Bacterial Growth Control Mechanisms Inferred from Multivariate Statistical Analysis of Single-Cell Measurements

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.

Microbial life in slow and stopped lanes

Microbes in nature often lack nutrients and face extreme or widely fluctuating temperatures, unlike microbes in growth-optimized settings in laboratories that much of the literature examines. Slowed or suspended lives are the norm for microbes. Studying them is important for understanding the consequences of climate change and for addressing fundamental questions about life: are there limits to how slowly a cell's life can progress, and how long cells can remain viable without self-replicating? Recent studies began addressing these questions with single-cell-level measurements and mathematical models. Emerging principles that govern slowed or suspended lives of cells - including lives of dormant spores and microbes at extreme temperatures - are re-defining discrete cellular states as continuums and revealing intracellular dynamics at new timescales. Nearly inactive, lifeless-appearing microbes are transforming our understanding of life.

Mechanisms of cell size regulation in slow-growing Escherichia coli cells: discriminating models beyond the adder

Under ideal conditions, Escherichia coli cells divide after adding a fixed cell size, a strategy known as the adder. This concept applies to various microbes and is often explained as the division that occurs after a certain number of stages, associated with the accumulation of precursor proteins at a rate proportional to cell size. However, under poor media conditions, E. coli cells exhibit a different size regulation. They are smaller and follow a sizer-like division strategy where the added size is inversely proportional to the size at birth. We explore three potential causes for this deviation: degradation of the precursor protein and two models where the propensity for accumulation depends on the cell size: a nonlinear accumulation rate, and accumulation starting at a threshold size termed the commitment size. These models fit the mean trends but predict different distributions given the birth size. To quantify the precision of the models to explain the data, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. We found that none of the models alone can consistently explain the data. However, the degradation model better explains the division strategy when cells are larger, whereas size-related models (power-law and commitment size) account for smaller cells. Our methodology proposes a data-based method in which different mechanisms can be tested systematically.

Effects of Molecular Noise on Cell Size Control

Cells employ control strategies to maintain a stable size. Dividing at a target size (the "sizer" strategy) is thought to produce the tightest size distribution. However, this result follows from phenomenological models that ignore the molecular mechanisms required to implement the strategy. Here we investigate a simple mechanistic model for exponentially growing cells whose division is triggered at a molecular abundance threshold. We find that size noise inherits the molecular noise and is consequently minimized not by the sizer but by the "adder" strategy, where a cell divides after adding a target amount to its birth size. We derive a lower bound on size noise that agrees with publicly available data from six microfluidic studies on Escherichia coli bacteria.

Coupling Cell Size Regulation and Proliferation Dynamics of C. glutamicum Reveals Cell Division Based on Surface Area

Single cells actively coordinate growth and division to regulate their size, yet how this size homeostasis at the single-cell level propagates over multiple generations to impact clonal expansion remains fundamentally unexplored. Classical timer models for cell proliferation (where the duration of the cell cycle is an independent variable) predict that the stochastic variation in colony size will increase monotonically over time. In stark contrast, implementing size control according to adder strategy (where on average a fixed size added from cell birth to division) leads to colony size variations that eventually decay to zero. While these results assume a fixed size of the colony-initiating progenitor cell, further analysis reveals that the magnitude of the intercolony variation in population number is sensitive to heterogeneity in the initial cell size. We validate these predictions by tracking the growth of isogenic microcolonies of Corynebacterium glutamicum in microfluidic chambers. Approximating their cell shape to a capsule, we observe that the degree of random variability in cell size is different depending on whether the cell size is quantified as per length, surface area, or volume, but size control remains an adder regardless of these size metrics. A comparison of the observed variability in the colony population with the predictions suggests that proliferation matches better with a cell division based on the cell surface. In summary, our integrated mathematical-experimental approach bridges the paradigms of single-cell size regulation and clonal expansion at the population levels. This innovative approach provides elucidation of the mechanisms of size homeostasis from the stochastic dynamics of colony size for rod-shaped microbes.

Bacterial cell-size changes resulting from altering the relative expression of Min proteins

The timing of cell division, and thus cell size in bacteria, is determined in part by the accumulation dynamics of the protein FtsZ, which forms the septal ring. FtsZ localization depends on membrane-associated Min proteins, which inhibit FtsZ binding to the cell pole membrane. Changes in the relative concentrations of Min proteins can disrupt FtsZ binding to the membrane, which in turn can delay cell division until a certain cell size is reached, in which the dynamics of Min proteins frees the cell membrane long enough to allow FtsZ ring formation. Here, we study the effect of Min proteins relative expression on the dynamics of FtsZ ring formation and cell size in individual Escherichia coli bacteria. Upon inducing overexpression of minE, cell size increases gradually to a new steady-state value. Concurrently, the time required to initiate FtsZ ring formation grows as the size approaches the new steady-state, at which point the ring formation initiates as early as before induction. These results highlight the contribution of Min proteins to cell size control, which may be partially responsible for the size fluctuations observed in bacterial populations, and may clarify how the size difference acquired during asymmetric cell division is offset.

Multigenerational memory in bacterial size control

Cells maintain a stable size as they grow and divide. Inspired by the available experimental data, most proposed models for size homeostasis assume size-control mechanisms that act on a timescale of one generation. Such mechanisms lead to short-lived autocorrelations in size fluctuations that decay within less than two generations. However, recent evidence from comparing sister lineages suggests that correlations in size fluctuations can persist for many generations. Here we develop a minimal model that explains these seemingly contradictory results. Our model proposes that different environments result in different control parameters, leading to distinct inheritance patterns. Multigenerational memory is revealed in constant environments but obscured when averaging over many different environments. Inferring the parameters of our model from Escherichia coli size data in microfluidic experiments, we recapitulate the observed statistics. Our paper elucidates the impact of the environment on cell homeostasis and growth and division dynamics.

Patterns of interdivision time correlations reveal hidden cell cycle factors

The time taken for cells to complete a round of cell division is a stochastic process controlled, in part, by intracellular factors. These factors can be inherited across cellular generations which gives rise to, often non-intuitive, correlation patterns in cell cycle timing between cells of different family relationships on lineage trees. Here, we formulate a framework of hidden inherited factors affecting the cell cycle that unifies known cell cycle control models and reveals three distinct interdivision time correlation patterns: aperiodic, alternator, and oscillator. We use Bayesian inference with single-cell datasets of cell division in bacteria, mammalian and cancer cells, to identify the inheritance motifs that underlie these datasets. From our inference, we find that interdivision time correlation patterns do not identify a single cell cycle model but generally admit a broad posterior distribution of possible mechanisms. Despite this unidentifiability, we observe that the inferred patterns reveal interpretable inheritance dynamics and hidden rhythmicity of cell cycle factors. This reveals that cell cycle factors are commonly driven by circadian rhythms, but their period may differ in cancer. Our quantitative analysis thus reveals that correlation patterns are an emergent phenomenon that impact cell proliferation and these patterns may be altered in disease.

Connecting single-cell ATP dynamics to overflow metabolism, cell growth, and the cell cycle in Escherichia coli

Adenosine triphosphate (ATP) is an abundant and essential metabolite that cells consume and regenerate in large amounts to support growth. Although numerous studies have inferred the intracellular concentration of ATP in bacterial cultures, what happens in individual bacterial cells under stable growth conditions is less clear. Here, we use the QUEEN-2m biosensor to quantify ATP dynamics in single Escherichia coli cells in relation to their growth rate, metabolism, cell cycle, and cell lineage. We find that ATP dynamics are more complex than expected from population studies and are associated with growth-rate variability. Under stable nutrient-rich condition, cells can display large fluctuations in ATP level that are partially coordinated with the cell cycle. Abrogation of aerobic acetate fermentation (overflow metabolism) through genetic deletion considerably reduces both the amplitude of ATP level fluctuations and the cell-cycle trend. Similarly, growth in media in which acetate fermentation is lower or absent results in the reduction of ATP level fluctuation and cell-cycle trend. This suggests that overflow metabolism exhibits temporal dynamics, which contributes to fluctuating ATP levels during growth. Remarkably, at the single-cell level, growth rate negatively correlates with the amplitude of ATP fluctuation for each tested condition, linking ATP dynamics to growth-rate heterogeneity in clonal populations. Our work highlights the importance of single-cell analysis in studying metabolism and its implication to phenotypic diversity and cell growth.

Methods to monitor bacterial growth and replicative rates at the single-cell level

The heterogeneity of bacterial growth and replicative rates within a population was proposed a century ago notably to explain the presence of bacterial persisters. The term "growth rate" at the single-cell level corresponds to the increase in size or mass of an individual bacterium while the "replicative rate" refers to its division capacity within a defined temporality. After a decades long hiatus, recent technical innovative approaches allow population growth and replicative rates heterogeneity monitoring at the single-cell level resuming in earnest. Among these techniques, the oldest and widely used is time-lapse microscopy, most recently combined with microfluidics. We also discuss recent fluorescence dilution methods informing only on replicative rates and best suited. Some new elegant single cell methods so far only sporadically used such as buoyant mass measurement and stable isotope probing have emerged. Overall, such tools are widely used to investigate and compare the growth and replicative rates of bacteria displaying drug-persistent behaviors to that of bacteria growing in specific ecological niches or collected from patients. In this review, we describe the current methods available, discussing both the type of queries these have been used to answer and the specific strengths and limitations of each method.

Simultant: simultaneous curve fitting of functions and differential equations using analytical gradient calculations

BACKGROUND

The initial step in comparing mathematical models to experimental data is to do a fit. This process can be complicated when either the mathematical models are not analytically solvable (e.g. because of nonlinear differential equations) or when the relation between data and models is complex (e.g. when some fitting parameters must be shared between many data sets).

RESULTS

We introduce Simultant, a software package that allows complex fitting setups to be easily defined using a simple graphical user interface. Fitting functions can be defined directly as mathematical expressions or indirectly as the solution to specified ordinary differential equations. Analytical gradients of these functions, including the solution of differential equations, are automatically calculated to provide fast fitting even for functions with many parameters. The software enables easy definition of complex fitting setups in which parameters can be shared across both data sets and models to allow simultaneous fits to be performed.

CONCLUSIONS

Simultant exploits differentiable programming and simplifies modern fitting approaches in a unified graphical interface.

Multiple timescales in bacterial growth homeostasis

In balanced exponential growth, bacteria maintain many properties statistically stable for a long time: cell size, cell cycle time, and more. As these are strongly coupled variables, it is not a-priori obvious which are directly regulated and which are stabilized through interactions. Here, we address this problem by separating timescales in bacterial single-cell dynamics. Disentangling homeostatic set points from fluctuations around them reveals that some variables, such as growth-rate, cell size and cycle time, are "sloppy" with highly volatile set points. Quantifying the relative contribution of environmental and internal sources, we find that sloppiness is primarily driven by the environment. Other variables such as fold-change define "stiff" combinations of coupled variables with robust set points. These results are manifested geometrically as a control manifold in the space of variables: set points span a wide range of values within the manifold, whereas out-of-manifold deviations are constrained. Our work offers a generalizable data-driven approach for identifying control variables in a multidimensional system.

A unified framework for measuring selection on cellular lineages and traits

Intracellular states probed by gene expression profiles and metabolic activities are intrinsically noisy, causing phenotypic variations among cellular lineages. Understanding the adaptive and evolutionary roles of such variations requires clarifying their linkage to population growth rates. Extending a cell lineage statistics framework, here we show that a population's growth rate can be expanded by the cumulants of a fitness landscape that characterize how fitness distributes in a population. The expansion enables quantifying the contribution of each cumulant, such as variance and skewness, to population growth. We introduce a function that contains all the essential information of cell lineage statistics, including mean lineage fitness and selection strength. We reveal a relation between fitness heterogeneity and population growth rate response to perturbation. We apply the framework to experimental cell lineage data from bacteria to mammalian cells, revealing distinct levels of growth rate gain from fitness heterogeneity across environments and organisms. Furthermore, third or higher order cumulants' contributions are negligible under constant growth conditions but could be significant in regrowing processes from growth-arrested conditions. We identify cellular populations in which selection leads to an increase of fitness variance among lineages in retrospective statistics compared to chronological statistics. The framework assumes no particular growth models or environmental conditions, and is thus applicable to various biological phenomena for which phenotypic heterogeneity and cellular proliferation are important.

Distinguishing different modes of growth using single-cell data

Collection of high-throughput data has become prevalent in biology. Large datasets allow the use of statistical constructs such as binning and linear regression to quantify relationships between variables and hypothesize underlying biological mechanisms based on it. We discuss several such examples in relation to single-cell data and cellular growth. In particular, we show instances where what appears to be ordinary use of these statistical methods leads to incorrect conclusions such as growth being non-exponential as opposed to exponential and vice versa. We propose that the data analysis and its interpretation should be done in the context of a generative model, if possible. In this way, the statistical methods can be validated either analytically or against synthetic data generated via the use of the model, leading to a consistent method for inferring biological mechanisms from data. On applying the validated methods of data analysis to infer cellular growth on our experimental data, we find the growth of length in E. coli to be non-exponential. Our analysis shows that in the later stages of the cell cycle the growth rate is faster than exponential.

Inferring simple but precise quantitative models of human oocyte and early embryo development

Macroscopic, phenomenological models are useful as concise framings of our understandings in fields from statistical physics to finance to biology. Constructing a phenomenological model for development would provide a framework for understanding the complicated, regulatory nature of oogenesis and embryogenesis. Here, we use a data-driven approach to infer quantitative, precise models of human oocyte maturation and pre-implantation embryo development, by analysing clinical in-vitro fertilization (IVF) data on 7399 IVF cycles resulting in 57 827 embryos. Surprisingly, we find that both oocyte maturation and early embryo development are quantitatively described by simple models with minimal interactions. This simplicity suggests that oogenesis and embryogenesis are composed of modular processes that are relatively siloed from one another. In particular, our analysis provides strong evidence that (i) pre-antral follicles produce anti-Müllerian hormone independently of effects from other follicles, (ii) oocytes mature to metaphase-II independently of the woman's age, her BMI and other factors, (iii) early embryo development is memoryless for the variables assessed here, in that the probability of an embryo transitioning from its current developmental stage to the next is independent of its previous stage. Our results both provide insight into the fundamentals of oogenesis and embryogenesis and have implications for the clinical IVF.

Cell size distribution of lineage data: analytic results and parameter inference

Recent advances in single-cell technologies have enabled time-resolved measurements of the cell size over several cell cycles. These data encode information on how cells correct size aberrations so that they do not grow abnormally large or small. Here, we formulate a piecewise deterministic Markov model describing the evolution of the cell size over many generations, for all three cell size homeostasis strategies (timer, sizer, and adder). The model is solved to obtain an analytical expression for the non-Gaussian cell size distribution in a cell lineage; the theory is used to understand how the shape of the distribution is influenced by the parameters controlling the dynamics of the cell cycle and by the choice of cell tracking protocol. The theoretical cell size distribution is found to provide an excellent match to the experimental cell size distribution of E. coli lineage data collected under various growth conditions.

Non-genetic inheritance restraint of cell-to-cell variation

Heterogeneity in physical and functional characteristics of cells (e.g. size, cycle time, growth rate, protein concentration) proliferates within an isogenic population due to stochasticity in intracellular biochemical processes and in the distribution of resources during divisions. Conversely, it is limited in part by the inheritance of cellular components between consecutive generations. Here we introduce a new experimental method for measuring proliferation of heterogeneity in bacterial cell characteristics, based on measuring how two sister cells become different from each other over time. Our measurements provide the inheritance dynamics of different cellular properties, and the 'inertia' of cells to maintain these properties along time. We find that inheritance dynamics are property specific and can exhibit long-term memory (∼10 generations) that works to restrain variation among cells. Our results can reveal mechanisms of non-genetic inheritance in bacteria and help understand how cells control their properties and heterogeneity within isogenic cell populations.