Noise in timing and precision of gene activities in a genetic cascade

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.

Lethal perturbation of an Escherichia coli regulatory network is triggered by a restriction-modification system's regulator and can be mitigated by excision of the cryptic prophage Rac

Bacterial gene regulatory networks orchestrate responses to environmental challenges. Horizontal gene transfer can bring in genes with regulatory potential, such as new transcription factors (TFs), and this can disrupt existing networks. Serious regulatory perturbations may even result in cell death. Here, we show the impact on Escherichia coli of importing a promiscuous TF that has adventitious transcriptional effects within the cryptic Rac prophage. A cascade of regulatory network perturbations occurred on a global level. The TF, a C regulatory protein, normally controls a Type II restriction-modification system, but in E. coli K-12 interferes with expression of the RacR repressor gene, resulting in de-repression of the normally-silent Rac ydaT gene. YdaT is a prophage-encoded TF with pleiotropic effects on E. coli physiology. In turn, YdaT alters expression of a variety of bacterial regulons normally controlled by the RcsA TF, resulting in deficient lipopolysaccharide biosynthesis and cell division. At the same time, insufficient RacR repressor results in Rac DNA excision, halting Rac gene expression due to loss of the replication-defective Rac prophage. Overall, Rac induction appears to counteract the lethal toxicity of YdaT. We show here that E. coli rewires its regulatory network, so as to minimize the adverse regulatory effects of the imported C TF. This complex set of interactions may reflect the ability of bacteria to protect themselves by having robust mechanisms to maintain their regulatory networks, and/or suggest that regulatory C proteins from mobile operons are under selection to manipulate their host's regulatory networks for their own benefit.

Generating dynamic gene expression patterns without the need for regulatory circuits

Synthetic biology has successfully advanced our ability to design and implement complex, time-varying genetic circuits to control the expression of recombinant proteins. However, these circuits typically require the production of regulatory genes whose only purpose is to coordinate expression of other genes. When designing very small genetic constructs, such as viral genomes, we may want to avoid introducing such auxiliary gene products while nevertheless encoding complex expression dynamics. To this end, here we demonstrate that varying only the placement and strengths of promoters, terminators, and RNase cleavage sites in a computational model of a bacteriophage genome is sufficient to achieve solutions to a variety of basic gene expression patterns. We discover these genetic solutions by computationally evolving genomes to reproduce desired gene expression time-course data. Our approach shows that non-trivial patterns can be evolved, including patterns where the relative ordering of genes by abundance changes over time. We find that some patterns are easier to evolve than others, and comparable expression patterns can be achieved via different genetic architectures. Our work opens up a novel avenue to genome engineering via fine-tuning the balance of gene expression and gene degradation rates.

Multiple transcription auto regulatory loops can act as robust oscillators and decision-making motifs

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We have shown that:

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Negative transcription auto regulation can speed up the response time at the cost of reduced steady state protein levels. Under strong binding conditions, one can increase the steady state protein level by increasing the gene copy number without a compromise on the response time.

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Multiple negative transcription autoregulatory motifs can be tuned for both the response time as well as steady state protein levels by varying the gene copy number.

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Multiple negative autoregulatory loops can act as robust genetic oscillators.

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Dual feedback motifs constructed with multiple negative and positive autoregulatory loop components can act as robust oscillators and bistable decision making units within the transcription factor networks.

Response time decides how fast a gene can react against an external signal at the transcription level in a signalling cascade. The steady state protein levels of the responding genes decide the coupling between two consecutive members of a signalling cascade. A negative autoregulatory loop (NARL) present in a transcription factor network can speed up the response time of the regulated gene at the cost of reduced steady state protein level. We present here a multi NARL motif which can be tuned for both the steady state protein level as well as response time in the required direction. Remarkably, there exists an optimum Hill coefficient n_opt≅4 at which the response time of the NARL motif is at minimum. When the Hill coefficient is n < n_opt, then under strong binding conditions, one can raise the steady state protein level by increasing the gene copy number with almost no change in the response time of the multi NARL motif. Using detailed computational analysis, we show that the coupled multi NARL and positive auto regulatory loop (PARL) motifs can act as an oscillator as well as decision making component which are robust against extrinsic fluctuations in the control parameters. We further demonstrate that the period of oscillation of the coupled multi NARL-PARL dual feedback oscillator can also be fine-tuned by the gene copy number apart from the inducer concentration. We finally demonstrate robustness of bistable dual feedback decision making motifs with multi autoregulatory loop component.

Encoding memory in tube diameter hierarchy of living flow network

The concept of memory is traditionally associated with organisms possessing a nervous system. However, even very simple organisms store information about past experiences to thrive in a complex environment-successfully exploiting nutrient sources, avoiding danger, and warding off predators. How can simple organisms encode information about their environment? We here follow how the giant unicellular slime mold Physarum polycephalum responds to a nutrient source. We find that the network-like body plan of the organism itself serves to encode the location of a nutrient source. The organism entirely consists of interlaced tubes of varying diameters. Now, we observe that these tubes grow and shrink in diameter in response to a nutrient source, thereby imprinting the nutrient's location in the tube diameter hierarchy. Combining theoretical model and experimental data, we reveal how memory is encoded: a nutrient source locally releases a softening agent that gets transported by the cytoplasmic flows within the tubular network. Tubes receiving a lot of softening agent grow in diameter at the expense of other tubes shrinking. Thereby, the tubes' capacities for flow-based transport get permanently upgraded toward the nutrient location, redirecting future decisions and migration. This demonstrates that nutrient location is stored in and retrieved from the networks' tube diameter hierarchy. Our findings explain how network-forming organisms like slime molds and fungi thrive in complex environments. We here identify a flow networks' version of associative memory-very likely of relevance for the plethora of living flow networks as well as for bioinspired design.

Stochastic fluctuations in apoptotic threshold of tumour cells can enhance apoptosis and combat fractional killing

Fractional killing, which is a significant impediment to successful chemotherapy, is observed even in a population of genetically identical cancer cells exposed to apoptosis-inducing agents. This phenomenon arises not from genetic mutation but from cell-to-cell variation in the activation timing and level of the proteins that regulates apoptosis. To understand the mechanism behind the phenomenon, we formulate complex fractional killing processes as a first-passage time (FPT) problem with a stochastically fluctuating boundary. Analytical calculations are performed for the FPT distribution in a toy model of stochastic p53 gene expression, where the cancer cell is killed only when the p53 expression level crosses an active apoptotic threshold. Counterintuitively, we find that threshold fluctuations can effectively enhance cellular killing by significantly decreasing the mean time that the p53 protein reaches the threshold level for the first time. Moreover, faster fluctuations lead to the killing of more cells. These qualitative results imply that fluctuations in threshold are a non-negligible stochastic source, and can be taken as a strategy for combating fractional killing of cancer cells.

Temporal development of Drosophila embryos is highly robust across a wide temperature range

Development is a process precisely coordinated in both space and time. Spatial precision has been quantified in a number of developmental systems, and such data have contributed significantly to our understanding of, for example, morphogen gradient interpretation. However, comparatively little quantitative analysis has been performed on timing and temporal coordination during development. Here, we use Drosophila to explore the temporal robustness of embryonic development within physiologically normal temperatures. We find that development is temporally very precise across a wide range of temperatures in the three Drosophila species investigated. However, we find temperature dependence in the timing of developmental events. A simple model incorporating history dependence can explain the developmental temporal trajectories. Interestingly, history dependence is temperature-specific, with either effective negative or positive feedback at different temperatures. We also find that embryos are surprisingly robust to shifting temperatures during embryogenesis. We further identify differences between tropical and temperate species, potentially due to different mechanisms regulating temporal development that depend on the local environment. Our data show that Drosophila embryonic development is temporally robust across a wide range of temperatures. This robustness shows interesting species-specific differences that are suggestive of different sensitivity to temperature fluctuations between Drosophila species.

Temporal order and precision of complex stress responses in individual bacteria

Sudden stress often triggers diverse, temporally structured gene expression responses in microbes, but it is largely unknown how variable in time such responses are and if genes respond in the same temporal order in every single cell. Here, we quantified timing variability of individual promoters responding to sublethal antibiotic stress using fluorescent reporters, microfluidics, and time-lapse microscopy. We identified lower and upper bounds that put definite constraints on timing variability, which varies strongly among promoters and conditions. Timing variability can be interpreted using results from statistical kinetics, which enable us to estimate the number of rate-limiting molecular steps underlying different responses. We found that just a few critical steps control some responses while others rely on dozens of steps. To probe connections between different stress responses, we then tracked the temporal order and response time correlations of promoter pairs in individual cells. Our results support that, when bacteria are exposed to the antibiotic nitrofurantoin, the ensuing oxidative stress and SOS responses are part of the same causal chain of molecular events. In contrast, under trimethoprim, the acid stress response and the SOS response are part of different chains of events running in parallel. Our approach reveals fundamental constraints on gene expression timing and provides new insights into the molecular events that underlie the timing of stress responses.

Noise⁻Seeded Developmental Pattern Formation in Filamentous Cyanobacteria

Under nitrogen-poor conditions, multicellular cyanobacteria such as Anabaena sp. PCC 7120 undergo a process of differentiation, forming nearly regular, developmental patterns of individual nitrogen-fixing cells, called heterocysts, interspersed between intervals of vegetative cells that carry out photosynthesis. Developmental pattern formation is mediated by morphogen species that can act as activators and inhibitors, some of which can diffuse along filaments. We survey the limitations of the classical, deterministic Turing mechanism that has been often invoked to explain pattern formation in these systems, and then, focusing on a simpler system governed by birth-death processes, we illustrate pedagogically a recently proposed paradigm that provides a much more robust description of pattern formation: stochastic Turing patterns. We emphasize the essential role that cell-to-cell differences in molecular numbers—caused by inevitable fluctuations in gene expression—play, the so called demographic noise, in seeding the formation of stochastic Turing patterns over a much larger region of parameter space, compared to their deterministic counterparts.

Temporal precision of regulated gene expression

Important cellular processes such as migration, differentiation, and development often rely on precise timing. Yet, the molecular machinery that regulates timing is inherently noisy. How do cells achieve precise timing with noisy components? We investigate this question using a first-passage-time approach, for an event triggered by a molecule that crosses an abundance threshold and that is regulated by either an accumulating activator or a diminishing repressor. We find that either activation or repression outperforms an unregulated strategy. The optimal regulation corresponds to a nonlinear increase in the amount of the target molecule over time, arises from a tradeoff between minimizing the timing noise of the regulator and that of the target molecule itself, and is robust to additional effects such as bursts and cell division. Our results are in quantitative agreement with the nonlinear increase and low noise of mig-1 gene expression in migrating neuroblast cells during Caenorhabditis elegans development. These findings suggest that dynamic regulation may be a simple and powerful strategy for precise cellular timing.

Cells control important processes with precise timing, even though their underlying molecular machinery is inherently imprecise. In the case of Caenorhabditis elegans development, migrating neuroblast cells produce a molecule until a certain abundance is reached, at which time the cells stop moving. Precise timing of this event is critical to C. elegans development, and here we investigate how it can be achieved. Specifically, we investigate regulation of the molecule production by either an accumulating activator or a diminishing repressor. Our results are consistent with the nonlinear increase and low noise of gene expression observed in the C. elegans cells.

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.

Modeling Cell-to-Cell Communication Networks Using Response-Time Distributions

Cell-to-cell communication networks have critical roles in coordinating diverse organismal processes, such as tissue development or immune cell response. However, compared with intracellular signal transduction networks, the function and engineering principles of cell-to-cell communication networks are far less understood. Major complications include: cells are themselves regulated by complex intracellular signaling networks; individual cells are heterogeneous; and output of any one cell can recursively become an additional input signal to other cells. Here, we make use of a framework that treats intracellular signal transduction networks as "black boxes" with characterized input-to-output response relationships. We study simple cell-to-cell communication circuit motifs and find conditions that generate bimodal responses in time, as well as mechanisms for independently controlling synchronization and delay of cell-population responses. We apply our modeling approach to explain otherwise puzzling data on cytokine secretion onset times in T cells. Our approach can be used to predict communication network structure using experimentally accessible input-to-output measurements and without detailed knowledge of intermediate steps.

Promoting Simultaneous Onset of Viral Gene Expression Among Cells Infected with Herpes Simplex Virus-1

Synchronous viral infection facilitates the study of viral gene expression, viral host interactions, and viral replication processes. However, the protocols for achieving synchronous infections were hardly ever tested in proper temporal resolution at the single-cell level. We set up a fluorescence-based, time lapse microscopy assay to study sources of variability in the timing of gene expression during herpes simplex virus-1 (HSV-1) infection. We found that with the common protocol, the onset of gene expression within different cells can vary by more than 3 h. We showed that simultaneous viral genome entry to the nucleus can be achieved with a derivative of the previously characterized temperature sensitive mutant tsB7, however, this did not improve gene expression synchrony. We found that elevating the temperature in which the infection is done and increasing the multiplicity of infection (MOI) significantly promoted simultaneous onset of viral gene expression among infected cells. Further, elevated temperature result in a decrease in the coefficient of variation (a standardized measure of dispersion) of viral replication compartments (RCs) sizes among cells as well as a slight increment of viral late gene expression synchrony. We conclude that simultaneous viral gene expression can be improved by simple modifications to the infection process and may reduce the effect of single-cell variability on population-based assays.

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.

The volumes and transcript counts of single cells reveal concentration homeostasis and capture biological noise

We present data on cell-to-cell variability (“‘noise”') of gene expression in human cells, obtained through a combination of single-molecule mRNA FISH and single-cell volume measurements. We find that noise in terms of mRNA numbers exceeds the noise in terms of mRNA concentration. This study provides an improved method to determine gene expression noise.

Transcriptional stochasticity can be measured by counting the number of mRNA molecules per cell. Cell-to-cell variability is best captured in terms of concentration rather than molecule counts, because reaction rates depend on concentrations. We combined single-molecule mRNA counting with single-cell volume measurements to quantify the statistics of both transcript numbers and concentrations in human cells. We compared three cell clones that differ only in the genomic integration site of an identical constitutively expressed reporter gene. The transcript number per cell varied proportionally with cell volume in all three clones, indicating concentration homeostasis. We found that the cell-to-cell variability in the mRNA concentration is almost exclusively due to cell-to-cell variation in gene expression activity, whereas the cell-to-cell variation in mRNA number is larger, due to a significant contribution of cell volume variability. We concluded that the precise relationship between transcript number and cell volume sets the biological stochasticity of living cells. This study highlights the importance of the quantitative measurement of transcript concentrations in studies of cell-to-cell variability in biology.

Theory on the dynamics of oscillatory loops in the transcription factor networks

We develop a detailed theoretical framework for various types of transcription factor gene oscillators. We further demonstrate that one can build genetic-oscillators which are tunable and robust against perturbations in the critical control parameters by coupling two or more independent Goodwin-Griffith oscillators through either -OR- or -AND- type logic. Most of the coupled oscillators constructed in the literature so far seem to be of -OR- type. When there are transient perturbations in one of the -OR- type coupled-oscillators, then the overall period of the system remains constant (period-buffering) whereas in case of -AND- type coupling the overall period of the system moves towards the perturbed oscillator. Though there is a period-buffering, the amplitudes of oscillators coupled through -OR- type logic are more sensitive to perturbations in the parameters associated with the promoter state dynamics than -AND- type. Further analysis shows that the period of -AND- type coupled dual-feedback oscillators can be tuned without conceding on the amplitudes. Using these results we derive the basic design principles governing the robust and tunable synthetic gene oscillators without compromising on their amplitudes.

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

The inherent stochastic nature of biochemical processes can drive differences in gene expression between otherwise identical cells. While cell-to-cell variability in gene expression has received much attention, randomness in timing of events has been less studied. We investigate event timing at the single-cell level in a simple system, the lytic pathway of the bacterial virus phage λ. In individual cells, lysis occurs on average at 65 min, with an s.d. of 3.5 min. Interestingly, mutations in the lysis protein, holin, alter both the lysis time (LT) mean and variance. In our analysis, LT is formulated as the first-passage time (FPT) for cellular holin levels to cross a critical threshold. Exact analytical formulae for the FPT moments are derived for stochastic gene expression models. These formulae reveal how holin transcription and translation efficiencies independently modulate the LT mean and variation. Analytical expressions for the LT moments are used to evaluate previously published single-cell LT data for λ phages with mutations in the holin sequence or its promoter. Our results show that stochastic holin expression is sufficient to account for the intercellular LT differences in both wild-type phages, and phage variants where holin transcription and the threshold for lysis have been experimentally altered. Finally, our analysis reveals regulatory motifs that enhance the robustness of lysis timing to cellular noise.

Transcription-factor-mediated DNA looping probed by high-resolution, single-molecule imaging in live E. coli cells

DNA looping mediated by transcription factors plays critical roles in prokaryotic gene regulation. The "genetic switch" of bacteriophage λ determines whether a prophage stays incorporated in the E. coli chromosome or enters the lytic cycle of phage propagation and cell lysis. Past studies have shown that long-range DNA interactions between the operator sequences O(R) and O(L) (separated by 2.3 kb), mediated by the λ repressor CI (accession number P03034), play key roles in regulating the λ switch. In vitro, it was demonstrated that DNA segments harboring the operator sequences formed loops in the presence of CI, but CI-mediated DNA looping has not been directly visualized in vivo, hindering a deep understanding of the corresponding dynamics in realistic cellular environments. We report a high-resolution, single-molecule imaging method to probe CI-mediated DNA looping in live E. coli cells. We labeled two DNA loci with differently colored fluorescent fusion proteins and tracked their separations in real time with ∼40 nm accuracy, enabling the first direct analysis of transcription-factor-mediated DNA looping in live cells. Combining looping measurements with measurements of CI expression levels in different operator mutants, we show quantitatively that DNA looping activates transcription and enhances repression. Further, we estimated the upper bound of the rate of conformational change from the unlooped to the looped state, and discuss how chromosome compaction may impact looping kinetics. Our results provide insights into transcription-factor-mediated DNA looping in a variety of operator and CI mutant backgrounds in vivo, and our methodology can be applied to a broad range of questions regarding chromosome conformations in prokaryotes and higher organisms.

Adaptive regulatory substitutions affect multiple stages in the life cycle of the bacteriophage φX174

BACKGROUND

Previously, we showed that adaptive substitutions in one of the three promoters of the bacteriophage φX174 improved fitness at high-temperature by decreasing transcript levels three- to four-fold. To understand how such an extreme change in gene expression might lead to an almost two-fold increase in fitness at the adaptive temperature, we focused on stages in the life cycle of the phage that occur before and after the initiation of transcription. For both the ancestral strain and two single-substitution strains with down-regulated transcription, we measured seven phenotypic components of fitness (attachment, ejection, eclipse, virion assembly, latent period, lysis rate and burst size) during a single cycle of infection at each of two temperatures. The lower temperature, 37°C, is the optimal temperature at which phages are cultivated in the lab; the higher temperature, 42°C, exerts strong selection and is the condition under which these substitutions arose in evolution experiments. We augmented this study by developing an individual-based stochastic model of this same life cycle to explore potential explanations for our empirical results.

RESULTS

Of the seven fitness parameters, three showed significant differences between strains that carried an adaptive substitution and the ancestor, indicating the presence of pleiotropy in regulatory evolution. 1) Eclipse was longer in the adaptive strains at both the optimal and high-temperature environments. 2) Lysis rate was greater in the adaptive strains at the high temperature. 3) Burst size for the mutants was double that of the ancestor at the high temperature, but half that at the lower temperature. Simulation results suggest that eclipse length and latent period variance can explain differences in burst sizes and fitness between the mutant and ancestral strains.

CONCLUSIONS

Down-regulating transcription affects several steps in the phage life cycle, and all of these occur after the initiation of transcription. We attribute the apparent tradeoff between delayed progeny production and faster progeny release to improved host resource utilization at high temperature.

Event timing at the single-cell level

The timing of a cellular event often hides critical information on the process leading to the event. Our ability to measure event times in single cells along with other quantities allow us to learn about the drivers of the timed process and its downstream effects. In this review, we cover different types of events that have been timed in single cells, methods to time such events and types of analysis that have been applied to event timings. We show how different timing distributions suggest different natures for the process. The statistical relations between the timing of different events may reveal how their respective processes are related biologically: Do they occur in sequence or in parallel? Are they independent or inter-dependent? Finally, quantifying morphological and molecular variables may help assess their contribution to the timing of an event and its related process.

Solitons and collapse in the λ-repressor protein

The enterobacteria lambda phage is a paradigm temperate bacteriophage. Its lysogenic and lytic life cycles echo competition between the DNA binding λ-repressor (CI) and CRO proteins. Here we scrutinize the structure, stability, and folding pathways of the λ-repressor protein, which controls the transition from the lysogenic to the lytic state. We first investigate the supersecondary helix-loop helix composition of its backbone. We use a discrete Frenet framing to resolve the backbone spectrum in terms of bond and torsion angles. Instead of four, there appears to be seven individual loops. We model the putative loops using an explicit soliton Ansatz. It is based on the standard soliton profile of the continuum nonlinear Schrödinger equation. The accuracy of the Ansatz far exceeds the B-factor fluctuation distance accuracy of the experimentally determined protein configuration. We then investigate the folding pathways and dynamics of the λ-repressor protein. We introduce a coarse-grained energy function to model the backbone in terms of the C(α) atoms and the side chains in terms of the relative orientation of the C(β) atoms. We describe the folding dynamics in terms of relaxation dynamics and find that the folded configuration can be reached from a very generic initial configuration. We conclude that folding is dominated by the temporal ordering of soliton formation. In particular, the third soliton should appear before the first and second. Otherwise, the DNA binding turn does not acquire its correct structure. We confirm the stability of the folded configuration by repeated heating and cooling simulations.

Stochastic delay accelerates signaling in gene networks

The creation of protein from DNA is a dynamic process consisting of numerous reactions, such as transcription, translation and protein folding. Each of these reactions is further comprised of hundreds or thousands of sub-steps that must be completed before a protein is fully mature. Consequently, the time it takes to create a single protein depends on the number of steps in the reaction chain and the nature of each step. One way to account for these reactions in models of gene regulatory networks is to incorporate dynamical delay. However, the stochastic nature of the reactions necessary to produce protein leads to a waiting time that is randomly distributed. Here, we use queueing theory to examine the effects of such distributed delay on the propagation of information through transcriptionally regulated genetic networks. In an analytically tractable model we find that increasing the randomness in protein production delay can increase signaling speed in transcriptional networks. The effect is confirmed in stochastic simulations, and we demonstrate its impact in several common transcriptional motifs. In particular, we show that in feedforward loops signaling time and magnitude are significantly affected by distributed delay. In addition, delay has previously been shown to cause stable oscillations in circuits with negative feedback. We show that the period and the amplitude of the oscillations monotonically decrease as the variability of the delay time increases.

Delay in gene regulatory networks often arises from the numerous sequential reactions necessary to create fully functional protein from DNA. While the molecular mechanisms behind protein production and maturation are known, it is still unknown to what extent the resulting delay affects signaling in transcriptional networks. In contrast to previous studies that have examined the consequences of fixed delay in gene networks, here we investigate how the variability of the delay time influences the resulting dynamics. The exact distribution of “transcriptional delay” is still unknown, and most likely greatly depends on both intrinsic and extrinsic factors. Nevertheless, we are able to deduce specific effects of distributed delay on transcriptional signaling that are independent of the underlying distribution. We find that the time it takes for a gene encoding a transcription factor to signal its downstream target decreases as the delay variability increases. We use queueing theory to derive a simple relationship describing this result, and use stochastic simulations to confirm it. The consequences of distributed delay for several common transcriptional motifs are also discussed.

On the minimization of fluctuations in the response times of autoregulatory gene networks

The temporal dynamics of the concentrations of several proteins are tightly regulated, particularly for critical nodes in biological networks such as transcription factors. An important mechanism to control transcription factor levels is through autoregulatory feedback loops where the protein can bind its own promoter. Here we use theoretical tools and computational simulations to further our understanding of transcription-factor autoregulatory loops. We show that the stochastic dynamics of feedback and mRNA synthesis can significantly influence the speed of response of autoregulatory genetic networks toward external stimuli. The fluctuations in the response-times associated with the accumulation of the transcription factor in the presence of negative or positive autoregulation can be minimized by confining the ratio of mRNA/protein lifetimes within 1:10. This predicted range of mRNA/protein lifetime agrees with ranges observed empirically in prokaryotes and eukaryotes. The theory can quantitatively and systematically account for the influence of regulatory element binding and unbinding dynamics on the transcription-factor concentration rise-times. The simulation results are robust against changes in several system parameters of the gene expression machinery.

Decision making in living cells: lessons from a simple system

The life cycle of bacteriophage lambda serves as a simplified paradigm for cell-fate decisions. The ongoing quantitative, high-resolution experimental investigation of this life cycle has produced some important insights in recent years. These insights have to do with the way cells choose among alternative fates, how they maintain long-term memory of their gene-expression state, and how they switch from one stable state to another. The recent studies have highlighted the role of spatiotemporal effects in cellular processes and the importance of distinguishing chemical stochasticity from possible hidden variables in cellular decision making.

Factors influencing lysis time stochasticity in bacteriophage λ

BACKGROUND

Despite identical genotypes and seemingly uniform environments, stochastic gene expression and other dynamic intracellular processes can produce considerable phenotypic diversity within clonal microbes. One trait that provides a good model to explore the molecular basis of stochastic variation is the timing of host lysis by bacteriophage (phage).

RESULTS

Individual lysis events of thermally-inducible λ lysogens were observed using a temperature-controlled perfusion chamber mounted on an inverted microscope. Both mean lysis time (MLT) and its associated standard deviation (SD) were estimated. Using the SD as a measure of lysis time stochasticity, we showed that lysogenic cells in controlled environments varied widely in lysis times, and that the level of lysis time stochasticity depended on allelic variation in the holin sequence, late promoter (pR') activity, and host growth rate. In general, the MLT was positively correlated with the SD. Both lower pR' activities and lower host growth rates resulted in larger SDs. Results from premature lysis, induced by adding KCN at different time points after lysogen induction, showed a negative correlation between the timing of KCN addition and lysis time stochasticity.

CONCLUSIONS

Taken together with results published by others, we conclude that a large fraction of λ lysis time stochasticity is the result of random events following the expression and diffusion of the holin protein. Consequently, factors influencing the timing of reaching critical holin concentrations in the cell membrane, such as holin production rate, strongly influence the mean lysis time and the lysis time stochasticity.

Timing of gene transcription in the galactose utilization system of Escherichia coli

In the natural environment, bacterial cells have to adjust their metabolism to alterations in the availability of food sources. The order and timing of gene expression are crucial in these situations to produce an appropriate response. We used the galactose regulation in Escherichia coli as a model system for understanding how cells integrate information about food availability and cAMP levels to adjust the timing and intensity of gene expression. We simulated the feast-famine cycle of bacterial growth by diluting stationary phase cells in fresh medium containing galactose as the sole carbon source. We followed the activities of six promoters of the galactose system as cells grew on and ran out of galactose. We found that the cell responds to a decreasing external galactose level by increasing the internal galactose level, which is achieved by limiting galactose metabolism and increasing the expression of transporters. We show that the cell alters gene expression based primarily on the current state of the cell and not on monitoring the level of extracellular galactose in real time. Some decisions have longer term effects; therefore, the current state does subtly encode the history of food availability. In summary, our measurements of timing of gene expression in the galactose system suggest that the system has evolved to respond to environments where future galactose levels are unpredictable rather than regular feast and famine cycles.

Simplified models of biological networks

The function of living cells is controlled by complex regulatory networks that are built of a wide diversity of interacting molecular components. The sheer size and intricacy of molecular networks of even the simplest organisms are obstacles toward understanding network functionality. This review discusses the achievements and promise of a bottom-up approach that uses well-characterized subnetworks as model systems for understanding larger networks. It highlights the interplay between the structure, logic, and function of various types of small regulatory circuits. The bottom-up approach advocates understanding regulatory networks as a collection of entangled motifs. We therefore emphasize the potential of negative and positive feedback, as well as their combinations, to generate robust homeostasis, epigenetics, and oscillations.

Stochastic gene expression as a molecular switch for viral latency

Stochastic 'noise' arises from random thermal fluctuations in the concentration of protein, RNA, or other molecules within the cell and is an unavoidable aspect of life at the single-cell level. Evidence is accumulating that this biochemical noise crucially influences cellular auto-regulatory circuits and can 'flip' genetic switches to drive probabilistic fate decisions in bacteria, viruses, cancer, and stem cells. Here, we review how stochastic gene expression in key auto-regulatory proteins can control fate determination between latency and productive replication in both phage-lambda and HIV-1. We highlight important new studies that synthetically manipulate auto-regulatory circuitry and noise, to bias HIV-1's ability to enter proviral latency. We argue that an appreciation of noise in gene expression may shed light on the mystery of animal virus latency and that strategies to manipulate noise may have impact on anti-viral therapeutics.

Kinetics of genetic switching into the state of bacterial competence

Nonlinear amplification of gene expression of master regulators is essential for cellular differentiation. Here we investigated determinants that control the kinetics of the genetic switching process from the vegetative state (B-state) to the competent state (K-state) of Bacillus subtilis, explicitly including the switching window which controls the probability for competence initiation in a cell population. For individual cells, we found that after initiation of switching, the levels of the master regulator [ComK](t) increased with sigmoid shape and saturation occurred at two distinct levels of [ComK]. We analyzed the switching kinetics into the state with highest [ComK] and found saturation after a switching period of length 1.4 +/- 0.3 h. The duration of the switching period was robust against variations in the gene regulatory network of the master regulator, whereas the saturation levels showed large variations between individual isogenic cells. We developed a nonlinear dynamics model, taking into account low-number stochastic effects. The model quantitatively describes the probability and timescale of switching at the single cell level and explains why the ComK level in the K-state is highly sensitive to extrinsic parameter variations. Furthermore, the model predicts a transition from stochastic to deterministic switching at increased production rates of ComK in agreement with experimental data.

Stochastic switching to competence

Distinct modes of gene expression enable isogenic populations of bacteria to maintain a diversity of phenotypes and to rapidly adapt to environmental changes. Competence development for DNA transformation in Bacillus subtilis has become a paradigm for a multimodal system which implements a genetic switch through a nonlinear positive feedback of a transcriptional master regulator. Recent advances in quantitative analysis at the single cell level in conjunction with mathematical modeling allowed a molecular level understanding of the switching probability between the noncompetent state and the competent state. It has been discovered that the genetic switching probability may be tuned by controlling noise in the transcription of the master regulator of competence, by timing of its expression, and by rewiring of the control circuit.

The stochastic nature of biochemical networks

Cell behaviour and the cellular environment are stochastic. Phenotypes vary across isogenic populations and in individual cells over time. Here we will argue that to understand the abilities of cells we need to understand their stochastic nature. New experimental techniques allow gene expression to be followed in single cells over time and reveal stochastic bursts of both mRNA and protein synthesis in many different types of organisms. Stochasticity has been shown to be exploited by bacteria and viruses to decide between different behaviours. In fluctuating environments, cells that respond stochastically can out-compete those that sense environmental changes, and stochasticity may even have contributed to chromosomal gene order. We will focus on advances in modelling stochasticity, in understanding its effects on evolution and cellular design, and on means by which it may be exploited in biotechnology and medicine.

Collective decision making in bacterial viruses

For many bacterial viruses, the choice of whether to kill host cells or enter a latent state depends on the multiplicity of coinfection. Here, we present a mathematical theory of how bacterial viruses can make collective decisions concerning the fate of infected cells. We base our theory on mechanistic models of gene regulatory dynamics. Unlike most previous work, we treat the copy number of viral genes as variable. Increasing the viral copy number increases the rate of transcription of viral mRNAs. When viral regulation of cell fate includes nonlinear feedback loops, very small changes in transcriptional rates can lead to dramatic changes in steady-state gene expression. Hence, we prove that deterministic decisions can be reached, e.g., lysis or latency, depending on the cellular multiplicity of infection within a broad class of gene regulatory models of viral decision-making. Comparisons of a parameterized version of the model with molecular studies of the decision structure in the temperate bacteriophage lambda are consistent with our conclusions. Because the model is general, it suggests that bacterial viruses can respond adaptively to changes in population dynamics, and that features of collective decision-making in viruses are evolvable life history traits.

Colored extrinsic fluctuations and stochastic gene expression

Stochasticity is both exploited and controlled by cells. Although the intrinsic stochasticity inherent in biochemistry is relatively well understood, cellular variation, or ‘noise', is predominantly generated by interactions of the system of interest with other stochastic systems in the cell or its environment. Such extrinsic fluctuations are nonspecific, affecting many system components, and have a substantial lifetime, comparable to the cell cycle (they are ‘colored'). Here, we extend the standard stochastic simulation algorithm to include extrinsic fluctuations. We show that these fluctuations affect mean protein numbers and intrinsic noise, can speed up typical network response times, and can explain trends in high-throughput measurements of variation. If extrinsic fluctuations in two components of the network are correlated, they may combine constructively (amplifying each other) or destructively (attenuating each other). Consequently, we predict that incoherent feedforward loops attenuate stochasticity, while coherent feedforwards amplify it. Our results demonstrate that both the timescales of extrinsic fluctuations and their nonspecificity substantially affect the function and performance of biochemical networks.

Host responses influence on the induction of lambda prophage

Inactivation of bacteriophage lambda CI repressor leads almost exclusively to lytic development. Prophage induction can be initiated either by DNA damage or by heat treatment of a temperature-sensitive repressor. These two treatments also cause a concurrent activation of either the host SOS or heat-shock stress responses respectively. We studied the effects of these two methods of induction on the lytic pathway by monitoring the activation of different lambda promoters, and found that the lambda genetic network co-ordinates information from the host stress response networks. Our results show that the function of the CII transcriptional activator, which facilitates the lysogenic developmental pathway, is not observed following either method of induction. Mutations in the cro gene restore the CII function irrespective of the induction method. Deletion of the heat-shock protease gene ftsH can also restore CII function following heat induction but not following SOS induction. Our findings highlight the importance of the elimination of CII function during induction as a way to ensure an efficient lytic outcome. We also show that, despite the common inhibitory effect on CII function, there are significant differences in the heat- and SOS-induced pathways leading to the lytic cascade.

Effects of molecular memory and bursting on fluctuations in gene expression

Many cellular components are present in such low numbers per cell that random births and deaths of individual molecules can cause substantial "noise" in concentrations. But biochemical events do not necessarily occur in single steps of individual molecules. Some processes are greatly randomized when synthesis or degradation occurs in large bursts of many molecules during a short time interval. Conversely, each birth or death of a macromolecule could involve several small steps, creating a memory between individual events. We present a generalized theory for stochastic gene expression, formulating the variance in protein abundance in terms of the randomness of the individual gene expression events. We show that common types of molecular mechanisms can produce gestation and senescence periods that reduce noise without requiring higher abundances, shorter lifetimes, or any concentration-dependent control loops. We also show that most single-cell experimental methods cannot distinguish between qualitatively different stochastic principles, although this in turn makes such methods better suited for identifying which components introduce fluctuations. Characterizing the random events that give rise to noise in concentrations instead requires dynamic measurements with single-molecule resolution.