Analysis and design of single-cell experiments to harvest fluctuation information while rejecting measurement noise.

Introduction: Despite continued technological improvements, measurement errors always reduce or distort the information that any real experiment can provide to quantify cellular dynamics. This problem is particularly serious for cell signaling studies to quantify heterogeneity in single-cell gene regulation, where important RNA and protein copy numbers are themselves subject to the inherently random fluctuations of biochemical reactions. Until now, it has not been clear how measurement noise should be managed in addition to other experiment design variables (e.g., sampling size, measurement times, or perturbation levels) to ensure that collected data will provide useful insights on signaling or gene expression mechanisms of interest. Methods: We propose a computational framework that takes explicit consideration of measurement errors to analyze single-cell observations, and we derive Fisher Information Matrix (FIM)-based criteria to quantify the information value of distorted experiments. Results and Discussion: We apply this framework to analyze multiple models in the context of simulated and experimental single-cell data for a reporter gene controlled by an HIV promoter. We show that the proposed approach quantitatively predicts how different types of measurement distortions affect the accuracy and precision of model identification, and we demonstrate that the effects of these distortions can be mitigated through explicit consideration during model inference. We conclude that this reformulation of the FIM could be used effectively to design single-cell experiments to optimally harvest fluctuation information while mitigating the effects of image distortion.

Visualization and modeling of inhibition of IL-1ß and TNF-α mRNA transcription at the single-cell level.

IL-1ß and TNF-α are canonical immune response mediators that play key regulatory roles in a wide range of inflammatory responses to both chronic and acute conditions. Here we employ an automated microscopy platform for the analysis of messenger RNA (mRNA) expression of IL-1ß and TNF-α at the single-cell level. The amount of IL-1ß and TNF-α mRNA expressed in a human monocytic leukemia cell line (THP-1) is visualized and counted using single-molecule fluorescent in-situ hybridization (smFISH) following exposure of the cells to lipopolysaccharide (LPS), an outer-membrane component of Gram-negative bacteria. We show that the small molecule inhibitors MG132 (a 26S proteasome inhibitor used to block NF-κB signaling) and U0126 (a MAPK Kinase inhibitor used to block CCAAT-enhancer-binding proteins C/EBP) successfully block IL-1ß and TNF-α mRNA expression. Based upon this single-cell mRNA expression data, we screened 36 different mathematical models of gene expression, and found two similar models that capture the effects by which the drugs U0126 and MG132 affect the rates at which the genes transition into highly activated states. When their parameters were informed by the action of each drug independently, both models were able to predict the effects of the combined drug treatment. From our data and models, we postulate that IL-1ß is activated by both NF-κB and C/EBP, while TNF-α is predominantly activated by NF-κB. Our combined single-cell experimental and modeling efforts show the interconnection between these two genes and demonstrates how the single-cell responses, including the distribution shapes, mean expression, and kinetics of gene expression, change with inhibition.

BAYESIAN INFERENCE OF STOCHASTIC REACTION NETWORKS USING MULTIFIDELITY SEQUENTIAL TEMPERED MARKOV CHAIN MONTE CARLO.

Stochastic reaction network models are often used to explain and predict the dynamics of gene regulation in single cells. These models usually involve several parameters, such as the kinetic rates of chemical reactions, that are not directly measurable and must be inferred from experimental data. Bayesian inference provides a rigorous probabilistic framework for identifying these parameters by finding a posterior parameter distribution that captures their uncertainty. Traditional computational methods for solving inference problems such as Markov Chain Monte Carlo methods based on classical Metropolis-Hastings algorithm involve numerous serial evaluations of the likelihood function, which in turn requires expensive forward solutions of the chemical master equation (CME). We propose an alternate approach based on a multifidelity extension of the Sequential Tempered Markov Chain Monte Carlo (ST-MCMC) sampler. This algorithm is built upon Sequential Monte Carlo and solves the Bayesian inference problem by decomposing it into a sequence of efficiently solved subproblems that gradually increase both model fidelity and the influence of the observed data. We reformulate the finite state projection (FSP) algorithm, a well-known method for solving the CME, to produce a hierarchy of surrogate master equations to be used in this multifidelity scheme. To determine the appropriate fidelity, we introduce a novel information-theoretic criteria that seeks to extract the most information about the ultimate Bayesian posterior from each model in the hierarchy without inducing significant bias. This novel sampling scheme is tested with high performance computing resources using biologically relevant problems.

Bayesian Estimation for Stochastic Gene Expression Using Multifidelity Models.

The finite state projection (FSP) approach to solving the chemical master equation has enabled successful inference of discrete stochastic models to predict single-cell gene regulation dynamics. Unfortunately, the FSP approach is highly computationally intensive for all but the simplest models, an issue that is highly problematic when parameter inference and uncertainty quantification takes enormous numbers of parameter evaluations. To address this issue, we propose two new computational methods for the Bayesian inference of stochastic gene expression parameters given single-cell experiments. We formulate and verify an adaptive delayed acceptance Metropolis-Hastings (ADAMH) algorithm to utilize with reduced Krylov-basis projections of the FSP. We then introduce an extension of the ADAMH into a hybrid scheme that consists of an initial phase to construct a reduced model and a faster second phase to sample from the approximate posterior distribution determined by the constructed model. We test and compare both algorithms to an adaptive Metropolis algorithm with full FSP-based likelihood evaluations on three example models and simulated data to show that the new ADAMH variants achieve substantial speedup in comparison to the full FSP approach. By reducing the computational costs of parameter estimation, we expect the ADAMH approach to enable efficient data-driven estimation for more complex gene regulation models.

An adaptive solution to the chemical master equation using tensors.

Solving the chemical master equation directly is difficult due to the curse of dimensionality. We tackle that challenge by a numerical scheme based on the quantized tensor train (QTT) format, which enables us to represent the solution in a compressed form that scales linearly with the dimension. We recast the finite state projection in this QTT framework and allow it to expand adaptively based on proven error criteria. The end result is a QTT-formatted matrix exponential that we evaluate through a combination of the inexact uniformization technique and the alternating minimal energy algorithm. Our method can detect when the equilibrium distribution is reached with an inexpensive test that exploits the structure of the tensor format. We successfully perform numerical tests on high-dimensional problems that had been out of reach for classical approaches.

Computational study of p53 regulation via the chemical master equation.

A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1-16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation. We report numerical results that demonstrate the extend of tackling the CME with this combination of tools.