The Chemical Fluctuation Theorem governing gene expression

Solving stochastic gene-expression models using queueing theory: A tutorial review

Stochastic models of gene expression are typically formulated using the chemical master equation, which can be solved exactly or approximately using a repertoire of analytical methods. Here, we provide a tutorial review of an alternative approach based on queueing theory that has rarely been used in the literature of gene expression. We discuss the interpretation of six types of infinite-server queues from the angle of stochastic single-cell biology and provide analytical expressions for the stationary and nonstationary distributions and/or moments of mRNA/protein numbers and bounds on the Fano factor. This approach may enable the solution of complex models that have hitherto evaded analytical solution.

Noisy Delay Denoises Biochemical Oscillators

Genetic oscillations are generated by delayed transcriptional negative feedback loops, wherein repressor proteins inhibit their own synthesis after a temporal production delay. This delay is distributed because it arises from a sequence of noisy processes, including transcription, translocation, translation, and folding. Because the delay determines repression timing and, therefore, oscillation period, it has been commonly believed that delay noise weakens oscillatory dynamics. Here, we demonstrate that noisy delay can surprisingly denoise genetic oscillators. Specifically, moderate delay noise improves the signal-to-noise ratio and sharpens oscillation peaks, all without impacting period and amplitude. We show that this denoising phenomenon occurs in a variety of well-studied genetic oscillators, and we use queueing theory to uncover the universal mechanisms that produce it.

Work distribution of a colloid in an elongational flow field and under Ornstein-Uhlenbeck noise

The study of thermodynamic properties of microscopic systems, such as a colloid in a fluid, has been of great interest to researchers since the discovery of the fluctuation theorem and associated laws of stochastic thermodynamics. However, most of these studies confine themselves to systems where effective fluctuations acting on the colloid are in the form of delta-correlated Gaussian white noise (GWN). In this study, instead, we look into the work distribution function when a colloid trapped in a harmonic potential moves from one position to another in a fluid medium with an elongational flow field where the effective fluctuations are given by the Ornstein-Uhlenbeck noise, a type of colored noise. We use path integrals to calculate this distribution function and compare and contrast its properties to the case with GWN. We find that the work distribution function turns out to be non-Gaussian as a result of the elongational flow field but continues to obey the fluctuation theorem in both types of noise. Further, we also look into the effects of the various system parameters on the behavior of work fluctuations and find that although the distribution tends to broaden with increasing noise intensity, increased correlation in fluctuations acts to oppose this effect. Additionally, the system is found to consume heat from the surroundings at early times and dissipate it into the media at later times. This study, therefore, is a step towards gaining a better understanding of the thermodynamic properties of colloidal systems under nonlinear complex flows that also display correlated fluctuations.

Stochastic modeling of the mRNA life process: A generalized master equation

The mRNA life cycle is a complex biochemical process, involving transcription initiation, elongation, termination, splicing, and degradation. Each of these molecular events is multistep and can create a memory. The effect of this molecular memory on gene expression is not clear, although there are many related yet scattered experimental reports. To address this important issue, we develop a general theoretical framework formulated as a master equation in the sense of queue theory, which can reduce to multiple previously studied gene models in limiting cases. This framework allows us to interpret experimental observations, extract kinetic parameters from experimental data, and identify how the mRNA kinetics vary under regulatory influences. Notably, it allows us to evaluate the influences of elongation processes on mature RNA distribution; e.g., we find that the non-exponential elongation time can induce the bimodal mRNA expression and there is an optimal elongation noise intensity such that the mature RNA noise achieves the lowest level. In a word, our framework can not only provide insight into complex mRNA life processes but also bridge a dialogue between theoretical studies and experimental data.

Steviol glycosides from Stevia rebaudiana Bertoni mitigate lipid metabolism abnormalities in diabetes by modulating selected gene expression - An in vivo study

In diabetes, in parallel to hyperglycaemia, elevated serum lipids are also diagnosed, representing a high-risk factor for coronary heart disease and cardiovascular complications. The objective of this study was to unravel the mechanisms that underlie the potential of steviol glycosides (stevioside or rebaudioside A) administered at two doses (500 or 2500 mg/kg body weight for 5 weeks) to regulate lipid metabolism. In this paper, the expression of selected genes responsible for glucose and lipid metabolism (Glut4, Pparγ, Cebpa, Fasn, Lpl and Egr1) in the peripheral tissues (adipose, liver and muscle tissue) was determined using quantitative real-time PCR method. It was found that the supplementation of steviol glycosides affected the expression of Glut4, Cebpa and Fasn genes, depending on the type of the glycoside and its dose, as well as the type of tissue, whish in part may explain the lipid-regulatory potential of steviol glycosides in hyperglycaemic conditions. Nevertheless, more in-depth studies, including human trials, are needed to confirm these effects, before steviol glycosides can be used in the therapy of type 2 diabetes.

On the thermodynamics of DNA methylation process

DNA methylation is an epigenetic mechanism that plays important roles in various biological processes including transcriptional and post-transcriptional regulation, genomic imprinting, aging, and stress response to environmental changes and disease. Consistent with thermodynamic principles acting within living systems and the application of maximum entropy principle, we propose a theoretical framework to understand and decode the DNA methylation process. A central tenet of this argument is that the probability density function of DNA methylation information-divergence summarizes the statistical biophysics underlying spontaneous methylation background and implicitly bears on the channel capacity of molecular machines conforming to Shannon's capacity theorem. On this theoretical basis, contributions from the molecular machine (enzyme) logical operations to Gibb entropy (S) and Helmholtz free energy (F) are intrinsic. Application to the estimations of S on datasets from Arabidopsis thaliana suggests that, as a thermodynamic state variable, individual methylome entropy is completely determined by the current state of the system, which in biological terms translates to a correspondence between estimated entropy values and observable phenotypic state. In patients with different types of cancer, results suggest that a significant information loss occurs in the transition from differentiated (healthy) tissues to cancer cells. This type of analysis may have important implications for early-stage diagnostics. The analysis of entropy fluctuations on experimental datasets revealed existence of restrictions on the magnitude of genome-wide methylation changes originating by organismal response to environmental changes. Only dysfunctional stages observed in the Arabidopsis mutant met1 and in cancer cells do not conform to these rules.

Systematic inference identifies a major source of heterogeneity in cell signaling dynamics: The rate-limiting step number

Identifying the sources of cell-to-cell variability in signaling dynamics is essential to understand drug response variability and develop effective therapeutics. However, it is challenging because not all signaling intermediate reactions can be experimentally measured simultaneously. This can be overcome by replacing them with a single random time delay, but the resulting process is non-Markovian, making it difficult to infer cell-to-cell heterogeneity in reaction rates and time delays. To address this, we developed an efficient and scalable moment-based Bayesian inference method (MBI) with a user-friendly computational package that infers cell-to-cell heterogeneity in the non-Markovian signaling process. We applied MBI to single-cell expression profiles from promoters responding to antibiotics and discovered a major source of cell-to-cell variability in antibiotic stress response: the number of rate-limiting steps in signaling cascades. This knowledge can help identify effective therapies that destroy all pathogenic or cancer cells, and the approach can be applied to precision medicine.

Chitosan Coating Enriched With Ruta graveolens L. Essential Oil Reduces Postharvest Anthracnose of Papaya (Carica papaya L.) and Modulates Defense-Related Gene Expression

Anthracnose of papaya (Carica papaya L.) caused by the fungus Colletotrichum spp. is one of the most economically important postharvest diseases. Coating with chitosan (CS) and Ruta graveolens essential oil (REO) might represent a novel eco-friendly method to prevent postharvest anthracnose infection. These compounds show both antimicrobial and eliciting activities, although the molecular mechanisms in papaya have not been investigated to date. In this study, the effectiveness of CS and REO alone and combined (CS-REO) on postharvest anthracnose of papaya fruit during storage were investigated, along with the expression of selected genes involved in plant defense mechanisms. Anthracnose incidence was reduced with CS, REO, and CS-REO emulsions after 9 days storage at 25°C, by 8, 21, and 37%, respectively, with disease severity reduced by 22, 29, and 44%, respectively. Thus, McKinney's decay index was reduced by 22, 30, and 44%, respectively. A protocol based on reverse transcription quantitative real-time PCR (RT-qPCR) was validated for 17 papaya target genes linked to signaling pathways that regulate plant defense, pathogenesis-related protein, cell wall-degrading enzymes, oxidative stress, abiotic stress, and the phenylpropanoid pathway. CS induced gene upregulation mainly at 6 h posttreatment (hpt) and 48 hpt, while REO induced the highest upregulation at 0.5 hpt, which then decreased over time. Furthermore, CS-REO treatment delayed gene upregulation by REO alone, from 0.5 to 6 hpt, and kept that longer over time. This study suggests that CS stabilizes the volatile and/or hydrophobic substances of highly reactive essential oils. The additive effects of CS and REO were able to reduce postharvest decay and affect gene expression in papaya fruit.

Neural network aided approximation and parameter inference of non-Markovian models of gene expression

Non-Markovian models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system's history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markovian models by the solutions of much simpler time-inhomogeneous Markovian models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markovian model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markovian models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

The common misuse of noise decomposition as applied to genetic systems

The noise-decomposition technique is applied in several fields, including genetic systems, optical images, recording, and navigation. In genetic systems, noise decomposition is usually achieved by using two reporters [Elowitz M.B., Levine A.J., Siggia E.D., Swain P·S., 2002. Stochastic gene expression in a single cell. Science 297, 1183-6.]. A reporter is a protein with fluorescence, an RNA hybridized with a fluorescent probe, or any other detectable intracellular component. If a reporter is constructed in addition to the original reporter, the system's stochasticity may change. Such phenomena became severe for genes in plasmids with a high copy number. By SSA (stochastic simulation algorithm), we observed an approximately 50% increment in the coefficient of variation while introducing additional reporters. Besides, if two reporters respond to the upstream element at a different time, the trunk noise (or extrinsic noise) cannot be accurately determined. This is because the "calculative trunk noise" changes along with the delay, though the real trunk noise does not. For RNA reporters, a 5-min transcriptional delay caused a calculative trunk noise that was 90% less than the real trunk noise. Fortunately, this problem is negligible when the degradation rate constant is low, and it is usually true in the case of the protein reporters. One can check the lifespan of the reporter before applying the noise-decomposition technique.

Queuing Models of Gene Expression: Analytical Distributions and Beyond

Activation of a gene is a multistep biochemical process, involving recruitments of transcription factors and histone kinases as well as modification of histones. Many of these intermediate reaction steps would have been unspecified by experiments. Therefore, classical two-state models of gene expression established based on the memoryless (or Markovian) assumption would not well describe the reality in gene expression. Recent experimental data have indicated that the inactive phases of gene promoters are differently distributed, showing strong memory. Here, we use a nonexponential waiting-time distribution to model the complex activation process of a gene, and then analyze a queuing model of stochastic transcription. We successfully derive the analytical expression of the stationary mRNA distribution, which provides insight into the effect of molecular memory created by complex activating events on the mRNA expression. We find that the reduction in the waiting-time noise may result in the increase in the mRNA noise, contrary to the previous conclusion. Based on the derived distribution, we also develop a method to infer the waiting-time distribution from a known mRNA distribution. Data analysis on a realistic example verifies the validity of this method.

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.

Molecular-memory-driven phenotypic switching in a genetic toggle switch without cooperative binding

A genetic toggle switch would involve multistep reaction processes (e.g., complex promoter activation), creating memories between individual reaction events. Revealing the effect of this molecular memory is important for understanding intracellular processes such as cellular decision making. We propose a generalized genetic toggle switch model and use a generalized chemical master equation theory to account for the memory effect. Interestingly, we find that molecular memory can induce bimodality in this memory system although the corresponding memoryless counterpart is not bimodal. This finding implies a plausible alternative mechanism for phenotypic switching that is driven by molecular memory rather than by ultrasensitivity or cooperative binding as shown in previous studies. We also find that unbalanced memories arising from the processes by which mutually inhibiting transcription factors are produced can give rise to asymmetric bimodality without changing the positions of two peaks in the bimodal protein distribution. Given the prevalence of molecular memory in gene regulation, our findings would provide insights into cell fate decisions in growth and development.

Different effects of fast and slow input fluctuations on output in gene regulation

An important task in the post-gene era is to understand the role of stochasticity in gene regulation. Here, we analyze a cascade model of stochastic gene expression, where the upstream gene stochastically generates proteins that regulate, as transcription factors, stochastic synthesis of the downstream output. We find that in contrast to fast input fluctuations that do not change the behavior of the downstream system qualitatively, slow input fluctuations can induce different modes of the distribution of downstream output and even stochastic focusing or defocusing of the downstream output level, although the regulatory protein follows the same distribution in both cases. This finding is counterintuitive but can have broad biological implications, e.g., slow input rather than fast fluctuations may both increase the survival probability of cells and enhance the sensitivity of intracellular regulation. In addition, we find that input fluctuations can minimize the output noise.

Markovian approaches to modeling intracellular reaction processes with molecular memory

Many cellular processes are governed by stochastic reaction events. These events do not necessarily occur in single steps of individual molecules, and, conversely, each birth or death of a macromolecule (e.g., protein) could involve several small reaction steps, creating a memory between individual events and thus leading to nonmarkovian reaction kinetics. Characterizing this kinetics is challenging. Here, we develop a systematic approach for a general reaction network with arbitrary intrinsic waiting-time distributions, which includes the stationary generalized chemical-master equation (sgCME), the stationary generalized Fokker-Planck equation, and the generalized linear-noise approximation. The first formulation converts a nonmarkovian issue into a markovian one by introducing effective transition rates (that explicitly decode the effect of molecular memory) for the reactions in an equivalent reaction network with the same substrates but without molecular memory. Nonmarkovian features of the reaction kinetics can be revealed by solving the sgCME. The latter 2 formulations can be used in the fast evaluation of fluctuations. These formulations can have broad applications, and, in particular, they may help us discover new biological knowledge underlying memory effects. When they are applied to generalized stochastic models of gene-expression regulation, we find that molecular memory is in effect equivalent to a feedback and can induce bimodality, fine-tune the expression noise, and induce switch.

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.

Discrimination of DNA Methylation Signal from Background Variation for Clinical Diagnostics

Advances in the study of human DNA methylation variation offer a new avenue for the translation of epigenetic research results to clinical applications. Although current approaches to methylome analysis have been helpful in revealing an epigenetic influence in major human diseases, this type of analysis has proven inadequate for the translation of these advances to clinical diagnostics. As in any clinical test, the use of a methylation signal for diagnostic purposes requires the estimation of an optimal cutoff value for the signal, which is necessary to discriminate a signal induced by a disease state from natural background variation. To address this issue, we propose the application of a fundamental signal detection theory and machine learning approaches. Simulation studies and tests of two available methylome datasets from autism and leukemia patients demonstrate the feasibility of this approach in clinical diagnostics, providing high discriminatory power for the methylation signal induced by disease, as well as high classification performance. Specifically, the analysis of whole biomarker genomic regions could suffice for a diagnostic, markedly decreasing its cost.

Frequency spectrum of chemical fluctuation: A probe of reaction mechanism and dynamics

Even in the steady-state, the number of biomolecules in living cells fluctuates dynamically, and the frequency spectrum of this chemical fluctuation carries valuable information about the dynamics of the reactions creating these biomolecules. Recent advances in single-cell techniques enable direct monitoring of the time-traces of the protein number in each cell; however, it is not yet clear how the stochastic dynamics of these time-traces is related to the reaction mechanism and dynamics. Here, we derive a rigorous relation between the frequency-spectrum of the product number fluctuation and the reaction mechanism and dynamics, starting from a generalized master equation. This relation enables us to analyze the time-traces of the protein number and extract information about dynamics of mRNA number and transcriptional regulation, which cannot be directly observed by current experimental techniques. We demonstrate our frequency spectrum analysis of protein number fluctuation, using the gene network model of luciferase expression under the control of the Bmal 1a promoter in mouse fibroblast cells. We also discuss how the dynamic heterogeneity of transcription and translation rates affects the frequency-spectra of the mRNA and protein number.

Transport dynamics of complex fluids

Thermal motion in complex fluids is a complicated stochastic process but ubiquitously exhibits initial ballistic, intermediate subdiffusive, and long-time diffusive motion, unless interrupted. Despite its relevance to numerous dynamical processes of interest in modern science, a unified, quantitative understanding of thermal motion in complex fluids remains a challenging problem. Here, we present a transport equation and its solutions, which yield a unified quantitative explanation of the mean-square displacement (MSD), the non-Gaussian parameter (NGP), and the displacement distribution of complex fluids. In our approach, the environment-coupled diffusion kernel and its time correlation function (TCF) are the essential quantities that determine transport dynamics and characterize mobility fluctuation of complex fluids; their time profiles are directly extractable from a model-free analysis of the MSD and NGP or, with greater computational expense, from the two-point and four-point velocity autocorrelation functions. We construct a general, explicit model of the diffusion kernel, comprising one unbound-mode and multiple bound-mode components, which provides an excellent approximate description of transport dynamics of various complex fluidic systems such as supercooled water, colloidal beads diffusing on lipid tubes, and dense hard disk fluid. We also introduce the concepts of intrinsic disorder and extrinsic disorder that have distinct effects on transport dynamics and different dependencies on temperature and density. This work presents an unexplored direction for quantitative understanding of transport and transport-coupled processes in complex disordered media.

Gene expression variability: the other dimension in transcriptome analysis

Transcriptome sequencing is a powerful technique to study molecular changes that underlie the differences in physiological conditions and disease progression. A typical question that is posed in such studies is finding genes with significant changes between sample groups. In this respect expression variability is regarded as a nuisance factor that is primarily of technical origin and complicates the data analysis. However, it is becoming apparent that the biological variation in gene expression might be an important molecular phenotype that can affect physiological parameters. In this review we explore the recent literature on technical and biological variability in gene expression, sources of expression variability, (epi-)genetic hallmarks, and evolutionary constraints in genes with robust and variable gene expression. We provide an overview of recent findings on effects of external cues, such as diet and aging, on expression variability and on other biological phenomena that can be linked to it. We discuss metrics and tools that were developed for quantification of expression variability and highlight the importance of future studies in this direction. To assist the adoption of expression variability analysis, we also provide a detailed description and computer code, which can easily be utilized by other researchers. We also provide a reanalysis of recently published data to highlight the value of the analysis method.