Bayesian inference of distributed time delay in transcriptional and translational regulation

Analytic delay distributions for a family of gene transcription models

Models intended to describe the time evolution of a gene network must somehow include transcription, the DNA-templated synthesis of RNA, and translation, the RNA-templated synthesis of proteins. In eukaryotes, the DNA template for transcription can be very long, often consisting of tens of thousands of nucleotides, and lengthy pauses may punctuate this process. Accordingly, transcription can last for many minutes, in some cases hours. There is a long history of introducing delays in gene expression models to take the transcription and translation times into account. Here we study a family of detailed transcription models that includes initiation, elongation, and termination reactions. We establish a framework for computing the distribution of transcription times, and work out these distributions for some typical cases. For elongation, a fixed delay is a good model provided elongation is fast compared to initiation and termination, and there are no sites where long pauses occur. The initiation and termination phases of the model then generate a nontrivial delay distribution, and elongation shifts this distribution by an amount corresponding to the elongation delay. When initiation and termination are relatively fast, the distribution of elongation times can be approximated by a Gaussian. A convolution of this Gaussian with the initiation and termination time distributions gives another analytic approximation to the transcription time distribution. If there are long pauses during elongation, because of the modularity of the family of models considered, the elongation phase can be partitioned into reactions generating a simple delay (elongation through regions where there are no long pauses), and reactions whose distribution of waiting times must be considered explicitly (initiation, termination, and motion through regions where long pauses are likely). In these cases, the distribution of transcription times again involves a nontrivial part and a shift due to fast elongation processes.

Identification of four mitochondria-related genes in sepsis based on RNA sequencing technology

OBJECTIVES

The purpose of this study was to identify and analyze the mitochondrial genes associated with sepsis patients in order to elucidate the underlying mechanism of sepsis immunity and provide new ideas for the clinical treatment of sepsis.

METHODS

The hospitalized cases of sepsis (n = 20) and systemic inflammatory response syndrome (SIRS) (n = 12) admitted to the Emergency Intensive Care Unit (EICU) of the Affiliated Hospital of Southwest Medical University from January 2019 to December 2019 were collected consecutively. RNA-seq was used to sequence the RNA (mRNA) of peripheral blood cells. Bioinformatics techniques were used to screen and identify differentially expressed RNAs, with an absolute value of fold change (FC) greater than or equal to 1.2 and a false discovery rate (FDR) less than 0.05. At the same time, mitochondrial genes were obtained from the MitoCarta 3.0 database. Differential genes were then intersected with mitochondrial genes. The resulting crossover genes were subjected to GO, KEGG, and PPI analysis. Subsequently, the GSE65682 dataset was downloaded from the GEO database for survival analysis to assess the prognostic value of core genes, and GSE67652 was downloaded for ROC curve analysis to validate the diagnostic value of core genes. Finally, the localization of core genes was clarified through 10X single-cell sequencing.

RESULTS

The crossing of 314 sepsis differential genes and 1136 mitochondrial genes yielded 28 genes. GO and KEGG analysis showed that the crossover genes were mainly involved in the mitochondrion, mitochondrial matrix, and mitochondrial inner membrane. Survival analysis screened four genes that were significantly negatively associated with the prognosis of sepsis, namely FIS1, FKBP8, GLRX5, and GUK1. A comparison of peripheral blood RNA-seq results between the sepsis group and the SIRS group showed that the expression levels of these four genes were significantly decreased in the sepsis group compared to the SIRS group. ROC curve analysis based on GSE67652 indicates these four genes' high sensitivity and specificity for sepsis detection. Additionally, single-cell RNA sequencing found that the core genes were mainly expressed in macrophages, T cells, and B cells.

CONCLUSIONS

Mitochondria-related genes (FIS1, FKBP8, GLRX5, GUK1) were underexpressed in the sepsis group, negatively correlated with survival, and mainly distributed in immune cells. This finding may guide studying the immune-related mechanisms of sepsis. This study protocol was reviewed by the Ethics Committee of the Affiliated Hospital of Southwest Medical University (ethics number: KY2018029), the clinical trial registration number is ChiCTR1900021261, and the registration date is February 4, 2019.

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.

Density physics-informed neural networks reveal sources of cell heterogeneity in signal transduction

The transduction time between signal initiation and final response provides valuable information on the underlying signaling pathway, including its speed and precision. Furthermore, multi-modality in a transduction-time distribution indicates that the response is regulated by multiple pathways with different transduction speeds. Here, we developed a method called density physics-informed neural networks (Density-PINNs) to infer the transduction-time distribution from measurable final stress response time traces. We applied Density-PINNs to single-cell gene expression data from sixteen promoters regulated by unknown pathways in response to antibiotic stresses. We found that promoters with slower signaling initiation and transduction exhibit larger cell-to-cell heterogeneity in response intensity. However, this heterogeneity was greatly reduced when the response was regulated by slow and fast pathways together. This suggests a strategy for identifying effective signaling pathways for consistent cellular responses to disease treatments. Density-PINNs can also be applied to understand other time delay systems, including infectious diseases.

Exact results for gene-expression models with general waiting-time distributions

Complex molecular details of transcriptional regulation can be coarse-grained by assuming that reaction waiting times for promoter-state transitions, the mRNA synthesis, and the mRNA degradation follow general distributions. However, how such a generalized two-state model is analytically solved is a long-standing issue. Here we first present analytical formulas of burst-size distributions for this model. Then, we derive an iterative equation for the mRNA moment-generating function, by which mRNA raw and binomial moments of any order can be conveniently calculated. The analytical results obtained in the special cases of phase-type waiting-time distributions not only provide insights into the mechanisms of complex transcriptional regulations but also bring conveniences for experimental data-based statistical inferences.

Inferring delays in partially observed gene regulation processes

MOTIVATION

Cell function is regulated by gene regulatory networks (GRNs) defined by protein-mediated interaction between constituent genes. Despite advances in experimental techniques, we can still measure only a fraction of the processes that govern GRN dynamics. To infer the properties of GRNs using partial observation, unobserved sequential processes can be replaced with distributed time delays, yielding non-Markovian models. Inference methods based on the resulting model suffer from the curse of dimensionality.

RESULTS

We develop a simulation-based Bayesian MCMC method employing an approximate likelihood for the efficient and accurate inference of GRN parameters when only some of their products are observed. We illustrate our approach using a two-step activation model: an activation signal leads to the accumulation of an unobserved regulatory protein, which triggers the expression of observed fluorescent proteins. With prior information about observed fluorescent protein synthesis, our method successfully infers the dynamics of the unobserved regulatory protein. We can estimate the delay and kinetic parameters characterizing target regulation including transcription, translation, and target searching of an unobserved protein from experimental measurements of the products of its target gene. Our method is scalable and can be used to analyze non-Markovian models with hidden components.

AVAILABILITY AND IMPLEMENTATION

Our code is implemented in R and is freely available with a simple example data at https://github.com/Mathbiomed/SimMCMC.

Distribution modeling quantifies collective TH cell decision circuits in chronic inflammation

Immune responses are tightly regulated by a diverse set of interacting immune cell populations. Alongside decision-making processes such as differentiation into specific effector cell types, immune cells initiate proliferation at the beginning of an inflammation, forming two layers of complexity. Here, we developed a general mathematical framework for the data-driven analysis of collective immune cell dynamics. We identified qualitative and quantitative properties of generic network motifs, and we specified differentiation dynamics by analysis of kinetic transcriptome data. Furthermore, we derived a specific, data-driven mathematical model for T helper 1 versus T follicular helper cell-fate decision dynamics in acute and chronic lymphocytic choriomeningitis virus infections in mice. The model recapitulates important dynamical properties without model fitting and solely by using measured response-time distributions. Model simulations predict different windows of opportunity for perturbation in acute and chronic infection scenarios, with potential implications for optimization of targeted immunotherapy.

Consistency across multi-omics layers in a drug-perturbed gut microbial community

Multi-omics analyses are used in microbiome studies to understand molecular changes in microbial communities exposed to different conditions. However, it is not always clear how much each omics data type contributes to our understanding and whether they are concordant with each other. Here, we map the molecular response of a synthetic community of 32 human gut bacteria to three non-antibiotic drugs by using five omics layers (16S rRNA gene profiling, metagenomics, metatranscriptomics, metaproteomics and metabolomics). We find that all the omics methods with species resolution are highly consistent in estimating relative species abundances. Furthermore, different omics methods complement each other for capturing functional changes. For example, while nearly all the omics data types captured that the antipsychotic drug chlorpromazine selectively inhibits Bacteroidota representatives in the community, the metatranscriptome and metaproteome suggested that the drug induces stress responses related to protein quality control. Metabolomics revealed a decrease in oligosaccharide uptake, likely caused by Bacteroidota depletion. Our study highlights how multi-omics datasets can be utilized to reveal complex molecular responses to external perturbations in microbial communities.

A general model-based causal inference method overcomes the curse of synchrony and indirect effect

To identify causation, model-free inference methods, such as Granger Causality, have been widely used due to their flexibility. However, they have difficulty distinguishing synchrony and indirect effects from direct causation, leading to false predictions. To overcome this, model-based inference methods that test the reproducibility of data with a specific mechanistic model to infer causality were developed. However, they can only be applied to systems described by a specific model, greatly limiting their applicability. Here, we address this limitation by deriving an easily testable condition for a general monotonic ODE model to reproduce time-series data. We built a user-friendly computational package, General ODE-Based Inference (GOBI), which is applicable to nearly any monotonic system with positive and negative regulations described by ODE. GOBI successfully inferred positive and negative regulations in various networks at both the molecular and population levels, unlike existing model-free methods. Thus, this accurate and broadly applicable inference method is a powerful tool for understanding complex dynamical systems.

Inference and uncertainty quantification of stochastic gene expression via synthetic models

Estimating uncertainty in model predictions is a central task in quantitative biology. Biological models at the single-cell level are intrinsically stochastic and nonlinear, creating formidable challenges for their statistical estimation which inevitably has to rely on approximations that trade accuracy for tractability. Despite intensive interest, a sweet spot in this trade-off has not been found yet. We propose a flexible procedure for uncertainty quantification in a wide class of reaction networks describing stochastic gene expression including those with feedback. The method is based on creating a tractable coarse-graining of the model that is learned from simulations, a synthetic model, to approximate the likelihood function. We demonstrate that synthetic models can substantially outperform state-of-the-art approaches on a number of non-trivial systems and datasets, yielding an accurate and computationally viable solution to uncertainty quantification in stochastic models of gene expression.

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.

PhysiPKPD: A pharmacokinetics and pharmacodynamics module for PhysiCell

Pharmacokinetics and pharmacodynamics (PKPD) are key considerations in any study of molecular therapies. It is thus imperative to factor their effects into any in silico model of biological tissue involving such therapies. Furthermore, creating a standardized and flexible framework will benefit the community by increasing access to such modules and enhancing their communicability. PhysiCell is an open-source physics-based cell simulator, i.e., a platform for modeling biological tissue, that is quickly being adopted and utilized by the mathematical biology community. We present here PhysiPKPD, an open-source PhysiCell-based package that allows users to include PKPD in PhysiCell models.

Availability & Implementation

The source code for PhysiPKPD is located here: https://github.com/drbergman/PhysiPKPD.

Inferring causality in biological oscillators

MOTIVATION

Fundamental to biological study is identifying regulatory interactions. The recent surge in time-series data collection in biology provides a unique opportunity to infer regulations computationally. However, when components oscillate, model-free inference methods, while easily implemented, struggle to distinguish periodic synchrony and causality. Alternatively, model-based methods test the reproducibility of time series given a specific model but require inefficient simulations and have limited applicability.

RESULTS

We develop an inference method based on a general model of molecular, neuronal and ecological oscillatory systems that merges the advantages of both model-based and model-free methods, namely accuracy, broad applicability and usability. Our method successfully infers the positive and negative regulations within various oscillatory networks, e.g. the repressilator and a network of cofactors at the pS2 promoter, outperforming popular inference methods.

AVAILABILITY AND IMPLEMENTATION

We provide a computational package, ION (Inferring Oscillatory Networks), that users can easily apply to noisy, oscillatory time series to uncover the mechanisms by which diverse systems generate oscillations. Accompanying MATLAB code under a BSD-style license and examples are available at https://github.com/Mathbiomed/ION. Additionally, the code is available under a CC-BY 4.0 License at https://doi.org/10.6084/m9.figshare.16431408.v1.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Hierarchical Bayesian models of transcriptional and translational regulation processes with delays

MOTIVATION

Simultaneous recordings of gene network dynamics across large populations have revealed that cell characteristics vary considerably even in clonal lines. Inferring the variability of parameters that determine gene dynamics is key to understanding cellular behavior. However, this is complicated by the fact that the outcomes and effects of many reactions are not observable directly. Unobserved reactions can be replaced with time delays to reduce model dimensionality and simplify inference. However, the resulting models are non-Markovian, and require the development of new inference techniques.

RESULTS

We propose a non-Markovian, hierarchical Bayesian inference framework for quantifying the variability of cellular processes within and across cells in a population. We illustrate our approach using a delayed birth-death process. In general, a distributed delay model, rather than a popular fixed delay model, is needed for inference, even if only mean reaction delays are of interest. Using in silico and experimental data we show that the proposed hierarchical framework is robust and leads to improved estimates compared to its non-hierarchical counterpart. We apply our method to data obtained using time-lapse microscopy and infer the parameters that describe the dynamics of protein production at the single cell and population level. The mean delays in protein production are larger than previously reported, have a coefficient of variation of around 0.2 across the population, and are not strongly correlated with protein production or growth rates.

AVAILABILITY AND IMPLEMENTATION

Accompanying code in Python is available at https://github.com/mvcortez/Bayesian-Inference.

CONTACT

kresimir.josic@gmail.com or jaekkim@kaist.ac.kr or cbskust@korea.ac.kr.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Incorporating age and delay into models for biophysical systems

In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this paper, we consider relaxing this assumption by incorporating age-dependent random time delays (distributed according to a given probability distribution) into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the 'ages' of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of partial differential equations (PDEs) in the large-volume limit, as opposed to ordinary differential equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. In order to describe the ideas, we use a simple transcription process as a running example. We, however, note that the methods developed in this paper apply to a wide class of biophysical systems.