Emergence of robust growth laws from optimal regulation of ribosome synthesis

Abstract

Bacteria must constantly adapt their growth to changes in nutrient availability; yet despite large-scale changes in protein expression associated with sensing, adaptation, and processing different environmental nutrients, simple growth laws connect the ribosome abundance and the growth rate. Here, we investigate the origin of these growth laws by analyzing the features of ribosomal regulation that coordinate proteome-wide expression changes with cell growth in a variety of nutrient conditions in the model organism Escherichia coli. We identify supply-driven feedforward activation of ribosomal protein synthesis as the key regulatory motif maximizing amino acid flux, and autonomously guiding a cell to achieve optimal growth in different environments. The growth laws emerge naturally from the robust regulatory strategy underlying growth rate control, irrespective of the details of the molecular implementation. The study highlights the interplay between phenomenological modeling and molecular mechanisms in uncovering fundamental operating constraints, with implications for endogenous and synthetic design of microorganisms.

Keywords: growth control; metabolic control; phenomenological model; resource allocation; synthetic biology.

Figure 1. Linear growth relations and minimal partitioning of the proteome

(A) Empirical relations between the ribosomal protein fraction and growth rate in exponentially growing Escherichia coli. Under changes in nutrient quality (filled symbols) or translational capacity (colored lines), the ribosomal protein fraction ϕ_R is a linear function of the growth rate λ. (B) The growth relations in (A), along with data on metabolic proteins responsible for coordinating carbon and nitrogen assimilation (You et al, 2013), suggest that a minimum partitioning of the proteome consists of three protein fractions (Scott et al, 2010): a growth rate-independent fraction ϕ_Q, a fraction including ribosome-affiliated proteins ϕ_R, and a metabolic fraction ϕ_P containing the remainder, including catabolic and anabolic enzymes. The growth rate dependence of the ribosome and metabolic proteins are constrained by the partitioning so that .

Figure 2. Amino acid flux balance and growth rate maximization

(A) In exponential growth, the amino acid consumption rate via protein synthesis, , must balance the supply rate via transport and biosynthesis, νϕ_P [equation (12)], to maintain a constant amino acid pool size. Using the proteome partitioning constraint that ribosomal protein fraction ϕ_R and metabolic protein fraction ϕ_P sum to a constant, (Fig 1B), the supply rate can be written as . The cell then must regulate the ribosomal protein fraction ϕ_R to both balance and maximize the flux through the system. (B) The ribosomal protein fraction ϕ_R determines the steady-state amino acid level a* (green solid line) and consequently the growth rate λ [equation (17)], when the amino acid flux is balanced. (C) The growth rate λ (green solid line) exhibits a unique maximum corresponding to an optimal size of the ribosomal protein fraction ϕ_R. The upper bound on the growth rate maximum occurs when the translational efficiency and nutritional efficiency are both maximal for a given nutrient environment, and (filled circle). (D) The optimal size of the ribosomal protein fraction ϕ_R depends upon the growth environment (filled circles), illustrated here by a change in the nutrient quality of the medium: poor nutrient ν₀ = 2.5/h (red solid line), good nutrient ν₀ = 3.3/h (blue solid line), and rich nutrient ν₀ = 5.8/h (green solid line). Dashed lines correspond to the empirical relations shown in Fig 1A, (black dashed line) and (colored dashed lines). The amino acid level for efficient peptide elongation K_γ = 10⁻⁴, and the level to trigger negative feedback inhibition of amino acid supply K_ν = 5K_γ = 5 × 10⁻⁴. The remaining parameters are γ₀ = 5.9/h, = 0.07 and = 0.55 (Scott et al, 2010).

Figure 3. Regulation of the ribosomal protein fraction ϕ_R

(A) Internal amino acid pools are kept in check by negative feedback inhibition (red block arrow) via regulation of protein expression (described by η_a) or allosteric inhibition (described by a decrease in efficacy k_a). Negative feedback inhibition is important to rapidly regain steady-state growth upon nutrient shift, but plays an auxiliary role in growth rate maximization. When internal amino acid pools increase, supply-driven activation of ribosomal protein synthesis (green arrow) increases the rate of consumption to restore flux balance. (B) If the amino acid level for efficient elongation (K_γ) and the level for negative feedback inhibition of amino acid supply (K_ν) are well separated, K_γ << K_ν, then the ribosomal protein fraction ϕ_R (blue solid line) is only weakly dependent on the steady-state amino acid level a* close to the optimal value (filled circle) (lower axis displays amino acid level in units of mass fraction, upper axis displays the corresponding level in units of concentration). The intersection of (blue line) and the ribosome synthesis function defines the steady state of the system (Supplementary Fig S2). A ribosome synthesis control function (dashed line) is shown passing through that yields the optimal ribosomal protein fraction and growth rate . Notice that any curve intersecting ϕ_R in the plateau (white region) will return a steady-state ribosomal protein fraction close to the optimum, . (C) Control functions that pass through this plateau provide autonomous optimal control of the ribosomal protein fraction over a range of nutrient conditions. The dark gray band illustrates the range of control functions that determine ribosomal protein fraction ϕ_R to within 10% of the optimum over a range of nutrient conditions. The colored lines and symbols correspond to those in Fig 2, with ν₀ = 2.5/h (red), ν₀ = 3.3/h (blue), and ν₀ = 5.8/h (green); K_γ = 10⁻⁴, and K_ν = 50K_γ = 5 × 10⁻³. Experimental estimates for K_γ, K_ν, and steady-state amino acid pool sizes are given in Supplementary Table S1 (illustrated in Supplementary Fig S3).

Figure 4. Schematic illustration of the growth model

The analysis identifies amino acid flux as a primary transaction during exponential growth, with supply rate proportional to the metabolic protein fraction and consumption through protein synthesis. If the amino acid pool becomes too large, negative feedback regulation attenuates the supply flux (red block arrow) and guarantees the system can reach a stable equilibrium. Supply-driven activation of ribosomal protein synthesis ensures optimal allocation of cellular resources by monitoring amino acid incorporation at the ribosome (green arrow)—the regulation is agnostic about the details of the supply. As a result, there is an inherent plasticity in the system. Specific catabolic pathways can be turned on and off depending upon the nutrient environment, with regulation of ribosomal protein synthesis automatically adjusting the rate of amino acid consumption to optimize growth rate. From an evolutionary perspective, the coarse-grained modularity in the system, with demand flux adjusted to the supply, allows innovative metabolic proteins and pathways to be swapped into the genome with robust regulation of ribosome synthesis ensuring maximal growth rate.