Linear Noise Approximation Research Articles

Analysis of a detailed multi-stage model of stochastic gene expression using queueing theory and model reduction

We introduce a biologically detailed, stochastic model of gene expression describing the multiple rate-limiting steps of transcription, nuclear pre-mRNA processing, nuclear mRNA export, cytoplasmic mRNA degradation and translation of mRNA into protein. The processes in sub-cellular compartments are described by an arbitrary number of processing stages, thus accounting for a significantly finer molecular description of gene expression than conventional models such as the telegraph, two-stage and three-stage models of gene expression. We use two distinct tools, queueing theory and model reduction using the slow-scale linear-noise approximation, to derive exact or approximate analytic expressions for the moments or distributions of nuclear mRNA, cytoplasmic mRNA and protein fluctuations, as well as lower bounds for their Fano factors in steady-state conditions. We use these to study the phase diagram of the stochastic model; in particular we derive parametric conditions determining three types of transitions in the properties of mRNA fluctuations: from sub-Poissonian to super-Poissonian noise, from high noise in the nucleus to high noise in the cytoplasm, and from a monotonic increase to a monotonic decrease of the Fano factor with the number of processing stages. In contrast, protein fluctuations are always super-Poissonian and show weak dependence on the number of mRNA processing stages. Our results delineate the region of parameter space where conventional models give qualitatively incorrect results and provide insight into how the number of processing stages, e.g. the number of rate-limiting steps in initiation, splicing and mRNA degradation, shape stochastic gene expression by modulation of molecular memory.

Noise propagation and amplification in microbial communities with different resistance mechanisms

Antimicrobial resistance is a growing concern in the field of microbiology. In a microbial community, the Susceptible Subpopulations (SSs) and the Non-genetically Resistant Subpopulations (NRSs) can switch between each other, and NRSs can mutate to Genetically Resistant Subpopulations (GRSs). In order to quantitatively describe the relationship between noise and signal amplification in the subpopulations, a mathematical model for three subpopulations is proposed in this paper. The gain fluctuation relation is obtained by linear noise approximation of the master equation. Then, through the simulation of these theoretical results, the following conclusions are found. First, the indirect effects of the interaction between subpopulations are much greater than the direct effects, which may be one of the reasons that the infection is difficult to cure. This is consistent with the findings of Zhou et al. and indicates the reliability of our new model. Second, the intrinsic net growth rate of SSs can be chosen as the adjustment parameter, which can effectively exploit or avoid the strong correlation between SSs and the two resistant subpopulations, including NRSs and GRSs. Third, unlike unidirectional cascades, additional intrinsic noise in GRSs can propagate to NRSs. Similarly, although there is no direct interaction between SSs and GRSs, additional intrinsic noise in GRSs can propagate to SSs, and vice versa. Finally, both total noise and adjusted noise exhibit similar amplification properties in this population. It is worth noting that the noise is consistently amplified from NRSs to GRSs. These findings may provide new insights into antimicrobial therapy.

Accelerating Bayesian inference for stochastic epidemic models using incidence data

We consider the case of performing Bayesian inference for stochastic epidemic compartment models, using incomplete time course data consisting of incidence counts that are either the number of new infections or removals in time intervals of fixed length. We eschew the most natural Markov jump process representation for reasons of computational efficiency, and focus on a stochastic differential equation representation. This is further approximated to give a tractable Gaussian process, that is, the linear noise approximation (LNA). Unless the observation model linking the LNA to data is both linear and Gaussian, the observed data likelihood remains intractable. It is in this setting that we consider two approaches for marginalising over the latent process: a correlated pseudo-marginal method and analytic marginalisation via a Gaussian approximation of the observation model. We compare and contrast these approaches using synthetic data before applying the best performing method to real data consisting of removal incidence of oak processionary moth nests in Richmond Park, London. Our approach further allows comparison between various competing compartment models.

Fitting stochastic epidemic models to gene genealogies using linear noise approximation.

Phylodynamics is a set of population genetics tools that aim at reconstructing demographic history of a population based on molecular sequences of individuals sampled from the population of interest. One important task in phylodynamics is to estimate changes in (effective) population size. When applied to infectious disease sequences such estimation of population size trajectories can provide information about changes in the number of infections. To model changes in the number of infected individuals, current phylodynamic methods use non-parametric approaches (e.g., Bayesian curve-fitting based on change-point models or Gaussian process priors), parametric approaches (e.g., based on differential equations), and stochastic modeling in conjunction with likelihood-free Bayesian methods. The first class of methods yields results that are hard to interpret epidemiologically. The second class of methods provides estimates of important epidemiological parameters, such as infection and removal/recovery rates, but ignores variation in the dynamics of infectious disease spread. The third class of methods is the most advantageous statistically, but relies on computationally intensive particle filtering techniques that limits its applications. We propose a Bayesian model that combines phylodynamic inference and stochastic epidemic models, and achieves computational tractability by using a linear noise approximation (LNA) - a technique that allows us to approximate probability densities of stochastic epidemic model trajectories. LNA opens the door for using modern Markov chain Monte Carlo tools to approximate the joint posterior distribution of the disease transmission parameters and of high dimensional vectors describing unobserved changes in the stochastic epidemic model compartment sizes (e.g., numbers of infectious and susceptible individuals). In a simulation study, we show that our method can successfully recover parameters of stochastic epidemic models. We apply our estimation technique to Ebola genealogies estimated using viral genetic data from the 2014 epidemic in Sierra Leone and Liberia.

Localization of Noise in Biochemical Networks.

Noise, or uncertainty in biochemical networks, has become an important aspect of many biological problems. Noise can arise and propagate from external factors and probabilistic chemical reactions occurring in small cellular compartments. For species survival, it is important to regulate such uncertainties in executing vital cell functions. Regulated noise can improve adaptability, whereas uncontrolled noise can cause diseases. Simulation can provide a detailed analysis of uncertainties, but parameters such as rate constants and initial conditions are usually unknown. A general understanding of noise dynamics from the perspective of network structure is highly desirable. In this study, we extended the previously developed law of localization for characterizing noise in terms of (co)variances and developed noise localization theory. With linear noise approximation, we can expand a biochemical network into an extended set of differential equations representing a fictitious network for pseudo-components consisting of variances and covariances, together with chemical species. Through localization analysis, perturbation responses at the steady state of pseudo-components can be summarized into a sensitivity matrix that only requires knowledge of network topology. Our work allows identification of buffering structures at the level of species, variances, and covariances and can provide insights into noise flow under non-steady-state conditions in the form of a pseudo-chemical reaction. We tested noise localization in various systems, and here we discuss its implications and potential applications. Results show that this theory is potentially applicable in discriminating models, scanning network topologies with interesting noise behavior, and designing and perturbing networks with the desired response.

Stochastic enzyme kinetics and the quasi-steady-state reductions: Application of the slow scale linear noise approximation à la Fenichel.

The linear noise approximation models the random fluctuations from the mean-field model of a chemical reaction that unfolds near the thermodynamic limit. Specifically, the fluctuations obey a linear Langevin equation up to order [Formula: see text], where [Formula: see text] is the size of the chemical system (usually the volume). In the presence of disparate timescales, the linear noise approximation admits a quasi-steady-state reduction referred to as the slow scale linear noise approximation (ssLNA). Curiously, the ssLNAs reported in the literature are slightly different. The differences in the reported ssLNAs lie at the mathematical heart of the derivation. In this work, we derive the ssLNA directly from geometric singular perturbation theory and explain the origin of the different ssLNAs in the literature. Moreover, we discuss the loss of normal hyperbolicity and we extend the ssLNA derived from geometric singular perturbation theory to a non-classical singularly perturbed problem. In so doing, we disprove a commonly-accepted qualifier for the validity of stochastic quasi-steady-state approximation of the Michaelis -Menten reaction mechanism.

Generalized fluctuation-dissipation theorem for non-Markovian reaction networks.

Intracellular biochemical networks often display large fluctuations in the molecule numbers or the concentrations of reactive species, making molecular approaches necessary for system descriptions. For Markovian reaction networks, the fluctuation-dissipation theorem (FDT) has been well established and extensively used in fast evaluation of fluctuations in reactive species. For non-Markovian reaction networks, however, the similar FDT has not been established so far. Here, we present a generalized FDT (gFDT) for a large class of non-Markovian reaction networks where general intrinsic-event waiting-time distributions account for the effect of intrinsic noise and general stochastic reaction delays represent the impact of extrinsic noise from environmental perturbations. The starting point is a generalized chemical master equation (gCME), which describes the probabilistic behavior of an equivalent Markovian reaction network and identifies the structure of the original non-Markovian reaction network in terms of stoichiometries and effective transition rates (extensions of common reaction propensity functions). From this formulation follows directly the solution of the linear noise approximation of the stationary gCME for all the components in the non-Markovian reaction network. While the gFDT can quickly trace noisy sources in non-Markovian reaction networks, example analysis verifies its effectiveness.

Inference for stochastic kinetic models from multiple data sources for joint estimation of infection dynamics from aggregate reports and virological data

Before the current pandemic, influenza and respiratory syncytial virus (RSV) were the leading etiological agents of seasonal acute respiratory infections (ARI) around the world. In this setting, medical doctors typically based the diagnosis of ARI on patients’ symptoms alone and did not routinely conduct virological tests necessary to identify individual viruses, limiting the ability to study the interaction between multiple pathogens and to make public health recommendations. We consider a stochastic kinetic model (SKM) for two interacting ARI pathogens circulating in a large population and an empirically-motivated background process for infections with other pathogens causing similar symptoms. An extended marginal sampling approach, based on the linear noise approximation to the SKM, integrates multiple data sources and additional model components. We infer the parameters defining the pathogens’ dynamics and interaction within a Bayesian model and explore the posterior trajectories of infections for each illness based on aggregate infection reports from six epidemic seasons collected by the state health department and a subset of virological tests from a sentinel program at a general hospital in San Luis Potosí, México. We interpret the results and make recommendations for future data collection strategies.

What Is the Trait d’Union between Retroactivity and Molecular Communication Performance Limits?

Information exchange is a critical process in all communication systems, including biological ones. Retroactivity describes the load that downstream modules apply to their upstream systems in biological circuits. The motivation behind this work is that of integrating retroactivity, a concept proper of biochemical circuits, with the metrics defined in Information Theory and Digital Communications. This paper focuses on studying the impact of retroactivity on different biological signaling system models, which present analogies with well-known telecommunication systems. The mathematical analysis is performed both in the high and low molecular counts regime, by mean of the Chemical Master Equation and the Linear Noise Approximation, respectively. The main goal of this work is to provide analytical tools to maximize the reliable information exchange across different biomolecular circuit models. Results highlight how, in general, retroactivity harms communication performance. This negative effect can be mitigated by adding to the signaling circuit an independent upstream system that connects with the same pool of downstream circuits.

Stationary and Time-Dependent Molecular Distributions in Slow-Fast Feedback Circuits

Biochemical reaction systems, in particular those that govern the expression of genes in single cells, involve discrete numbers of molecules and are inherently stochastic. This study concerns reaction systems with two molecular species and an arbitrary number of reactions, the rates of which conform to a scaling of the classical type in the first species (but not the second). For limiting values of the scaling parameter, the first species is highly abundant, evolves slowly, and buffers the relatively fast fluctuation of the second, scarce, species. The scale separation facilitates the construction of asymptotic approximations to the molecular distributions: the large-time behavior is described by a WKB approximation, which is further resolved into a tractable mixture of metastable modes; the earlier-time dynamics are described by quasi-stationary and linear noise approximations. The paper presents the theory and the algebraic recipes that are required to implement it. The framework is applied on two reaction systems with positive feedback: a delayed feedback circuit and a gene autoregulation model with cooperative binding of the protein to the promoter. For the former, the approximation scheme results into a mixture of Gaussian/Poisson modes for the inactive/active protein distribution. For the latter, the analysis elucidates the effects of bursty gene expression, promoter responsivity, and protein sequestration on bimodality of protein distributions.

Parameter inference for stochastic biochemical models from perturbation experiments parallelised at the single cell level.

Understanding and characterising biochemical processes inside single cells requires experimental platforms that allow one to perturb and observe the dynamics of such processes as well as computational methods to build and parameterise models from the collected data. Recent progress with experimental platforms and optogenetics has made it possible to expose each cell in an experiment to an individualised input and automatically record cellular responses over days with fine time resolution. However, methods to infer parameters of stochastic kinetic models from single-cell longitudinal data have generally been developed under the assumption that experimental data is sparse and that responses of cells to at most a few different input perturbations can be observed. Here, we investigate and compare different approaches for calculating parameter likelihoods of single-cell longitudinal data based on approximations of the chemical master equation (CME) with a particular focus on coupling the linear noise approximation (LNA) or moment closure methods to a Kalman filter. We show that, as long as cells are measured sufficiently frequently, coupling the LNA to a Kalman filter allows one to accurately approximate likelihoods and to infer model parameters from data even in cases where the LNA provides poor approximations of the CME. Furthermore, the computational cost of filtering-based iterative likelihood evaluation scales advantageously in the number of measurement times and different input perturbations and is thus ideally suited for data obtained from modern experimental platforms. To demonstrate the practical usefulness of these results, we perform an experiment in which single cells, equipped with an optogenetic gene expression system, are exposed to various different light-input sequences and measured at several hundred time points and use parameter inference based on iterative likelihood evaluation to parameterise a stochastic model of the system.

Criticality enhances the multilevel reliability of stimulus responses in cortical neural networks.

Cortical neural networks exhibit high internal variability in spontaneous dynamic activities and they can robustly and reliably respond to external stimuli with multilevel features–from microscopic irregular spiking of neurons to macroscopic oscillatory local field potential. A comprehensive study integrating these multilevel features in spontaneous and stimulus–evoked dynamics with seemingly distinct mechanisms is still lacking. Here, we study the stimulus–response dynamics of biologically plausible excitation–inhibition (E–I) balanced networks. We confirm that networks around critical synchronous transition states can maintain strong internal variability but are sensitive to external stimuli. In this dynamical region, applying a stimulus to the network can reduce the trial-to-trial variability and shift the network oscillatory frequency while preserving the dynamical criticality. These multilevel features widely observed in different experiments cannot simultaneously occur in non-critical dynamical states. Furthermore, the dynamical mechanisms underlying these multilevel features are revealed using a semi-analytical mean-field theory that derives the macroscopic network field equations from the microscopic neuronal networks, enabling the analysis by nonlinear dynamics theory and linear noise approximation. The generic dynamical principle revealed here contributes to a more integrative understanding of neural systems and brain functions and incorporates multimodal and multilevel experimental observations. The E–I balanced neural network in combination with the effective mean-field theory can serve as a mechanistic modeling framework to study the multilevel neural dynamics underlying neural information and cognitive processes.

Modeling noise propagation in time-delayed auto-inhibitory genetic circuits

The abundance of specific protein molecules in genetically identical cell populations exposed to the same external environment can show remarkable cell-to-cell variations as biochemical reactions are inherently stochastic and occur with low numbers of molecular copies. Such variations in gene products are commonly known as gene expression noise. One of the mechanisms for cells to reduce such noise is auto-regulatory negative feedback (auto-inhibition), commonly found across organisms. This auto-inhibition is subjected to unavoidable time-delays associated with transcriptional and translational processes. Sufficient time-delays and strong auto-inhibition can generate sustained oscillations in gene products, which is a common mechanism for precise timekeeping in many biomolecular clocks. While the importance of time-delays in the generation of oscillations is well appreciated, its role in stochastic dynamics is not well understood in the absence of sustained oscillations. Here, we investigate the interplay between the feedback strength and the time-delay to study the noise propagation in the non-oscillatory regime using linear stability analysis, the linear noise approximation, and stochastic simulations. From a simple auto-regulatory model with one protein species (no delay), we systematically introduce one-step and two-step time-delays by incorporating intermediate dynamics with additional second and third species, respectively. Interestingly, the negative feedback in the presence of time-delay can show counterintuitive noise behavior to our common perception about its role as a noise buffer.

Bayesian inference in epidemics: linear noise analysis.

This paper offers a qualitative insight into the convergence of Bayesian parameter inference in a setup which mimics the modeling of the spread of a disease with associated disease measurements. Specifically, we are interested in the Bayesian model's convergence with increasing amounts of data under measurement limitations. Depending on how weakly informative the disease measurements are, we offer a kind of 'best case' as well as a 'worst case' analysis where, in the former case, we assume that the prevalence is directly accessible, while in the latter that only a binary signal corresponding to a prevalence detection threshold is available. Both cases are studied under an assumed so-called linear noise approximation as to the true dynamics. Numerical experiments test the sharpness of our results when confronted with more realistic situations for which analytical results are unavailable.

Effects of bidirectional phenotype switching on signal noise in a bacterial community.

Cells can sense and process various signals. Noise is inevitable in the cell signaling system. In a bacterial community, the mutual conversion between normal cells and persistent cells forms a bidirectional phenotype switching cascade, in which either one can be used as an upstream signal and the other as a downstream signal. In order to quantitatively describe the relationship between noise and signal amplification of each phenotype, the gain-fluctuation relationship is obtained by using the linear noise approximation of the master equation. Through the simulation of these theoretical formulas, it is found that the bidirectional phenotype switching can directly generate interconversion noise which is usually very small and almost negligible. In particular, the bidirectional phenotype switching can provide a global fluctuating environment, which will not only affect the values of noise and covariance, but also generate additional intrinsic noise. The additional intrinsic noise in each phenotype is the main part of the total noise and can be transmitted to the other phenotype. The transmitted noise is also a powerful supplement to the total noise. Therefore, the indirect impact of bidirectional phenotype switching is far greater than its direct impact, which may be one of the reasons why chronic infections caused by persistent cells are refractory to treat.

A Language for Modeling and Optimizing Experimental Biological Protocols

Automation is becoming ubiquitous in all laboratory activities, moving towards precisely defined and codified laboratory protocols. However, the integration between laboratory protocols and mathematical models is still lacking. Models describe physical processes, while protocols define the steps carried out during an experiment: neither cover the domain of the other, although they both attempt to characterize the same phenomena. We should ideally start from an integrated description of both the model and the steps carried out to test it, to concurrently analyze uncertainties in model parameters, equipment tolerances, and data collection. To this end, we present a language to model and optimize experimental biochemical protocols that facilitates such an integrated description, and that can be combined with experimental data. We provide probabilistic semantics for our language in terms of Gaussian processes (GPs) based on the linear noise approximation (LNA) that formally characterizes the uncertainties in the data collection, the underlying model, and the protocol operations. In a set of case studies, we illustrate how the resulting framework allows for automated analysis and optimization of experimental protocols, including Gibson assembly protocols.

Information transmission in a two-step cascade: interplay of activation and repression.

We present an information-theoretic formalism to study signal transduction in four architectural variants of a model two-step cascade with increasing input population. Our results categorize these four types into two classes depending upon the effect of activation and repression on mutual information, net synergy, and signal-to-noise ratio. Using the Gaussian framework and linear noise approximation, we derive the analytic expressions for these metrics to establish their underlying relationships in terms of the biochemical parameters. We also verify our approximations through stochastic simulations.

A linear noise approximation for stochastic epidemic models fit to partially observed incidence counts.

Stochastic epidemic models (SEMs) fit to incidence data are critical to elucidating outbreak dynamics, shaping response strategies, and preparing for future epidemics. SEMs typically represent counts of individuals in discrete infection states using Markov jump processes (MJPs), but are computationally challenging as imperfect surveillance, lack of subject-level information, and temporal coarseness of the data obscure the true epidemic. Analytic integration over the latent epidemic process is impossible, and integration via Markov chain Monte Carlo (MCMC) is cumbersome due to the dimensionality and discreteness of the latent state space. Simulation-based computational approaches can address the intractability of the MJP likelihood, but are numerically fragile and prohibitively expensive for complex models. A linear noise approximation (LNA) that approximates the MJP transition density with a Gaussian density has been explored for analyzing prevalence data in large-population settings, but requires modification for analyzing incidence counts without assuming that the data are normally distributed. We demonstrate how to reparameterize SEMs to appropriately analyze incidence data, and fold the LNA into a data augmentation MCMC framework that outperforms deterministic methods, statistically, and simulation-based methods, computationally. Our framework is computationally robust when the model dynamics are complex and applies to a broad class of SEMs. We evaluate our method in simulations that reflect Ebola, influenza, and SARS-CoV-2 dynamics, and apply our method to national surveillance counts from the 2013-2015 West Africa Ebolaoutbreak.

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations.

The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise. We find that the solution of the chemical master equation-including static extrinsic noise-exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a novel condition that generalizes concentration homeostasis in deterministic systems to higher order moments and distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks. When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise models.

MicroRNA Based Feedforward Control of Intrinsic Gene Expression Noise.

Intrinsic noise, which arises in gene expression at low copy numbers, can be controlled by diverse regulatory motifs, including feedforward loops. Here, we study an example of a feedforward control system based on the interaction between an mRNA molecule and an antagonistic microRNA molecule encoded by the same gene, aiming to quantify the variability (or noise) in molecular copy numbers. Using linear noise approximation, we show that the mRNA noise is sub-Poissonian in case of non-bursty transcription, and exhibits a nonmonotonic response both to the species natural lifetime ratio and to the strength of the antagonistic interaction. Additionally, we use the Chemical Reaction Network Theory to prove that the mRNA copy number distribution is Poissonian in the absence of spontaneous mRNA decay channel. In case of transcriptional bursts, we show that feedforward control can attenuate the super-Poissonian gene-expression noise that is due to bursting, and that the effect is more considerable at the protein than at the mRNA level. Our results indicate that the strong coupling between mRNA and microRNA in the sense of burst stoichiometry and also of timing of production events renders the microRNA based feedforward motif an effective mechanism for the control of gene expression noise.