The Hands-On Growth Laws Theory Cookbook

TL;DR

Each section provides detailed, bare-bone models to start working in each area, from basic steady-state growth to variable environments and focusing on different key layers relevant to biosynthesis, transcription, translation, nutrient sensing and protein degradation, links between cell cycle and growth.

Abstract

This tutorial covers the emerging field of coarse-grained cellular growth modeling, and aims to bridge the gap between theoretical foundations and practical application. By adopting an original "cookbook" approach, it is designed to offer a hands-on guide for constructing and analyzing different key aspects of cellular growth, focusing on available results for bacteria and beyond. The tutorial is structured as a series of step-by-step "recipes", and covers essential concepts, recent literature, and key challenges. It aims to empower a broad audience, from students to seasoned researchers, to replicate, extend, and innovate in this scientific area. Specifically, each section provides detailed, bare-bone models to start working in each area, from basic steady-state growth to variable environments and focusing on different key layers relevant to biosynthesis, transcription, translation, nutrient sensing and protein degradation, links between cell cycle and growth, ending with ecological insights.

Figures (10)

• Figure 1: Overview of the topics covered in this tutorial. The topics covered in this tutorial begin with the classical proteome allocation theory (Sec. \ref{['sec2:classical']}), which introduces the phenomenology of growth laws. Building on this foundation, Sec. \ref{['sec3:translation']} explores the dependence of the translation elongation rate and ribosomal activity on growth, providing a mechanistic interpretation of these trends based on ribosome recruitment. Section \ref{['sec4:transcription']} further extends the framework by incorporating transcriptional dynamics and traffic theory, while Sec. \ref{['sec5:degradation']} refines the model to account for slow-growth regimes by including protein degradation. Subsequently, Sec. \ref{['sec6:mecha_regulation']} examines the mechanistic regulation of ribogenesis in both prokaryotes and eukaryotes, offering a molecular perspective on the emergence of growth laws. Section \ref{['sec7:shifts']} then illustrates how to model cellular behavior in fluctuating environments. The final three sections address broader topics beyond the classical theory: density homeostasis and volume–mass coupling (Sec. \ref{['sec8:density']}); the interplay between growth laws and cell-cycle progression (Sec. \ref{['sec9:cell_cycle']}); and, finally, the insights gained from a community and ecological perspective (Sec. \ref{['sec10:communities']}).
• Figure 2: Linear relationships between growth rate and ribosome allocation (RNA/protein ratio or $\phi_\text{R}$), observed under different conditions. (a) Variation in nutrient quality alters growth rate and ribosome allocation. (b) Sub-lethal doses of the translation-inhibiting antibiotic chloramphenicol also affect ribosome allocation and growth rate. (c) The burden of over-expressing an unnecessary protein leads to reduced growth. The squares in panels b and c represent identical conditions (no induction of unnecessary proteins and no addition of chloramphenicol), enabling direct comparison of the plots. Panels (a)-(c) are inspired by and adapted from Ref. scott_interdependence_2010. (d) While proteome sectors remain stable during exponential balanced growth, their relative allocation varies across different environmental conditions.
• Figure 3: Translation elongation rate and ribosome activity as function of growth rate. (a) The ribosome elongation rate $\varepsilon$ as a function of $\phi_\text{R}$. The values of $\varepsilon$, obtained by varying nutrient conditions and under sub-lethal antibiotic dosage (chloramphenicol), collapse when plotted as a function of $\phi_\text{R}$, see Dai2016. (b) The active ribosome fraction $f_\text{a}$ as a function of the growth rate and $\phi_\text{R}$, predicted by Eq. (\ref{['eq:fa2']}).This figure is inspired by and adapted from Ref. Dai2016.
• Figure 4: Scheme and prediction of the transcription-translation model. (a) Scheme of the model recipe considering transcription and translation initiation rates $\alpha^\text{TX}$ and $\alpha^\text{TL}$ as dependent on the availability of free RNAPs and ribosomes, respectively. mRNAs are synthesized with a flux $J^\text{TX}$, then either degraded at rate $\delta$ or used as templates for translation. Both unbound RNAPs and ribosomes can be sequestered by inactivation factors, reducing their availability for gene expression. (b) Illustration of the mRNA growth law balakrishnan_principles_2022, showing the roughly linear relationship between total mRNA concentration and growth rate. (c) While the total RNAP concentration remains approximately constant across conditions, sequestration factors modulate the fraction of RNAPs actively engaged in transcription balakrishnan_principles_2022. Panel (b) is inspired by and adapted from Ref. balakrishnan_principles_2022.
• Figure 5: Protein degradation and ribosome activity as function of growth rate. (a) The growth rate dependence of the degradation rate $\eta$ for the bacterium E. coli. The x-axis represents the steady state growth rate $\lambda$, while the y-axis the degradation rate $\eta$. Data points are sketched from calabrese_protein_2022, the original data can be found in Pine1973. (b) The active ribosome franction $f_\text{a}$ as a function of the growth rate and $\phi_\text{R}$ for a model including degradation (blue upper line) and for one not including it (grey lower line). (c) Scheme of the degradation process. The process degrades proteins possibly to (I) produce energy, (II) obtain amino acids under starvation conditions or (III) function as a quality control, to degrade misfolded or damaged proteins. This figure is inspired by and adapted from Ref. calabrese_protein_2022.