Stochastic holin expression can account for lysis time variation in the bacteriophage λ

Accounting for cellular-level variation in lysis: implications for virus-host dynamics

Viral impacts on microbial populations depend on interaction phenotypes-including viral traits spanning the adsorption rate, latent period, and burst size. The latent period is a key viral trait in lytic infections. Defined as the time from viral adsorption to viral progeny release, the latent period of bacteriophage is conventionally inferred via one-step growth curves in which the accumulation of free virus is measured over time in a population of infected cells. Developed more than 80 years ago, one-step growth curves do not account for cellular-level variability in the timing of lysis, potentially biasing inference of viral traits. Here, we use nonlinear dynamical models to understand how individual-level variation of the latent period impacts virus-host dynamics. Our modeling approach shows that inference of the latent period via one-step growth curves is systematically biased-generating estimates of shorter latent periods than the underlying population-level mean. The bias arises because variability in lysis timing at the cellular level leads to a fraction of early burst events, which are interpreted, artefactually, as an earlier mean time of viral release. We develop a computational framework to estimate latent period variability from joint measurements of host and free virus populations. Our computational framework recovers both the mean and variance of the latent period within simulated infections including realistic measurement noise. This work suggests that reframing the latent period as a distribution to account for variability in the population will improve the study of viral traits and their role in shaping microbial populations.IMPORTANCEQuantifying viral traits-including the adsorption rate, burst size, and latent period-is critical to characterize viral infection dynamics and develop predictive models of viral impacts across scales from cells to ecosystems. Here, we revisit the gold standard of viral trait estimation-the one-step growth curve-to assess the extent to which assumptions at the core of viral infection dynamics lead to ongoing and systematic biases in inferences of viral traits. We show that latent period estimates obtained via one-step growth curves systematically underestimate the mean latent period and, in turn, overestimate the rate of viral killing at population scales. By explicitly incorporating trait variability into a dynamical inference framework that leverages both virus and host time series, we provide a practical route to improve estimates of the mean and variance of viral traits across diverse virus-microbe systems.

Molecular mechanisms of precise timing in cell lysis

Many biological systems exhibit precise timing of events, and one of the most known examples is cell lysis, which is a process of breaking bacterial host cells in the virus infection cycle. However, the underlying microscopic picture of precise timing remains not well understood. We present a novel theoretical approach to explain the molecular mechanisms of effectively deterministic dynamics in biological systems. Our hypothesis is based on the idea of stochastic coupling between relevant underlying biophysical and biochemical processes that lead to noise cancellation. To test this hypothesis, we introduced a minimal discrete-state stochastic model to investigate how holin proteins produced by bacteriophages break the inner membranes of gram-negative bacteria. By explicitly solving this model, the dynamic properties of cell lysis are fully evaluated, and theoretical predictions quantitatively agree with available experimental data for both wild-type and holin mutants. It is found that the observed threshold-like behavior is a result of the balance between holin proteins entering the membrane and leaving the membrane during the lysis. Theoretical analysis suggests that the cell lysis achieves precise timing for wild-type species by maximizing the number of holins in the membrane and narrowing their spatial distribution. In contrast, for mutated species, these conditions are not satisfied. Our theoretical approach presents a possible molecular picture of precise dynamic regulation in intrinsically random biological processes.

Estimates of differential toxin expression governing heterogeneous intracellular lifespans of Streptococcus pneumoniae

Following invasion of the host cell, pore-forming toxins secreted by pathogens compromise vacuole integrity and expose the microbe to diverse intracellular defence mechanisms. However, the quantitative correlation between toxin expression levels and consequent pore dynamics, fostering the intracellular life of pathogens, remains largely unexplored. In this study, using Streptococcus pneumoniae and its secreted pore-forming toxin pneumolysin (Ply) as a model system, we explored various facets of host-pathogen interactions in the host cytosol. Using time-lapse fluorescence imaging, we monitored pore formation dynamics and lifespans of different pneumococcal subpopulations inside host cells. Based on experimental histograms of various event timescales such as pore formation time, vacuolar death or cytosolic escape time and total degradation time, we developed a mathematical model based on first-passage processes that could correlate the event timescales to intravacuolar toxin accumulation. This allowed us to estimate Ply production rate, burst size and threshold Ply quantities that trigger these outcomes. Collectively, we present a general method that illustrates a correlation between toxin expression levels and pore dynamics, dictating intracellular lifespans of pathogens.

Stochastic chemical kinetics of cell fate decision systems: From single cells to populations and back

Stochastic chemical kinetics is a widely used formalism for studying stochasticity of chemical reactions inside single cells. Experimental studies of reaction networks are generally performed with cells that are part of a growing population, yet the population context is rarely taken into account when models are developed. Models that neglect the population context lose their validity whenever the studied system influences traits of cells that can be selected in the population, a property that naturally arises in the complex interplay between single-cell and population dynamics of cell fate decision systems. Here, we represent such systems as absorbing continuous-time Markov chains. We show that conditioning on non-absorption allows one to derive a modified master equation that tracks the time evolution of the expected population composition within a growing population. This allows us to derive consistent population dynamics models from a specification of the single-cell process. We use this approach to classify cell fate decision systems into two types that lead to different characteristic phases in emerging population dynamics. Subsequently, we deploy the gained insights to experimentally study a recurrent problem in biology: how to link plasmid copy number fluctuations and plasmid loss events inside single cells to growth of cell populations in dynamically changing environments.

Single-Cell Approach Reveals Intercellular Heterogeneity in Phage-Producing Capacities

Bacteriophage burst size is the average number of phage virions released from infected bacterial cells, and its magnitude depends on the duration of an intracellular progeny accumulation phase. Burst size is often measured at the population level, not the single-cell level, and consequently, statistical moments are not commonly available. In this study, we estimated the bacteriophage lambda (λ) single-cell burst size mean and variance following different intracellular accumulation period durations by employing Escherichia coli lysogens bearing lysis-deficient λ prophages. Single lysogens can be isolated and chemically lysed at desired times following prophage induction to quantify progeny intracellular accumulation within individual cells. Our data showed that λ phage burst size initially increased exponentially with increased lysis time (i.e., period between induction and chemical lysis) and then saturated at longer lysis times. We also demonstrated that cell-to-cell variation, or "noise," in lysis timing did not contribute significantly to burst size noise. The burst size noise remained constant with increasing mean burst size. The most likely explanation for the experimentally observed constant burst size noise was that cell-to-cell differences in burst size originated from intercellular heterogeneity in cellular capacities to produce phages. The mean burst size measured at different lysis times was positively correlated to cell volume, which may determine the cellular phage production capacity. However, experiments controlling for cell size indicated that there are other factors in addition to cell size that determine this cellular capacity. IMPORTANCE Phages produce offspring by hijacking a cell's replicative machinery. Previously, it was noted that the variation in the number of phages produced by single infected cells far exceeded cell size variation. It was hypothesized that this variation is a consequence of variation in the timing of host cell lysis. Here, we show that cell-to-cell variation in lysis timing does not significantly contribute to the burst size variation. We suggest that the constant burst size variation across different host lysis times results from cell-to-cell differences in capacity to produce phages. We found that the mean burst size measured at different lysis times was positively correlated to cell volume, which may determine the cellular phage production capacity. However, experiments controlling for cell size indicated that there are other factors in addition to cell size that determine this cellular capacity.

Bacteriophage-encoded lethal membrane disruptors: Advances in understanding and potential applications

Holins and spanins are bacteriophage-encoded membrane proteins that control bacterial cell lysis in the final stage of the bacteriophage reproductive cycle. Due to their efficient mechanisms for lethal membrane disruption, these proteins are gaining interest in many fields, including the medical, food, biotechnological, and pharmaceutical fields. However, investigating these lethal proteins is challenging due to their toxicity in bacterial expression systems and the resultant low protein yields have hindered their analysis compared to other cell lytic proteins. Therefore, the structural and dynamic properties of holins and spanins in their native environment are not well-understood. In this article we describe recent advances in the classification, purification, and analysis of holin and spanin proteins, which are beginning to overcome the technical barriers to understanding these lethal membrane disrupting proteins, and through this, unlock many potential biotechnological applications.

An Optimal Lysis Time Maximizes Bacteriophage Fitness in Quasi-Continuous Culture

Optimality models have a checkered history in evolutionary biology. While optimality models have been successful in providing valuable insight into the evolution of a wide variety of biological traits, a common objection is that optimality models are overly simplistic and ignore organismal genetics. We revisit evolutionary optimization in the context of a major bacteriophage life history trait, lysis time. Lysis time refers to the period spanning phage infection of a host cell and its lysis, whereupon phage progenies are released. Lysis time, therefore, directly determines phage fecundity assuming progeny assembly does not exhaust host resources prior to lysis. Noting that previous tests of lysis time optimality rely on batch culture, we implemented a quasi-continuous culture system to observe productivity of a panel of isogenic phage λ genotypes differing in lysis time. We report that under our experimental conditions, λ phage productivity is maximized around optimal lysis times ranging from 60 to 100 min, and λ wildtype strain falls within this range. It would appear that natural selection on phage λ lysis time uncovered a set of genetic solutions that optimized progeny production in its ecological milieu relative to alternative genotypes. We discuss this finding in light of recent results that lysis time variation is also minimized in the strains with lysis times closer to the λ wild-type strain. IMPORTANCE Optimality theory presents the idea that natural selection acts on organismal traits to produce genotypes that maximize organismal survival and reproduction. As such, optimality theory is a valuable tool in guiding our understanding of the genetic constraints and tradeoffs organisms experience as their genotypes are selected to produce optimal solutions to biological problems. However, optimality theory is often critiqued as being overly simplistic and ignoring the roles of chance and history in the evolution of organismal traits. We show here that the wild-type genotype of a popular laboratory model organism, the bacteriophage λ, produces a phenotype for a major life history trait, lysis time, that maximizes the reproductive success of bearers of that genotype relative to other possible genotypes. This result demonstrates, as is rarely shown experimentally, that natural selection can achieve optimal solutions to ecological challenges.

Exact Distribution of Threshold Crossing Times for Protein Concentrations: Implication for Biological Timekeeping

Stochastic protein accumulation up to some concentration threshold sets the timing of many cellular physiological processes. Here we obtain the exact distribution of first threshold crossing times of protein concentration, in either Laplace or time domain, and its associated cumulants: mean, variance, and skewness. The distribution is asymmetric, and its skewness nonmonotonically varies with the threshold. We study lysis times of E. coli cells for holin gene mutants of bacteriophage-λ and find a good match with theory. Mutants requiring higher holin thresholds show more skewed lysis time distributions as predicted. The theory also predicts a linear relationship between infection delay time and host doubling time for lytic viruses, that has recently been experimentally observed.

Conjugative plasmid transfer is limited by prophages but can be overcome by high conjugation rates

Antibiotic resistance spread via plasmids is a serious threat to successfully fight infections and makes understanding plasmid transfer in nature crucial to prevent the rise of antibiotic resistance. Studies addressing the dynamics of plasmid conjugation have yet neglected one omnipresent factor: prophages (viruses integrated into bacterial genomes), whose activation can kill host and surrounding bacterial cells. To investigate the impact of prophages on conjugation, we combined experiments and mathematical modelling. Using Escherichia coli, prophage λ and the multidrug-resistant plasmid RP4 we find that prophages can substantially limit the spread of conjugative plasmids. This inhibitory effect was strongly dependent on environmental conditions and bacterial genetic background. Our empirically parameterized model reproduced experimental dynamics of cells acquiring either the prophage or the plasmid well but could only reproduce the number of cells acquiring both elements by assuming complex interactions between conjugative plasmids and prophages in sequential infections. Varying phage and plasmid infection parameters over empirically realistic ranges revealed that plasmids can overcome the negative impact of prophages through high conjugation rates. Overall, the presence of prophages introduces an additional death rate for plasmid carriers, the magnitude of which is determined in non-trivial ways by the environment, the phage and the plasmid. This article is part of the theme issue 'The secret lives of microbial mobile genetic elements'.

Biological Characterization and Evolution of Bacteriophage T7-△holin During the Serial Passage Process

Bacteriophage T7 gene 17.5 coding for the only known holin is one of the components of its lysis system, but the holin activity in T7 is more complex than a single gene, and evidence points to the existence of additional T7 genes with holin activity. In this study, a T7 phage with a gene 17.5 deletion (T7-△holin) was rescued and its biological characteristics and effect on cell lysis were determined. Furthermore, the genomic evolution of mutant phage T7-△holin during serial passage was assessed by whole-genome sequencing analysis. It was observed that deletion of gene 17.5 from phage T7 delays lysis time and enlarges the phage burst size; however, this biological characteristic recovered to normal lysis levels during serial passage. Scanning electron microscopy showed that the two opposite ends of E. coli BL21 cells swell post-T7-△holin infection rather than drilling holes on cell membrane when compared with T7 wild-type infection. No visible progeny phage particle accumulation was observed inside the E. coli BL21 cells by transmission electron microscopy. Following serial passage of T7-△holin from the 1st to 20th generations, the mRNA levels of gene 3.5 and gene 19.5 were upregulated and several mutation sites were discovered, especially two missense mutations in gene 19.5, which indicate a potential contribution to the evolution of the T7-△holin. Although the burst size of T7-△holin increased, high titer cultivation of T7-△holin was not achieved by optimizing the culture process. Accordingly, these results suggest that gene 19.5 is a potential lysis-related component that needs to be studied further and that the T7-△holin strain with its gene 17.5 deletion is not adequate to establish the high-titer phage cultivation process.

From Bench to Keyboard and Back Again: A Brief History of Lambda Phage Modeling

Cellular decision making is the process whereby cells choose one developmental pathway from multiple possible ones, either spontaneously or due to environmental stimuli. Examples in various cell types suggest an almost inexhaustible plethora of underlying molecular mechanisms. In general, cellular decisions rely on the gene regulatory network, which integrates external signals to drive cell fate choice. The search for general principles of such a process benefits from appropriate biological model systems that reveal how and why certain gene regulatory mechanisms drive specific cellular decisions according to ecological context and evolutionary outcomes. In this article, we review the historical and ongoing development of the phage lambda lysis-lysogeny decision as a model system to investigate all aspects of cellular decision making. The unique generality, simplicity, and richness of phage lambda decision making render it a constant source ofmathematical modeling-aided inspiration across all of biology. We discuss the origins and progress of quantitative phage lambda modeling from the 1950s until today, as well as its possible future directions. We provide examples of how modeling enabled methods and theory development, leading to new biological insights by revealing gaps in the theory and pinpointing areas requiring further experimental investigation. Overall, we highlight the utility of theoretical approaches both as predictive tools, to forecast the outcome of novel experiments, and as explanatory tools, to elucidate the natural processes underlying experimental data.

Bacteriophage-encoded enzymes destroying bacterial cell membranes and walls, and their potential use as antimicrobial agents

Appearance of pathogenic bacteria resistant to most, if not all, known antibiotics is currently one of the most significant medical problems. Therefore, development of novel antibacterial therapies is crucial for efficient treatment of bacterial infections in the near future. One possible option is to employ enzymes, encoded by bacteriophages, which cause destruction of bacterial cell membranes and walls. Bacteriophages use such enzymes to destroy bacterial host cells at the final stage of their lytic development, in order to ensure effective liberation of progeny virions. Nevertheless, to use such bacteriophage-encoded proteins in medicine and/or biotechnology, it is crucial to understand details of their biological functions and biochemical properties. Therefore, in this review article, we will present and discuss our current knowledge on the processes of bacteriophage-mediated bacterial cell lysis, with special emphasis on enzymes involved in them. Regulation of timing of the lysis is also discussed. Finally, possibilities of the practical use of these enzymes as antibacterial agents will be underlined and perspectives of this aspect will be presented.

Protein hourglass: Exact first passage time distributions for protein thresholds

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of Escherichia coli infected with bacteriophage λ. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time (FPT) of the protein synthesis system, which is a stochastic quantity. The first few moments of the distribution of first passage times were known earlier, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on λ-phage lysis time distributions. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. Furthermore, we study the noise in the precision of the first passage times described by coefficient of variation (CV) of the distribution as a function of the protein threshold value. We show that under conditions of positive self-regulation, the CV declines monotonically with increasing protein threshold, while under conditions of linear negative self-regulation, there is an optimal protein threshold that minimizes the noise in the first passage times.

Optimum Threshold Minimizes Noise in Timing of Intracellular Events

How the noisy expression of regulatory proteins affects timing of intracellular events is an intriguing fundamental problem that influences diverse cellular processes. Here we use the bacteriophage λ to study event timing in individual cells where cell lysis is the result of expression and accumulation of a single protein (holin) in the Escherichia coli cell membrane up to a critical threshold level. Site-directed mutagenesis of the holin gene generated phage variants that vary in their lysis times from 30 to 190 min. Observation of the lysis times of single cells reveals an intriguing finding-the noise in lysis timing first decreases with increasing lysis time to reach a minimum and then sharply increases at longer lysis times. A mathematical model with stochastic expression of holin together with dilution from cell growth was sufficient to explain the non-monotonic noise profile and identify holin accumulation thresholds that generate precision in lysis timing.

Control strategies for the timing of intracellular events

While the timing of intracellular events is essential for many cellular processes, gene expression inside a single cell can exhibit substantial cell-to-cell variability, raising the question of how cells ensure precision in event timing despite such stochasticity. We address this question by analyzing a biologically reasonable model of gene expression in the context of first passage time (FPT), focusing on two experimentally measurable statistics: mean FPT (MFPT) and timing variability (TV). We show that (1) transcriptional burst size (BS) and burst frequency (BF) can minimize the TV; (2) translational BS monotonically reduces the MFPT to a nonzero low bound; (3) the timescale of promoter kinetics can minimize both the MFPT and the TV, depending on the ratio of the on-switching rate over the off-switching rate; and (4) positive feedback regulation of any form can all minimize the TV, whereas negative feedback regulation of transcriptional BF or BS always enhances the TV. These control strategies can have broad implications for diverse cellular processes relying on precise temporal triggering of events.

BAYESIAN INFERENCE OF STOCHASTIC REACTION NETWORKS USING MULTIFIDELITY SEQUENTIAL TEMPERED MARKOV CHAIN MONTE CARLO

Stochastic reaction network models are often used to explain and predict the dynamics of gene regulation in single cells. These models usually involve several parameters, such as the kinetic rates of chemical reactions, that are not directly measurable and must be inferred from experimental data. Bayesian inference provides a rigorous probabilistic framework for identifying these parameters by finding a posterior parameter distribution that captures their uncertainty. Traditional computational methods for solving inference problems such as Markov Chain Monte Carlo methods based on classical Metropolis-Hastings algorithm involve numerous serial evaluations of the likelihood function, which in turn requires expensive forward solutions of the chemical master equation (CME). We propose an alternate approach based on a multifidelity extension of the Sequential Tempered Markov Chain Monte Carlo (ST-MCMC) sampler. This algorithm is built upon Sequential Monte Carlo and solves the Bayesian inference problem by decomposing it into a sequence of efficiently solved subproblems that gradually increase both model fidelity and the influence of the observed data. We reformulate the finite state projection (FSP) algorithm, a well-known method for solving the CME, to produce a hierarchy of surrogate master equations to be used in this multifidelity scheme. To determine the appropriate fidelity, we introduce a novel information-theoretic criteria that seeks to extract the most information about the ultimate Bayesian posterior from each model in the hierarchy without inducing significant bias. This novel sampling scheme is tested with high performance computing resources using biologically relevant problems.

Modulation of first-passage time for gene expression via asymmetric cell division

How to balance the size of exponentially growing cells has always been a focus of biologists. Recent experiments have uncovered that the cell is divided into two daughter cells only when the level of time-keeper protein reaches a fixed threshold and cell division in prokaryote is not completely symmetric. The timing of cell division is essentially random because gene expression is stochastic, but cells seen to manage to have precise timing of cell division events. Although the inter-cellular variability of gene expression has attracted much attention, the randomness of event timing has been rarely studied. In our analysis, the timing of cell division is formulated as the first-passage time (denoted by FPT) for time-keeper protein’s level to cross a critical threshold firstly, we derive exact analytical formulae for the mean and noise of FPT based on stochastic gene expression model with asymmetric cell division. The results of numerical simulation show that the regulatory factors (division rate, newborn cell size, exponential growth rate and threshold) have significant influence on the mean and noise of FPT. We also show that both the increase of division rate and newborn cell size could reduce the mean of FPT and increase the noise of FPT, the larger the exponential growth rate is, the smaller the mean and noise of FPT will be; and the larger the threshold value is, the higher the mean of FPT is and the lower the noise is. In addition, compared with symmetric division, asymmetric division can reduce the mean of FPT and improve the noise of FPT. In summary, our results provide insight into the relationship between regulatory factors and FPT and reveal that asymmetric division is an effective mechanism to shorten the mean of FPT.

First-passage time statistics of stochastic transcription process for time-dependent reaction rates

Transcription in gene expression is an intrinsically noisy process which involves production and degradation of mRNAs. An important quantity to describe this stochastic process is the first-passage time (FPT), i.e., the time taken by mRNAs to reach a particular threshold. The process of transcription can be modelled as a simple birth-death process, assuming that the promoter is always in an active state and to encode the stochastic environment we consider the transcription rate to be time dependent. This generalization is suitable to capture bursty mRNA dynamics usually modelled as an ON-Off model and simplifies the calculation of FPT statistics for a cell population. We study the role of periodic modulation of the transcription rate on different moments of FPT distribution of a population of cells. Our calculation shows that for sinusoidal modulation there exists an extremal value of mean FPT as a function of the time period and phase of the transcription signal. However, for the square wave modulation of transcription rates simulation results show that the extremal value of the MFPT behaves monotonically with the variation of the phase.

Added value of autoregulation and multi-step kinetics of transcription initiation

Bacterial gene expression regulation occurs mostly during transcription, which has two main rate-limiting steps: the close complex formation, when the RNA polymerase binds to an active promoter, and the subsequent open complex formation, after which it follows elongation. Tuning these steps' kinetics by the action of e.g. transcription factors, allows for a wide diversity of dynamics. For example, adding autoregulation generates single-gene circuits able to perform more complex tasks. Using stochastic models of transcription kinetics with empirically validated parameter values, we investigate how autoregulation and the multi-step transcription initiation kinetics of single-gene autoregulated circuits can be combined to fine-tune steady state mean and cell-to-cell variability in protein expression levels, as well as response times. Next, we investigate how they can be jointly tuned to control complex behaviours, namely, time counting, switching dynamics and memory storage. Overall, our finding suggests that, in bacteria, jointly regulating a single-gene circuit's topology and the transcription initiation multi-step dynamics allows enhancing complex task performance.

Phage-Antibiotic Synergy via Delayed Lysis

When phages infect bacteria cultured in the presence of sublethal doses of antibiotics, the sizes of the phage plaques are significantly increased. This phenomenon is known as phage-antibiotic synergy (PAS). In this study, the observation of PAS was extended to a wide variety of bacterium-phage pairs using different classes of antibiotics. PAS was shown in both Gram-positive and Gram-negative bacteria. Cells stressed with β-lactam antibiotics filamented or swelled extensively, resulting in an increase in phage production. PAS was also sometimes observed in the presence of other classes of antibiotics with or without bacterial filamentation. The addition of antibiotics induced recA expression in various bacteria, but a recA deletion mutant strain of Escherichia coli also showed filamentation and PAS in the presence of quinolone antibiotics. The phage adsorption efficiency did not change in the presence of the antibiotics when the cell surfaces were enlarged as they filamented. Increases in the production of phage DNA and mRNAs encoding phage proteins were observed in these cells, with only a limited increase in protein production. The data suggest that PAS is the product of a prolonged period of particle assembly due to delayed lysis. The increase in the cell surface area far exceeded the increase in phage holin production in the filamented host cells, leading to a relatively limited availability of intracellular holins for aggregating and forming holes in the host membrane. Reactive oxygen species (ROS) stress also led to an increased production of phages, while heat stress resulted in only a limited increase in phage production.IMPORTANCE Phage-antibiotic synergy (PAS) has been reported for a decade, but the underlying mechanism has never been vigorously investigated. This study shows the presence of PAS from a variety of phage-bacterium-antibiotic pairings. We show that increased phage production resulted directly from a lysis delay caused by the relative shortage of holin in filamented bacterial hosts in the presence of sublethal concentrations of stress-inducing substances, such as antibiotics and reactive oxygen species (ROS).

Modulation of first-passage time for bursty gene expression via random signals

The stochastic nature of cell-specific signal molecules (such as transcription factor, ribosome, etc.) and the intrinsic stochastic nature of gene expression process result in cell-to-cell variations at protein levels. Increasing experimental evidences suggest that cell phenotypic variations often depend on the accumulation of some special proteins. Hence, a natural and fundamental question is: How does input signal affect the timing of protein count up to a given threshold? To this end, we study effects of input signal on the first-passage time (FPT), the time at which the number of proteins crosses a given threshold. Input signal is distinguished into two types: constant input signal and random input signal, regulating only burst frequency (or burst size) of gene expression. Firstly, we derive analytical formulae for FPT moments in each case of constant signal regulation and random signal regulation. Then, we find that random input signal tends to increases the mean and noise of FPT compared with constant input signal. Finally, we observe that different regulation ways of random signal have different effects on FPT, that is, burst size modulation tends to decrease the mean of FPT and increase the noise of FPT compared with burst frequency modulation. Our findings imply a fundamental mechanism that random fluctuating environment may prolong FPT. This can provide theoretical guidance for studies of some cellular key events such as latency of HIV and lysis time of bacteriophage λ. In conclusion, our results reveal impacts of external signal on FPT and aid understanding the regulation mechanism of gene expression.

Modeling the probability distribution of the bacterial burst size via a game-theoretic approach

Based on previous studies, empirical distribution of the bacterial burst size varies even in a population of isogenic bacteria. Since bacteriophage progenies increase linearly with time, it is the lysis time variation that results in the bacterial burst size variations. Here, the burst size variation is computationally modeled by considering the lysis time decisions as a game. Each player in the game is a bacteriophage that has initially infected and lysed its host bacterium. Also, the payoff of each burst size strategy is the average number of bacteria that are solely infected by the bacteriophage progenies after lysis. For calculating the payoffs, a new version of ball and bin model with time dependent occupation probabilities (TDOP) is proposed. We show that Nash equilibrium occurs for a range of mixed burst size strategies that are chosen and played by bacteriophages, stochastically. Moreover, it is concluded that the burst size variations arise from choosing mixed lysis strategies by each player. By choosing the lysis time and also the burst size stochastically, the released bacteriophage progenies infect a portion of host bacteria in environment and avoid extinction. The probability distribution of the mixed burst size strategies is also identified.

Infection by bacteriophage lambda: an evolving paradigm for cellular individuality

Since the earliest days of molecular biology, bacteriophage lambda has served to illuminate cellular function. Among its many roles, lambda infection is a paradigm for phenotypic heterogeneity among genetically identical cells. Early studies attributed this cellular individuality to random biochemical fluctuations, or 'noise'. More recently, however, attention has turned to the role played by deterministic hidden variables in driving single-cell behavior. Here, I briefly describe how studies in lambda are driving the shift in our understanding of cellular heterogeneity, allowing us to better appreciate the precision at which cells function.

Stochastic timing in gene expression for simple regulatory strategies

Timing is essential for many cellular processes, from cellular responses to external stimuli to the cell cycle and circadian clocks. Many of these processes are based on gene expression. For example, an activated gene may be required to reach in a precise time a threshold level of expression that triggers a specific downstream process. However, gene expression is subject to stochastic fluctuations, naturally inducing an uncertainty in this threshold-crossing time with potential consequences on biological functions and phenotypes. Here, we consider such ‘timing fluctuations’ and we ask how they can be controlled. Our analytical estimates and simulations show that, for an induced gene, timing variability is minimal if the threshold level of expression is approximately half of the steady-state level. Timing fluctuations can be reduced by increasing the transcription rate, while they are insensitive to the translation rate. In presence of self-regulatory strategies, we show that self-repression reduces timing noise for threshold levels that have to be reached quickly, while self-activation is optimal at long times. These results lay a framework for understanding stochasticity of endogenous systems such as the cell cycle, as well as for the design of synthetic trigger circuits.

First-passage time approach to controlling noise in the timing of intracellular events

In the noisy cellular environment, gene products are subject to inherent random fluctuations in copy numbers over time. How cells ensure precision in the timing of key intracellular events despite such stochasticity is an intriguing fundamental problem. We formulate event timing as a first-passage time problem, where an event is triggered when the level of a protein crosses a critical threshold for the first time. Analytical calculations are performed for the first-passage time distribution in stochastic models of gene expression. Derivation of these formulas motivates an interesting question: Is there an optimal feedback strategy to regulate the synthesis of a protein to ensure that an event will occur at a precise time, while minimizing deviations or noise about the mean? Counterintuitively, results show that for a stable long-lived protein, the optimal strategy is to express the protein at a constant rate without any feedback regulation, and any form of feedback (positive, negative, or any combination of them) will always amplify noise in event timing. In contrast, a positive feedback mechanism provides the highest precision in timing for an unstable protein. These theoretical results explain recent experimental observations of single-cell lysis times in bacteriophage [Formula: see text] Here, lysis of an infected bacterial cell is orchestrated by the expression and accumulation of a stable [Formula: see text] protein up to a threshold, and precision in timing is achieved via feedforward rather than feedback control. Our results have broad implications for diverse cellular processes that rely on precise temporal triggering of events.

Effects of cell-cycle-dependent expression on random fluctuations in protein levels

Expression of many genes varies as a cell transitions through different cell-cycle stages. How coupling between stochastic expression and cell cycle impacts cell-to-cell variability (noise) in the level of protein is not well understood. We analyse a model where a stable protein is synthesized in random bursts, and the frequency with which bursts occur varies within the cell cycle. Formulae quantifying the extent of fluctuations in the protein copy number are derived and decomposed into components arising from the cell cycle and stochastic processes. The latter stochastic component represents contributions from bursty expression and errors incurred during partitioning of molecules between daughter cells. These formulae reveal an interesting trade-off: cell-cycle dependencies that amplify the noise contribution from bursty expression also attenuate the contribution from partitioning errors. We investigate the existence of optimum strategies for coupling expression to the cell cycle that minimize the stochastic component. Intriguingly, results show that a zero production rate throughout the cell cycle, with expression only occurring just before cell division, minimizes noise from bursty expression for a fixed mean protein level. By contrast, the optimal strategy in the case of partitioning errors is to make the protein just after cell division. We provide examples of regulatory proteins that are expressed only towards the end of the cell cycle, and argue that such strategies enhance robustness of cell-cycle decisions to the intrinsic stochasticity of gene expression.

A mechanistic stochastic framework for regulating bacterial cell division

How exponentially growing cells maintain size homeostasis is an important fundamental problem. Recent single-cell studies in prokaryotes have uncovered the adder principle, where cells add a fixed size (volume) from birth to division, irrespective of their size at birth. To mechanistically explain the adder principle, we consider a timekeeper protein that begins to get stochastically expressed after cell birth at a rate proportional to the volume. Cell-division time is formulated as the first-passage time for protein copy numbers to hit a fixed threshold. Consistent with data, the model predicts that the noise in division timing increases with size at birth. Intriguingly, our results show that the distribution of the volume added between successive cell-division events is independent of the newborn cell size. This was dramatically seen in experimental studies, where histograms of the added volume corresponding to different newborn sizes collapsed on top of each other. The model provides further insights consistent with experimental observations: the distribution of the added volume when scaled by its mean becomes invariant of the growth rate. In summary, our simple yet elegant model explains key experimental findings and suggests a mechanism for regulating both the mean and fluctuations in cell-division timing for controlling size.

Genetically Determined Variation in Lysis Time Variance in the Bacteriophage φX174

Researchers in evolutionary genetics recently have recognized an exciting opportunity in decomposing beneficial mutations into their proximal, mechanistic determinants. The application of methods and concepts from molecular biology and life history theory to studies of lytic bacteriophages (phages) has allowed them to understand how natural selection sees mutations influencing life history. This work motivated the research presented here, in which we explored whether, under consistent experimental conditions, small differences in the genome of bacteriophage φX174 could lead to altered life history phenotypes among a panel of eight genetically distinct clones. We assessed the clones' phenotypes by applying a novel statistical framework to the results of a serially sampled parallel infection assay, in which we simultaneously inoculated each of a large number of replicate host volumes with ∼1 phage particle. We sequentially plated the volumes over the course of infection and counted the plaques that formed after incubation. These counts served as a proxy for the number of phage particles in a single volume as a function of time. From repeated assays, we inferred significant, genetically determined heterogeneity in lysis time and burst size, including lysis time variance. These findings are interesting in light of the genetic and phenotypic constraints on the single-protein lysis mechanism of φX174. We speculate briefly on the mechanisms underlying our results, and we discuss the potential importance of lysis time variance in viral evolution.

Quantifying gene expression variability arising from randomness in cell division times

The level of a given mRNA or protein exhibits significant variations from cell-to-cell across a homogeneous population of living cells. Much work has focused on understanding the different sources of noise in the gene-expression process that drive this stochastic variability in gene-expression. Recent experiments tracking growth and division of individual cells reveal that cell division times have considerable inter-cellular heterogeneity. Here we investigate how randomness in the cell division times can create variability in population counts. We consider a model by which mRNA/protein levels in a given cell evolve according to a linear differential equation and cell divisions occur at times spaced by independent and identically distributed random intervals. Whenever the cell divides the levels of mRNA and protein are halved. For this model, we provide a method for computing any statistical moment (mean, variance, skewness, etcetera) of the mRNA and protein levels. The key to our approach is to establish that the time evolution of the mRNA and protein statistical moments is described by an upper triangular system of Volterra equations. Computation of the statistical moments for physiologically relevant parameter values shows that randomness in the cell division process can be a major factor in driving difference in protein levels across a population of cells.

Holins in bacteria, eukaryotes, and archaea: multifunctional xenologues with potential biotechnological and biomedical applications

Holins form pores in the cytoplasmic membranes of bacteria for the primary purpose of releasing endolysins that hydrolyze the cell wall and induce cell death. Holins are encoded within bacteriophage genomes, where they promote cell lysis for virion release, and within bacterial genomes, where they serve a diversity of potential or established functions. These include (i) release of gene transfer agents, (ii) facilitation of programs of differentiation such as those that allow sporulation and spore germination, (iii) contribution to biofilm formation, (iv) promotion of responses to stress conditions, and (v) release of toxins and other proteins. There are currently 58 recognized families of holins and putative holins with members exhibiting between 1 and 4 transmembrane α-helical spanners, but many more families have yet to be discovered. Programmed cell death in animals involves holin-like proteins such as Bax and Bak that may have evolved from bacterial holins. Holin homologues have also been identified in archaea, suggesting that these proteins are ubiquitous throughout the three domains of life. Phage-mediated cell lysis of dual-membrane Gram-negative bacteria also depends on outer membrane-disrupting "spanins" that function independently of, but in conjunction with, holins and endolysins. In this minireview, we provide an overview of their modes of action and the first comprehensive summary of the many currently recognized and postulated functions and uses of these cell lysis systems. It is anticipated that future studies will result in the elucidation of many more such functions and the development of additional applications.