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

A generalized adder for cell size homeostasis: Effects on stochastic clonal proliferation

Measurements of cell size dynamics have revealed phenomenological principles by which individual cells control their size across diverse organisms. One of the emerging paradigms of cell size homeostasis is the adder, where the cell cycle duration is established such that the cell size increase from birth to division is independent of the newborn cell size. We provide a mechanistic formulation of the adder, considering that cell size follows any arbitrary nonexponential growth law. Our results show that the main requirement to obtain an adder regardless of the growth law (the time derivative of cell size) is that cell cycle regulators are produced at a rate proportional to the growth law, and cell division is triggered when these molecules reach a prescribed threshold level. Among the implications of this generalized adder, we investigate fluctuations in the proliferation of single-cell-derived colonies. Considering exponential cell size growth, random fluctuations in clonal size show a transient increase and then eventually decay to zero over time (i.e., clonal populations become asymptotically more similar). In contrast, several forms of nonexponential cell size dynamics (with adder-based cell size control) yield qualitatively different results: clonal size fluctuations monotonically increase over time, reaching a nonzero value. These results characterize the interplay between cell size homeostasis at the single-cell level and clonal proliferation at the population level, explaining the broad fluctuations in clonal sizes seen in barcoded human cell lines.

Unveiling microbial single-cell growth dynamics under rapid periodic oxygen oscillations

Microbial metabolism and growth are tightly linked to oxygen (O2). Microbes experience fluctuating O2 levels in natural environments; however, our understanding of how cells respond to fluctuating O2 over various time scales remains limited due to challenges in observing microbial growth at single-cell resolution under controlled O2 conditions and in linking individual cell growth with the specific O2 microenvironment. We performed time-resolved microbial growth analyses at single-cell resolution under a temporally controlled O2 supply. A multilayer microfluidic device was developed, featuring a gas supply above a cultivation layer, separated by a thin membrane enabling efficient gas transfer. This platform allows microbial cultivation under constant, dynamic, and oscillating O2 conditions. Automated time-lapse microscopy and deep-learning-based image analysis provide access to spatiotemporally resolved growth data at the single-cell level. O2 switching within tens of seconds, coupled with precise microenvironment monitoring, allows us to accurately correlate cellular growth with local O2 concentrations. Growing Escherichia coli microcolonies subjected to varying O2 oscillation periods show distinct growth dynamics characterized by response and recovery phases. The comprehensive growth data and insights gained from our unique platform are a crucial step forward to systematically study cell response and adaptation to fluctuating O2 environments at single-cell resolution.

The design of unit cells by combining the self-reproduction systems and metabolic cushioning loads

Recently, we published a comprehensive theoretical analysis of the self-reproduction processes in proto-cells (doubling of their components) composed of different combinations of cellular subsystems. In this paper, we extend the detailed analysis of structural and functional peculiarities of self-reproduction processes to unit cells of the Cooper-Helmstetter-Donachie cell cycle theory. We show that: 1. Our modelling framework allows to calculate physiological parameters (numbers of cell components, flux patterns, cellular composition, etc.) of unit cells, including also unit cell mass that determines the DNA replication initiation conditions. 2. Unit cells might have additional cell (cushioning) components that are responsible not only for carrying out various special functions, but also for regulating cell size and stabilizing the growth of cells. 3. The optimal productivity of the synthesis of cushioning components (useful cellular load) is observed at doubling time approximately two times longer than the minimal doubling time of the unit cells.

Eukaryotic cell size regulation and its implications for cellular function and dysfunction

Depending on cell type, environmental inputs, and disease, the cells in the human body can have widely different sizes. In recent years, it has become clear that cell size is a major regulator of cell function. However, we are only beginning to understand how the optimization of cell function determines a given cell's optimal size. Here, we review currently known size control strategies of eukaryotic cells and the intricate link of cell size to intracellular biomolecular scaling, organelle homeostasis, and cell cycle progression. We detail the cell size-dependent regulation of early development and the impact of cell size on cell differentiation. Given the importance of cell size for normal cellular physiology, cell size control must account for changing environmental conditions. We describe how cells sense environmental stimuli, such as nutrient availability, and accordingly adapt their size by regulating cell growth and cell cycle progression. Moreover, we discuss the correlation of pathological states with misregulation of cell size and how for a long time this was considered a downstream consequence of cellular dysfunction. We review newer studies that reveal a reversed causality, with misregulated cell size leading to pathophysiological phenotypes such as senescence and aging. In summary, we highlight the important roles of cell size in cellular function and dysfunction, which could have major implications for both diagnostics and treatment in the clinic.

A Generalized Adder mechanism for Cell Size Homeostasis: Implications for Stochastic Dynamics of Clonal Proliferation

Measurements of cell size dynamics have revealed phenomenological principles by which individual cells control their size across diverse organisms. One of the emerging paradigms of cell size homeostasis is the adder, where the cell cycle duration is established such that the cell size increase from birth to division is independent of the newborn cell size. We provide a mechanistic formulation of the adder considering that cell size follows any arbitrary non-exponential growth law. Our results show that the main requirement to obtain an adder regardless of the growth law (the time derivative of cell size) is that cell cycle regulators are produced at a rate proportional to the growth law and cell division is triggered when these molecules reach a prescribed threshold level. Among the implications of this generalized adder, we investigate fluctuations in the proliferation of single-cell derived colonies. Considering exponential cell size growth, random fluctuations in clonal size show a transient increase and then eventually decay to zero over time (i.e., clonal populations become asymptotically more similar). In contrast, several forms of non-exponential cell size dynamics (with adder-based cell size control) yield qualitatively different results: clonal size fluctuations monotonically increase over time reaching a non-zero value. These results characterize the interplay between cell size homeostasis at the single-cell level and clonal proliferation at the population level, explaining the broad fluctuations in clonal sizes seen in barcoded human cell lines.

Phenotypic and metabolic adaptations of Rhodococcus cerastii strain IEGM 1243 to separate and combined effects of diclofenac and ibuprofen

Introduction

The increasing use of non-steroidal anti-inflammatory drugs (NSAIDs) has raised concerns regarding their environmental impact. To address this, understanding the effects of NSAIDs on bacteria is crucial for bioremediation efforts in pharmaceutical-contaminated environments. The primary challenge in breaking down persistent compounds lies not in the biochemical pathways but in capacity of bacteria to surmount stressors.

Methods

In this study, we examined the biodegradative activity, morphological and physiological changes, and ultrastructural adaptations of Rhodococcus cerastii strain IEGM 1243 when exposed to ibuprofen, diclofenac, and their mixture.

Results and Discussion

Our findings revealed that R. cerastii IEGM 1243 exhibited moderate biodegradative activity towards the tested NSAIDs. Cellular respiration assay showed higher metabolic activity in the presence of NSAIDs, indicating their influence on bacterial metabolism. Furthermore, catalase activity in R. cerastii IEGM 1243 exposed to NSAIDs showed an initial decrease followed by fluctuations, with the most significant changes observed in the presence of DCF and the NSAID mixture, likely influenced by bacterial growth phases, active NSAID degradation, and the formation of multicellular aggregates, suggesting potential intercellular synergy and task distribution within the bacterial community. Morphometric analysis demonstrated alterations in size, shape, and surface roughness of cells exposed to NSAIDs, with a decrease in surface area and volume, and an increase in surface area-to-volume ratio (SA/V). Moreover, for the first time, transmission electron microscopy confirmed the presence of lipid inclusions, polyphosphates, and intracellular membrane-like structures in the ibuprofen-treated cells.

Conclusion

These results provide valuable insights into the adaptive responses of R. cerastii IEGM 1243 to NSAIDs, shedding light on the possible interaction between bacteria and pharmaceutical compounds in the environment.

Self-reproduction and doubling time limits of different cellular subsystems

Ribosomes which can self-replicate themselves practically autonomously in beneficial physicochemical conditions have been recognized as the central organelles of cellular self-reproduction processes. The challenge of cell design is to understand and describe the rates and mechanisms of self-reproduction processes of cells as of coordinated functioning of ribosomes and the enzymatic networks of different functional complexity that support those ribosomes. We show that doubling times of proto-cells (ranging from simplest replicators up to those reaching the size of E. coli) increase rather with the number of different cell component species than with the total numbers of cell components. However, certain differences were observed between cell components in increasing the doubling times depending on the types of relationships between those cell components and ribosomes. Theoretical limits of doubling times of the self-reproducing proto-cells determined by the molecular parameters of cell components and cell processes were in the range between 6-40 min.

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.

Scaling of nuclear numbers and their spatial arrangement in skeletal muscle cell size regulation

Many cells display considerable functional plasticity and depend on the regulation of numerous organelles and macromolecules for their maintenance. In large cells, organelles also need to be carefully distributed to supply the cell with essential resources and regulate intracellular activities. Having multiple copies of the largest eukaryotic organelle, the nucleus, epitomizes the importance of scaling gene products to large cytoplasmic volumes in skeletal muscle fibers. Scaling of intracellular constituents within mammalian muscle fibers is, however, poorly understood, but according to the myonuclear domain hypothesis, a single nucleus supports a finite amount of cytoplasm and is thus postulated to act autonomously, causing the nuclear number to be commensurate with fiber volume. In addition, the orderly peripheral distribution of myonuclei is a hallmark of normal cell physiology, as nuclear mispositioning is associated with impaired muscle function. Because underlying structures of complex cell behaviors are commonly formalized by scaling laws and thus emphasize emerging principles of size regulation, the work presented herein offers more of a unified conceptual platform based on principles from physics, chemistry, geometry, and biology to explore cell size-dependent correlations of the largest mammalian cell by means of scaling.

Dynamic proteome trade-offs regulate bacterial cell size and growth in fluctuating nutrient environments

Bacteria dynamically regulate cell size and growth to thrive in changing environments. While previous studies have characterized bacterial growth physiology at steady-state, a quantitative understanding of bacterial physiology in time-varying environments is lacking. Here we develop a quantitative theory connecting bacterial growth and division rates to proteome allocation in time-varying nutrient environments. In such environments, cell size and growth are regulated by trade-offs between prioritization of biomass accumulation or division, resulting in decoupling of single-cell growth rate from population growth rate. Specifically, bacteria transiently prioritize biomass accumulation over production of division machinery during nutrient upshifts, while prioritizing division over growth during downshifts. When subjected to pulsatile nutrient concentration, we find that bacteria exhibit a transient memory of previous metabolic states due to the slow dynamics of proteome reallocation. This allows for faster adaptation to previously seen environments and results in division control which is dependent on the time-profile of fluctuations.

Fission yeast obeys a linear size law under nutrient titration

Steady-state cell size and geometry depend on growth conditions. Here, we use an experimental setup based on continuous culture and single-cell imaging to study how cell volume, length, width and surface-to-volume ratio vary across a range of growth conditions including nitrogen and carbon titration, the choice of nitrogen source, and translation inhibition. Overall, we find cell geometry is not fully determined by growth rate and depends on the specific mode of growth rate modulation. However, under nitrogen and carbon titrations, we observe that the cell volume and the growth rate follow the same linear scaling.

Super-exponential growth and stochastic size dynamics in rod-like bacteria

Proliferating bacterial cells exhibit stochastic growth and size dynamics, but the regulation of noise in bacterial growth and morphogenesis remains poorly understood. A quantitative understanding of morphogenetic noise control, and how it changes under different growth conditions, would provide better insights into cell-to-cell variability and intergenerational fluctuations in cell physiology. Using multigenerational growth and width data of single Escherichia coli and Caulobacter crescentus cells, we deduce the equations governing growth and size dynamics of rod-like bacterial cells. Interestingly, we find that both E. coli and C. crescentus cells deviate from exponential growth within the cell cycle. In particular, the exponential growth rate increases during the cell cycle irrespective of nutrient or temperature conditions. We propose a mechanistic model that explains the emergence of super-exponential growth from autocatalytic production of ribosomes coupled to the rate of cell elongation and surface area synthesis. Using this new model and statistical inference on large datasets, we construct the Langevin equations governing cell growth and size dynamics of E. coli cells in different nutrient conditions. The single-cell level model predicts how noise in intragenerational and intergenerational processes regulate variability in cell morphology and generation times, revealing quantitative strategies for cellular resource allocation and morphogenetic noise control in different growth conditions.

Cellular resource allocation strategies for cell size and shape control in bacteria

Bacteria are highly adaptive microorganisms that thrive in a wide range of growth conditions via changes in cell morphologies and macromolecular composition. How bacterial morphologies are regulated in diverse environmental conditions is a long-standing question. Regulation of cell size and shape implies control mechanisms that couple the growth and division of bacteria to their cellular environment and macromolecular composition. In the past decade, simple quantitative laws have emerged that connect cell growth to proteomic composition and the nutrient availability. However, the relationships between cell size, shape, and growth physiology remain challenging to disentangle and unifying models are lacking. In this review, we focus on regulatory models of cell size control that reveal the connections between bacterial cell morphology and growth physiology. In particular, we discuss how changes in nutrient conditions and translational perturbations regulate the cell size, growth rate, and proteome composition. Integrating quantitative models with experimental data, we identify the physiological principles of bacterial size regulation, and discuss the optimization strategies of cellular resource allocation for size control.

Effects of antibiotics on bacterial cell morphology and their physiological origins

Characterizing the physiological response of bacterial cells to antibiotic treatment is crucial for the design of antibacterial therapies and for understanding the mechanisms of antibiotic resistance. While the effects of antibiotics are commonly characterized by their minimum inhibitory concentrations or the minimum bactericidal concentrations, the effects of antibiotics on cell morphology and physiology are less well characterized. Recent technological advances in single-cell studies of bacterial physiology have revealed how different antibiotic drugs affect the physiological state of the cell, including growth rate, cell size and shape, and macromolecular composition. Here, we review recent quantitative studies on bacterial physiology that characterize the effects of antibiotics on bacterial cell morphology and physiological parameters. In particular, we present quantitative data on how different antibiotic targets modulate cellular shape metrics including surface area, volume, surface-to-volume ratio, and the aspect ratio. Using recently developed quantitative models, we relate cell shape changes to alterations in the physiological state of the cell, characterized by changes in the rates of cell growth, protein synthesis and proteome composition. Our analysis suggests that antibiotics induce distinct morphological changes depending on their cellular targets, which may have important implications for the regulation of cellular fitness under stress.

Growth-rate-dependent and nutrient-specific gene expression resource allocation in fission yeast

Cellular resources are limited and their relative allocation to gene expression programmes determines physiological states and global properties such as the growth rate. Here, we determined the importance of the growth rate in explaining relative changes in protein and mRNA levels in the simple eukaryote Schizosaccharomyces pombe grown on non-limiting nitrogen sources. Although expression of half of fission yeast genes was significantly correlated with the growth rate, this came alongside wide-spread nutrient-specific regulation. Proteome and transcriptome often showed coordinated regulation but with notable exceptions, such as metabolic enzymes. Genes positively correlated with growth rate participated in every level of protein production apart from RNA polymerase II-dependent transcription. Negatively correlated genes belonged mainly to the environmental stress response programme. Critically, metabolic enzymes, which represent ∼55-70% of the proteome by mass, showed mostly condition-specific regulation. In summary, we provide a rich account of resource allocation to gene expression in a simple eukaryote, advancing our basic understanding of the interplay between growth-rate-dependent and nutrient-specific gene expression.

Applications of Coarse-Grained Models in Metabolic Engineering

Mathematical modeling is a promising tool for better understanding of cellular processes. In recent years, the development of coarse-grained models has gained attraction since these simple models are able to capture and describe a broad range of growth conditions. Coarse-grained models often comprise only two cellular components, a low molecular component as representative for central metabolism and energy generation and a macromolecular component, representing the entire proteome. A framework is presented that presents a strict mass conservative model for bacterial growth during a biotechnological production process. After providing interesting properties for the steady-state solution, applications are presented 1) for a production process of an amino acid and 2) production of a metabolite from central metabolism.

Protein concentration fluctuations in the high expression regime: Taylor's law and its mechanistic origin

Protein concentration in a living cell fluctuates over time due to noise in growth and division processes. In the high expression regime, variance of the protein concentration in a cell was found to scale with the square of the mean, which belongs to a general phenomenon called Taylor's law (TL). To understand the origin for these fluctuations, we measured protein concentration dynamics in single E. coli cells from a set of strains with a variable expression of fluorescent proteins. The protein expression is controlled by a set of constitutive promoters with different strength, which allows to change the expression level over 2 orders of magnitude without introducing noise from fluctuations in transcription regulators. Our data confirms the square TL, but the prefactor A has a cell-to-cell variation independent of the promoter strength. Distributions of the normalized protein concentration for different promoters are found to collapse onto the same curve. To explain these observations, we used a minimal mechanistic model to describe the stochastic growth and division processes in a single cell with a feedback mechanism for regulating cell division. In the high expression regime where extrinsic noise dominates, the model reproduces our experimental results quantitatively. By using a mean-field approximation in the minimal model, we showed that the stochastic dynamics of protein concentration is described by a Langevin equation with multiplicative noise. The Langevin equation has a scale invariance which is responsible for the square TL. By solving the Langevin equation, we obtained an analytical solution for the protein concentration distribution function that agrees with experiments. The solution shows explicitly how the prefactor A depends on strength of different noise sources, which explains its cell-to-cell variability. By using this approach to analyze our single-cell data, we found that the noise in production rate dominates the noise from cell division. The deviation from the square TL in the low expression regime can also be captured in our model by including intrinsic noise in the production rate.

Quantitative Examination of Five Stochastic Cell-Cycle and Cell-Size Control Models for Escherichia coli and Bacillus subtilis

We examine five quantitative models of the cell-cycle and cell-size control in Escherichia coli and Bacillus subtilis that have been proposed over the last decade to explain single-cell experimental data generated with high-throughput methods. After presenting the statistical properties of these models, we test their predictions against experimental data. Based on simple calculations of the defining correlations in each model, we first dismiss the stochastic Helmstetter-Cooper model and the Initiation Adder model, and show that both the Replication Double Adder (RDA) and the Independent Double Adder (IDA) model are more consistent with the data than the other models. We then apply a recently proposed statistical analysis method and obtain that the IDA model is the most likely model of the cell cycle. By showing that the RDA model is fundamentally inconsistent with size convergence by the adder principle, we conclude that the IDA model is most consistent with the data and the biology of bacterial cell-cycle and cell-size control. Mechanistically, the Independent Adder Model is equivalent to two biological principles: (i) balanced biosynthesis of the cell-cycle proteins, and (ii) their accumulation to a respective threshold number to trigger initiation and division.

Threshold accumulation of a constitutive protein explains E. coli cell-division behavior in nutrient upshifts

Despite a boost of recent progress in dynamic single-cell measurements and analyses in Escherichia coli, we still lack a mechanistic understanding of the determinants of the decision to divide. Specifically, the debate is open regarding the processes linking growth and chromosome replication to division and on the molecular origin of the observed "adder correlations," whereby cells divide, adding roughly a constant volume independent of their initial volume. In order to gain insight into these questions, we interrogate dynamic size-growth behavior of single cells across nutrient upshifts with a high-precision microfluidic device. We find that the division rate changes quickly after nutrients change, much before growth rate goes to a steady state, and in a way that adder correlations are robustly conserved. Comparison of these data to simple mathematical models falsifies proposed mechanisms, where replication-segregation or septum completions are the limiting step for cell division. Instead, we show that the accumulation of a putative constitutively expressed "P-sector divisor" protein explains the behavior during the shift.

Cell Size Is Coordinated with Cell Cycle by Regulating Initiator Protein DnaA in E. coli

Sixty years ago, bacterial cell size was found to be an exponential function of growth rate. Fifty years ago, a more general relationship was proposed, in which cell mass was equal to the initiation mass multiplied by 2 to the power of the ratio of the total time of C and D periods to the doubling time. This relationship has recently been experimentally confirmed by perturbing doubling time, C period, D period, or initiation mass. However, the underlying molecular mechanism remains unclear. Here, we developed a theoretical model for initiator protein DnaA mediating DNA replication initiation in Escherichia coli. We introduced an initiation probability function for competitive binding of DnaA-ATP and DnaA-ADP at oriC. We established a kinetic description of regulatory processes (e.g., expression regulation, titration, inactivation, and reactivation) of DnaA. Cell size as a spatial constraint also participates in the regulation of DnaA. By simulating DnaA kinetics, we obtained a regular DnaA oscillation coordinated with cell cycle and a converged cell size that matches replication initiation frequency to the growth rate. The relationship between the simulated cell size and growth rate, C period, D period, or initiation mass reproduces experimental results. The model also predicts how DnaA number and initiation mass vary with perturbation parameters, comparable with experimental data. The results suggest that 1) when growth rate, C period, or D period changes, the regulation of DnaA determines the invariance of initiation mass; 2) ppGpp inhibition of replication initiation may be important for the growth rate independence of initiation mass because three possible mechanisms therein produce different DnaA dynamics, which is experimentally verifiable; and 3) perturbation of some DnaA regulatory process causes a changing initiation mass or even an abnormal cell cycle. This study may provide clues for concerted control of cell size and cell cycle in synthetic biology.