Correlation of the rate of protein synthesis and the third power of the RNA : protein ratio in Escherichia coli and Mycobacterium tuberculosis

Definition of slow growth.

Traditionally (see, for example, Wayne & Kubica, 1986) mycobacteria are classified as either fast-growing or slow-growing according to whether colonies appear on a solid medium within 5 days (fast-growers) or longer than 5 days (slow-growers). Bacteria growing optimally are defined according to μ, their specific growth rate (Fig. 1⇓); slow growth spans the range μ⩽0·14 h⁻¹ (t_D⩾5 h), fast growth span the range μ>0·14 h⁻¹<0·7 h⁻¹ (t_D 1–5 h). Faster-growing bacteria (μ>0·7 h⁻¹ t_D<1 h) are classified as ultra-fast growers.

A feature of ultra-fast growth is that genome replication spans more than one cell division cycle (Cooper & Helmstetter, 1968; Helmstetter & Cooper, 1968) giving rise to newborn cells that have more than one genome equivalent per cell. In contrast, newborn cells of fast- and slow-growing bacteria, including mycobacteria (Hiriyanna & Ramakrishnan, 1986), are thought to have a single genome that is replicated within the cell division cycle. When growth conditions are less favourable (μ<0·7 h⁻¹), ultra-fast-growers resemble fast-growers and slow-growers, with genome replication being started and completed within the cell division cycle (Fig. 1⇑).

Definitions, axioms and assumptions.

The analysis presented below concerns exponentially growing cell cultures of E. coli for which it is known that the conditions of growth govern both the specific growth rate μ and macromolecular composition (DNA : protein : RNA). Specifically, different growth rates correlate with different macromolecular compositions of cell cultures (see, for example, Bremer & Dennis, 1996). In contrast, when the specific growth rate was 0·2 h⁻¹ or less, it was noted that the macromolecular content changed very little. Cultures had minimal contents of protein and RNA as judged by the DNA : protein and DNA : RNA ratios (Jacobsen, 1974 cited by Ingraham et al., 1983). For cells of minimal protein and RNA contents, the growth rates are designated μ_min; the subscript _min denotes that protein and RNA contents are minimal. These cells are thought to have an excess of ribosomes over the apparent demand for protein synthesis, as was shown for a Vibrio sp. by Flärdh et al. (1992). Cultures characterized by μ_min are not included in the analysis. However, the concept of minimal cells could be relevant to dormant mycobacterial cells. (Symbols are defined in Table 1⇓).

At any time, t, during the exponential growth of a culture the mass of protein, p_t, is given by equation (1) where p₀ is the mass of protein at t=0.

The mass of protein, p_t, is the product of the number of cells, n_t, present at time t and the average mass of protein m_p(av) per cell. Although the number of cells per culture increases exponentially their age distribution remains unchanged (Powell, 1956). A cell's age, a, ranges from a=0 (a newborn cell) to a=1 (a cell about to divide), where a=t/t_D; t=0 for newborn cells and t=t_D for cells about to divide. Hence, the average amount of protein per cell is given by equation (2).

Figure image not available in archive

where n_a is the number of cells aged a; n_a=0 is the number of cells aged a=0; m_p(a) is the amount of protein per cell aged a. Evaluation of equation (2) reveals (Ingraham et al., 1983) that m_p(av) is equal to m_p(a=0)/ln2 [1·4 m_p(a=0)] where m_p(a=0) is the mass of protein per newborn cell. Other average cell properties are similarly defined; for example, v_(av) (fl or μm³), average dry cell mass m_dc(av), average dry cell mass RNA m_RNA(av) and average dry cell mass DNA m_DNA(av). In each case, mass is measured in femtograms (fg). Each of these properties, which serve to define an ‘average’ cell, has a constant value for a particular set of growth conditions because the age distribution of cells remains unchanged throughout exponential growth.

A culture, which at time t comprises n_t cells, accumulates protein at an instantaneous rate specified by equation (3), which is the differential of equation (1).

Substitution for p_t=n_t•m_p(av) on the right-hand side of equation (3) leads to equation (4)

Figure image not available in archive

Rearrangement of equation (4) leads to the identities shown in equation (5), which also defines ω_p(av), the specific protein synthesis rate (fg protein synthesized per cell per hour).

Figure image not available in archive

In other words, dp_t/dt is equal to the product of n_t and ω_p(av).

It is thought that the amount of protein per cell increases exponentially over the range t=0 for a newborn cell to t=t_D when the cells divide (Cooper, 1988). Hence, m_p(t), the amount of protein per cell at time t, is given by equation (6) where m_p(t=0) is the amount of protein per newborn cell.

Over the same period (t=0 to t=t_D) the specific protein synthesis rate [ω_p(t)] at time t is given by equation (7).

When m_p(t)=m_p(av), equation (7) may be written as equation (8).

Figure image not available in archive

Thus, equations (3) and (8) are related through n_t [equation (5)], which further illustrates the notion that the properties of a cell culture are determined by the properties of average cells. Thus, both m_p(av) and ω_p(av) are parameters that usefully describe the exponential growth of a cell culture.

Macromolecular composition of E. coli as a function of the specific growth rate.

For cells growing at a rate in the range 0·42–1·73 doublings h⁻¹ (Table 2⇓), the mass of protein [m_p(av)] and dry cell mass [m_dc(av)] increase concomitantly with growth rate; that is, m_p(av) appears to be in constant proportion, β_m, of m_dc(av). The average cell volume [v_(av)] will be directly proportional to m_dc(av) and hence, to m_p(av) if the fraction of cell mass due to water, β_aq, and the buoyant density, ρ, are independent of growth rate. A value for β_aq of 0·7 was established for E. coli growing optimally in glucose minimal medium (Ingraham et al., 1983); this value appears to be accepted for E. coli cells in general. The buoyant density, ρ, of E. coli was found to lie within the range 1·09±0·015 g cell mass (ml cell volume)⁻¹ [or pg fl⁻¹ (or μm³)] for cells with growth rates of 0·4–3·0 h⁻¹ and to remain constant during the growth cycle (Woldringh et al., 1981); and was found to be independent of the composition of the growth medium below 1000 mosM (Baldwin et al., 1994). Thus, as a first approximation, ρ may be regarded as a constant. The parameters ρ, m_dc(av) (or m_p(av)/β_m) and v_(av) are related by equations (9a) and (9b).

Figure image not available in archive

The concentration of protein [m_p(av)/v_(av)], which was estimated to be 0·16 pg fl⁻¹ (or μm³), is thought to be not only constant during the growth cycle but also to be independent of growth rate (see also Bremer & Dennis, 1996); that is, both β_m and β_aq are also thought to be independent of growth rate.

Whereas the concentration of protein appears to be independent of the specific growth rate, the mass of protein per cell is not. The relationship between m_p(av) and μ [equation (10)] is illustrated in Fig. 2⇓.

where α=4·1×10⁻³ (fg protein)⁻¹ h⁻¹.

The plot of RNA : protein ratios [m_RNAav/m_p(av)] against μ (Fig. 3⇓) has features in common with the plot of m_p(av) against μ (Fig. 2⇑). A linear correlation [equation (11)] of μ with m_RNA(av)/m_p(av) was found (see data of Table 2⇑).

where b=4·37 h⁻¹. The intercept on the abscissa raises the possibility that there is a limiting value of m_RNAav/m_p(av)≈0·08 for viable cells.

Whereas the RNA : protein ratio was found to be directly proportional to the specific growth rate when cells were grown optimally, the specific protein synthesis rate ω_p(av) [equation (5)] was not (Fig. 4⇓). Both the specific protein synthesis rate and hence, the associated rate of consumption of energy were found to increase as the specific growth rate increased. The dependence of the specific protein synthesis rate on the RNA : protein ratio (Fig. 5⇓a) was similar to the plot shown in Fig. 4⇓; the similarities in the two graphs are to be expected since μ and m_RNAav/m_p(av) are linearly related [equation (11)]. The specific protein synthesis rate was found to increase approximately eight-fold for a doubling of the RNA : protein ratio (Fig. 5a⇓), which suggests that ω_p(av) is proportional to the third power of the ratio m_RNA(av)/m_p(av). This conclusion is supported by the plot of log [ω_p(av)] against log [m_RNA(av)/m_p(av)], which was found to have a slope of 3·2 (Fig. 5b⇓), in accord with a third power relationship. The plot of the specific protein synthesis rate against the third power of the RNA : protein ratio (Fig. 5c⇓) was found to be linear [equation (12)] with a gradient θ=7·5×10³ fg protein synthesized h⁻¹.

Growth and macromolecular composition of M. tuberculosis.

The era of the study of the macromolecular composition of bacteria (see, for example, Maaløe & Kjeldgaard, 1966) was centred around 1960–1970 and such studies are now unfashionable. The only study of the chemical composition of a mycobacterium was carried out by Winder & Rooney (1970), who reported a careful and comprehensive study of Mycobacterium bovis BCG, a member of the M. tuberculosis complex that is very closely related to M. tuberculosis (Brosch et al., 2000). The mycobacterium was grown in a standard medium and the generation time was approximately 24 h. The results presented in Table 3⇓ were obtained from the data of Winder & Rooney (1970) by making a number of simple calculations based on the definitions of the units used, and current data for the size of the genome, the average number of genomes per cell (footnote * of Table 3⇓) and the size of the rRNA moiety of ribosomes (Cole et al., 1998). The numbers of cells per millilitre, m_DNA(av), m_RNA(av), m_p(av) and v_(av) were calculated. Thus, Table 3⇓ summarizes all the information that is available for the macromolecular composition of a member of the M. tuberculosis complex in particular and mycobacteria in general.

The limited data for M. bovis BCG are presented in Figs 2⇑–5. In Fig. 2⇑, a plot of μ against m_p(av), the datum for M. bovis BCG (curve II) does not conform with the data for E. coli (curve I). More data were needed for M. bovis in order to explain why this should be the case. In contrast, in Figs 3⇑–5 data for the mycobacterium were found to form a smooth curve with data for E. coli, suggesting that the associated equations (11) and (12) may also apply to M. bovis BCG growing optimally. This was confirmed, as may be seen from the following example. Very similar protein contents [m_p(av)≈156 fg] were reported for E. coli (t_D=1 h) grown in a glycerol medium (Table 2⇑) and for M. bovis BCG (t_D=24 h) growing exponentially (Table 3⇑). In contrast, the corresponding RNA : protein ratios of E. coli (Table 2⇑) and M. bovis BCG (Table 3⇑) were found to be 0·25 and 0·085, respectively, almost a three-fold difference. Substitution of the RNA : protein ratios in equation (11) yielded values of μ of 0·73 and 0·025 h⁻¹, respectively, for E. coli and M. bovis BCG; the observed values were 0·69 and 0·029 h⁻¹, respectively. The RNA : protein ratios were also substituted in equation (12) in order to calculate the specific protein synthesis rate [ω_p(av)]. The calculated specific protein synthesis rates were 117 and 4·6 fg protein synthesized per cell per hour, respectively, for E. coli and M. bovis BCG; the observed values, calculated by means of equation (5), were 108 and 4·5 fg protein synthesized per cell per hour, respectively. Thus, equations (11) and (12) which were derived for E. coli apply also to M. bovis BCG. The ability of these equations to describe both E. coli and M. bovis BCG emphasizes that the concentration of RNA which is measured by RNA : protein ratio is an informative cell parameter.

The macromolecular composition of stationary phase cells (day 8 of Table 3⇑) provides a guide to the properties of minimal cells of the mycobacterium. When appropriate conversion factors (Table 3⇑) were applied to data for M. bovis BCG as to data for E. coli, the following picture emerged. Minimal cells of E. coli were estimated to have an average of 6200 ribosomes, a concentration of 8200 ribosomes per femtolitre of cell volume. In contrast, minimal cells of M. bovis BCG were estimated to have an average of 2200ribosomes per cell, a concentration of approx. 4500 ribosomes per femtolitre of cell volume, which is close to the limit inferred for viable cells from the intercept on the RNA : protein axis in Fig. 3⇑.