Dynamical Allocation of Cellular Resources as an Optimal Control Problem: Novel Insights into Microbial Growth Strategies

Microbial physiology exhibits growth laws that relate the macromolecular composition of the cell to the growth rate. Recent work has shown that these empirical regularities can be derived from coarse-grained models of resource allocation. While these studies focus on steady-state growth, such conditions are rarely found in natural habitats, where microorganisms are continually challenged by environmental fluctuations. The aim of this paper is to extend the study of microbial growth strategies to dynamical environments, using a self-replicator model. We formulate dynamical growth maximization as an optimal control problem that can be solved using Pontryagin’s Maximum Principle. We compare this theoretical gold standard with different possible implementations of growth control in bacterial cells. We find that simple control strategies enabling growth-rate maximization at steady state are suboptimal for transitions from one growth regime to another, for example when shifting bacterial cells to a medium supporting a higher growth rate. A near-optimal control strategy in dynamical conditions is shown to require information on several, rather than a single physiological variable. Interestingly, this strategy has structural analogies with the regulation of ribosomal protein synthesis by ppGpp in the enterobacterium Escherichia coli. It involves sensing a mismatch between precursor and ribosome concentrations, as well as the adjustment of ribosome synthesis in a switch-like manner. Our results show how the capability of regulatory systems to integrate information about several physiological variables is critical for optimizing growth in a changing environment.

The values derived below are summarized in S1 Table. e M By definition, e M is the effective turnover of the metabolic macroreaction producing precursors from external substrates, obtained by dividing the reaction rate v M by the enzyme concentration m (Eq. 6). The unit of e M is min −1 , and can be decomposed as follows: .
Note that e M = k M s/(K M + s) where k M is a rate constant, indicating the maximal rate of conversion of external nutrients to precursor metabolites. e M will thus vary with the concentration s of the external nutrients and the kind of nutrient. For example, the precursor mass that can be produced from 1 g of glucose is higher than that produced from 1 g of acetate.
How can we find a typical value for k M , and thus for e M (both have the same order of magnitude if we suppose that the reaction is not operating far below saturation, that is, e M ≈ k M )? A reasonable estimate for k M can be obtained from the turnover numbers of reactions involved in the synthesis of charged tRNA, since the latter are directly consumed by the most abundant part of the gene expression machinery, the ribosomes. Ref.
[1] provides a typical value for such a reaction, catalyzed by glutaminyl-tRNA synthetase: k cat,GlnRS = 3.2 s -1 , indicating that on average 3.2 glutaminyl-tRNA molecules are produced per glutaminyl-tRNA synthetase molecule per second. After conversion to mass units using molar weight from [2], this yields k cat,GlnRS = 3.2 · 147 64.4 · 10 3 ≈ 10 −3 g of glutaminyl-tRNA · g of enzyme −1 · s −1 . We therefore take k M ≈ 3. To obtain an order of magnitude for the mass of macromolecules, we focus on proteins since they are the most abundant macromolecules in the cell [3]. The dimensional analysis of k R thus becomes: The values in the last equality are available from the literature [3,4,5,6]. We obtain k R ≈ 10 · 100 10 6 · 3600 ≈ 3.6 h −1 . This value is comparable with the translational capacity k T , in µg of protein per µg of ribosomal protein per hour, given by Scott et al.

K R
A value for the parameter K R , representing the half-saturation constant of macromolecular synthesis, is more difficult to obtain from the literature. However, assuming that ribosomes operate close to saturation (80% over a range of growth rates [3]), we find that K R ≈ 0.25 p, with p the total amino acid concentration. The total concentration of amino acids in the cell is around 150 mmol L -1 [8], which with a mean molecular weight of 118.9 g mol -1 for amino acids [5], yields a mass concentration of 17.8 g L -1 . These considerations led to the following order of magnitude for K R : β β is the inverse of the cellular density of macromolecules, which has been shown constant during balanced growth over a large range of growth rates [9], and there is some data suggesting that β varies little during growth transitions as well [10]. From [11,12] we take the following typical value for β: β ≈ 1 300 ≈ 0.003 L g −1 .

E M and K
From the values of the parameter in the dimensional model, one can deduce the parameters in the nondimensional model used in the simulations: 3.6 = 1 , K = β K R = 3 · 10 −3 · 1 = 0.003.