Interpretable and tractable models of transcriptional noise for the rational design of single-molecule quantification experiments

Spectral neural approximations for models of transcriptional dynamics

The advent of high-throughput transcriptomics provides an opportunity to advance mechanistic understanding of transcriptional processes and their connections to cellular function at an unprecedented, genome-wide scale. These transcriptional systems, which involve discrete stochastic events, are naturally modeled using chemical master equations (CMEs), which can be solved for probability distributions to fit biophysical rates that govern system dynamics. While CME models have been used as standards in fluorescence transcriptomics for decades to analyze single-species RNA distributions, there are often no closed-form solutions to CMEs that model multiple species, such as nascent and mature RNA transcript counts. This has prevented the application of standard likelihood-based statistical methods for analyzing high-throughput, multi-species transcriptomic datasets using biophysical models. Inspired by recent work in machine learning to learn solutions to complex dynamical systems, we leverage neural networks and statistical understanding of system distributions to produce accurate approximations to a steady-state bivariate distribution for a model of the RNA life cycle that includes nascent and mature molecules. The steady-state distribution to this simple model has no closed-form solution and requires intensive numerical solving techniques: our approach reduces likelihood evaluation time by several orders of magnitude. We demonstrate two approaches, whereby solutions are approximated by 1) learning the weights of kernel distributions with constrained parameters or 2) learning both weights and scaling factors for parameters of kernel distributions. We show that our strategies, denoted by kernel weight regression and parameter-scaled kernel weight regression, respectively, enable broad exploration of parameter space and can be used in existing likelihood frameworks to infer transcriptional burst sizes, RNA splicing rates, and mRNA degradation rates from experimental transcriptomic data.

Biophysically interpretable inference of cell types from multimodal sequencing data

Multimodal, single-cell genomics technologies enable simultaneous measurement of multiple facets of DNA and RNA processing in the cell. This creates opportunities for transcriptome-wide, mechanistic studies of cellular processing in heterogeneous cell populations, such as regulation of cell fate by transcriptional stochasticity or tumor proliferation through aberrant splicing dynamics. However, current methods for determining cell types or 'clusters' in multimodal data often rely on ad hoc approaches to balance or integrate measurements, and assumptions ignoring inherent properties of the data. To enable interpretable and consistent cell cluster determination, we present meK-means (mechanistic K-means) which integrates modalities through a unifying model of transcription to learn underlying, shared biophysical states. With meK-means we can cluster cells with nascent and mature mRNA measurements, utilizing the causal, physical relationships between these modalities. This identifies shared transcription dynamics across cells, which induce the observed molecule counts, and provides an alternative definition for 'clusters' through the governing parameters of cellular processes.

Transcriptional bursting: from fundamentals to novel insights

Transcription occurs as irregular bursts in a very wide range of systems, including numerous different species and many genes within these. In this review, we examine the underlying theories, discuss how these relate to experimental measurements, and explore some of the discrepancies that have emerged among various studies. Finally, we consider more recent works that integrate novel concepts, such as the involvement of biomolecular condensates in enhancer-promoter interactions and their effects on the dynamics of transcriptional bursting.

Global transcription regulation revealed from dynamical correlations in time-resolved single-cell RNA sequencing

Single-cell transcriptomics reveals significant variations in transcriptional activity across cells. Yet, it remains challenging to identify mechanisms of transcription dynamics from static snapshots. It is thus still unknown what drives global transcription dynamics in single cells. We present a stochastic model of gene expression with cell size- and cell cycle-dependent rates in growing and dividing cells that harnesses temporal dimensions of single-cell RNA sequencing through metabolic labeling protocols and cel lcycle reporters. We develop a parallel and highly scalable approximate Bayesian computation method that corrects for technical variation and accurately quantifies absolute burst frequency, burst size, and degradation rate along the cell cycle at a transcriptome-wide scale. Using Bayesian model selection, we reveal scaling between transcription rates and cell size and unveil waves of gene regulation across the cell cycle-dependent transcriptome. Our study shows that stochastic modeling of dynamical correlations identifies global mechanisms of transcription regulation. A record of this paper's transparent peer review process is included in the supplemental information.

Biophysical modeling with variational autoencoders for bimodal, single-cell RNA sequencing data

Here we present biVI, which combines the variational autoencoder framework of scVI with biophysical models describing the transcription and splicing kinetics of RNA molecules. We demonstrate on simulated and experimental single-cell RNA sequencing data that biVI retains the variational autoencoder's ability to capture cell type structure in a low-dimensional space while further enabling genome-wide exploration of the biophysical mechanisms, such as system burst sizes and degradation rates, that underlie observations.

Inferring Stochastic Rates from Heterogeneous Snapshots of Particle Positions

Many imaging techniques for biological systems-like fixation of cells coupled with fluorescence microscopy-provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Balanced implicit Patankar-Euler methods for positive solutions of stochastic differential equations of biological regulatory systems

Stochastic differential equations (SDEs) are a powerful tool to model fluctuations and uncertainty in complex systems. Although numerical methods have been designed to simulate SDEs effectively, it is still problematic when numerical solutions may be negative, but application problems require positive simulations. To address this issue, we propose balanced implicit Patankar-Euler methods to ensure positive simulations of SDEs. Instead of considering the addition of balanced terms to explicit methods in existing balanced methods, we attempt the deletion of possible negative terms from the explicit methods to maintain positivity of numerical simulations. The designed balanced terms include negative-valued drift terms and potential negative diffusion terms. The proposed method successfully addresses the issue of divisions with very small denominators in our recently designed stochastic Patankar method. Stability analysis shows that the balanced implicit Patankar-Euler method has much better stability properties than our recently designed composite Patankar-Euler method. Four SDE systems are used to examine the effectiveness, accuracy, and convergence properties of balanced implicit Patankar-Euler methods. Numerical results suggest that the proposed balanced implicit Patankar-Euler method is an effective and efficient approach to ensure positive simulations when any appropriate stepsize is used in simulating SDEs of biological regulatory systems.

kallisto, bustools, and kb-python for quantifying bulk, single-cell, and single-nucleus RNA-seq

The term "RNA-seq" refers to a collection of assays based on sequencing experiments that involve quantifying RNA species from bulk tissue, from single cells, or from single nuclei. The kallisto, bustools, and kb-python programs are free, open-source software tools for performing this analysis that together can produce gene expression quantification from raw sequencing reads. The quantifications can be individualized for multiple cells, multiple samples, or both. Additionally, these tools allow gene expression values to be classified as originating from nascent RNA species or mature RNA species, making this workflow amenable to both cell-based and nucleus-based assays. This protocol describes in detail how to use kallisto and bustools in conjunction with a wrapper, kb-python, to preprocess RNA-seq data.

Quantifying and correcting bias in transcriptional parameter inference from single-cell data

The snapshot distribution of mRNA counts per cell can be measured using single-molecule fluorescence in situ hybridization or single-cell RNA sequencing. These distributions are often fit to the steady-state distribution of the two-state telegraph model to estimate the three transcriptional parameters for a gene of interest: mRNA synthesis rate, the switching on rate (the on state being the active transcriptional state), and the switching off rate. This model assumes no extrinsic noise, i.e., parameters do not vary between cells, and thus estimated parameters are to be understood as approximating the average values in a population. The accuracy of this approximation is currently unclear. Here, we develop a theory that explains the size and sign of estimation bias when inferring parameters from single-cell data using the standard telegraph model. We find specific bias signatures depending on the source of extrinsic noise (which parameter is most variable across cells) and the mode of transcriptional activity. If gene expression is not bursty then the population averages of all three parameters are overestimated if extrinsic noise is in the synthesis rate; underestimation occurs if extrinsic noise is in the switching on rate; both underestimation and overestimation can occur if extrinsic noise is in the switching off rate. We find that some estimated parameters tend to infinity as the size of extrinsic noise approaches a critical threshold. In contrast when gene expression is bursty, we find that in all cases the mean burst size (ratio of the synthesis rate to the switching off rate) is overestimated while the mean burst frequency (the switching on rate) is underestimated. We estimate the size of extrinsic noise from the covariance matrix of sequencing data and use this together with our theory to correct published estimates of transcriptional parameters for mammalian genes.

Inferring stochastic rates from heterogeneous snapshots of particle positions

Many imaging techniques for biological systems - like fixation of cells coupled with fluorescence microscopy - provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Studying stochastic systems biology of the cell with single-cell genomics data

Recent experimental developments in genome-wide RNA quantification hold considerable promise for systems biology. However, rigorously probing the biology of living cells requires a unified mathematical framework that accounts for single-molecule biological stochasticity in the context of technical variation associated with genomics assays. We review models for a variety of RNA transcription processes, as well as the encapsulation and library construction steps of microfluidics-based single-cell RNA sequencing, and present a framework to integrate these phenomena by the manipulation of generating functions. Finally, we use simulated scenarios and biological data to illustrate the implications and applications of the approach.

Assessing Markovian and Delay Models for Single-Nucleus RNA Sequencing

The serial nature of reactions involved in the RNA life-cycle motivates the incorporation of delays in models of transcriptional dynamics. The models couple a transcriptional process to a fairly general set of delayed monomolecular reactions with no feedback. We provide numerical strategies for calculating the RNA copy number distributions induced by these models, and solve several systems with splicing, degradation, and catalysis. An analysis of single-cell and single-nucleus RNA sequencing data using these models reveals that the kinetics of nuclear export do not appear to require invocation of a non-Markovian waiting time.

Biophysically Interpretable Inference of Cell Types from Multimodal Sequencing Data

Multimodal, single-cell genomics technologies enable simultaneous capture of multiple facets of DNA and RNA processing in the cell. This creates opportunities for transcriptome-wide, mechanistic studies of cellular processing in heterogeneous cell types, with applications ranging from inferring kinetic differences between cells, to the role of stochasticity in driving heterogeneity. However, current methods for determining cell types or 'clusters' present in multimodal data often rely on ad hoc or independent treatment of modalities, and assumptions ignoring inherent properties of the count data. To enable interpretable and consistent cell cluster determination from multimodal data, we present meK-Means (mechanistic K-Means) which integrates modalities and learns underlying, shared biophysical states through a unifying model of transcription. In particular, we demonstrate how meK-Means can be used to cluster cells from unspliced and spliced mRNA count modalities. By utilizing the causal, physical relationships underlying these modalities, we identify shared transcriptional kinetics across cells, which induce the observed gene expression profiles, and provide an alternative definition for 'clusters' through the governing parameters of cellular processes.

The specious art of single-cell genomics

Dimensionality reduction is standard practice for filtering noise and identifying relevant features in large-scale data analyses. In biology, single-cell genomics studies typically begin with reduction to 2 or 3 dimensions to produce "all-in-one" visuals of the data that are amenable to the human eye, and these are subsequently used for qualitative and quantitative exploratory analysis. However, there is little theoretical support for this practice, and we show that extreme dimension reduction, from hundreds or thousands of dimensions to 2, inevitably induces significant distortion of high-dimensional datasets. We therefore examine the practical implications of low-dimensional embedding of single-cell data and find that extensive distortions and inconsistent practices make such embeddings counter-productive for exploratory, biological analyses. In lieu of this, we discuss alternative approaches for conducting targeted embedding and feature exploration to enable hypothesis-driven biological discovery.

Studying stochastic systems biology of the cell with single-cell genomics data

Recent experimental developments in genome-wide RNA quantification hold considerable promise for systems biology. However, rigorously probing the biology of living cells requires a unified mathematical framework that accounts for single-molecule biological stochasticity in the context of technical variation associated with genomics assays. We review models for a variety of RNA transcription processes, as well as the encapsulation and library construction steps of microfluidics-based single-cell RNA sequencing, and present a framework to integrate these phenomena by the manipulation of generating functions. Finally, we use simulated scenarios and biological data to illustrate the implications and applications of the approach.

Biophysical modeling with variational autoencoders for bimodal, single-cell RNA sequencing data

We motivate and present biVI, which combines the variational autoencoder framework of scVI with biophysically motivated, bivariate models for nascent and mature RNA distributions. While previous approaches to integrate bimodal data via the variational autoencoder framework ignore the causal relationship between measurements, biVI models the biophysical processes that give rise to observations. We demonstrate through simulated benchmarking that biVI captures cell type structure in a low-dimensional space and accurately recapitulates parameter values and copy number distributions. On biological data, biVI provides a scalable route for identifying the biophysical mechanisms underlying gene expression. This analytical approach outlines a generalizable strategy for treating multimodal datasets generated by high-throughput, single-cell genomic assays.

The telegraph process is not a subordinator

Investigations of transcriptional models by Amrhein et al. outline a strategy for connecting steady-state distributions to process dynamics. We clarify its limitations: the strategy holds for a very narrow class of processes, which excludes an example given by the authors.

Interpretable and tractable models of transcriptional noise for the rational design of single-molecule quantification experiments

The question of how cell-to-cell differences in transcription rate affect RNA count distributions is fundamental for understanding biological processes underlying transcription. Answering this question requires quantitative models that are both interpretable (describing concrete biophysical phenomena) and tractable (amenable to mathematical analysis). This enables the identification of experiments which best discriminate between competing hypotheses. As a proof of principle, we introduce a simple but flexible class of models involving a continuous stochastic transcription rate driving a discrete RNA transcription and splicing process, and compare and contrast two biologically plausible hypotheses about transcription rate variation. One assumes variation is due to DNA experiencing mechanical strain, while the other assumes it is due to regulator number fluctuations. We introduce a framework for numerically and analytically studying such models, and apply Bayesian model selection to identify candidate genes that show signatures of each model in single-cell transcriptomic data from mouse glutamatergic neurons.