Optimal proteome allocation and the temperature dependence of microbial growth laws

Although the effect of temperature on microbial growth has been widely studied, the role of proteome allocation in bringing about temperature-induced changes remains elusive. To tackle this problem, we propose a coarse-grained model of microbial growth, including the processes of temperature-sensitive protein unfolding and chaperone-assisted (re)folding. We determine the proteome sector allocation that maximizes balanced growth rate as a function of nutrient limitation and temperature. Calibrated with quantitative proteomic data for Escherichia coli, the model allows us to clarify general principles of temperature-dependent proteome allocation and formulate generalized growth laws. The same activation energy for metabolic enzymes and ribosomes leads to an Arrhenius increase in growth rate at constant proteome composition over a large range of temperatures, whereas at extreme temperatures resources are diverted away from growth to chaperone-mediated stress responses. Our approach points at risks and possible remedies for the use of ribosome content to characterize complex ecosystems with temperature variation.

where v m f , v r f , and v q f are the folding rates per unit of biomass (in g/(g prot·h)) of metabolic proteins, house-keeping proteins, and ribosomal proteins, respectively.
• Protein unfolding: where v m u , v r u , and v q u are the unfolding rates per unit of biomass (in g/(g prot·h)) of metabolic proteins, house-keeping proteins, and ribosomal proteins, respectively.

Mass balance
Given this set of reactions, we obtain the following mass-balanced system: (1)

Kinetics
Assuming that the mass of precursors is negligible compared to that of proteins, we define the mass fractions (in g/g prot) as follows: The rates for precursor and protein synthesis are taken as Michaelis-Menten functions: Folding and unfolding rates are represented by mass-action kinetics: Finally, we can compute the specific growth rate µ: Actually, given that biomass corresponds by definition to proteins, the specific growth rate equals the rate of protein synthesis per unit of biomass.

Full kinetic model
We get by simple derivation the dynamics of the mass fractions: Quasi-steady-state approximation (QSSA) The folding/unfolding rates are much faster than the synthesis rates of precursors and proteins [1]. Therefore, the system can be decomposed into two time-scales: the first five equations are slow, while the last three ones are fast.
Using the quasi-steady-state assumption [2], the system can be reduced to the dynamics of the slow system, taking the fast system at equilibrium. Neglecting the small terms (v r << v f , v u ), the equilibrium of the fast system is given by Using the aforementioned kinetic expressions, the fast equilibrium v m f = v m u gives: Recalling that m = m u + m f , we finally get: Similarly, we obtain for the other protein sectors: With this approximation, we obtain the QSSA model presented in the main text. Plugging in the kinetic expressions, the system reads: Comparison of the full model and the QSSA model Finally, we compare the optimal solutions of System (4) (obtained analytically) and of System (2) (obtained numerically), using the parameter set given in Tab. 2 of the main article. Given that only the ratio k f /k u has been estimated, the value of k f has to be fixed. We consider two cases: 100. This corresponds to the difference between the mean rates of protein folding and synthesis [1], which supports our slow-fast approximation.
1, corresponding to a worst case with no time-scale separation.
In the first case, the equilibria of the QSSA system are almost identical to those of the complete system ( Supplementary Fig. 5B). In the second case, the equilibria are different, but the trends remain similar. The QSSA allows to obtain a simpler model (much more tractable for mathematical analysis) with almost the same optimal equilibria, which supports our approach. Nonetheless, given the wide time scale distribution of protein folding rates [1], the QSSA model should be used with caution in other situations, e.g., to study short-term response and regulation.  Contrary to the model predictions, the chaperone content in (A) increases only at high, but not at low temperatures. Actually, different proteins are involved for cold and heat stresses. Cold-induced proteins have been observed in [3] and [4] (e.g., protein G41.2 shown in (B)), but their functions remain unknown or do not correspond to chaperones. Since these studies, other cold-induced proteins have been identified with a role of chaperone for RNA/DNA or proteins [5]. The global trend of the chaperone sector is thus in line with the model predictions, when aggregating both cold and heat-shock chaperones.