A unifying autocatalytic network-based framework for bacterial growth laws.

Recently discovered simple quantitative relations, known as bacterial growth laws, hint at the existence of simple underlying principles at the heart of bacterial growth. In this work, we provide a unifying picture of how these known relations, as well as relations that we derive, stem from a universal autocatalytic network common to all bacteria, facilitating balanced exponential growth of individual cells. We show that the core of the cellular autocatalytic network is the transcription-translation machinery-in itself an autocatalytic network comprising several coupled autocatalytic cycles, including the ribosome, RNA polymerase, and transfer RNA (tRNA) charging cycles. We derive two types of growth laws per autocatalytic cycle, one relating growth rate to the relative fraction of the catalyst and its catalysis rate and the other relating growth rate to all the time scales in the cycle. The structure of the autocatalytic network generates numerous regimes in state space, determined by the limiting components, while the number of growth laws can be much smaller. We also derive a growth law that accounts for the RNA polymerase autocatalytic cycle, which we use to explain how growth rate depends on the inducible expression of the rpoB and rpoC genes, which code for the RpoB and C protein subunits of RNA polymerase, and how the concentration of rifampicin, which targets RNA polymerase, affects growth rate without changing the RNA-to-protein ratio. We derive growth laws for tRNA synthesis and charging and predict how growth rate depends on temperature, perturbation to ribosome assembly, and membrane synthesis.

Cellular interactions constrain tumor growth.

A tumor is made up of a heterogeneous collection of cell types, all competing on a fitness landscape mediated by microenvironmental conditions that dictate their interactions. Despite the fact that much is known about cell signaling, cellular cooperation, and the functional constraints that affect cellular behavior, the specifics of how these constraints (and the range over which they act) affect the macroscopic tumor growth laws that govern total volume, mass, and carrying capacity remain poorly understood. We develop a statistical mechanics approach that focuses on the total number of possible states each cell can occupy and show how different assumptions on correlations of these states give rise to the many different macroscopic tumor growth laws used in the literature. Although it is widely understood that molecular and cellular heterogeneity within a tumor is a driver of growth, here we emphasize that focusing on the functional coupling of states at the cellular level is what determines macroscopic growth characteristics.

Close Encounters of the Cell Kind: The Impact of Contact Inhibition on Tumour Growth and Cancer Models.

Cancer is a complex phenomenon, and the sheer variation in behaviour across different types renders it difficult to ascertain underlying biological mechanisms. Experimental approaches frequently yield conflicting results for myriad reasons, and mathematical modelling of cancer is a vital tool to explore what we cannot readily measure, and ultimately improve treatment and prognosis. Like experiments, models are underpinned by certain biological assumptions, variation of which can lead to divergent predictions. An outstanding and important question concerns contact inhibition of proliferation (CIP), the observation that proliferation ceases when cells are spatially confined by their neighbours. CIP is a characteristic of many healthy adult tissues, but it remains unclear to which extent it holds in solid tumours, which exhibit regions of hyper-proliferation, and apparent breakdown of CIP. What precisely occurs in tumour tissue remains an open question, which mathematical modelling can help shed light on. In this perspective piece, we explore the implications of different hypotheses and available experimental evidence to elucidate the implications of these scenarios. We also outline how erroneous conclusions about the nature of tumour growth may be arrived at by looking selectively at biological data in isolation, and how this might be circumvented.

Phenomenological models as effective tools to discover cellular design principles.

Microbes have proved useful to us in many different ways. To utilize microbes, we have mostly focused on maximizing growth, to improve yield of chemicals derived from the microbes. However, to truly tap into their potential, we should also aim to understand microbial physiology. We present a historical perspective of the developments in the field of Microbial Biotechnology, focusing on how the growth-modelling approaches have changed. Starting from simple empirical growth models, we have evolved towards mechanistic and phenomenological models which use molecular and physiological details to drastically improve prediction power of these models. Lastly, we explore the as of yet unsolved questions in microbial physiology, and discuss how the ability to monitor microbial growth at single cell resolution using the lab-on-a-chip technologies is uncovering previously unobservable causal principles underlying microbial growth.

Bacterial growth laws reflect the evolutionary importance of energy efficiency.

We are interested in the balance of energy and protein synthesis in bacterial growth. How has evolution optimized this balance? We describe an analytical model that leverages extensive literature data on growth laws to infer the underlying fitness landscape and to draw inferences about what evolution has optimized in Escherichia coli. Is E. coli optimized for growth speed, energy efficiency, or some other property? Experimental data show that at its replication speed limit, E. coli produces about four mass equivalents of nonribosomal proteins for every mass equivalent of ribosomes. This ratio can be explained if the cell's fitness function is the the energy efficiency of cells under fast growth conditions, indicating a tradeoff between the high energy costs of ribosomes under fast growth and the high energy costs of turning over nonribosomal proteins under slow growth. This model gives insight into some of the complex nonlinear relationships between energy utilization and ribosomal and nonribosomal production as a function of cell growth conditions.

Using stochastic cell division and death to probe minimal units of cellular replication.

The invariant cell initiation mass measured in bacterial growth experiments has been interpreted as a minimal unit of cellular replication. Here we argue that the existence of such minimal units induces a coupling between the rates of stochastic cell division and death. To probe this coupling we tracked live and dead cells in Escherichia coli populations treated with a ribosome-targeting antibiotic. We find that the growth exponent from macroscopic cell growth or decay measurements can be represented as the difference of microscopic first-order cell division and death rates. The boundary between cell growth and decay, at which the number of live cells remains constant over time, occurs at the minimal inhibitory concentration (MIC) of the antibiotic. This state appears macroscopically static but is microscopically dynamic: division and death rates exactly cancel at MIC but each is remarkably high, reaching 60% of the antibiotic-free division rate. A stochastic model of cells as collections of minimal replicating units we term 'widgets' reproduces both steady-state and transient features of our experiments. Sub-cellular fluctuations of widget numbers stochastically drive each new daughter cell to one of two alternate fates, division or death. First-order division or death rates emerge as eigenvalues of a stationary Markov process, and can be expressed in terms of the widget's molecular properties. High division and death rates at MIC arise due to low mean and high relative fluctuations of widget number. Isolating cells at the threshold of irreversible death might allow molecular characterization of this minimal replication unit.

Matching metabolic supply to demand optimizes microbial growth.

Recent research has strengthened the notion that microbes allocate their biosynthetic capacity to maximize the growth rate, λ. Yet many microbes can grow substantially faster after laboratory evolution. Chure and Cremer advance a resource-allocation model, which they derive from first principles, that offers resolution to this conundrum.

A coarse-grained resource allocation model of carbon and nitrogen metabolism in unicellular microbes.

Coarse-grained resource allocation models (C-GRAMs) are simple mathematical models of cell physiology, where large components of the macromolecular composition are abstracted into single entities. The dynamics and steady-state behaviour of such models provides insights on optimal allocation of cellular resources and have explained experimentally observed cellular growth laws, but current models do not account for the uptake of compound sources of carbon and nitrogen. Here, we formulate a C-GRAM with nitrogen and carbon pathways converging on biomass production, with parametrizations accounting for respirofermentative and purely respiratory growth. The model describes the effects of the uptake of sugars, ammonium and/or compound nutrients such as amino acids on the translational resource allocation towards proteome sectors that maximized the growth rate. It robustly recovers cellular growth laws including the Monod law and the ribosomal growth law. Furthermore, we show how the growth-maximizing balance between carbon uptake, recycling, and excretion depends on the nutrient environment. Lastly, we find a robust linear correlation between the ribosome fraction and the abundance of amino acid equivalents in the optimal cell, which supports the view that simple regulation of translational gene expression can enable cells to achieve an approximately optimal growth state.

Containment effort reduction and regrowth patterns of the Covid-19 spreading.

In all countries the political decisions aim to achieve an almost stable configuration with a small number of new infected individuals per day due to Covid-19. When such a condition is reached, the containment effort is usually reduced in favor of a gradual reopening of the social life and of the various economical sectors. However, in this new phase, the infection spread restarts and, moreover, possible mutations of the virus give rise to a large specific growth rate of the infected people. Therefore, a quantitative analysis of the regrowth pattern is very useful. We discuss a macroscopic approach which, on the basis of the collected data in the first lockdown, after few days from the beginning of the new phase, outlines different scenarios of the Covid-19 diffusion for longer time. The purpose of this paper is a demonstration-of-concept: one takes simple growth models, considers the available data and shows how the future trend of the spread can be obtained. The method applies a time dependent carrying capacity, analogously to many macroscopic growth laws in biology, economics and population dynamics. The illustrative cases of France, Italy and United Kingdom are analyzed.

Fundamental limits on the rate of bacterial growth and their influence on proteomic composition.

Despite abundant measurements of bacterial growth rate, cell size, and protein content, we lack a rigorous understanding of what sets the scale of these quantities and when protein abundances should (or should not) depend on growth rate. Here, we estimate the basic requirements and physical constraints on steady-state growth by considering key processes in cellular physiology across a collection of Escherichia coli proteomic data covering ≈4,000 proteins and 36 growth rates. Our analysis suggests that cells are predominantly tuned for the task of cell doubling across a continuum of growth rates; specific processes do not limit growth rate or dictate cell size. We present a model of proteomic regulation as a function of nutrient supply that reconciles observed interdependences between protein synthesis, cell size, and growth rate and propose that a theoretical inability to parallelize ribosomal synthesis places a firm limit on the achievable growth rate. A record of this paper's transparent peer review process is included in the supplemental information.

Nucleation and growth of cholesteric collagen tactoids: A time-series statistical analysis based on integration of direct numerical simulation (DNS) and long short-term memory recurrent neural network (LSTM-RNN).

HYPOTHESIS: Liquid-crystalline phase separation by nucleation and growth (NG) is a crucial step in the formation of collagen-based biomaterials. However, the fundamental mechanisms are not completely understood for chiral lyotropic colloidal mesogens such as collagen. METHODOLOGY: To capture the dynamics of NG under a quenching process into the biphasic equilibrium zone, we use direct numerical simulation based on the time-dependent Ginzburg-Landau model allowing minimization of the total free energy comprised of five key contributions: phase separation (Flory-Huggins), ordering (Landau-de Gennes), chiral orientational elasticity (Frank-Oseen-Mermin), interfacial and coupling effects. LSTM-RNN is applied as a surrogate model to greatly enrich the results. Significant correlations are established using Symbolic Regression. FINDINGS: We quantify the NG boundaries existing in the collagen phase diagram that has recently been developed and validated by our thermodynamic model (Khadem and Rey, 2019 [1]). We characterize the three NG stages (induction, nucleation, and coarsening) in terms of tactoids' shape, morphology, growth laws, and population across the NG zone. Wide-range generic correlations are developed, revealing the quench depth dependence of NG characteristics and connecting the sequential NG stages. We confirm experimental observations on time-dependent growth law exponent changes from an initial n≈0.5 for the mass transfer limited regime to n≈1 for the volume-driven phase ordering regime upon increasing quench depth during the nucleation period and having exclusively a value of n≈0.5 for the coarsening period regardless of quench depth. We lastly uncover the underlying physics behind the NG phenomena.

Patterns of the COVID-19 pandemic spread around the world: exponential versus power laws.

We have analysed the COVID-19 epidemic data of more than 174 countries (excluding China) in the period between 22 January and 28 March 2020. We found that some countries (such as the USA, the UK and Canada) follow an exponential epidemic growth, while others (like Italy and several other European countries) show a power law like growth. Regardless of the best fitting law, many countries can be shown to follow a common trajectory that is similar to Italy (the epicentre at the time of analysis), but with varying degrees of delay. We found that countries with 'younger' epidemics, i.e. countries where the epidemic started more recently, tend to exhibit more exponential like behaviour, while countries that were closer behind Italy tend to follow a power law growth. We hypothesize that there is a universal growth pattern of this infection that starts off as exponential and subsequently becomes more power law like. Although it cannot be excluded that this growth pattern is a consequence of social distancing measures, an alternative explanation is that it is an intrinsic epidemic growth law, dictated by a spatially distributed community structure, where the growth in individual highly mixed communities is exponential but the longer term, local geographical spread (in the absence of global mixing) results in a power law. This is supported by computer simulations of a metapopulation model that gives rise to predictions about the growth dynamics that are consistent with correlations found in the epidemiological data. Therefore, seeing a deviation from straight exponential growth may be a natural progression of the epidemic in each country. On the practical side, this indicates that (i) even in the absence of strict social distancing interventions, exponential growth is not an accurate predictor of longer term infection spread, and (ii) a deviation from exponential spread and a reduction of estimated doubling times do not necessarily indicate successful interventions, which are instead indicated by a transition to a reduced power or by a deviation from power law behaviour.

Quantitative insights into the cyanobacterial cell economy.

Phototrophic microorganisms are promising resources for green biotechnology. Compared to heterotrophic microorganisms, however, the cellular economy of phototrophic growth is still insufficiently understood. We provide a quantitative analysis of light-limited, light-saturated, and light-inhibited growth of the cyanobacterium Synechocystis sp. PCC 6803 using a reproducible cultivation setup. We report key physiological parameters, including growth rate, cell size, and photosynthetic activity over a wide range of light intensities. Intracellular proteins were quantified to monitor proteome allocation as a function of growth rate. Among other physiological acclimations, we identify an upregulation of the translational machinery and downregulation of light harvesting components with increasing light intensity and growth rate. The resulting growth laws are discussed in the context of a coarse-grained model of phototrophic growth and available data obtained by a comprehensive literature search. Our insights into quantitative aspects of cyanobacterial acclimations to different growth rates have implications to understand and optimize photosynthetic productivity.

A model of optimal protein allocation during phototrophic growth.

Photoautotrophic growth depends upon an optimal allocation of finite cellular resources to diverse intracellular processes. Commitment of a certain mass fraction of the proteome to a specific cellular function typically reduces the proteome available for other cellular functions. Here, we develop a semi-quantitative kinetic model of cyanobacterial phototrophic growth to describe such trade-offs of cellular protein allocation. The model is based on coarse-grained descriptions of key cellular processes, in particular carbon uptake, metabolism, photosynthesis, and protein translation. The model is parameterized using literature data and experimentally obtained growth curves. Of particular interest are the resulting cyanobacterial growth laws as fundamental characteristics of cellular growth. We show that the model gives rise to similar growth laws as observed for heterotrophic organisms, with several important differences due to the distinction between light energy and carbon uptake. We discuss recent experimental data supporting the model results and show that coarse-grained growth models have implications for our understanding of the limits of phototrophic growth and bridge a gap between molecular physiology and ecology.

A Bacterial Growth Law out of Steady State.

Bacterial growth follows simple laws in constant conditions. However, bacteria in nature often face fluctuating environments. We therefore ask whether there are growth laws that apply to changing environments. We derive a law for upshifts using an optimal resource-allocation model: the post-shift growth rate equals the geometrical mean of the pre-shift growth rate and the growth rate on saturating carbon. We test this using chemostat and batch culture experiments, as well as previous data from several species. The increase in growth rate after an upshift indicates that ribosomes have spare capacity (SC). We demonstrate theoretically that SC has the cost of slow steady-state growth but is beneficial after an upshift because it prevents large overshoots in intracellular metabolites and allows rapid response to change. We also provide predictions for downshifts. The present study quantifies the optimal degree of SC, which rises the slower the growth rate, and suggests that SC can be precisely regulated.

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

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.

Unified approach to growth and aging in biological, technical and biotechnical systems.

Complex systems, in many different scientific sectors, show coarse-grain properties with simple growth laws with respect to fundamental microscopic algorithms. We propose a classification scheme of growth laws which includes human aging, tumor (and/or tissue) growth, logistic and generalized logistic growth and the aging of technical devices. The proposed classification permits to evaluate the aging/failure of combined new bio-technical "manufactured products", where part of the system evolves in time according to biological-mortality laws and part according to technical device behaviors. Moreover it suggests a direct relation between the mortality leveling-off for humans and technical devices and the observed small cure probability for large tumors.