Optimum Threshold Minimizes Noise in Timing of Intracellular Events

Elucidating Physicochemical Features of Holin Proteins Responsible for Bacterial Cell Lysis

Bacterial resistance to conventional antibiotics stimulated the development of so-called "phage therapies" that rely on cell lysis, which is a process of destroying bacterial cells due to their infections by bacterial viruses. For λ bacteriophages, it is known that the critical role in this process is played by holin proteins that aggregate in cellular membranes before breaking them apart. While multiple experimental studies probed various aspects of cell lysis, the underlying molecular mechanisms remain not well understood. Here we investigate what physicochemical properties of holin proteins are the most relevant for these processes by employing statistical correlation analysis of cell lysis dynamics for different experimentally observed mutant species. Our findings reveal significant correlations between various physicochemical features and cell lysis dynamics. Notably, we uncover a strong inverse correlation between local hydrophobicity and cell lysis times, underscoring the crucial role of hydrophobic interactions in membrane disruption. Stimulated by these observations, a predictive model capable of explicitly estimating cell lysis times for any holin protein mutants based on their mean hydrophobicity values is developed. Our study not only provides important microscopic insights into cell lysis phenomena but also proposes specific routes to optimize medical and biotechnological applications of bacteriophages.

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.

A stochastic vs deterministic perspective on the timing of cellular events

Cells are the fundamental units of life, and like all life forms, they change over time. Changes in cell state are driven by molecular processes; of these many are initiated when molecule numbers reach and exceed specific thresholds, a characteristic that can be described as "digital cellular logic". Here we show how molecular and cellular noise profoundly influence the time to cross a critical threshold-the first-passage time-and map out scenarios in which stochastic dynamics result in shorter or longer average first-passage times compared to noise-less dynamics. We illustrate the dependence of the mean first-passage time on noise for a set of exemplar models of gene expression, auto-regulatory feedback control, and enzyme-mediated catalysis. Our theory provides intuitive insight into the origin of these effects and underscores two important insights: (i) deterministic predictions for cellular event timing can be highly inaccurate when molecule numbers are within the range known for many cells; (ii) molecular noise can significantly shift mean first-passage times, particularly within auto-regulatory genetic feedback circuits.

Characterizing the role of autaptic feedback in enhancing precision of neuronal firing times

In a chemical synapse, information flow occurs via the release of neurotransmitters from a presynaptic neuron that triggers an Action potential (AP) in the postsynaptic neuron. At its core, this occurs via the postsynaptic membrane potential integrating neurotransmitter-induced synaptic currents, and AP generation occurs when potential reaches a critical threshold. This manuscript investigates feedback implementation via an autapse, where the axon from the postsynaptic neuron forms an inhibitory synapse onto itself. Using a stochastic model of neuronal synaptic transmission, we formulate AP generation as a first-passage time problem and derive expressions for both the mean and noise of AP-firing times. Our analytical results supported by stochastic simulations identify parameter regimes where autaptic feedback transmission enhances the precision of AP firing times consistent with experimental data. These noise attenuating regimes are intuitively based on two orthogonal mechanisms - either expanding the time window to integrate noisy upstream signals; or by linearizing the mean voltage increase over time. Interestingly, we find regimes for noise amplification that specifically occur when the inhibitory synapse has a low probability of release for synaptic vesicles. In summary, this work explores feedback modulation of the stochastic dynamics of autaptic neurotransmission and reveals its function of creating more regular AP firing patterns.

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.

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.

Diverse role of decoys on emergence and precision of oscillations in a biomolecular clock

Biomolecular clocks are key drivers of oscillatory dynamics in diverse biological processes including cell-cycle regulation, circadian rhythms, and pattern formation during development. A minimal clock implementation is based on the classical Goodwin oscillator, in which a repressor protein inhibits its own synthesis via time-delayed negative feedback. Clock motifs, however, do not exist in isolation; its components are open to interacting with the complex environment inside cells. For example, there are ubiquitous high-affinity binding sites along the genome, known as decoys, where transcription factors such as repressor proteins can potentially interact. This binding affects the availability of transcription factors and has often been ignored in theoretical studies. How does such genomic decoy binding impact the clock's robustness and precision? To address this question, we systematically analyze deterministic and stochastic models of the Goodwin oscillator in the presence of reversible binding of the repressor to a finite number of decoy sites. Our analysis reveals that the relative stability of decoy-bound repressors compared to the free repressor plays distinct roles on the emergence and precision of oscillations. Interestingly, active degradation of the bound repressor can induce sustained oscillations that are otherwise absent without decoys. In contrast, decoy abundances can kill oscillation dynamics if the bound repressor is protected from degradation. Taking into account low copy-number fluctuations in clock components, we show that the degradation of the bound repressors enhances precision by attenuating noise in both the amplitude and period of oscillations. Overall, these results highlight the versatile role of otherwise hidden decoys in shaping the stochastic dynamics of biological clocks and emphasize the importance of synthetic decoys in designing robust clocks.

The role of dynamic phenotypes in cancer

The question of whether cancer recurrence is mediated by a process that is exclusively Darwinian or that involves both Darwinian and Lamarckian processes is long standing and far from answered. The major open question is the origin of variation, whether it relays exclusively on stable, mostly genetic, mechanisms or whether it can also involve dynamic processes. Recent evidence with single-cell epigenomic and transcriptomic profiling and measurement of phenotypes in colonies indicate that several phenotypes quickly change with a few cell divisions. Most importantly, cell fitness under basal as well as in the presence of chemotherapeutic agents changes considerably over short periods of time and this dynamic is reduced by epigenetic modulators. These studies contribute to establish the dynamic nature of fitness and are key for the interplay between cancer cell dynamics and stable genetic and epigenetic alterations in the survival of a few cancer cells after therapy.

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.