Bayesian inference of biochemical kinetic parameters using the linear noise approximation

Background Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data. Results We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo. Conclusion The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods

Inferring rate coefficents of biochemical reactions from noisy data with KInfer

Dynamical models of inter- and intra-cellular processes contain the rate constants of the biochemical reactions. These kinetic parameters are often not accessible directly through experiments, but they can be inferred from time-resolved data. Time resolved data, that is, measurements of reactant concentration at series of time points, are usually affected by different types of error, whose source can be both experimental and biological. The noise in the input data makes the estimation of the model parameters a very difficult task, as if the inference method is not sufficiently robust to the noise, the resulting estimates are not reliable. Therefore "noise-robust" methods that estimate rate constants with the maximum precision and accuracy are needed. In this report we present the probabilistic generative model of parameter inference implemented by the software prototype KInfer and we show the ability of this tool of estimating the rate coefficients of models of biochemical network with a good accuracy even from very noisy input data

Determining Physical Mechanisms of Gene Expression Regulation from Single Cell Gene Expression Data.

Many genes are expressed in bursts, which can contribute to cell-to-cell heterogeneity. It is now possible to measure this heterogeneity with high throughput single cell gene expression assays (single cell qPCR and RNA-seq). These experimental approaches generate gene expression distributions which can be used to estimate the kinetic parameters of gene expression bursting, namely the rate that genes turn on, the rate that genes turn off, and the rate of transcription. We construct a complete pipeline for the analysis of single cell qPCR data that uses the mathematics behind bursty expression to develop more accurate and robust algorithms for analyzing the origin of heterogeneity in experimental samples, specifically an algorithm for clustering cells by their bursting behavior (Simulated Annealing for Bursty Expression Clustering, SABEC) and a statistical tool for comparing the kinetic parameters of bursty expression across populations of cells (Estimation of Parameter changes in Kinetics, EPiK). We applied these methods to hematopoiesis, including a new single cell dataset in which transcription factors (TFs) involved in the earliest branchpoint of blood differentiation were individually up- and down-regulated. We could identify two unique sub-populations within a seemingly homogenous group of hematopoietic stem cells. In addition, we could predict regulatory mechanisms controlling the expression levels of eighteen key hematopoietic transcription factors throughout differentiation. Detailed information about gene regulatory mechanisms can therefore be obtained simply from high throughput single cell gene expression data, which should be widely applicable given the rapid expansion of single cell genomics.This work was supported by: Royal Society Research Fellowship, Marshall Scholarship, Medical Research Council, the Leukemia and Lymphoma Society and core support grants from the Wellcome Trust to the Cambridge Institute for Medical Research and the Wellcome Trust and MRC Cambridge Stem Cell Institute.This is the final version of the article. It first appeared from the Public Library of Science via http://dx.doi.org/10.1371/journal.pcbi.100507

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterising stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealisations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with MATLAB implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community

Global parameter estimation methods for stochastic biochemical systems

<p>Abstract</p> <p>Background</p> <p>The importance of stochasticity in cellular processes having low number of molecules has resulted in the development of stochastic models such as chemical master equation. As in other modelling frameworks, the accompanying rate constants are important for the end-applications like analyzing system properties (e.g. robustness) or predicting the effects of genetic perturbations. Prior knowledge of kinetic constants is usually limited and the model identification routine typically includes parameter estimation from experimental data. Although the subject of parameter estimation is well-established for deterministic models, it is not yet routine for the chemical master equation. In addition, recent advances in measurement technology have made the quantification of genetic substrates possible to single molecular levels. Thus, the purpose of this work is to develop practical and effective methods for estimating kinetic model parameters in the chemical master equation and other stochastic models from single cell and cell population experimental data.</p> <p>Results</p> <p>Three parameter estimation methods are proposed based on the maximum likelihood and density function distance, including probability and cumulative density functions. Since stochastic models such as chemical master equations are typically solved using a Monte Carlo approach in which only a finite number of Monte Carlo realizations are computationally practical, specific considerations are given to account for the effect of finite sampling in the histogram binning of the state density functions. Applications to three practical case studies showed that while maximum likelihood method can effectively handle low replicate measurements, the density function distance methods, particularly the cumulative density function distance estimation, are more robust in estimating the parameters with consistently higher accuracy, even for systems showing multimodality.</p> <p>Conclusions</p> <p>The parameter estimation methodologies described in this work have provided an effective and practical approach in the estimation of kinetic parameters of stochastic systems from either sparse or dense cell population data. Nevertheless, similar to kinetic parameter estimation in other modelling frameworks, not all parameters can be estimated accurately, which is a common problem arising from the lack of complete parameter identifiability from the available data.</p

Modelling capture efficiency of single-cell RNA-sequencing data improves inference of transcriptome-wide burst kinetics

Motivation: Gene expression is characterized by stochastic bursts of transcription that occur at brief and random periods of promoter activity. The kinetics of gene expression burstiness differs across the genome and is dependent on the promoter sequence, among other factors. Single-cell RNA sequencing (scRNA-seq) has made it possible to quantify the cell-to-cell variability in transcription at a global genome-wide level. However, scRNA-seq data are prone to technical variability, including low and variable capture efficiency of transcripts from individual cells. // Results: Here, we propose a novel mathematical theory for the observed variability in scRNA-seq data. Our method captures burst kinetics and variability in both the cell size and capture efficiency, which allows us to propose several likelihood-based and simulation-based methods for the inference of burst kinetics from scRNA-seq data. Using both synthetic and real data, we show that the simulation-based methods provide an accurate, robust and flexible tool for inferring burst kinetics from scRNA-seq data. In particular, in a supervised manner, a simulation-based inference method based on neural networks proves to be accurate and useful when applied to both allele and nonallele-specific scRNA-seq data. // Availability and implementation: The code for Neural Network and Approximate Bayesian Computation inference is available at https://github.com/WT215/nnRNA and https://github.com/WT215/Julia_ABC, respectively