Using conditional independence tests to elucidate causal links in cell cycle regulation in Escherichia coli

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.

Empirical methods that provide physical descriptions of dynamic cellular processes

We review empirical methods that can be used to provide physical descriptions of dynamic cellular processes during development and disease. Our focus will be nonspatial descriptions and the inference of underlying interaction networks including cell-state lineages, gene regulatory networks, and molecular interactions in living cells. Our overarching questions are: How much can we learn from just observing? To what degree is it possible to infer causal and/or precise mathematical relationships from observations? We restrict ourselves to data sets arising from only observations, or experiments in which minimal perturbations have taken place to facilitate observation of the systems as they naturally occur. We discuss analysis perspectives in order from those offering the least descriptive power but requiring the least assumptions such as statistical associations. We end with those that are most descriptive, but require stricter assumptions and more previous knowledge of the systems such as causal inference and dynamical systems approaches. We hope to provide and encourage the use of a wide array of options for quantitative cell biologists to learn as much as possible from their observations at all stages of understanding of their system of interest. Finally, we provide our own recipe of how to empirically determine quantitative relationships and growth laws from live-cell microscopy data, the resultant predictions of which can then be verified with perturbation experiments. We also include an extended supplement that describes further inference algorithms and theory for the interested reader.

Criticality in the Luria-Delbrück Model with an Arbitrary Mutation Rate

The Luria-Delbrück model is a classic model of population dynamics with random mutations, that has been used historically to prove that random mutations drive evolution. In typical scenarios, the relevant mutation rate is exceedingly small, and mutants are counted only at the final time point. Here, inspired by recent experiments on DNA repair, we study a mathematical model that is formally equivalent to the Luria-Delbrück setup, with the repair rate p playing the role of mutation rate, albeit taking on large values, of order unity per cell division. We find that although at large times the fraction of repaired cells approaches one, the variance of the number of repaired cells undergoes a phase transition: when p>1/2 the variance decreases with time, but, intriguingly, for p<1/2 even though the fraction of repaired cells approaches 1, the variance in the number of repaired cells increases with time. Analyzing DNA-repair experiments, we find that in order to explain the data the model should also take into account the probability of a successful repair process once it is initiated. Taken together, our work shows how the study of variability can lead to surprising phase transitions as well as provide biological insights into the process of DNA repair.

Determining the rate-limiting processes for cell division in Escherichia coli

A critical cell cycle checkpoint for most bacteria is the onset of constriction when the septal peptidoglycan synthesis starts. According to the current understanding, the arrival of FtsN to midcell triggers this checkpoint in Escherichia coli. Recent structural and in vitro data suggests that recruitment of FtsN to the Z-ring leads to a conformational switch in actin-like FtsA, which links FtsZ protofilaments to the cell membrane and acts as a hub for the late divisome proteins. Here, we investigate this putative pathway using in vivo measurements and stochastic cell cycle modeling at moderately fast growth rates. Quantitatively upregulating protein concentrations and determining the resulting division timings shows that FtsN and FtsA numbers are not rate-limiting for the division in E. coli. However, at higher overexpression levels, they affect divisions: FtsN by accelerating and FtsA by inhibiting them. At the same time, we find that the FtsZ numbers in the cell are one of the rate-limiting factors for cell divisions in E. coli. Altogether, these findings suggest that instead of FtsN, accumulation of FtsZ in the Z-ring is one of the main drivers of the onset of constriction in E. coli at faster growth rates.

Deciphering the Relationship between Cell Growth and Cell Cycle in Individual Escherichia coli Cells by Flow Cytometry

Accurate coordination of chromosome replication and cell division is essential for cellular processes, yet the regulatory mechanisms governing the bacterial cell cycle remain contentious. The lack of quantitative data connecting key cell cycle players at the single-cell level across large samples hinders consensus. Employing high-throughput flow cytometry, we quantitatively correlated the expression levels of key cell cycle proteins (FtsZ, MreB, and DnaA) with DNA content in individual bacteria. Our findings reveal distinct correlations depending on the chromosome number (CN), specifically whether CN ≤2 or ≥4, unveiling a mixed regulatory scenario in populations where CN of 2 or 4 coexist. We observed function-dependent regulations for these key proteins across nonoverlapping division cycles and various nutrient conditions. Notably, a logarithmic relationship between total protein content and replication origin number across nutrient conditions suggests a unified mechanism governing cell cycle progression, confirming the applicability of Schaechter's growth law to cells with CN ≥4. For the first time, we established a proportional relationship between the synthesis rates of key cell cycle proteins and chromosome dynamics in cells with CN ≥4. Drug experiments highlighted CN 2 and 4 as pivotal turning points influencing cellular resource allocation. This high-throughput, single-cell analysis provides interconnected quantitative insights into key molecular events, facilitating a predictive understanding of the relationship between cell growth and cell cycle.

Coinfecting phages impede each other's entry into the cell

The developmental choice made by temperate phages, between cell death (lysis) and viral dormancy (lysogeny), is influenced by the relative abundance of viruses and hosts in the environment. The paradigm for this abundance-driven decision is phage lambda of E. coli, whose propensity to lysogenize increases with the number of viruses coinfecting the same bacterium. It is believed that lambda uses this number to infer whether phages or bacteria outnumber each other. However, this interpretation is premised on an accurate mapping between the extracellular phage-to-bacteria ratio and the intracellular multiplicity of infection (MOI). Here, we show this premise to be faulty. By simultaneously labeling phage capsids and genomes, we find that, while the number of phages landing on each cell reliably samples the population ratio, the number of phages entering the cell does not. Single-cell infections, performed in a microfluidic device and interpreted using a stochastic model, reveal that the probability and rate of phage entry decrease with the number of adsorbed phages. This decrease reflects an MOI-dependent perturbation to host physiology caused by phage attachment, as evidenced by compromised membrane integrity and loss of membrane potential. The dependence of entry dynamics on the surrounding medium results in a strong impact on the infection outcome, while the protracted entry of coinfecting phages increases the heterogeneity in infection outcome at a given MOI. Our findings in lambda, and similar results we obtained for phages T5 and P1, demonstrate the previously unappreciated role played by entry dynamics in determining the outcome of bacteriophage infection.

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.

Apparent simplicity and emergent robustness in the control of the Escherichia coli cell cycle

To examine how bacteria achieve robust cell proliferation across diverse conditions, we developed a method that quantifies 77 cell morphological, cell cycle, and growth phenotypes of a fluorescently labeled Escherichia coli strain and >800 gene deletion derivatives under multiple nutrient conditions. This approach revealed extensive phenotypic plasticity and deviating mutant phenotypes were often nutrient dependent. From this broad phenotypic landscape emerged simple and robust unifying rules (laws) that connect DNA replication initiation, nucleoid segregation, FtsZ ring formation, and cell constriction to specific aspects of cell size (volume, length, or added length) at the population level. Furthermore, completion of cell division followed the initiation of cell constriction after a constant time delay across strains and nutrient conditions, identifying cell constriction as a key control point for cell size determination. Our work provides a population-level description of the governing principles by which E. coli integrates cell cycle processes and growth rate with cell size to achieve its robust proliferative capability. A record of this paper's transparent peer review process is included in the supplemental information.

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.

Gene expression in growing cells: A biophysical primer

Cell growth and gene expression, two essential elements of all living systems, have long been the focus of biophysical interrogation. Advances in experimental single-cell methods have invigorated theoretical studies into these processes. However, until recently, there was little dialog between the two areas of study. In particular, most theoretical models for gene regulation assumed gene activity to be oblivious to the progression of the cell cycle between birth and division. But, in fact, there are numerous ways in which the periodic character of all cellular observables can modulate gene expression. The molecular factors required for transcription and translation-RNA polymerase, transcription factors, ribosomes-increase in number during the cell cycle, but are also diluted due to the continuous increase in cell volume. The replication of the genome changes the dosage of those same cellular players but also provides competing targets for regulatory binding. Finally, cell division reduces their number again, and so forth. Stochasticity is inherent to all these biological processes, manifested in fluctuations in the synthesis and degradation of new cellular components as well as the random partitioning of molecules at each cell division event. The notion of gene expression as stationary is thus hard to justify. In this review, we survey the emerging paradigm of cell-cycle regulated gene expression, with an emphasis on the global expression patterns rather than gene-specific regulation. We discuss recent experimental reports where cell growth and gene expression were simultaneously measured in individual cells, providing first glimpses into the coupling between the two, and motivating several questions. How do the levels of gene expression products - mRNA and protein - scale with the cell volume and cell-cycle progression? What are the molecular origins of the observed scaling laws, and when do they break down to yield non-canonical behavior? What are the consequences of cell-cycle dependence for the heterogeneity ("noise") in gene expression within a cell population? While the experimental findings, not surprisingly, differ among genes, organisms, and environmental conditions, several theoretical models have emerged that attempt to reconcile these differences and form a unifying framework for understanding gene expression in growing cells.

Mid-cell migration of the chromosomal terminus is coupled to origin segregation in Escherichia coli

Bacterial chromosomes are dynamically and spatially organised within cells. In slow-growing Escherichia coli, the chromosomal terminus is initially located at the new pole and must therefore migrate to midcell during replication to reproduce the same pattern in the daughter cells. Here, we use high-throughput time-lapse microscopy to quantify this transition, its timing and its relationship to chromosome segregation. We find that terminus centralisation is a rapid discrete event that occurs ~25 min after initial separation of duplicated origins and ~50 min before the onset of bulk nucleoid segregation but with substantial variation between cells. Despite this variation, its movement is tightly coincident with the completion of origin segregation, even in the absence of its linkage to the divisome, suggesting a coupling between these two events. Indeed, we find that terminus centralisation does not occur if origin segregation away from mid-cell is disrupted, which results in daughter cells having an inverted chromosome organisation. Overall, our study quantifies the choreography of origin-terminus positioning and identifies an unexplored connection between these loci, furthering our understanding of chromosome segregation in this bacterium.

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: precursor protein degradation, nonlinear accumulation rate, and a threshold size termed the commitment size. These models fit mean trends but predict different distributions given the birth size. To validate these models, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. the degradation model could explain the division strategy for media where cells are larger, while the commitment size model could account for smaller cells. The power-law model, finally, better fits the data at intermediate regimes.

CO-INFECTING PHAGES IMPEDE EACH OTHER'S ENTRY INTO THE CELL

Bacteriophage lambda tunes its propensity to lysogenize based on the number of viral genome copies inside the infected cell. Viral self-counting is believed to serve as a way of inferring the abundance of available hosts in the environment. This interpretation is premised on an accurate mapping between the extracellular phage-to-bacteria ratio and the intracellular multiplicity of infection (MOI). However, here we show this premise to be untrue. By simultaneously labeling phage capsids and genomes, we find that, while the number of phages landing on each cell reliably samples the population ratio, the number of phages entering the cell does not. Single-cell infections, followed in a microfluidic device and interpreted using a stochastic model, reveal that the probability and rate of individual phage entries decrease with MOI. This decrease reflects an MOI-dependent perturbation to host physiology caused by phage landing, evidenced by compromised membrane integrity and loss of membrane potential. The dependence of phage entry dynamics on the surrounding medium is found to result in a strong impact of environmental conditions on the infection outcome, while the protracted entry of co-infecting phages increases the cell-to-cell variability in infection outcome at a given MOI. Our findings demonstrate the previously unappreciated role played by entry dynamics in determining the outcome of bacteriophage infection.

Regulatory elements coordinating initiation of chromosome replication to the Escherichia coli cell cycle

Escherichia coli coordinates replication and division cycles by initiating replication at a narrow range of cell sizes. By tracking replisomes in individual cells through thousands of division cycles in wild-type and mutant strains, we were able to compare the relative importance of previously described control systems. We found that accurate triggering of initiation does not require synthesis of new DnaA. The initiation size increased only marginally as DnaA was diluted by growth after dnaA expression had been turned off. This suggests that the conversion of DnaA between its active ATP- and inactive ADP-bound states is more important for initiation size control than the total free concentration of DnaA. In addition, we found that the known ATP/ADP converters DARS and datA compensate for each other, although the removal of them makes the initiation size more sensitive to the concentration of DnaA. Only disruption of the regulatory inactivation of DnaA mechanism had a radical impact on replication initiation. This result was corroborated by the finding that termination of one round of replication correlates with the next initiation at intermediate growth rates, as would be the case if RIDA-mediated conversion from DnaA-ATP to DnaA-ADP abruptly stops at termination and DnaA-ATP starts accumulating.

Mycobacterium tuberculosis grows linearly at the single-cell level with larger variability than model organisms

The ability of bacterial pathogens to regulate growth is crucial to control homeostasis, virulence, and drug response. Yet, we do not understand the growth and cell cycle behaviors of Mycobacterium tuberculosis (Mtb), a slow-growing pathogen, at the single-cell level. Here, we use time-lapse imaging and mathematical modeling to characterize these fundamental properties of Mtb. Whereas most organisms grow exponentially at the single-cell level, we find that Mtb exhibits a unique linear growth mode. Mtb growth characteristics are highly variable from cell-to-cell, notably in their growth speeds, cell cycle timing, and cell sizes. Together, our study demonstrates that growth behavior of Mtb diverges from what we have learned from model bacteria. Instead, Mtb generates a heterogeneous population while growing slowly and linearly. Our study provides a new level of detail into how Mtb grows and creates heterogeneity, and motivates more studies of growth behaviors in bacterial pathogens.

Recollections of a Helmstetter Disciple

Nearly fifty years ago, it became possible to construct E. coli minichromosomes using recombinant DNA technology. These very small replicons, comprising the unique replication origin of the chromosome oriC coupled to a drug resistance marker, provided new opportunities to study the regulation of bacterial chromosome replication, were key to obtaining the nucleotide sequence information encoded into oriC and were essential for the development of a ground-breaking in vitro replication system. However, true authenticity of the minichromosome model system required that they replicate during the cell cycle with chromosome-like timing specificity. I was fortunate enough to have the opportunity to construct E. coli minichromosomes in the laboratory of Charles Helmstetter and, for the first time, measure minichromosome cell cycle regulation. In this review, I discuss the evolution of this project along with some additional studies from that time related to the DNA topology and segregation properties of minichromosomes. Despite the significant passage of time, it is clear that large gaps in our understanding of oriC regulation still remain. I discuss some specific topics that continue to be worthy of further study.