Learning rates and states from biophysical time series: a Bayesian approach to model selection and single-molecule FRET data

Abstract

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.

Figure 1

A single (synthetic) FRET trace analyzed by ME and ML. The trace contains three hidden states. (A) (Upper) Idealized traces inferred by ME when K = 1, K = 3, and K = 5, as well as the corresponding log(evidence) for the inference. The data are underresolved when K = 1, but for both K = 3 and K = 5, the correct number of states is populated. (Lower) Idealized traces inferred by ML when K = 1, K = 3, and K = 5, as well as the corresponding log(likelihood). Inferences when K = 1 and K = 3 are the same as for ME, but the data are overfit when K = 5. (B) The log (evidence) from ME (black) and log likelihood from ML (gray) for 1 ≤ K ≤ 10. The evidence is correctly maximized for K = 3, but the likelihood increases monotonically.

Figure 2

Comparison of ME and ML as a function of increasing hidden-state noise. Fast-transitioning (hidden-state mean lifetime of four time steps) and slow-transitioning (hidden-state mean lifetime of 15 time steps) traces were created and analyzed separately. Each data point represents the average value taken over 100 traces. (Upper left) $p\left(\right|\widehat{\mathbf{z}}|=|{\mathbf{z}}_0\left|\right)$: the probability in any trace of inferring the correct number of states. (Upper right) $p(\widehat{\mathbf{z}}={\mathbf{z}}_0)$: the probability in any trace at any time that a transition is inferred given that a transition actually occurred. (Lower left) Sensitivity to true transitions: the fraction of time the correct FRET state was inferred during FRET trajectories. (Lower right) Specificity of inferred transitions: the probability in any trace at any time that no transition is inferred given that no transition actually occurred. Error bars on all plots were omitted for clarity and because the data plotted represent mean success rates for Bernoulli processes (and, therefore, determine the variances of the data as well).

Figure 3

Analysis of the smFRET_L1–tRNAf_mid state. (A) A representative smFRET_L1–tRNA trace idealized by ME, taken from the 50-ms exposure time data set. Both the observed data (blue) and idealized path (red) are shown. Individual data points, real and idealized, are shown as Xs. To emphasize the data at or near f_mid, the Xs are enlarged and the observed and idealized data are shown in black and green, respectively. (B) Bar graph of the percentages of transitions to or from the f_mid state under 25 ms, 50 ms, and 100 ms CCD integration time. (C) Normalized population histograms of dwell time spent at the f_mid state under 25 ms, 50 ms, and 100 ms CCD integration time.