Learning Rates and States from Biophysical Time Series: A Bayesian Approach to Model Selection and Single-Molecule FRET Data

Jonathan E. Bronson; Jingyi Fei; Jake M. Hofman; Ruben L. Gonzalez,; Chris H. Wiggins

Source-linked AI summary

Learning Rates and States from Biophysical Time Series: A Bayesian Approach to Model Selection and Single-Molecule FRET Data

Jonathan E. Bronson, Jingyi Fei, Jake M. Hofman, Ruben L. Gonzalez,, Chris H. Wiggins

arXiv:0907.3156v1Jul 20, 2009q-bio.QMq-bio.BM

TL;DR

The paper addresses how to infer both parameters and unknown conformational-state counts from smFRET time series without relying on overfitting-prone model selection. It applies maximum evidence with variational Bayes, finding improved state-selection consistency and useful posterior and trace inference across biophysical time-series examples.

Problem
Unknown numbers of conformational states make smFRET model selection difficult, while maximum-likelihood comparison can overfit when model complexity is not fixed.
Method
The paper uses maximum evidence with variational Bayes to select state-number models and approximate parameter and hidden-state posteriors.
Results
Variational maximum-evidence inference showed comparable or superior accuracy, substantially less post-processing, and reduced overfitting relative to maximum likelihood in the analyzed smFRET data.
Takeaways & Limitations
The framework supports individual-trace state identification and kinetic analysis and can generalize to biological time series with unknown numbers of molecular states.
Takeaways & Limitations
A detected intermediate state may reflect camera time-averaging artifacts rather than a biologically relevant conformation.

Abstract

from arXiv · show

Time series data provided by single-molecule Forster resonance energy transfer (sm-FRET) 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 or how many of such states exist requires a careful approach to the problem of model selection, here meaning discriminating 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 inference presents a tremendous computational challenge, which we here address by exploiting an approximation technique termed variational Bayes. 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; 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

1 Introduction

The paper frames smFRET time series as a route to infer both biophysical parameters and the number of molecular conformational states. It proposes maximum evidence and variational Bayes to perform model selection while analyzing temporal data.

smFRET time series can support inference of mechanical, probabilistic, and kinetic parameters in increasingly complex biological systems.
The central model-selection task is revealing how many enzymatic conformational states are present in individual smFRET data.
FRET efficiency decreases with fluorophore distance, allowing paired donor and acceptor intensities to serve as a proxy for molecular separation.
A time-varying FRET ratio reflects the geometric relationship between fluorophores and can be modeled using hidden molecular conformations.
The manuscript develops variational Bayes to estimate parameters while selecting the optimal number of hidden-state values.

2 Parameter and model selection

The paper contrasts maximum likelihood, which can overfit when model complexity is unknown, with Bayesian maximum evidence, which selects models by integrating over parameter uncertainty. Variational Bayes makes evidence and posterior inference tractable for hidden-state time-series models.

Maximum likelihood is useful for estimating parameters under a fixed model but generally overfits when competing model complexities must be compared.
2.2 Maximum evidence inference: Maximum evidence selects the model with the highest marginal likelihood while explicitly penalizing excessive complexity through parameter integration.
2.2 Maximum evidence inference: Model selection introduces K as an index over candidate models, including the number of conformational states, and chooses K* by maximizing p(y|K).
2.2 Maximum evidence inference: The posterior distribution p(ϑ|y,K) represents uncertainty over parameter settings after observing data and combines likelihood and prior information through Bayes’ rule.
2.3 Variational approximate inference: Variational Bayes approximates both the evidence and posterior when exact computation is analytically or numerically intractable.
2.3 Variational approximate inference: The variational method uses a conditionally independent approximating distribution and iterative updates to make inference tractable.
2.3 Variational approximate inference: Expectation-maximization procedures can depend on initialization, and practitioners may use one chosen starting condition instead of optimizing over restarts.

3 Statistical inference and FRET

The smFRET analysis uses hidden Markov models to represent molecular conformations as latent states generating observed FRET ratios. Variational inference estimates state and parameter distributions, while idealized traces support state counting and transition-rate analysis.

3.1 Hidden Markov modeling: An HMM models the observed FRET ratio as conditionally dependent on a hidden conformational process with K possible states.
3.1 Hidden Markov modeling: The HMM parameter set includes initial-state probabilities, a K × K transition matrix, and Gaussian means and precisions for state emissions.
3.1 Hidden Markov modeling: The model evidence is obtained by multiplying likelihood and priors and marginalizing over hidden states and parameters.
3.1 Hidden Markov modeling: Prior parameter settings had little discernible effect on inference across the experiments and tested prior ranges.
3.1 Hidden Markov modeling: The forward-backward algorithm makes variational evidence computation feasible with O(K^2T) operations rather than direct summation requiring O(K^T).
3.2 Rates from states: Because state labels vary across individually fitted traces, transition rates are inferred from idealized traces rather than directly from parameter distributions.

4 Numerical experiments

Numerical experiments compare maximum evidence (ME) with maximum likelihood (ML) on synthetic smFRET traces whose true states are known. ME generally selects the correct state number more reliably, avoids overfitting, and is especially stronger for fast transitions.

Maximum likelihood vs maximum evidence: ME correctly retained three states when five were allowed, whereas ML overfit the same synthetic trace.The trace was generated with K0 = 3 states centered at 0.41, 0.61, and 0.81 FRET.
Maximum likelihood vs maximum evidence: The evidence peaked at K = 3, while likelihood increased as additional states were permitted.This illustrates why evidence supports model selection when model complexity is unknown.
Statistical validation: Across 2,000 traces, ME overfit 1 and underfit 232, whereas ML overfit 767 and underfit 391.ME overfit essentially none of the individual traces, while ML overfit 38%.
Statistical validation: ME inferred the correct number of states more often than ML except for the noisiest fast-transitioning traces.Both methods performed better on low-noise traces, while ME underfitting was concentrated at noise above 0.09 FRET.
Statistical validation: For fast transitions, ME inferred true trajectories 1.5–1.6 times better and detected transitions with 2.7–12.5-fold greater sensitivity.For slow transitions, the methods performed roughly equally on the other trajectory probabilities.
Statistical validation: ME was at least as accurate as ML for slow transitions and more accurate for fast transitions when the state number was unknown.The stronger fast-transition performance is relevant to detecting transient biophysical states.

5 Results

Experimental smFRET analyses show that ME and ML can agree on transition rates while differing in inferred state numbers. ME identifies short-lived intermediate-like states, including a blur state attributable primarily to time averaging, whereas ML often suppresses them.

Experimental state and rate inference: For L1–L9 data, ML inferred four or five states in 20.1% ± 3.7% of traces, compared with 0.9% ± 0.5% for ME.Both methods showed the expected two FRET states plus a photobleached state, but ML required more post-processing for transition-rate extraction.
Experimental state and rate inference: ME and ML produced very good overall agreement for k_close and k_open, with indistinguishable values in the slower-transitioning data.Their values differed slightly for the relatively fast-transitioning PMN_fMet+EFG data.
Experimental state and rate inference: ME inferred a third smFRET_L1−tRNA state at f_mid ≡ 0.35 FRET, while ML inferred only the low and high states.Approximately 46% of ME-analyzed transitions involved f_mid, and approximately 75% of f_mid observations lasted one integration-time point.
Experimental state and rate inference: The putative f_mid state could represent either a short-lived intermediate conformation or a CCD time-binning artifact.A transition halfway through a 50 msec bin can average low- and high-state photons into a midpoint datum.
Experimental state and rate inference: Changing integration time tests the two explanations by comparing transition involvement and f_mid dwell duration.A true intermediate predicts integration-time-independent dwell duration, whereas time averaging predicts integration-time-dependent transition involvement.
Experimental state and rate inference: The f_mid transition frequency increased with integration time, while its consecutive-point dwell duration remained remarkably insensitive.These observations support interpreting f_mid primarily as a camera-blurring artifact, termed the blur state.
Experimental state and rate inference: ME’s detection of the blur state aligns with synthetic results showing stronger detection of true states, particularly in fast-transitioning data.ML’s suppression of blur states may also cause it to overlook bona fide short-lived intermediate FRET states.

6 Conclusions

ME with variational Bayes selects the number of smFRET states for individual traces while estimating parameters and idealized trajectories. The analyses report improved accuracy, reduced overfitting and post-processing, and applicability beyond smFRET.

Variational Bayes provides q*, an estimate of the true-parameter and idealized-trace posterior, enabling kinetic-parameter analysis for individual traces.
ME idealized trajectories are visually similar to ML trajectories and achieve comparable or superior accuracy on synthetic data emulating experiments.
ME usually infers the correct state count, reducing post-processing because similarly valued states within a trace need not be combined.
ME detected a short-lived blur state attributed to camera time averaging, while ML did not identify it; such detections may reveal missed biological intermediates.
The method is presented as applicable to biological time series beyond smFRET, including motor trajectories, binding studies, and molecular dynamics simulations.

C Proof of variational relation

The appendix derives a variational relation connecting model evidence, free energy, and KL divergence. This establishes evidence approximation as optimization over a tractable distribution q.

The proof begins from the evidence p(y|K) and rewrites it using normalized probability distributions and conditional probability.
Separating logarithmic terms identifies a KL divergence between q(z,ϑ) and the posterior p(z,ϑ|y,K), alongside a negative free-energy term.
The resulting variational relation is completed in Eq. 6, with the free-energy construction providing its statistical-physics interpretation.
Because KL divergence is non-negative, free energy is bounded by log-evidence, so evidence approximation becomes the search for q closest to the posterior in KL sense.

S.1 Methods

The methods compare ML and ME inference settings for smFRET traces and describe how transition rates are extracted from idealized trajectories. ML uses fixed state settings, whereas ME evaluates multiple state counts.

ML inference settings: ML analyses use Kmax + 2 states, with Kmax set from the true synthetic-state count or to 3 for experimental data, without added complexity control.
ML inference settings: ML parameters were obtained from one initial parameter guess, despite expectation-maximization converging only to a local optimum.
ML inference settings: For real-valued emissions with inferred state widths, ML optimization is ill-posed because assigning one observation to a zero-uncertainty state is infinitely likely.
ME inference settings: ME evaluates K = 1, 2, …, Kmax + 2 using 25 synthetic-data or 100 experimental-data initializations, retaining parameters from the best optimization.
Rate extraction: Transition rates are extracted by histogramming idealized traces, defining acceptable FRET ranges, recording dwell times, and fitting their cumulative distributions.

S.1.4 Generating synthetic data

Synthetic traces use a two-dimensional Gaussian-output HMM designed to approximate experimental fluorophore data rather than exactly match the scalar emission model. Noise is varied across a broad range, with Bayesian priors specified for model parameters.

Synthetic traces are generated from a hidden Markov model with 2D Gaussian outputs representing two fluorophore colors.
The two color means sum to 1000 in every state, while variances and covariance are sampled from uniform distributions tied to the means.
Noise is increased by multiplying each hidden state's covariance matrix by ten log-linearly spaced constants from 1 to 100, producing 1D FRET noise of approximately 0.02 < σ < 1.4.
For model evidence, the initial-state vector and transition-matrix rows use Dirichlet distributions, while each mean-precision pair uses a Gaussian-Gamma distribution.
The parameter-distribution hyperparameters are chosen to be weakly influential and consistent with experimental data, favoring equally occupied and equally transitionable hidden states.

S.2.3 Sensitivity to hyperparameter settings

ME inference was tested against changes in hyperparameter settings using synthetic and experimental traces. Transition-rate estimates remained stable under the tested prior changes, while state-identification sensitivity varied with trace difficulty.

Sensitivity analysis: Halving and doubling hyperparameters were used to recompute model evidence for synthetic two- and three-state traces.The prior on each Gaussian mean was held fixed because its value was set to 0.5.
Synthetic traces: The tested sensitivity analyses covered fast- and slow-transitioning two-state traces and fast- and slow-transitioning three-state traces.These cases correspond to the supplementary synthetic analyses shown in Figures S.1–S.4.
Experimental data: ME transition-rate estimates stayed within error of one another under default and strongly diagonal transition-matrix priors.The diagonal prior encoded a roughly tenfold preference for remaining in the current state over transitioning.
Reporting boundary: Rates were reported as averages and standard deviations from three or four independent data sets, without photobleaching correction.This reporting convention applies to the associated transition-rate measurements.

S.3 Synthetic validation – 2 and 4 state traces

Synthetic analyses of two- and four-state traces produced results qualitatively similar to the three-state example. Inference became less accurate at lower noise as additional states made the FRET levels more closely spaced.

Synthetic validation: Two- and four-state synthetic traces showed results qualitatively similar to those from the three-state analysis.The traces included fast- and slow-transitioning two-state cases and fast-transitioning four-state cases.
Resolution with state number: Inference accuracy decreased at lower noise levels as more FRET states were added to the traces.Additional states were more closely spaced and therefore harder to resolve.

S.4 Blur state TDPs

Supplementary analyses examined synthetic two- and four-state traces and transition-density plots for smFRET data. The supplied passages identify the analyzed trace sets and ME blur-state correction but do not state a substantive TDP outcome.

Synthetic traces: Synthetic results were separately reported for two-state and four-state traces.The supplied figure labels identify the two-state results in Figure S.5 and the four-state results in Figure S.6.
Blur-state handling: The ME analysis included correction for the presence of a blur state.The supplied passage gives no further description of the correction procedure.
Transition-density plots: Transition-density plots compared smFRET data derived from ME and ML analyses across CCD integration times.The plots use starting and ending FRET values as axes and estimate transition density with kernel density estimation.