## Abstract

In bacteria, the rate of cell proliferation and the level of gene expression are intimately intertwined. Elucidating these relations is important both for understanding the physiological functions of endogenous genetic circuits and for designing robust synthetic systems. We describe a phenomenological study that reveals intrinsic constraints governing the allocation of resources toward protein synthesis and other aspects of cell growth. A theory incorporating these constraints can accurately predict how cell proliferation and gene expression affect one another, quantitatively accounting for the effect of translation-inhibiting antibiotics on gene expression and the effect of gratuitous protein expression on cell growth. The use of such empirical relations, analogous to phenomenological laws, may facilitate our understanding and manipulation of complex biological systems before underlying regulatory circuits are elucidated.

Systems biology is as an integrative approach to connect molecular-level mechanisms to cell-level behavior (*1*). Many studies have characterized the impact of molecular circuits and networks on cellular physiology (*1*, *2*), but less is known about the impact of cellular physiology on the functions of molecular networks (*3*–*5*). Endogenous and synthetic genetic circuits can be strongly affected by the physiological states of the organism, resulting in unpredictable outcomes (*4*, *6*–*8*). Consequently, both the understanding and implementation of molecular control are predicated on distinguishing global physiological constraints from specific regulatory interactions.

For bacterial cells under steady-state exponential growth, the rate of cell proliferation (the “growth rate”) is an important characteristic of the physiological state. It is well known that the macromolecular composition (e.g., the mass fractions of protein, RNA, and DNA) of bacterial cells under exponential growth depends on the growth medium predominantly through the growth rate allowed by the nutritional content of the medium (*9*, *10*). Such growth rate dependencies inevitably affect the expression of individual genes (*4*, *11*) because protein synthesis is directly dependent on the cell’s ribosome content. The latter is reflected by the RNA/protein ratio. In *Escherichia coli*, most of the RNA (~85%) is rRNA folded in ribosomes (*10*, *11*). A predictive understanding of the impact of growth physiology on gene expression therefore first requires an understanding of the cell’s allocation of cellular resources to ribosome synthesis (manifested by the RNA/protein ratio) at different growth rates.

For exponentially growing *E. coli* cells (*10*, *12*), the RNA/protein ratio *r* is linearly correlated with the specific growth rate λ [ = (ln 2)/doubling time] (Fig. 1A). The correlation is described mathematically as
*r*_{0} is the vertical intercept and κ_{t} is the inverse of the slope (table S1). This linear correlation holds for various *E. coli* strains growing in medium that supports fast to moderately slow growth [e.g., 20 min to ~2 hours per doubling (*11*)], and it appears to be quite universal; similar linear correlations have been observed in many other microbes, including slow-growing unicellular eukaryotes (fig. S1). As suggested long ago from mass-balance considerations (*11*) and elaborated in (*13*), this linear correlation is expected if the ribosomes are growth-limiting and are engaged in translation at a constant rate, with the phenomenological parameter κ_{t} predicted to be proportional to the rate of protein synthesis. Consistent with the prediction, data on RNA/protein ratios from slow-translation mutants of *E. coli* K-12 (triangles in Fig. 1B) also exhibited linear correlations with the growth rate λ, but with steeper slopes than the parent strain (circles), which have smaller κ_{t}. Moreover, the corresponding κ_{t} values correlated linearly with the directly measured speed of translational elongation (*14*) (Fig. 1B, inset). Consequently, we call κ_{t} the “translational capacity” of the organism.

Translation can be inhibited in a graded manner by exposing cells to sublethal doses of a translation-inhibiting antibiotic. The RNA/protein ratios obtained for wild-type cells grown in medium with a fixed nutrient source and various amounts of chloramphenicol (Fig. 1B, light blue circles) were consistent with data obtained for the isogenic translational mutants grown in medium with the same nutrient but no antibiotic (light blue triangles). Surprisingly, these data revealed another linear correlation between *r* and λ (Fig. 1B, dashed line), given by*r*_{max} is the vertical intercept and κ_{n} is the inverse slope. Such a linear correlation was obtained for cells grown with each of the six nutrient sources studied (Fig. 2A and table S3). The correlation described by Eq. 2 has been observed in cells subjected to numerous other means of imposing translational limitation (fig. S2).

From Fig. 2A and the best-fit values of the parameters *r*_{max} and κ_{n} (table S4), we observed that the parameter κ_{n} exhibited a strong, positive correlation with the growth rate of cells in drug-free medium (fig. S3A). Thus, κ_{n} reflects the nutrient quality and is referred to as the “nutritional capacity” of the organism in a medium [see eq. S18 in (*13*) for a molecular interpretation of κ_{n}]. In contrast, the vertical intercept *r*_{max} depended only weakly on the composition of the growth medium (fig. S3B). Qualitatively, the increase of the RNA/protein ratio *r* with increasing degree of translational inhibition can be seen as a compensation for the reduced translational capacity, implemented possibly through the relief of repression of rRNA synthesis by the alarmone ppGpp (*15*), in response to the buildup of intracellular amino acid pools resulting from slow translation. Because *r*_{max} is the (extrapolated) maximal RNA/protein ratio as translation capacity is reduced toward zero, its weak dependence on the quality of the nutrients suggests a common limit in the allocation of cellular resources toward ribosome synthesis.

The simplest model connecting ribosome abundance to gene expression assumes that the total protein content of the cell (called the proteome) is composed of two classes: ribosome-affiliated “class R” proteins (with mass fraction ϕ_{R}), and “others” (with mass fraction 1 – ϕ_{R}) (*5*, *16*). But the maximum allocation to the R-class proteins as derived from the value of *r*_{max}, *13*) for the conversion factor ρ]. This suggests that the “other” proteins can be further subdivided minimally into two classes (Fig. 2B): “class Q” of mass fraction ϕ_{Q}, which is not affected translational inhibition, and the remainder, “class P” of mass fraction ϕ_{P}, with ϕ_{P} → 0 as *17*). Because ϕ_{P} + ϕ_{Q} + ϕ_{R} = 1, we must have _{P} and ϕ_{R}. Together with Eq. 2, the model predicts_{n}. The growth rate independence of protein abundance may be maintained through negative autoregulation (*4*) (fig. S4). Unregulated (or “constitutively expressed”) proteins belong instead to the P-class and can be used to test the prediction of Eq. 4: Expression of β-galactosidase driven by a constitutive promoter (ϕ_{Z}, mass of β-galactosidase per total protein mass) in cells grown under different degrees of chloramphenicol inhibition indeed correlated linearly with λ for each nutrient source studied (Fig. 2C), and the slopes of these correlations (colored lines) agree quantitatively with the nutritional capacity κ_{n} (fig. S5, A and B) as predicted by Eq. 4.

Although the correlations (Eqs. 2 and 4) were revealed by growth with antibiotics, their forms do not pertain specifically to translational inhibition. Equation 4 may be interpreted as a manifestation of P-class proteins providing the nutrients needed for growth [eqs. S15 to S18 in (*13*)], just as Eq. 1 is a reflection of R-class proteins providing the protein synthesis needed for growth (Fig. 3A). For different combinations of the nutritional and translational capacities (κ_{n}, κ_{t}), efficient resource allocation requires that the abundance of P- and R-class proteins be adjusted so that the rate of nutrient influx provided by P (via import or biosynthesis) matches the rate of protein synthesis achievable by R (Fig. 3B), while simultaneously satisfying the constraint of Eq. 3 (Fig. 3C). We can derive the resulting allocation mathematically by postulating that λ, ϕ_{R} (or *r*), and ϕ_{P} are analytical functions of the variables κ_{t} and κ_{n} that respectively capture all molecular details of translation and nutrition (analogous to state variables in thermodynamics). The mathematics is identical to the description of an electric circuit with two resistors (fig. S6), with Eqs. 1 and 4 being analogous to Ohm’s law. Solving these equations simultaneously leads to the Michaelis-Menten relation known empirically for the dependence of cell growth on nutrient level (*18*)
_{c}(κ_{t}) = κ_{t}⋅(*r*_{max} – *r*_{0}) ≈ 2.85 hour^{–1} (based on the average *r*_{max}) corresponds well to the doubling time of ~20 min for typical *E. coli* strains in rich media. Moreover, Eq. 5 quantitatively accounts for the correlation observed between growth rate λ and nutritional capacity κ_{n} (fig. S3A).

This theory can be inverted to predict the effect of protein expression on cell growth. Unnecessary protein expression leads to diminished growth (*19*). Understanding the origin of this growth inhibition is of value in efforts to increase the yield of heterologous protein in bacteria (*20*) and to understand the fitness benefit of gene regulation (*21*, *22*). Aside from protein-specific toxicity, several general causes of growth inhibition have been suggested, including diversion of metabolites (*23*), competition among sigma factors for RNA polymerases (*24*), and competition among mRNA for ribosomes (*19*, *25*).

We modeled the expression of unnecessary protein (of mass fraction ϕ_{U}) as an additional (neutral) component of the proteome that effectively causes a reduction of *r*_{max} to *r*_{max} – ϕ_{U}/ρ (Fig. 4A). Equation 5 then predicts a linear reduction of the growth rate,
_{c} = ρ · (*r*_{max} – *r*_{0}) ≈ 0.48. The prediction quantitatively described the observed growth defect caused by inducible expression of β-galactosidase (Fig. 4B), as well as previous results obtained for various proteins and expression vectors (Fig. 4C) (*19*, *26*), without any adjustable parameters. These results suggest that growth reduction is a simple consequence of ribosome allocation subject to the constraints of Eqs. 1, 3, and 4.

Robust empirical correlations of the RNA/protein ratio with the growth rate (Figs. 1A and 2A and figs. S1 and S2) revealed underlying constraints of cellular resource allocation and led to the formulation of a simple growth theory that provided quantitative predictions and unifying descriptions of many important but seemingly unrelated aspects of bacterial physiology. Like Ohm’s law, which greatly expedited the design of electrical circuits well before electricity was understood microscopically, the empirical correlations described here may be viewed as microbial “growth laws,” the use of which may facilitate our understanding of the operation and design of complex biological systems well before all the underlying regulatory circuits are elucidated at the molecular level.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/330/6007/1099/DC1

Materials and Methods

SOM Text

Figs. S1 to S6

Tables S1 to S7

References

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
See supporting material on
*Science*Online. - ↵
- ↵
- ↵
- ↵ A particular protein species may belong to multiple classes; sometimes this is a result of expression from multiple promoters that are differently regulated.
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- We thank H. Bremer, L. Csonka, P. Dennis, M. Ehrenberg, P. Geiduschek, M. Schaechter, A. Tadmor, and members of the Hwa lab for suggestions and discussions; D. Hughes for the Sm mutant strains; and P.-h. Lee and B. Willumsen for the use of unpublished data. Supported by NIH grant RO1GM77298, NSF grant MCB0746581, and the NSF-supported Center for Theoretical Biological Physics (grant PHY0822283). M.S. was supported by a Natural Sciences and Engineering Research Council of Canada fellowship.