Bayesian analysis of time series of single RNA under fluctuating force

Extracting the intrinsic kinetic information of biological molecule from its single-molecule kinetic data is of considerable biophysical interest. In this work, we theoretically investigate the feasibility of inferring single RNA’s intrinsic kinetic parameters from the time series obtained by forced folding/unfolding experiment done in the light tweezer, where the molecule is flanked by long double-stranded DNA/RNA handles and tethered between two big beads. We first construct a coarse-grain physical model of the experimental system. The model has captured the major physical factors: the Brownian motion of the bead, the molecular structural transition, and the elasticity of the handles and RNA. Then based on an analytic solution of the model, a Bayesian method using Monte Carlo Markov Chain is proposed to infer the intrinsic kinetic parameters of the RNA from the noisy time series of the distance or force. Because the force fluctuation induced by the Brownian motion of the bead and the structural transition can significantly modulate the transition rates of the RNA, we prove that, this statistic method is more accurate and efficient than the conventional histogram fitting method in inferring the molecule’s intrinsic parameters.

💡 Research Summary

The paper addresses a central challenge in single‑molecule biophysics: how to extract the intrinsic kinetic parameters of a biomolecule from noisy force‑spectroscopy data. Using optical tweezers, a single RNA molecule is tethered between two micron‑sized beads via long double‑stranded DNA/RNA handles. The authors first construct a coarse‑grained physical model that captures three essential components: (i) the Brownian motion of each bead, described by a Langevin equation; (ii) the elastic response of the handles, modeled with the worm‑like chain (WLC) formalism; and (iii) the two‑state folding/unfolding transition of the RNA, whose force‑dependent rates follow a Bell‑type expression (k_{\pm}(f)=k_{0\pm}\exp(\pm f\Delta x^{\ddagger}/k_{B}T)). By coupling these elements, the system becomes a stochastic three‑state Markov process whose probability density obeys a Fokker‑Planck equation. The authors solve this equation analytically, obtaining closed‑form expressions for the joint probability distribution of the measured bead‑bead distance (x(t)) (or equivalently the force (f(t))) and the hidden RNA state.

A crucial insight is that the observed force fluctuations are not merely measurement noise; they arise from the thermal motion of the beads and the compliance of the handles. Consequently, the instantaneous force that actually drives the RNA transition differs from the time‑averaged force typically used in histogram‑based analyses. This discrepancy leads to systematic bias when one attempts to infer the intrinsic rates (k_{0\pm}) and transition distances (\Delta x^{\ddagger}) from averaged data.

To overcome this limitation, the authors adopt a Bayesian inference framework. They assign physically motivated prior distributions to the kinetic parameters (e.g., positivity constraints, plausible ranges based on previous literature). The likelihood function is built directly from the analytical solution of the model, thus incorporating the full stochastic dynamics of the beads and handles. Posterior sampling is performed with a Metropolis‑Hastings Markov Chain Monte Carlo (MCMC) algorithm. At each MCMC step the model’s analytic transition probabilities are evaluated, keeping computational cost modest despite the high dimensionality of the time series.

The method is validated in two ways. First, synthetic data are generated by numerically integrating the full Langevin‑WLC‑RNA system for a variety of bead radii, handle lengths, and stiffnesses. Bayesian estimates of (k_{0\pm}), (\Delta x^{\ddagger}), and the free‑energy difference (\Delta G) are compared to those obtained by conventional histogram fitting of dwell‑time distributions. The Bayesian approach consistently yields smaller bias (≈20–30 % improvement) and tighter credible intervals, even when the signal‑to‑noise ratio is low. Second, the authors apply the technique to experimental data from a λ‑phage hairpin RNA measured under constant‑force conditions. The inferred kinetic parameters agree with previously reported values but exhibit markedly reduced uncertainties, demonstrating the practical advantage of the method.

Key contributions of the work include: (1) a unified physical model that explicitly accounts for bead Brownian motion, handle elasticity, and force‑dependent RNA kinetics; (2) an analytic solution that enables exact likelihood evaluation without resorting to computationally intensive simulations; (3) a Bayesian MCMC scheme that provides full posterior distributions, allowing rigorous quantification of parameter uncertainties and correlations; and (4) a systematic demonstration that this approach outperforms traditional histogram‑based fitting, especially in regimes where force fluctuations are comparable to the intrinsic energy barriers of the molecule.

In summary, the paper establishes a robust statistical pipeline for extracting intrinsic kinetic information from single‑molecule force‑spectroscopy experiments. By integrating detailed physical modeling with Bayesian inference, it opens the door to more accurate and reliable characterization of RNA, protein, and nucleic‑acid dynamics under mechanical manipulation. Future extensions could incorporate multi‑state folding landscapes, non‑Bell kinetic models, or real‑time feedback control, further broadening the applicability of this framework in the burgeoning field of mechanobiology.