Protein hourglass: Exact first passage time distributions for protein thresholds

How Transcriptional Bursting and mRNA Production Affect Precise Timing of Cell Lysis Phenomena

Bacterial viruses infect bacterial cells and stimulate the production of holin proteins that accumulate in the cellular membranes. When the number of such proteins reaches a threshold, the membrane permeabilizes and the cell is destroyed in the process known as cell lysis. Experimental studies indicate that cell lysis occurs at specific times, although the underlying molecular mechanisms of such precise timing remain not well understood. Recently, a theoretical framework has been introduced to explain these phenomena as a coupling between stochastic processes of holins accumulation in the membrane and breaking the membrane that leads to threshold behavior. However, this approach does not account for many biologically important processes in cell lysis. In this work, we investigated the role of transcriptional bursting and mRNA production on the dynamics of cell lysis. The original stochastic framework is extended, allowing us to evaluate the cell lysis dynamics under more realistic biological conditions using analytical calculations and Monte Carlo computer simulations. It is shown explicitly that the random processes of transcription bursting and mRNA production do not affect the threshold-like dynamics of cell lysis, although they influence the absolute values of the maximal thresholds and their distributions. It is also found that the effect of mRNA production is generally stronger than the effect due to transcriptional bursting. Physical-chemical arguments to explain these observations are presented. Thus, our theoretical analysis suggests that the precise timing of cell lysis is a robust phenomenon despite involving multiple random biochemical processes. Our theoretical approach clarifies some important mechanistic aspects of complex biological processes of cell lysis.

Microscopic origin of the spatial and temporal precision in biological systems

All living systems display remarkable spatial and temporal precision, despite operating in intrinsically fluctuating environments. It is even more surprising given that biological phenomena are regulated by multiple chemical reactions that are also random. Although the underlying molecular mechanisms of surprisingly high precision in biology remain not well understood, a novel theoretical picture that relies on the coupling of relevant stochastic processes has recently been proposed and applied to explain different phenomena. To illustrate this approach, in this review, we discuss two systems that exhibit precision control: spatial regulation in bacterial cell size and temporal regulation in the timing of cell lysis by λ bacteriophage. In cell-size regulation, it is argued that a balance between stochastic cell growth and cell division processes leads to a narrow distribution of cell sizes. In cell lysis, it is shown that precise timing is due to the coupling of holin protein accumulation and the breakage of the cellular membrane. The stochastic coupling framework also allows us to explicitly evaluate dynamic properties for both biological systems, eliminating the need to utilize the phenomenological concept of thresholds. Excellent agreement with experimental observations is observed, supporting the proposed theoretical ideas. These observations also suggest that the stochastic coupling method captures the important aspects of molecular mechanisms of precise cellular regulation, providing a powerful new tool for more advanced investigations of complex biological phenomena.

Optimizing search processes in systems with state toggling: Exact condition delimiting the efficacy of stochastic resetting strategy

Will the strategy of resetting help a stochastic process to reach its target efficiently, with its environment continually toggling between a strongly favorable and an unfavorable (or weakly favorable) state? A diffusive run-and-tumble motion, transport of molecular motors on or off a filament, and motion under flashing optical traps are special examples of such state toggling. For any general process with toggling under Poisson reset, we derive a mathematical condition for continuous transitions where the advantage rendered by resetting vanishes. For the case of diffusive motion with linear potentials of unequal strength, we present exact solutions, which reveal that there is quite generically a re-entrance of the advantage of resetting as a function of the strength of the strongly favorable potential. This result is shown to be valid for quadratic potential traps by using the general condition of transition.

Exact distributions of threshold crossing times of proteins under post-transcriptional regulation by small RNAs

The timings of several cellular events like cell lysis, cell division, or pore formation in endosomes are regulated by the time taken for the relevant proteins to cross a threshold in number or concentration. Since protein synthesis is stochastic, the threshold crossing time is a first passage problem. The exact distributions of these first passage processes have been obtained recently for unregulated and autoregulated genes. Many proteins are however regulated by post-transcriptional regulation, controlled by small noncoding RNAs (sRNAs). Certain mathematical models of gene expression with post-transcriptional sRNA regulation have been recently exactly mapped to models without sRNA regulation. Utilizing this mapping and the exact distributions, we calculate exact results on fluctuations (full distribution, all cumulants, and characteristic times) of protein threshold crossing times in the presence of sRNA regulation. We derive two interesting predictions from these exact results. We show that the size of the fluctuation of the threshold crossing times have a nonmonotonic U-shaped behavior as a function of the rates of binding and unbinding of the sRNA-mRNA complex. Thus there are optimal parameters that minimize noise. Furthermore, the fluctuations in models with sRNA regulation may be higher or lower compared to the model without regulation, depending on the mean protein burst size.

Nature of barriers determines first passage times in heterogeneous media

Intuition suggests that passage times across a region increase with the number of barriers along the path. Can this fail depending on the nature of the barrier? To probe this fundamental question, we exactly solve for the first passage time in general d-dimensions for diffusive transport through a spatially patterned array of obstacles - either entropic or energetic, depending on the nature of the obstacles. For energetic barriers, we show that first passage times vary non-monotonically with the number of barriers, while for entropic barriers it increases monotonically. This non-monotonicity for energetic barriers is further reflected in the behaviour of effective diffusivity as well. We then design a simple experiment where a robotic bug navigates in a heterogeneous environment through a spatially patterned array of obstacles to validate our predictions. Finally, using numerical simulations, we show that this non-monotonic behaviour for energetic barriers is general and extends to even super-diffusive transport.

Continuous gated first-passage processes

Gated first-passage processes, where completion depends on both hitting a target and satisfying additional constraints, are prevalent across various fields. Despite their significance, analytical solutions to basic problems remain unknown, e.g. the detection time of a diffusing particle by a gated interval, disk, or sphere. In this paper, we elucidate the challenges posed by continuous gated first-passage processes and present a renewal framework to overcome them. This framework offers a unified approach for a wide range of problems, including those with single-point, half-line, and interval targets. The latter have so far evaded exact solutions. Our analysis reveals that solutions to gated problems can be obtained directly from the ungated dynamics. This, in turn, reveals universal properties and asymptotic behaviors, shedding light on cryptic intermediate-time regimes and refining the notion of high-crypticity for continuous-space gated processes. Moreover, we extend our formalism to higher dimensions, showcasing its versatility and applicability. Overall, this work provides valuable insights into the dynamics of continuous gated first-passage processes and offers analytical tools for studying them across diverse domains.

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.

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.