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.

The chemical Langevin equation for biochemical systems in dynamic environments.

Modeling and simulation of complex biochemical reaction networks form cornerstones of modern biophysics. Many of the approaches developed so far capture temporal fluctuations due to the inherent stochasticity of the biophysical processes, referred to as intrinsic noise. Stochastic fluctuations, however, predominantly stem from the interplay of the network with many other-and mostly unknown-fluctuating processes, as well as with various random signals arising from the extracellular world; these sources contribute extrinsic noise. Here, we provide a computational simulation method to probe the stochastic dynamics of biochemical systems subject to both intrinsic and extrinsic noise. We develop an extrinsic chemical Langevin equation (CLE)-a physically motivated extension of the CLE-to model intrinsically noisy reaction networks embedded in a stochastically fluctuating environment. The extrinsic CLE is a continuous approximation to the chemical master equation (CME) with time-varying propensities. In our approach, noise is incorporated at the level of the CME, and it can account for the full dynamics of the exogenous noise process, irrespective of timescales and their mismatches. We show that our method accurately captures the first two moments of the stationary probability density when compared with exact stochastic simulation methods while reducing the computational runtime by several orders of magnitude. Our approach provides a method that is practical, computationally efficient, and physically accurate to study systems that are simultaneously subject to a variety of noise sources.

Noise distorts the epigenetic landscape and shapes cell-fate decisions.

The Waddington epigenetic landscape has become an iconic representation of the cellular differentiation process. Recent single-cell transcriptomic data provide new opportunities for quantifying this originally conceptual tool, offering insight into the gene regulatory networks underlying cellular development. While many methods for constructing the landscape have been proposed, by far the most commonly employed approach is based on computing the landscape as the negative logarithm of the steady-state probability distribution. Here, we use simple models to highlight the complexities and limitations that arise when reconstructing the potential landscape in the presence of stochastic fluctuations. We consider how the landscape changes in accordance with different stochastic systems and show that it is the subtle interplay between the deterministic and stochastic components of the system that ultimately shapes the landscape. We further discuss how the presence of noise has important implications for the identifiability of the regulatory dynamics from experimental data. A record of this paper's transparent peer review process is included in the supplemental information.

Pathway dynamics can delineate the sources of transcriptional noise in gene expression.

Single-cell expression profiling opens up new vistas on cellular processes. Extensive cell-to-cell variability at the transcriptomic and proteomic level has been one of the stand-out observations. Because most experimental analyses are destructive we only have access to snapshot data of cellular states. This loss of temporal information presents significant challenges for inferring dynamics, as well as causes of cell-to-cell variability. In particular, we typically cannot separate dynamic variability from within cells ('intrinsic noise') from variability across the population ('extrinsic noise'). Here, we make this non-identifiability mathematically precise, allowing us to identify new experimental set-ups that can assist in resolving this non-identifiability. We show that multiple generic reporters from the same biochemical pathways (e.g. mRNA and protein) can infer magnitudes of intrinsic and extrinsic transcriptional noise, identifying sources of heterogeneity. Stochastic simulations support our theory, and demonstrate that 'pathway-reporters' compare favourably to the well-known, but often difficult to implement, dual-reporter method.

In biology, seemingly random variation within or between cells can have significant effects on a number of cellular processes, like how cells divide and develop. For example, how often a gene is switched on, or 'expressed', can randomly fluctuate over time. This 'noise' may lead to a cell having slightly more of a particular molecule, causing it to behave differently to other cells in the population. However, it is currently unclear how this random variation is created and controlled in cells, and what effect this has on biological systems as a whole. When a gene is expressed, its sequence typically gets copied in to a molecule called mRNA, which is then processed and used to build the protein encoded by the gene. By measuring the levels of mRNA molecules in individual cells, researchers have been able to investigate how gene expression varies within populations. These experiments are carried out on dead cells at a single point in time, and mathematical models are then applied to detect noise in the molecular data. This approach, however, precludes how noise changes over time, making it difficult to determine the source of cell-to-cell variability. In particular, whether the variation detected is the result of genuine random molecular changes (intrinsic noise), or external factors ­ such as temperature and pH ­ fluctuating in the cells environment (extrinsic noise). Here, Ham et al. have built on previous mathematical models to identify a new approach for investigating the source of molecular noise. They found that for any given gene it is impossible to understand what causes its activity levels to vary just from data on its mRNA levels. Instead, information on other molecules that are affected by expression of the gene (termed 'pathway reporters') can provide a clearer picture of whether molecular variability is the result of intrinsic or extrinsic noise. The mathematical models developed by Ham et al. reveal what can and cannot be learned about noise from gene expression data. Furthermore, pathway-reporters are easier to measure experimentally than other reporters that are typically used to study the origins and effects of cell-to-cell variability. These findings could help researchers design single cell experiments that are better for studying noise, leading to a deeper understanding of how different types of variation impact cell biology.

Exactly solvable models of stochastic gene expression.

Stochastic models are key to understanding the intricate dynamics of gene expression. However, the simplest models that only account for active and inactive states of a gene fail to capture common observations in both prokaryotic and eukaryotic organisms. Here, we consider multistate models of gene expression that generalize the canonical Telegraph process and are capable of capturing the joint effects of transcription factors, heterochromatin state, and DNA accessibility (or, in prokaryotes, sigma-factor activity) on transcript abundance. We propose two approaches for solving classes of these generalized systems. The first approach offers a fresh perspective on a general class of multistate models and allows us to "decompose" more complicated systems into simpler processes, each of which can be solved analytically. This enables us to obtain a solution of any model from this class. Next, we develop an approximation method based on a power series expansion of the stationary distribution for an even broader class of multistate models of gene transcription. We further show that models from both classes cannot have a heavy-tailed distribution in the absence of extrinsic noise. The combination of analytical and computational solutions for these realistic gene expression models also holds the potential to design synthetic systems and control the behavior of naturally evolved gene expression systems in guiding cell-fate decisions.

Extrinsic Noise and Heavy-Tailed Laws in Gene Expression.

Noise in gene expression is one of the hallmarks of life at the molecular scale. Here we derive analytical solutions to a set of models describing the molecular mechanisms underlying transcription of DNA into RNA. Our ansatz allows us to incorporate the effects of extrinsic noise-encompassing factors external to the transcription of the individual gene-and discuss the ramifications for heterogeneity in gene product abundance that has been widely observed in single cell data. Crucially, we are able to show that heavy-tailed distributions of RNA copy numbers cannot result from the intrinsic stochasticity in gene expression alone, but must instead reflect extrinsic sources of variability.