Time series data provided by single-molecule Förster resonance energy transfer (smFRET) experiments offer the opportunity to infer not only model parameters describing molecular complexes, e.g., rate constants, but also information about the model itself, e.g., the number of conformational states. Resolving whether such states exist or how many of them exist requires a careful approach to the problem of model selection, here meaning discrimination among models with differing numbers of states. The most straightforward approach to model selection generalizes the common idea of maximum likelihood--selecting the most likely parameter values--to maximum evidence: selecting the most likely model. In either case, such an inference presents a tremendous computational challenge, which we here address by exploiting an approximation technique termed variational Bayesian expectation maximization. We demonstrate how this technique can be applied to temporal data such as smFRET time series; show superior statistical consistency relative to the maximum likelihood approach; compare its performance on smFRET data generated from experiments on the ribosome; and illustrate how model selection in such probabilistic or generative modeling can facilitate analysis of closely related temporal data currently prevalent in biophysics. Source code used in this analysis, including a graphical user interface, is available open source via http://vbFRET.sourceforge.net.

Determining the mechanism by which tRNAs rapidly and precisely transit through the ribosomal A, P, and E sites during translation remains a major goal in the study of protein synthesis. Here, we report the real-time dynamics of the L1 stalk, a structural element of the large ribosomal subunit that is implicated in directing tRNA movements during translation. Within pretranslocation ribosomal complexes, the L1 stalk exists in a dynamic equilibrium between open and closed conformations. Binding of elongation factor G (EF-G) shifts this equilibrium toward the closed conformation through one of at least two distinct kinetic mechanisms, where the identity of the P-site tRNA dictates the kinetic route that is taken. Within posttranslocation complexes, L1 stalk dynamics are dependent on the presence and identity of the E-site tRNA. Collectively, our data demonstrate that EF-G and the L1 stalk allosterically collaborate to direct tRNA translocation from the P to the E sites, and suggest a model for the release of E-site tRNA.uring the elongation phase of protein synthesis, the ribosome repetitively cycles through three principal steps: (i) selection of an aminoacyl-transfer RNA (tRNA) at the ribosomal A site (1), (ii) peptidyl transfer from the P site-bound peptidyl-tRNA to the A site-bound aminoacyl-tRNA (2), and (iii) translocation of the messenger RNA (mRNA)-tRNA complex by one codon, effectively moving the P-and A-site tRNAs into the E-and P-sites, respectively (3). Perhaps the most dynamic steps of this cycle are the precisely directed mRNA and tRNA movements that occur during translocation (3-5). Although this step of the elongation cycle is promoted by elongation factor G (EF-G), numerous biochemical (6), structural (7,8), and Förster resonance energy transfer (FRET) (9-15) studies have provided strong evidence that the peptidyl transfer step of the elongation cycle spontaneously triggers an EF-Gindependent structural rearrangement of the ribosomal pretranslocation (PRE) complex that involves movements of the ribosome-bound tRNAs from their classical to their hybridbound configurations (6-10, 12), movement of the ribosomal L1 stalk from an open to a closed conformation (7,8,13,15), and a counterclockwise, ratchet-like rotation of the small ribosomal subunit relative to the large subunit (7,8,11,14).Single-molecule FRET (smFRET) investigations have proven a powerful means for directly investigating the conformational dynamics of PRE complexes. Aided by X-ray and cryogenic electron microscopy (cryo-EM)-derived structural models, several groups have reported kinetic studies of tRNA and ribosome movements within PRE complexes (9, 10, 12-15). tRNA-tRNA smFRET (smFRET tRNA-tRNA ) experiments initially revealed that upon peptidyl transfer, tRNAs enter into a classicalĥ ybrid dynamic equilibrium within PRE complexes (9, 10, 12). More recently, we have used an L1 stalk-tRNA smFRET (smFRET L1-tRNA ) signal to demonstrate that upon peptidyl transfer, a direct L1 stalk-tRNA contact is reversibly established (denoted as L1YtR...

Many single-molecule experiments aim to characterize biomolecular processes in terms of kinetic models that specify the rates of transition between conformational states of the biomolecule. Estimation of these rates often requires analysis of a population of molecules, in which the conformational trajectory of each molecule is represented by a noisy, time-dependent signal trajectory. Although hidden Markov models (HMMs) may be used to infer the conformational trajectories of individual molecules, estimating a consensus kinetic model from the population of inferred conformational trajectories remains a statistically difficult task, as inferred parameters vary widely within a population. Here, we demonstrate how a recently developed empirical Bayesian method for HMMs can be extended to enable a more automated and statistically principled approach to two widely occurring tasks in the analysis of single-molecule fluorescence resonance energy transfer (smFRET) experiments: 1), the characterization of changes in rates across a series of experiments performed under variable conditions; and 2), the detection of degenerate states that exhibit the same FRET efficiency but differ in their rates of transition. We apply this newly developed methodology to two studies of the bacterial ribosome, each exemplary of one of these two analysis tasks. We conclude with a discussion of model-selection techniques for determination of the appropriate number of conformational states. The code used to perform this analysis and a basic graphical user interface front end are available as open source software.

Translocation of transfer RNAs (tRNAs) through the ribosome during protein synthesis involves large-scale structural rearrangements of the ribosome and the ribosome-bound tRNAs that are accompanied by extensive and dynamic remodeling of tRNA-ribosome interactions. The contributions that rearranging individual tRNA-ribosome interactions make to directing tRNA movements during translocation, however, remain largely unknown. To address this question, we have used single-molecule fluorescence resonance energy transfer to characterize the dynamics of ribosomal pre-translocation (PRE) complex analogs carrying either wild-type or systematically mutagenized tRNAs. Our data reveal how specific tRNA-ribosome interactions regulate the rate with which the PRE complex rearranges into a critical, on-pathway translocation intermediate and how these interactions control the stability of the resulting configuration. More interestingly, our results suggest that the conformational flexibility of the tRNA molecule itself plays a crucial role in directing the structural dynamics of the PRE complex during translocation.

BackgroundThe recent explosion of experimental techniques in single molecule biophysics has generated a variety of novel time series data requiring equally novel computational tools for analysis and inference. This article describes in general terms how graphical modeling may be used to learn from biophysical time series data using the variational Bayesian expectation maximization algorithm (VBEM). The discussion is illustrated by the example of single-molecule fluorescence resonance energy transfer (smFRET) versus time data, where the smFRET time series is modeled as a hidden Markov model (HMM) with Gaussian observables. A detailed description of smFRET is provided as well.ResultsThe VBEM algorithm returns the model’s evidence and an approximating posterior parameter distribution given the data. The former provides a metric for model selection via maximum evidence (ME), and the latter a description of the model’s parameters learned from the data. ME/VBEM provide several advantages over the more commonly used approach of maximum likelihood (ML) optimized by the expectation maximization (EM) algorithm, the most important being a natural form of model selection and a well-posed (non-divergent) optimization problem.ConclusionsThe results demonstrate the utility of graphical modeling for inference of dynamic processes in single molecule biophysics.

Chemical complementation was used to make a transcription factor circuit capable of performing complex Boolean logic.

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