1·73 h^–1). Thus, E. coli growing at its maximum rate (µ=1·73 h^–1) 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 h^–1, 1·00 h^–1, 1·40 h^–1 and 1·70 h^–1. 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 i_RNAP/i_lim) 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 (t_D=C*+D*) the number of genome equivalents per cell, g_(av), increases to two or more, with a concomitant increase in the number, n_rrn(av), of rrn operons per cell [see equation (14)]:

The rate of ribosome synthesis needed to support the maximum specific growth rate of 1·73 h^–1 could not be achieved without increasing n_rrn(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 h^–1 to 1·73 h^–1 (see Table 3). Empirically, linear plots were obtained by plotting µ² against m_dc(av) or v_(av) (see Fig. 1), which are summarized by equations (15a) and (15b):

View larger version (10K): [in this window] [in a new window]   Fig. 1. Correlations between specific growth rate squared and average dry cell mass. I, , E. coli (see Table 3); II, , S. coelicolor (see Table 4).

By definition, the slope, q=4·2x10^–3, 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 (n_rrn(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:

The term _F relates to a particular growth medium (see Appendix).

The RNA concentration (m_RNA(av)/m_dc(av)) reflects the ribosome concentration needed to achieve a particular value of µ. The relation between µ and the product of (m_RNA(av)/m_dc(av)) and _aa is illustrated in Fig. 2 and is described by equation (10b), where =1x10^–4, in accord with the theoretical value. The relations between µ and (m_RNA(av)/m_dc(av)), and between µ and m_RNA(av)/v_(av), which are illustrated in Fig. 3 are described by equations (17a) and (17b):

View larger version (14K): [in this window] [in a new window]   Fig. 2. Correlations between specific growth rate and the product of RNA concentration and the peptide chain elongation rate [see equation (10b)]. , E. coli (see Table 3); , S. coelicolor (see Table 4).

View larger version (16K): [in this window] [in a new window]   Fig. 3. Correlations between specific growth rate and concentration of RNA (mRNA(av)/mdc(av)) or (mRNA(av)/v(av)). I, , E. coli (see Table 3). II, , S. coelicolor (see Table 4). , M. bovis (see Table 5).

View larger version (17K): [in this window] [in a new window]   Fig. 4. Correlations between specific protein synthesis rate and the third power of RNA concentration [see equation (12)]. (a) I, , E. coli (see Table 3); , M. bovis BCG (see Table 5). (b) II, , S. coelicolor (see Table 4).

View larger version (35K): [in this window] [in a new window]   Fig. 5. Virtual schematic cells summarizing the genomic properties and transcription/translation activities of ultra­fast­, fast­ and slow­growing bacteria proliferating at their maximum rates µmax. (a) E. coli B/r (µmax=1·73 h–1); (b) S. coelicolor A3(2) (µmax=0·30 h–1); (c) M. bovis BCG (µmax=0·029 h–1). (i) Summary of genomic properties and the number of genome equivalents per ‘average’ cell. The properties of the genome, namely the number of rrn operons and the number of ‘average’ ORFs (see Table 1), are presented within the square brackets. The number of genome equivalents per ‘average’ cell is indicated outside the brackets on the lower right hand side. Striped bars represent the rrn operon; stippled bars represent the ‘average’ ORF. (ii) Transcription/translation activity of an ‘average’ ORF is represented schematically within the square brackets in the form of a fibril diagram. The size (bp) of an ‘average’ ORF is indicated and also the locations of 5' ends of nascent mRNA transcripts. Nascent polypeptide chains are not shown. The proportion of non­programmed ribosomes per ORF is indicated by the relative number of free ribosomes. The number of ‘average’ ORFs being transcribed/translated at any instant is shown by the number outside the square brackets on the lower right hand side. Stippled bars represent the ‘average’ ORF; vertical black bars along the ORF represent RNAP holoenzyme; black circles represent ribosomes; and the lines joining these circles represent nascent mRNA. (iii) Rate of synthesis of ‘average’ proteins. The rates of synthesis of ‘average’ proteins per fibril is indicated within the square brackets. The number of fibrils synthesizing protein is given outside the square brackets on the lower right­hand side.

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 (n_g(av)) are provisional and more precise values may lead to an adjustment of dry cell mass (m_dc(av)), protein content (m_p(av)), etc., but would not affect ratios such as m_RNA(av)/m_dc(av). Equations (6a) and (6b) were used to analyse features of rRNA synthesis. In cells grown optimally (µ_max=0·3 h^–1) 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 h^–1=19 nucleotides s^–1. In contrast with E. coli, the rate of initiation of pre­rRNA synthesis was found to be high (i_RNAP/i_lim1), indicating that, when growing optimally, S. coelicolor A3(2) has little spare capacity for pre­rRNA synthesis.

The correlations between µ² and m_dc(av) (or v_(av)), which are summarized by equations (19a) and (19b), are presented in Fig. 1.

It was estimated that =320 fg and m_dc(av)*=430 fg (µ=0·3 h^–1), 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 (m_RNA(av)/m_dv(av)) and the peptide chain elongation rate (_aa) is illustrated in Fig. 2 and is described by equation (10b), where =1x10^–4, 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):

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)]:

Properties of cells of S. coelicolor A3(2) and E. coli B/r synthesizing protein at similar rates (51 and 42 fg h^–1, respectively) were compared. It was found that cells of S. coelicolor A3(2) (µ=0·30 h^–1) had a higher number (22 000 ribosomes) and a higher concentration (18 200 ribosomes fl^–1) of ribosomes than cells of E. coli (µ=0·42 h^–1), which had 6800 ribosomes (11 000 ribosomes fl^–1). Application of equation (9) reveals that the peptide chain elongation rate _aa/3600 s h^–h is lower for S. coelicolor A3(2) than for E. coli B/r; namely 3·17 and 14·0 amino acid residues per ribosome s^–1 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 h^–1) near to its maximum rate (µ0·35 h^–1) 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, t_rrn, i_RNAP, 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 (t_D=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 (m_RNA(av)/m_dc(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 (m_RNA(av)/m_dc(av)) derived by means of equation (17a) for µ=0·029 h^–1 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 m_RNA(av)/m_dc(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 m_RNA(av)/m_dc(av)=0·042 fg RNA per fg dry cell mass and _aa=2 amino acids incorporated into protein per ribosome s^–1 into the equation led to a value of µ=0·030 h^–1 compared with the observed value of 0·029 h^–1.

The activity of rrn operons is described by _rRNA and i_RNAP. A value of _rRNA/3600 s h^–1=12 residues s^–1 was made on the basis of the assumption that _rRNA=6_aa residues h^–1. 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 h^–1=12·2 nucleotides s^–1 for pre­rRNA. An average value of i_RNAP=12 RNAP complexes per rrn operon was calculated from _R(av)/n_rrn(av) for _rRNA/3600 s h^–1=12·2 nucleotides s^–1. The rate of initiation of transcription of rrn operons of M. bovis BCG is inferred to be close to 12 % of the theoretical maximum (i_RNAP/i_lim=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, t_rrn*·60 min h^–1=7·6 min, t_ap*·60 min h^–1=2·8 min, and _E/3600 s h^–1=0·64x10⁵ ATP equivalents s^–1. 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 n_rrn(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 µ_max calculated for E. coli possessing either one or two rrn operons per genome, namely 0·69 h^–1 and 1·00 h^–1 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 DNA–DNA 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 h^–1 (t_D=4 h) at 30 °C (Clark & Shepard, 1963), which compares with 0·69 h^–1 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 µ_max of 0·13 h^–1 (t_D=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 µ_max of 0·28 h^–1 (t_D=2·5 h) at 37 °C (Gonzalez­y­Merchand et al., 1999), compared with t_D=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 µ_max=0·029 h^–1 to µ_max=1·73 h^–1 (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 (t_rrn*·60 min h^–1=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 h^–1). In contrast, the predicted rates for rRNA synthesis and for synthesis of an average protein are relatively slow, namely, t_rrn*·60 min h^–1=4·8 min and t_ap*·60 mins h^–1=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.

	APPENDIX

TOP ABSTRACT INTRODUCTION THEORETICAL ANALYSES RESULTS AND DISCUSSION APPENDIX REFERENCES

 
Correlation between specific growth rate and the average number of rrn operons per cell
When a bacterial culture is growing exponentially both protein content and RNA content increase exponentially. The specific protein synthesis rate, _p(av), of an average cell is defined by equation (1A) (all symbols are defined in Table 2):

Rearrangement leads to equation (2A) [equation (10a) of the Theoretical Analyses section]:

However, n_R(av)/n_aa(av) is proportional to m_RNA(av)/m_dc(av), leading to equation (3A) [equation (10b) of the Theoretical Analyses section], where =_R·_rRNA·(M_r[aa(av)]/M_r[nuc(av)])/(_m·l_rRNA); m_p(av)=_m·m_dc(av); n_aa(av)=m_p(av)/(m_aa(av)/N_A) and n_R(av)=_rRNA·[m_RNA(av)/l_rRNA·M_r[nuc(av)]/N_A)], where M_r[aa(av)] is the average mass of an amino acid residue, N_A is Avogadro's constant, _rRNA is the fraction of RNA that is rRNA, l_rRNA is the number of nucleotides per ribosome, and M_r[nuc(av)] is the average mass of a nucleotide.

For E. coli the constant has a numerical value of 1x10^–4.

The specific ribosome synthesis rate is defined by equation (4A):

Implicit in this equation is the assumption that all rrn operons are equally active in rRNA synthesis. Rearrangement of equation (4A) leads to equation (5A):

Multiplying equation (2A) by equation (5A) leads to equation (6A):

When particular values, , and , are assigned to i_RNAP, _rRNA and n_aa(av), respectively, equation (6A) reduces to equation (7A), where .