Analyzing Single-Molecule Protein Transportation Experiments via Hierarchical Hidden Markov Models

Panel Data Analysis via Mechanistic Models

Panel data, also known as longitudinal data, consist of a collection of time series. Each time series, which could itself be multivariate, comprises a sequence of measurements taken on a distinct unit. Mechanistic modeling involves writing down scientifically motivated equations describing the collection of dynamic systems giving rise to the observations on each unit. A defining characteristic of panel systems is that the dynamic interaction between units should be negligible. Panel models therefore consist of a collection of independent stochastic processes, generally linked through shared parameters while also having unit-specific parameters. To give the scientist flexibility in model specification, we are motivated to develop a framework for inference on panel data permitting the consideration of arbitrary nonlinear, partially observed panel models. We build on iterated filtering techniques that provide likelihood-based inference on nonlinear partially observed Markov process models for time series data. Our methodology depends on the latent Markov process only through simulation; this plug-and-play property ensures applicability to a large class of models. We demonstrate our methodology on a toy example and two epidemiological case studies. We address inferential and computational issues arising due to the combination of model complexity and dataset size. Supplementary materials for this article are available online.

Bayesian-Estimated Hierarchical HMMs Enable Robust Analysis of Single-Molecule Kinetic Heterogeneity

Single-molecule kinetic experiments allow the reaction trajectories of individual biomolecules to be directly observed, eliminating the effects of population averaging and providing a powerful approach for elucidating the kinetic mechanisms of biomolecular processes. A major challenge to the analysis and interpretation of these experiments, however, is the kinetic heterogeneity that almost universally complicates the recorded single-molecule signal versus time trajectories (i.e., signal trajectories). Such heterogeneity manifests as changes and/or differences in the transition rates that are observed within individual signal trajectories or across a population of signal trajectories. Because characterizing kinetic heterogeneity can provide critical mechanistic information, we have developed a computational method that effectively and comprehensively enables such analysis. To this end, we have developed a computational algorithm and software program, hFRET, that uses the variational approximation for Bayesian inference to estimate the parameters of a hierarchical hidden Markov model, thereby enabling robust identification and characterization of kinetic heterogeneity. Using simulated signal trajectories, we demonstrate the ability of hFRET to accurately and precisely characterize kinetic heterogeneity. In addition, we use hFRET to analyze experimentally recorded signal trajectories reporting on the conformational dynamics of ribosomal pre-translocation (PRE) complexes. The results of our analyses demonstrate that PRE complexes exhibit kinetic heterogeneity, reveal the physical origins of this heterogeneity, and allow us to expand the current model of PRE complex dynamics. The methods described here can be applied to signal trajectories generated using any type of signal and can be easily extended to the analysis of signal trajectories exhibiting more complex kinetic behaviors. Moreover, variations of our approach can be easily developed to integrate kinetic data obtained from different experimental constructs and/or from molecular dynamics simulations of a biomolecule of interest.

Analyzing dwell times with the Generalized Method of Moments

The Generalized Method of Moments (GMM) is a statistical method for the analysis of samples from random processes. First developed for the analysis of econometric data, the method is here formulated to extract hidden kinetic parameters from measurements of single molecule dwell times. Our method is based on the analysis of cumulants of the measured dwell times. We develop a general form of an objective function whose minimization can return estimates of decay parameters for any number of intermediates directly from the data. We test the performance of our technique using both simulated and experimental data. We also compare the performance of our method to nonlinear least-squares minimization (NL-LSQM), a commonly-used technique for analysis of single molecule dwell times. Our findings indicate that the GMM performs comparably to NL-LSQM over most of the parameter range we explore. It offers some benefits compared with NL-LSQM in that it does not require binning, exhibits slightly lower bias and variance with small sample sizes (N<20), and is somewhat superior in identifying fast decay times with these same low count data sets. Additionally, a comparison with the Classical Method of Moments (CMM) shows that the CMM can fail in many cases, whereas the GMM always returns estimates. Our results show that the GMM can be a useful tool and complements standard approaches to analysis of single molecule dwell times.

Single-Molecule Analysis beyond Dwell Times: Demonstration and Assessment in and out of Equilibrium

We present a simple and robust technique for extracting kinetic rate models and thermodynamic quantities from single-molecule time traces. Single-molecule analysis of complex kinetic sequences (SMACKS) is a maximum-likelihood approach that resolves all statistically relevant rates and also their uncertainties. This is achieved by optimizing one global kinetic model based on the complete data set while allowing for experimental variations between individual trajectories. In contrast to dwell-time analysis, which is the current standard method, SMACKS includes every experimental data point, not only dwell times. As a result, it works as well for long trajectories as for an equivalent set of short ones. In addition, the previous systematic overestimation of fast over slow rates is solved. We demonstrate the power of SMACKS on the kinetics of the multidomain protein Hsp90 measured by single-molecule Förster resonance energy transfer. Experiments in and out of equilibrium are analyzed and compared to simulations, shedding new light on the role of Hsp90's ATPase function. SMACKS resolves accurate rate models even if states cause indistinguishable signals. Thereby, it pushes the boundaries of single-molecule kinetics beyond those of current methods.

Sparse Regression Based Structure Learning of Stochastic Reaction Networks from Single Cell Snapshot Time Series

Stochastic chemical reaction networks constitute a model class to quantitatively describe dynamics and cell-to-cell variability in biological systems. The topology of these networks typically is only partially characterized due to experimental limitations. Current approaches for refining network topology are based on the explicit enumeration of alternative topologies and are therefore restricted to small problem instances with almost complete knowledge. We propose the reactionet lasso, a computational procedure that derives a stepwise sparse regression approach on the basis of the Chemical Master Equation, enabling large-scale structure learning for reaction networks by implicitly accounting for billions of topology variants. We have assessed the structure learning capabilities of the reactionet lasso on synthetic data for the complete TRAIL induced apoptosis signaling cascade comprising 70 reactions. We find that the reactionet lasso is able to efficiently recover the structure of these reaction systems, ab initio, with high sensitivity and specificity. With only < 1% false discoveries, the reactionet lasso is able to recover 45% of all true reactions ab initio among > 6000 possible reactions and over 10²⁰⁰⁰ network topologies. In conjunction with information rich single cell technologies such as single cell RNA sequencing or mass cytometry, the reactionet lasso will enable large-scale structure learning, particularly in areas with partial network structure knowledge, such as cancer biology, and thereby enable the detection of pathological alterations of reaction networks. We provide software to allow for wide applicability of the reactionet lasso.

Virtually all biological processes are driven by biochemical reactions. However, their quantitative description in terms of stochastic chemical reaction networks is often precluded by the computational difficulty of structure learning, i.e. the identification of biologically active reaction networks among the combinatorially many possible topologies. This work describes the reactionet lasso, a structure learning approach that takes advantage of novel, information-rich single cell data and a tractable problem formulation to achieve structure learning for problem instances hundreds of orders of magnitude larger than previously reported. This approach opens the prospect of obtaining quantitative and predictive reaction models in many areas of biology and medicine, and in particular areas such as cancer biology, which are characterized by significant system alterations and many unknown reactions.