Architectural underpinnings of stochastic intergenerational homeostasis.

Living systems are naturally complex and adaptive and offer unique insights into the strategies for achieving and sustaining stochastic homeostasis in different conditions. Here we focus on homeostasis in the context of stochastic growth and division of individual bacterial cells. We take advantage of high-precision long-term dynamical data that have recently been used to extract emergent simplicities and to articulate empirical intra- and intergenerational scaling laws governing these stochastic dynamics. From these data, we identify the core motif in the mechanistic coupling between division and growth, which naturally yields these precise rules, thus also bridging the intra- and intergenerational phenomenologies. By developing and utilizing techniques for solving a broad class of first-passage processes, we derive the exact analytic necessary and sufficient condition for sustaining stochastic intergenerational cell-size homeostasis within this framework. Furthermore, we provide predictions for the precision kinematics of cell-size homeostasis and the shape of the interdivision time distribution, which are compellingly borne out by the high-precision data. Taken together, these results provide insights into the functional architecture of control systems that yield robust yet flexible stochastic homeostasis.

Scaling of stochastic growth and division dynamics: A comparative study of individual rod-shaped cells in the Mother Machine and SChemostat platforms.

Microfluidic platforms enable long-term quantification of stochastic behaviors of individual bacterial cells under precisely controlled growth conditions. Yet, quantitative comparisons of physiological parameters and cell behaviors of different microorganisms in different experimental and device modalities is not available due to experiment-specific details affecting cell physiology. To rigorously assess the effects of mechanical confinement, we designed, engineered, and performed side-by-side experiments under otherwise identical conditions in the Mother Machine (with confinement) and the SChemostat (without confinement), using the latter as the ideal comparator. We established a protocol to cultivate a suitably engineered rod-shaped mutant of Caulobacter crescentus in the Mother Machine and benchmarked the differences in stochastic growth and division dynamics with respect to the SChemostat. While the single-cell growth rate distributions are remarkably similar, the mechanically confined cells in the Mother Machine experience a substantial increase in interdivision times. However, we find that the division ratio distribution precisely compensates for this increase, which in turn reflects identical emergent simplicities governing stochastic intergenerational homeostasis of cell sizes across device and experimental configurations, provided the cell sizes are appropriately mean-rescaled in each condition. Our results provide insights into the nature of the robustness of the bacterial growth and division machinery.

Emergent Spatiotemporal Organization in Stochastic Intracellular Transport Dynamics.

The interior of a living cell is an active, fluctuating, and crowded environment, yet it maintains a high level of coherent organization. This dichotomy is readily apparent in the intracellular transport system of the cell. Membrane-bound compartments called endosomes play a key role in carrying cargo, in conjunction with myriad components including cargo adaptor proteins, membrane sculptors, motor proteins, and the cytoskeleton. These components coordinate to effectively navigate the crowded cell interior and transport cargo to specific intracellular locations, even though the underlying protein interactions and enzymatic reactions exhibit stochastic behavior. A major challenge is to measure, analyze, and understand how, despite the inherent stochasticity of the constituent processes, the collective outcomes show an emergent spatiotemporal order that is precise and robust. This review focuses on this intriguing dichotomy, providing insights into the known mechanisms of noise suppression and noise utilization in intracellular transport processes, and also identifies opportunities for future inquiry.

Deterministic early endosomal maturations emerge from a stochastic trigger-and-convert mechanism.

Endosomal maturation is critical for robust and timely cargo transport to specific cellular compartments. The most prominent model of early endosomal maturation involves a phosphoinositide-driven gain or loss of specific proteins on individual endosomes, emphasising an autonomous and stochastic description. However, limitations in fast, volumetric imaging long hindered direct whole cell-level measurements of absolute numbers of maturation events. Here, we use lattice light-sheet imaging and bespoke automated analysis to track individual very early (APPL1-positive) and early (EEA1-positive) endosomes over the entire population, demonstrating that direct inter-endosomal contact drives maturation between these populations. Using fluorescence lifetime, we show that this endosomal interaction is underpinned by asymmetric binding of EEA1 to very early and early endosomes through its N- and C-termini, respectively. In combination with agent-based simulation which supports a 'trigger-and-convert' model, our findings indicate that APPL1- to EEA1-positive maturation is driven not by autonomous events but by heterotypic EEA1-mediated interactions, providing a mechanism for temporal and population-level control of maturation.

Emergent periodicity in the collective synchronous flashing of fireflies.

In isolation from their peers, Photinus carolinus fireflies flash with no intrinsic period between successive bursts. Yet, when congregating into large mating swarms, these fireflies transition into predictability, synchronizing with their neighbors with a rhythmic periodicity. Here we propose a mechanism for emergence of synchrony and periodicity, and formulate the principle in a mathematical framework. Remarkably, with no fitting parameters, analytic predictions from this simple principle and framework agree strikingly well with data. Next, we add further sophistication to the framework using a computational approach featuring groups of random oscillators via integrate-and-fire interactions controlled by a tunable parameter. This agent-based framework of P. carolinus fireflies interacting in swarms of increasing density also shows quantitatively similar phenomenology and reduces to the analytic framework in the appropriate limit of the tunable coupling strength. We discuss our findings and note that the resulting dynamics follow the style of a decentralized follow-the-leader synchronization, where any of the randomly flashing individuals may take the role of the leader of any subsequent synchronized flash burst.

Cellular learning: Habituation sans neurons in a unicellular organism.

Stentor coeruleus cells stochastically switch between non-responsive (contracted) and responsive (extended) states. Learning is accomplished via habituation, in which the internal model is updated to reflect the current environment by tuning the transition rates according to the time series properties of mechanical stimuli.

Beyond the average: An updated framework for understanding the relationship between cell growth, DNA replication, and division in a bacterial system.

Our understanding of the bacterial cell cycle is framed largely by population-based experiments that focus on the behavior of idealized average cells. Most famously, the contributions of Cooper and Helmstetter help to contextualize the phenomenon of overlapping replication cycles observed in rapidly growing bacteria. Despite the undeniable value of these approaches, their necessary reliance on the behavior of idealized average cells masks the stochasticity inherent in single-cell growth and physiology and limits their mechanistic value. To bridge this gap, we propose an updated and agnostic framework, informed by extant single-cell data, that quantitatively accounts for stochastic variations in single-cell dynamics and the impact of medium composition on cell growth and cell cycle progression. In this framework, stochastic timers sensitive to medium composition impact the relationship between cell cycle events, accounting for observed differences in the relationship between cell cycle events in slow- and fast-growing cells. We conclude with a roadmap for potential application of this framework to longstanding open questions in the bacterial cell cycle field.

Physical bioenergetics: Energy fluxes, budgets, and constraints in cells.

Cells are the basic units of all living matter which harness the flow of energy to drive the processes of life. While the biochemical networks involved in energy transduction are well-characterized, the energetic costs and constraints for specific cellular processes remain largely unknown. In particular, what are the energy budgets of cells? What are the constraints and limits energy flows impose on cellular processes? Do cells operate near these limits, and if so how do energetic constraints impact cellular functions? Physics has provided many tools to study nonequilibrium systems and to define physical limits, but applying these tools to cell biology remains a challenge. Physical bioenergetics, which resides at the interface of nonequilibrium physics, energy metabolism, and cell biology, seeks to understand how much energy cells are using, how they partition this energy between different cellular processes, and the associated energetic constraints. Here we review recent advances and discuss open questions and challenges in physical bioenergetics.

Phenomenology of stochastic exponential growth.

Stochastic exponential growth is observed in a variety of contexts, including molecular autocatalysis, nuclear fission, population growth, inflation of the universe, viral social media posts, and financial markets. Yet literature on modeling the phenomenology of these stochastic dynamics has predominantly focused on one model, geometric Brownian motion (GBM), which can be described as the solution of a Langevin equation with linear drift and linear multiplicative noise. Using recent experimental results on stochastic exponential growth of individual bacterial cell sizes, we motivate the need for a more general class of phenomenological models of stochastic exponential growth, which are consistent with the observation that the mean-rescaled distributions are approximately stationary at long times. We show that this behavior is not consistent with GBM, instead it is consistent with power-law multiplicative noise with positive fractional powers. Therefore, we consider this general class of phenomenological models for stochastic exponential growth, provide analytical solutions, and identify the important dimensionless combination of model parameters, which determines the shape of the mean-rescaled distribution. We also provide a prescription for robustly inferring model parameters from experimentally observed stochastic growth trajectories.

Intergenerational continuity of cell shape dynamics in Caulobacter crescentus.

We investigate the intergenerational shape dynamics of single Caulobacter crescentus cells using a novel combination of imaging techniques and theoretical modeling. We determine the dynamics of cell pole-to-pole lengths, cross-sectional widths, and medial curvatures from high accuracy measurements of cell contours. Moreover, these shape parameters are determined for over 250 cells across approximately 10000 total generations, which affords high statistical precision. Our data and model show that constriction is initiated early in the cell cycle and that its dynamics are controlled by the time scale of exponential longitudinal growth. Based on our extensive and detailed growth and contour data, we develop a minimal mechanical model that quantitatively accounts for the cell shape dynamics and suggests that the asymmetric location of the division plane reflects the distinct mechanical properties of the stalked and swarmer poles. Furthermore, we find that the asymmetry in the division plane location is inherited from the previous generation. We interpret these results in terms of the current molecular understanding of shape, growth, and division of C. crescentus.

Mixed Poisson distributions in exact solutions of stochastic autoregulation models.

In this paper we study the interplay between stochastic gene expression and system design using simple stochastic models of autoactivation and autoinhibition. Using the Poisson representation, a technique whose particular usefulness in the context of nonlinear gene regulation models we elucidate, we find exact results for these feedback models in the steady state. Further, we exploit this representation to analyze the parameter spaces of each model, determine which dimensionless combinations of rates are the shape determinants for each distribution, and thus demarcate where in the parameter space qualitatively different behaviors arise. These behaviors include power-law-tailed distributions, bimodal distributions, and sub-Poisson distributions. We also show how these distribution shapes change when the strength of the feedback is tuned. Using our results, we reexamine how well the autoinhibition and autoactivation models serve their conventionally assumed roles as paradigms for noise suppression and noise exploitation, respectively.

Scaling laws governing stochastic growth and division of single bacterial cells.

Uncovering the quantitative laws that govern the growth and division of single cells remains a major challenge. Using a unique combination of technologies that yields unprecedented statistical precision, we find that the sizes of individual Caulobacter crescentus cells increase exponentially in time. We also establish that they divide upon reaching a critical multiple (≈ 1.8) of their initial sizes, rather than an absolute size. We show that when the temperature is varied, the growth and division timescales scale proportionally with each other over the physiological temperature range. Strikingly, the cell-size and division-time distributions can both be rescaled by their mean values such that the condition-specific distributions collapse to universal curves. We account for these observations with a minimal stochastic model that is based on an autocatalytic cycle. It predicts the scalings, as well as specific functional forms for the universal curves. Our experimental and theoretical analysis reveals a simple physical principle governing these complex biological processes: a single temperature-dependent scale of cellular time governs the stochastic dynamics of growth and division in balanced growth conditions.

Universality in stochastic exponential growth.

Recent imaging data for single bacterial cells reveal that their mean sizes grow exponentially in time and that their size distributions collapse to a single curve when rescaled by their means. An analogous result holds for the division-time distributions. A model is needed to delineate the minimal requirements for these scaling behaviors. We formulate a microscopic theory of stochastic exponential growth as a Master Equation that accounts for these observations, in contrast to existing quantitative models of stochastic exponential growth (e.g., the Black-Scholes equation or geometric Brownian motion). Our model, the stochastic Hinshelwood cycle (SHC), is an autocatalytic reaction cycle in which each molecular species catalyzes the production of the next. By finding exact analytical solutions to the SHC and the corresponding first passage time problem, we uncover universal signatures of fluctuations in exponential growth and division. The model makes minimal assumptions, and we describe how more complex reaction networks can reduce to such a cycle. We thus expect similar scalings to be discovered in stochastic processes resulting in exponential growth that appear in diverse contexts such as cosmology, finance, technology, and population growth.

Stochasticity of gene products from transcriptional pulsing.

Transcriptional pulsing has been observed in both prokaryotes and eukaryotes and plays a crucial role in cell-to-cell variability of protein and mRNA numbers. An important issue is how the time constants associated with episodes of transcriptional bursting and mRNA and protein degradation rates lead to different cellular mRNA and protein distributions, starting from the transient regime leading to the steady state. We address this by deriving and then investigating the exact time-dependent solution of the master equation for a transcriptional pulsing model of mRNA distributions. We find a plethora of results. We show that, among others, bimodal and long-tailed (power-law) distributions occur in the steady state as the rate constants are varied over biologically significant time scales. Since steady state may not be reached experimentally we present results for the time evolution of the distributions. Because cellular behavior is determined by proteins, we also investigate the effect of the different mRNA distributions on the corresponding protein distributions using numerical simulations.