Exponential trajectories, cell size fluctuations, and the adder property in bacteria follow from simple chemical dynamics and division control

Multistable Protocells Can Aid the Evolution of Prebiotic Autocatalytic Sets

We present a simple mathematical model that captures the evolutionary capabilities of a prebiotic compartment or protocell. In the model, the protocell contains an autocatalytic set whose chemical dynamics is coupled to the growth-division dynamics of the compartment. Bistability in the dynamics of the autocatalytic set results in a protocell that can exist with two distinct growth rates. Stochasticity in chemical reactions plays the role of mutations and causes transitions from one growth regime to another. We show that the system exhibits 'natural selection', where a 'mutant' protocell in which the autocatalytic set is active arises by chance in a population of inactive protocells, and then takes over the population because of its higher growth rate or 'fitness'. The work integrates three levels of dynamics: intracellular chemical, single protocell, and population (or ecosystem) of protocells.

First-passage-time statistics of growing microbial populations carry an imprint of initial conditions

In exponential population growth, variability in the timing of individual division events and environmental factors (including stochastic inoculation) compound to produce variable growth trajectories. In several stochastic models of exponential growth we show power-law relationships that relate variability in the time required to reach a threshold population size to growth rate and inoculum size. Population-growth experiments in E. coli and S. aureus with inoculum sizes ranging between 1 and 100 are consistent with these relationships. We quantify how noise accumulates over time, finding that it encodes-and can be used to deduce-information about the early growth rate of a population.

Bacterial Replication Initiation as Precision Control by Protein Counting

Balanced biosynthesis is the hallmark of bacterial cell physiology, where the concentrations of stable proteins remain steady. However, this poses a conceptual challenge to modeling the cell-cycle and cell-size controls in bacteria, as prevailing concentration-based eukaryote models are not directly applicable. In this study, we revisit and significantly extend the initiator-titration model, proposed 30 years ago, and we explain how bacteria precisely and robustly control replication initiation based on the mechanism of protein copy-number sensing. Using a mean-field approach, we first derive an analytical expression of the cell size at initiation based on three biological mechanistic control parameters for an extended initiator-titration model. We also study the stability of our model analytically and show that initiation can become unstable in multifork replication conditions. Using simulations, we further show that the presence of the conversion between active and inactive initiator protein forms significantly represses initiation instability. Importantly, the two-step Poisson process set by the initiator titration step results in significantly improved initiation synchrony with C V ~ 1 / N scaling rather than the standard 1 / N scaling in the Poisson process, where N is the total number of initiators required for initiation. Our results answer two long-standing questions in replication initiation: (i) Why do bacteria produce almost two orders of magnitude more DnaA, the master initiator proteins, than required for initiation? (ii) Why does DnaA exist in active (DnaA-ATP) and inactive (DnaA-ADP) forms if only the active form is competent for initiation? The mechanism presented in this work provides a satisfying general solution to how the cell can achieve precision control without sensing protein concentrations, with broad implications from evolution to the design of synthetic cells.

Replication initiation in bacteria: precision control based on protein counting

Balanced biosynthesis is the hallmark of bacterial cell physiology, where the concentrations of stable proteins remain steady. However, this poses a conceptual challenge to modeling the cell-cycle and cell-size controls in bacteria, as prevailing concentration-based eukaryote models are not directly applicable. In this study, we revisit and significantly extend the initiator-titration model, proposed thirty years ago, and explain how bacteria precisely and robustly control replication initiation based on the mechanism of protein copy-number sensing. Using a mean-field approach, we first derive an analytical expression of the cell size at initiation based on three biological mechanistic control parameters for an extended initiator-titration model. We also study the stability of our model analytically and show that initiation can become unstable in multifork replication conditions. Using simulations, we further show that the presence of the conversion between active and inactive initiator protein forms significantly represses initiation instability. Importantly, the two-step Poisson process set by the initiator titration step results in significantly improved initiation synchrony with C V ~ 1 / N scaling rather than the standard 1 / N scaling in the Poisson process, where N is the total number of initiators required for initiation. Our results answer two long-standing questions in replication initiation: (1) Why do bacteria produce almost two orders of magnitude more DnaA, the master initiator proteins, than required for initiation? (2) Why does DnaA exist in active (DnaA-ATP) and inactive (DnaA-ADP) forms if only the active form is competent for initiation? The mechanism presented in this work provides a satisfying general solution to how the cell can achieve precision control without sensing protein concentrations, with broad implications from evolution to the design of synthetic cells.

Interplay of Cellular mRNA, miRNA and Viral miRNA during Infection of a Cell

The understanding of the kinetics of gene expression in cells infected by viruses is currently limited. As a rule, the corresponding models do not take viral microRNAs (miRNAs) into account. Such RNAs are, however, operative during the replication of some viruses, including, e.g., herpesvirus. To clarify the kinetics of this category (with emphasis on the information available for herpesvirus), I introduce a generic model describing the transient interplay of cellular mRNA, protein, miRNA and viral miRNA. In the absence of viral miRNA, the cellular miRNA is considered to suppress the populations of mRNA and protein due to association with mRNA and subsequent degradation. During infection, the viral miRNA suppresses the population of cellular miRNA and via this pathway makes the mRNA and protein populations larger. This effect becomes appreciable with the progress of intracellular viral replication. Using biologically reasonable parameters, I investigate the corresponding mean-field kinetics and show the scale of the effect of viral miRNAs on cellular miRNA and mRNA. The scale of fluctuations of the populations of these species is illustrated as well by employing Monte Carlo simulations.

The correlation between cell and nucleus size is explained by an eukaryotic cell growth model

In eukaryotes, the cell volume is observed to be strongly correlated with the nuclear volume. The slope of this correlation depends on the cell type, growth condition, and the physical environment of the cell. We develop a computational model of cell growth and proteome increase, incorporating the kinetics of amino acid import, protein/ribosome synthesis and degradation, and active transport of proteins between the cytoplasm and the nucleoplasm. We also include a simple model of ribosome biogenesis and assembly. Results show that the cell volume is tightly correlated with the nuclear volume, and the cytoplasm-nucleoplasm transport rates strongly influence the cell growth rate as well as the cell/nucleus volume ratio (C/N ratio). Ribosome assembly and the ratio of ribosomal proteins to mature ribosomes also influence the cell volume and the cell growth rate. We find that in order to regulate the cell growth rate and the cell/nucleus volume ratio, the cell must optimally control groups of kinetic and transport parameters together, which could explain the quantitative roles of canonical growth pathways. Finally, although not explicitly demonstrated in this work, we point out that it is possible to construct a detailed proteome distribution using our model and RNAseq data, provided that a quantitative cell division mechanism is known.

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.

Bacterial cell proliferation: from molecules to cells

Bacterial cell proliferation is highly efficient, both because bacteria grow fast and multiply with a low failure rate. This efficiency is underpinned by the robustness of the cell cycle and its synchronization with cell growth and cytokinesis. Recent advances in bacterial cell biology brought about by single-cell physiology in microfluidic chambers suggest a series of simple phenomenological models at the cellular scale, coupling cell size and growth with the cell cycle. We contrast the apparent simplicity of these mechanisms based on the addition of a constant size between cell cycle events (e.g. two consecutive initiation of DNA replication or cell division) with the complexity of the underlying regulatory networks. Beyond the paradigm of cell cycle checkpoints, the coordination between the DNA and division cycles and cell growth is largely mediated by a wealth of other mechanisms. We propose our perspective on these mechanisms, through the prism of the known crosstalk between DNA replication and segregation, cell division and cell growth or size. We argue that the precise knowledge of these molecular mechanisms is critical to integrate the diverse layers of controls at different time and space scales into synthetic and verifiable models.

A unifying autocatalytic network-based framework for bacterial growth laws

Bacterial cells contain various autocatalytic cycles, e.g., the ribosome cycle, where ribosomes translate ribosomal proteins that subsequently self-assemble to form new ribosomes. Here, we show that the transcription–translation machinery couples all cellular autocatalytic cycles, resulting in balanced exponential growth. Each autocatalytic cycle generates two types of growth laws. We derive the RNA polymerase (RNAP) growth law based on the RNAP autocatalytic cycle, where RNAPs transcribe messenger RNAs (mRNAs) of its constituent Rpo protein subunits. Before degrading, these mRNAs catalyze Rpo proteins employing ribosomes. The Rpo proteins subsequently self-assemble, forming new RNAPs, thus completing the cycle. Contrary to ribosome growth law, a reduction in growth rate due to shortage in RNAPs occurs without affecting the ribosomal protein mass fraction.

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.

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.

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

Universal observations in Biology are sometimes described as “laws”. In E. coli, experimental studies performed over the past six decades have revealed major growth laws relating ribosomal mass fraction and cell size to the growth rate. Because they formalize complex emerging principles in biology, growth laws have been instrumental in shaping our understanding of bacterial physiology. Here, we discovered a novel size law that connects cell size to the inverse of the metabolic proteome mass fraction and the active fraction of ribosomes. We used a simple whole-cell coarse-grained model of cell physiology that combines the proteome allocation theory and the structural model of cell division. This integrated model captures all available experimental data connecting the cell proteome composition, ribosome activity, division size and growth rate in response to nutrient quality, antibiotic treatment and increased protein burden. Finally, a stochastic extension of the model explains non-trivial correlations observed in single cell experiments including the adder principle. This work provides a simple and robust theoretical framework for studying the fundamental principles of cell size determination in unicellular organisms.

Bacteria respond to environmental changes by adjusting their molecular composition, cell size and growth rate. This plasticity is thought to result from years of evolution and to be at least in part optimal for bacterial physiology. Over the past decades, quantitative studies of bacterial growth have revealed simple phenomenological relationships, called “growth laws”, which link cell size and cell composition to the growth rate. Simplified mathematical models of cell physiology are useful tools to gain quantitative understanding of the molecular mechanisms that underlie growth laws. For instance, these models helped explaining how optimal allocation of cellular resource to physiological processes and pathways governs the cell molecular composition in response to specific environmental conditions. In this study, we have extended and integrated existing mathematical models and used experimental data from several recent studies to understand the co-regulation of cell composition, cell size and the cellular growth rate. The model predictions uncovered a novel “size law” that links cell size to the levels of metabolic proteins and the fraction of active ribosomes present in the cell. This work provides a useful theoretical tool and a quantitative basis for understanding mechanistically bacterial physiology as a function of external conditions.