Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model

Noisy voter models in switching environments

We study the stationary states of variants of the noisy voter model, subject to fluctuating parameters or external environments. Specifically, we consider scenarios in which the herding-to-noise ratio switches randomly and on different timescales between two values. We show that this can lead to a phase in which polarized and heterogeneous states exist. Second, we analyze a population of noisy voters subject to groups of external influencers, and show how multipeak stationary distributions emerge. Our work is based on a combination of individual-based simulations, analytical approximations in terms of a piecewise-deterministic Markov processes (PDMP), and on corrections to this process capturing intrinsic stochasticity in the linear-noise approximation. We also propose a numerical scheme to obtain the stationary distribution of PDMPs with three environmental states and linear velocity fields.

Optimal control of bioproduction in the presence of population heterogeneity

Cell-to-cell variability, born of stochastic chemical kinetics, persists even in large isogenic populations. In the study of single-cell dynamics this is typically accounted for. However, on the population level this source of heterogeneity is often sidelined to avoid the inevitable complexity it introduces. The homogeneous models used instead are more tractable but risk disagreeing with their heterogeneous counterparts and may thus lead to severely suboptimal control of bioproduction. In this work, we introduce a comprehensive mathematical framework for solving bioproduction optimal control problems in the presence of heterogeneity. We study population-level models in which such heterogeneity is retained, and propose order-reduction approximation techniques. The reduced-order models take forms typical of homogeneous bioproduction models, making them a useful benchmark by which to study the importance of heterogeneity. Moreover, the derivation from the heterogeneous setting sheds light on parameter selection in ways a direct homogeneous outlook cannot, and reveals the source of approximation error. With view to optimally controlling bioproduction in microbial communities, we ask the question: when does optimising the reduced-order models produce strategies that work well in the presence of population heterogeneity? We show that, in some cases, homogeneous approximations provide remarkably accurate surrogate models. Nevertheless, we also demonstrate that this is not uniformly true: overlooking the heterogeneity can lead to significantly suboptimal control strategies. In these cases, the heterogeneous tools and perspective are crucial to optimise bioproduction.

Revisiting moment-closure methods with heterogeneous multiscale population models

Stochastic chemical kinetics at the single-cell level give rise to heterogeneous populations of cells even when all individuals are genetically identical. This heterogeneity can lead to nonuniform behaviour within populations, including different growth characteristics, cell-fate dynamics, and response to stimuli. Ultimately, these diverse behaviours lead to intricate population dynamics that are inherently multiscale: the population composition evolves based on population-level processes that interact with stochastically distributed single-cell states. Therefore, descriptions that account for this heterogeneity are essential to accurately model and control chemical processes. However, for real-world systems such models are computationally expensive to simulate, which can make optimisation problems, such as optimal control or parameter inference, prohibitively challenging. Here, we consider a class of multiscale population models that incorporate population-level mechanisms while remaining faithful to the underlying stochasticity at the single-cell level and the interplay between these two scales. To address the complexity, we study an order-reduction approximations based on the distribution moments. Since previous moment-closure work has focused on the single-cell kinetics, extending these techniques to populations models prompts us to revisit old observations as well as tackle new challenges. In this extended multiscale context, we encounter the previously established observation that the simplest closure techniques can lead to non-physical system trajectories. Despite their poor performance in some systems, we provide an example where these simple closures outperform more sophisticated closure methods in accurately, efficiently, and robustly solving the problem of optimal control of bioproduction in a microbial consortium model.

Beyond the chemical master equation: Stochastic chemical kinetics coupled with auxiliary processes

The chemical master equation and its continuum approximations are indispensable tools in the modeling of chemical reaction networks. These are routinely used to capture complex nonlinear phenomena such as multimodality as well as transient events such as first-passage times, that accurately characterise a plethora of biological and chemical processes. However, some mechanisms, such as heterogeneous cellular growth or phenotypic selection at the population level, cannot be represented by the master equation and thus have been tackled separately. In this work, we propose a unifying framework that augments the chemical master equation to capture such auxiliary dynamics, and we develop and analyse a numerical solver that accurately simulates the system dynamics. We showcase these contributions by casting a diverse array of examples from the literature within this framework and applying the solver to both match and extend previous studies. Analytical calculations performed for each example validate our numerical results and benchmark the solver implementation.

Populations of genetically identical cells tend to exhibit remarkable variability. This seemingly counter-intuitive observation has broad and fascinating implications, and has thus been a focal point of biological modeling. Many important processes act on this cellular heterogeneity at the population level, leading to an intricate coupling between the single-cell and the population-level dynamics. For example, selection pressures or growth rates may depend crucially on the expression of a particular gene (or gene family). Classical single-cell modeling approaches, such as the chemical master equation, can accurately describe the mechanisms driving cellular noise, however, they cannot encapsulate how the aforementioned auxiliary processes affect the population composition. In this work, we propose a unifying framework that extends the classical chemical master equation to faithfully capture the single-cell variability alongside the population-level evolution. We develop, analyse, and showcase an open-source numerical tool to simulate the dynamics of such populations in time. The tool is designed for straightforward use by a non-technical audience: a high-level description of the underlying chemical and population-level processes suffices to simulate complex system dynamics. Simultaneously, we retain high customisability of the underlying mathematical representation for the more advanced user. Ultimately, the unifying framework and the associated computational tool open new horizons in the study of how fundamental microscopic dynamics give rise to complex macroscopic phenomena.

Analytic solutions for stochastic hybrid models of gene regulatory networks

Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have become a popular alternative. The evolution of the corresponding probability density functions is given by a coupled system of hyperbolic PDEs. This system has Markovian nature but its hyperbolic structure makes it difficult to apply standard functional analytical methods. We are able to prove convergence towards the stationary solution and determine such equilibrium explicitly by combining abstract methods from the theory of positive operators and elementary ideas from potential analysis.

Optogenetic Control Reveals Differential Promoter Interpretation of Transcription Factor Nuclear Translocation Dynamics

Gene expression is thought to be affected not only by the concentration of transcription factors (TFs) but also the dynamics of their nuclear translocation. Testing this hypothesis requires direct control of TF dynamics. Here, we engineer CLASP, an optogenetic tool for rapid and tunable translocation of a TF of interest. Using CLASP fused to Crz1, we observe that, for the same integrated concentration of nuclear TF over time, changing input dynamics changes target gene expression: pulsatile inputs yield higher expression than continuous inputs, or vice versa, depending on the target gene. Computational modeling reveals that a dose-response saturating at low TF input can yield higher gene expression for pulsatile versus continuous input, and that multi-state promoter activation can yield the opposite behavior. Our integrated tool development and modeling approach characterize promoter responses to Crz1 nuclear translocation dynamics, extracting quantitative features that may help explain the differential expression of target genes.

CLASP is a modular optogenetic strategy to control the nuclear localization of transcription factors (TFs) and elicit gene expression from their cognate promoters. CLASP control of Crz1 nuclear localization, coupled with computational modeling, revealed how promoters can differentially decode dynamic transcription factor signals. The integrated strategy of CLASP development and modeling presents a generalized approach to causally investigate the transcriptional consequences of dynamic TF nuclear shuttling.

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach

Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Autoregulatory feedback loops are one of the most common network motifs. A wide variety of stochastic models have been constructed to understand how the fluctuations in protein numbers in these loops are influenced by the kinetic parameters of the main biochemical steps. These models differ according to 1) which subcellular processes are explicitly modeled, 2) the modeling methodology employed (discrete, continuous, or hybrid), and 3) whether they can be analytically solved for the steady-state distribution of protein numbers. We discuss the assumptions and properties of the main models in the literature, summarize our current understanding of the relationship between them, and highlight some of the insights gained through modeling.

Single-cell stochastic gene expression kinetics with coupled positive-plus-negative feedback

Here we investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. A triphasic stochastic bifurcation is observed upon increasing the ratio of the positive and negative feedback strengths, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two types of macroscopic limits of the discrete chemical master equation model: the Kurtz limit applies to proteins with large burst frequencies and the Lévy limit applies to proteins with large burst sizes. We derive the analytic steady-state distributions of the protein abundance in a coupled gene circuit for both the discrete model and its two macroscopic limits, generalizing the results obtained by Liu et al. [Chaos 26, 043108 (2016)CHAOEH1054-150010.1063/1.4947202]. We also obtain the analytic time-dependent protein distribution for the classical Friedman-Cai-Xie random bursting model [Friedman, Cai, and Xie, Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302]. Our analytic results are further applied to study the structure of gene expression noise in a coupled gene circuit, and a complete decomposition of noise in terms of five different biophysical origins is provided.

Exact and efficient hybrid Monte Carlo algorithm for accelerated Bayesian inference of gene expression models from snapshots of single-cell transcripts

Single cells exhibit a significant amount of variability in transcript levels, which arises from slow, stochastic transitions between gene expression states. Elucidating the nature of these states and understanding how transition rates are affected by different regulatory mechanisms require state-of-the-art methods to infer underlying models of gene expression from single cell data. A Bayesian approach to statistical inference is the most suitable method for model selection and uncertainty quantification of kinetic parameters using small data sets. However, this approach is impractical because current algorithms are too slow to handle typical models of gene expression. To solve this problem, we first show that time-dependent mRNA distributions of discrete-state models of gene expression are dynamic Poisson mixtures, whose mixing kernels are characterized by a piecewise deterministic Markov process. We combined this analytical result with a kinetic Monte Carlo algorithm to create a hybrid numerical method that accelerates the calculation of time-dependent mRNA distributions by 1000-fold compared to current methods. We then integrated the hybrid algorithm into an existing Monte Carlo sampler to estimate the Bayesian posterior distribution of many different, competing models in a reasonable amount of time. We demonstrate that kinetic parameters can be reasonably constrained for modestly sampled data sets if the model is known a priori. If there are many competing models, Bayesian evidence can rigorously quantify the likelihood of a model relative to other models from the data. We demonstrate that Bayesian evidence selects the true model and outperforms approximate metrics typically used for model selection.

Scaling methods for accelerating kinetic Monte Carlo simulations of chemical reaction networks

Various kinetic Monte Carlo algorithms become inefficient when some of the population sizes in a system are large, which gives rise to a large number of reaction events per unit time. Here, we present a new acceleration algorithm based on adaptive and heterogeneous scaling of reaction rates and stoichiometric coefficients. The algorithm is conceptually related to the commonly used idea of accelerating a stochastic simulation by considering a subvolume λΩ (0 < λ < 1) within a system of interest, which reduces the number of reaction events per unit time occurring in a simulation by a factor 1/λ at the cost of greater error in unbiased estimates of first moments and biased overestimates of second moments. Our new approach offers two unique benefits. First, scaling is adaptive and heterogeneous, which eliminates the pitfall of overaggressive scaling. Second, there is no need for an a priori classification of populations as discrete or continuous (as in a hybrid method), which is problematic when discreteness of a chemical species changes during a simulation. The method requires specification of only a single algorithmic parameter, N_c, a global critical population size above which populations are effectively scaled down to increase simulation efficiency. The method, which we term partial scaling, is implemented in the open-source BioNetGen software package. We demonstrate that partial scaling can significantly accelerate simulations without significant loss of accuracy for several published models of biological systems. These models characterize activation of the mitogen-activated protein kinase ERK, prion protein aggregation, and T-cell receptor signaling.

Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes

Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime. We illustrate the utility of the PDMP on a non-trivial dynamical system by analysing the properties of a titration-based oscillator in the non-adiabatic limit. We first show how to transform the underlying chemical master equation into a PDMP where the slow transitions between promoter states are stochastic, but whose rates depend upon the faster deterministic dynamics of the transcription factors regulated by these promoters. We show that the PDMP accurately describes the observed periods of stochastic cycles in activator and repressor-based titration oscillators. We then generalize our PDMP analysis to more complicated versions of titration-based oscillators to explain how multiple binding sites lengthen the period and improve coherence. Last, we show how noise-induced oscillation previously observed in a titration-based oscillator arises from non-adiabatic and discrete binding events at the promoter site.

Generalizing Gillespie's Direct Method to Enable Network-Free Simulations

Gillespie's direct method for stochastic simulation of chemical kinetics is a staple of computational systems biology research. However, the algorithm requires explicit enumeration of all reactions and all chemical species that may arise in the system. In many cases, this is not feasible due to the combinatorial explosion of reactions and species in biological networks. Rule-based modeling frameworks provide a way to exactly represent networks containing such combinatorial complexity, and generalizations of Gillespie's direct method have been developed as simulation engines for rule-based modeling languages. Here, we provide both a high-level description of the algorithms underlying the simulation engines, termed network-free simulation algorithms, and how they have been applied in systems biology research. We also define a generic rule-based modeling framework and describe a number of technical details required for adapting Gillespie's direct method for network-free simulation. Finally, we briefly discuss potential avenues for advancing network-free simulation and the role they continue to play in modeling dynamical systems in biology.

A stochastic and dynamical view of pluripotency in mouse embryonic stem cells

Pluripotent embryonic stem cells are of paramount importance for biomedical sciences because of their innate ability for self-renewal and differentiation into all major cell lines. The fateful decision to exit or remain in the pluripotent state is regulated by complex genetic regulatory networks. The rapid growth of single-cell sequencing data has greatly stimulated applications of statistical and machine learning methods for inferring topologies of pluripotency regulating genetic networks. The inferred network topologies, however, often only encode Boolean information while remaining silent about the roles of dynamics and molecular stochasticity inherent in gene expression. Herein we develop a framework for systematically extending Boolean-level network topologies into higher resolution models of networks which explicitly account for the promoter architectures and gene state switching dynamics. We show the framework to be useful for disentangling the various contributions that gene switching, external signaling, and network topology make to the global heterogeneity and dynamics of transcription factor populations. We find the pluripotent state of the network to be a steady state which is robust to global variations of gene switching rates which we argue are a good proxy for epigenetic states of individual promoters. The temporal dynamics of exiting the pluripotent state, on the other hand, is significantly influenced by the rates of genetic switching which makes cells more responsive to changes in extracellular signals.

Emergent Lévy behavior in single-cell stochastic gene expression

Single-cell gene expression is inherently stochastic; its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter tends to infinity. We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the Lévy limit provides a theoretical foundation for an empirical equation proposed in N. Friedman et al., Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302. Furthermore, we clarify the biochemical implications and ranges of applicability for various macroscopic limits and calculate a comprehensive analytic expression for the protein concentration distribution in autoregulatory gene networks. The relationship between our work and modern population genetics is discussed.

Simplification of Markov chains with infinite state space and the mathematical theory of random gene expression bursts

Here we develop an effective approach to simplify two-time-scale Markov chains with infinite state spaces by removal of states with fast leaving rates, which improves the simplification method of finite Markov chains. We introduce the concept of fast transition paths and show that the effective transitions of the reduced chain can be represented as the superposition of the direct transitions and the indirect transitions via all the fast transition paths. Furthermore, we apply our simplification approach to the standard Markov model of single-cell stochastic gene expression and provide a mathematical theory of random gene expression bursts. We give the precise mathematical conditions for the bursting kinetics of both mRNAs and proteins. It turns out that random bursts exactly correspond to the fast transition paths of the Markov model. This helps us gain a better understanding of the physics behind the bursting kinetics as an emergent behavior from the fundamental multiscale biochemical reaction kinetics of stochastic gene expression.

Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise

This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0). While in the multiplicative noise case, non-Gaussianity index value α = 1 is a separating point or line for both the MFET and the FEP. We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in "regulating" the likelihood of gene transcription by tuning asymmetric stable Lévy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.

Efficient and flexible implementation of Langevin simulation for gene burst production

Gene expression involves bursts of production of both mRNA and protein, and the fluctuations in their number are increased due to such bursts. The Langevin equation is an efficient and versatile means to simulate such number fluctuation. However, how to include these mRNA and protein bursts in the Langevin equation is not intuitively clear. In this work, we estimated the variance in burst production from a general gene expression model and introduced such variation in the Langevin equation. Our approach offers different Langevin expressions for either or both transcriptional and translational bursts considered and saves computer time by including many production events at once in a short burst time. The errors can be controlled to be rather precise (<2%) for the mean and <10% for the standard deviation of the steady-state distribution. Our scheme allows for high-quality stochastic simulations with the Langevin equation for gene expression, which is useful in analysis of biological networks.

Stochastic fluctuations can reveal the feedback signs of gene regulatory networks at the single-molecule level

Understanding the relationship between spontaneous stochastic fluctuations and the topology of the underlying gene regulatory network is of fundamental importance for the study of single-cell stochastic gene expression. Here by solving the analytical steady-state distribution of the protein copy number in a general kinetic model of stochastic gene expression with nonlinear feedback regulation, we reveal the relationship between stochastic fluctuations and feedback topology at the single-molecule level, which provides novel insights into how and to what extent a feedback loop can enhance or suppress molecular fluctuations. Based on such relationship, we also develop an effective method to extract the topological information of a gene regulatory network from single-cell gene expression data. The theory is demonstrated by numerical simulations and, more importantly, validated quantitatively by single-cell data analysis of a synthetic gene circuit integrated in human kidney cells.

Bursting noise in gene expression dynamics: linking microscopic and mesoscopic models

The dynamics of short-lived mRNA results in bursts of protein production in gene regulatory networks. We investigate the propagation of bursting noise between different levels of mathematical modelling and demonstrate that conventional approaches based on diffusion approximations can fail to capture bursting noise. An alternative coarse-grained model, the so-called piecewise deterministic Markov process (PDMP), is seen to outperform the diffusion approximation in biologically relevant parameter regimes. We provide a systematic embedding of the PDMP model into the landscape of existing approaches, and we present analytical methods to calculate its stationary distribution and switching frequencies.

Lévy-noise-induced transport in a rough triple-well potential

Rough energy landscape and noisy environment are two common features in many subjects, such as protein folding. Due to the wide findings of bursting or spiking phenomenon in biology science, small diffusions mixing large jumps are adopted to model the noisy environment that can be properly described by Lévy noise. We combine the Lévy noise with the rough energy landscape, modeled by a potential function superimposed by a fast oscillating function, and study the transport of a particle in a rough triple-well potential excited by Lévy noise, rather than only small perturbations. The probabilities of a particle staying in the middle well are considered under different amplitudes of roughness to find out how roughness affects the steady-state probability density function. Variations in the mean first passage time from the middle well to the right well have been investigated with respect to Lévy parameters and amplitudes of the roughness. In addition, we have examined the influences of roughness on the splitting probabilities of the first escape from the middle well. We uncover that the roughness can enhance significantly the first escape of a particle from the middle well, especially for different skewness parameters, but weak differences are found for stability index and noise intensity on the probabilities a particle staying in the middle well and splitting probability to the right.

Gene expression noise is affected differentially by feedback in burst frequency and burst size

Inside individual cells, expression of genes is stochastic across organisms ranging from bacterial to human cells. A ubiquitous feature of stochastic expression is burst-like synthesis of gene products, which drives considerable intercellular variability in protein levels across an isogenic cell population. One common mechanism by which cells control such stochasticity is negative feedback regulation, where a protein inhibits its own synthesis. For a single gene that is expressed in bursts, negative feedback can affect the burst frequency or the burst size. In order to compare these feedback types, we study a piecewise deterministic model for gene expression of a self-regulating gene. Mathematically tractable steady-state protein distributions are derived and used to compare the noise suppression abilities of the two feedbacks. Results show that in the low noise regime, both feedbacks are similar in term of their noise buffering abilities. Intriguingly, feedback in burst size outperforms the feedback in burst frequency in the high noise regime. Finally, we discuss various regulatory strategies by which cells implement feedback to control burst sizes of expressed proteins at the level of single cells.

The Switch in a Genetic Toggle System with Lévy Noise

A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the system induced by Gaussian noise has been intensively investigated, but Gaussian noise cannot incorporate large bursts typically occurring in real experiments. This paper aims to examine effects of variations from one protein imposed by a non-Gaussian Lévy noise, which is able to describe even large jumps, on the coherent switch and the on/off switch via the steady-state probability density, the joint steady-state probability density, and the mean first passage time. We find that a large burst of one protein due to the Lévy noises can induce coherent switches even with small noise intensities in contrast to the Gaussian case which requires large intensities for this. The influences of the stability index, skewness parameter and noise intensity on the on/off switch are analyzed, leading to an adjustment of the concentrations of both proteins and a decision which stable point to stay most. The mean first passage times show complex effects under Lévy noise, especially the stability index and skewness parameter. Our results also imply that the presence of non-Gaussian Lévy noises has fundamentally changed the escape mechanism in such a system compared with Gaussian noise.