Analytic derivation of bacterial growth laws from a simple model of intracellular chemical dynamics

Abstract

Experiments have found that the growth rate and certain other macroscopic properties of bacterial cells in steady-state cultures depend upon the medium in a surprisingly simple manner; these dependencies are referred to as ‘growth laws’. Here we construct a dynamical model of interacting intracellular populations to understand some of the growth laws. The model has only three population variables: an amino acid pool, a pool of enzymes that transport an external nutrient and produce the amino acids, and ribosomes that catalyze their own and the enzymes’ production from the amino acids. We assume that the cell allocates its resources between the enzyme sector and the ribosomal sector to maximize its growth rate. We show that the empirical growth laws follow from this assumption and derive analytic expressions for the phenomenological parameters in terms of the more basic model parameters. Interestingly, the maximization of the growth rate of the cell as a whole implies that the cell allocates resources to the enzyme and ribosomal sectors in inverse proportion to their respective ‘efficiencies’. The work introduces a mathematical scheme in which the cellular growth rate can be explicitly determined and shows that two large parameters, the number of amino acid residues per enzyme and per ribosome, are useful for making approximations.

A bacterial size law revealed by a coarse-grained model of cell physiology

“…Based on these findings, they proposed a phenomenological model of proteome allocation that extended the first growth law to these orthogonal types of growth rate modulation (Scott et al 2010). More mechanistic coarse-grained models predicting both the growth rate and the coarsegrained proteome as a function of growth conditions have also been proposed confirming and extending this finding (Marr 1991;Molenaar et al 2009;Weisse et al 2015;Maitra & Dill 2015;Bosdriesz et al 2015;Maitra & Dill 2016;Pandey & Jain 2016;Giordano et al 2016;Liao et al 2017; Thomas et al 2018;Sharma et al 2018;Pandey et al 2018).…”

“…Our model unifies previous efforts to understand proteome allocation(Molenaar et al 2009;Scott et al 2010;Scott et al 2014;Weisse et al 2015;Pandey & Jain 2016) and division control(Fantes 1977;Taheri-Araghi, Bradde, et al 2015;Basan et al 2015;Ghusinga et al 2016). While other theoretical models(Scott et al 2010;Scott et al 2014;Weisse et al 2015;Pandey & Jain 2016) have also captured proteome allocation data, our model stands out because: 1) it uses a minimal set of parameters (similar to the 'rheostat' model,(Scott et al 2014)); 2) it is holistic with respect to cell composition and cell size; 3) it does not neglect the mass fraction of protein precursors. Despite its simplicity and low parameter number, our model quantitatively reproduces experimental data on proteome allocation (Figure 1B) and average cell size (Figure 3C) for the three types of growth rate modulation (change of nutrient quality, chloramphenicol-mediated ribosome inactivation and expression of useless proteins).…”

“…As in previous work, the model (Figure 1, see Methods for a mathematical description) considers only two types of molecular components: protein precursors (noted A) and proteins (Molenaar et al 2009;Weisse et al 2015;Pandey & Jain 2016). Proteins are split between four different classes (or sectors): transport and metabolism proteins (referred to as only metabolic sector for brevity) (E), ribosomal proteins (R), housekeeping proteins (Q), and division proteins (X).…”

How to deal with parameters for whole-cell modelling

“…While real cells will be much more complicated, these simple model systems have successfully provided insight into fundamental cell physiology, e.g. processes affecting microbial growth rates [12,13,15,16].…”

Exponential trajectories, cell size fluctuations and the adder property in bacteria follow from simple chemical dynamics and division control

“…We will later (in section IV D) give results when h has a non-trivial Z dependence arising from auto-regulation of Z (e.g., cooperativity). The PTR model [50] has been independently extended by [51] to include a sector whose dynamics is given by (5) and that triggers division upon reaching a population threshold. Their treatment differs from ours in various respects, in particular, that the new sector affects the local dynamics of the PTR sector through the volume.…”