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Quantitative and Predictive Folding Models from Limited Single-Molecule Data Using Simulation-Based Inference
Phys. Rev. Lett. 137, 088402 – Published 19 August, 2026
DOI: https://doi.org/10.1103/yltb-6jkj
Abstract
Single-molecule force spectroscopy resolves folding dynamics one molecule at a time, but extracting quantitative free-energy landscapes typically requires extensive datasets and careful instrument calibration to disentangle the molecule from linker and apparatus artifacts. We introduce a simulation-based inference framework that combines physics-based modeling with deep learning to recover the Bayesian posterior of a folding model directly from a single short trajectory. A 2-s constant-force measurement of a DNA hairpin is sufficient to reconstruct the folding landscape, matching deconvolution baselines that require 20–100 times more data and eliminating separate linker or instrument characterization. The same approach recovers four metastable states of a riboswitch aptamer from a single 5-s trajectory. Every parameter, including diffusion coefficients and linker stiffness, is included in the calibrated posterior. The current implementation assumes one-dimensional Markovian dynamics, but more complex models can be substituted within the same framework, opening single-molecule force spectroscopy to high-throughput parallel experiments and to systems where extensive data collection is impractical.
Physics Subject Headings (PhySH)
- Biochemistry
- Biomolecular processes
- Biomolecular self-assembly
- Chemical kinetics, dynamics & catalysis
- Diffusion
- Noise-induced transitions
- Polymer conformation changes
- Protein dynamics, structure & function
- Stochastic inference
- Biomolecules
- Macromolecules
- Nucleic acids
- Artificial neural networks
- Markovian processes
- Single molecule techniques
Article Text
Supplemental Material
References (55)
- K. A. Dill and J. L. MacCallum, The protein-folding problem, 50 years on, Science 338, 1042 (2012).
- K. A. Dill and H. S. Chan, From Levinthal to pathways to funnels, Nat. Struct. Biol. 4, 10 (1997).
- J. Buchner and T. Kiefhaber, Protein Folding Handbook (Wiley-VCH, Weinheim, 2005), Vol. 3.
- R. Petrosyan, A. Narayan, and M. T. Woodside, Single-molecule force spectroscopy of protein folding, J. Mol. Biol. 433 (2021).
- K. C. Neuman and A. Nagy, Single-molecule force spectroscopy: Optical tweezers, magnetic tweezers and atomic force microscopy, Nat. Methods 5, 491 (2008).
- G. Hummer and A. Szabo, Free energy reconstruction from nonequilibrium single-molecule pulling experiments, Proc. Natl. Acad. Sci. U.S.A. 98, 3658 (2001).
- M. T. Woodside, P. C. Anthony, W. M. Behnke-Parks, K. Larizadeh, D. Herschlag, and S. M. Block, Direct measurement of the full, sequence-dependent folding landscape of a nucleic acid, Science 314, 1001 (2006).
- O. K. Dudko, G. Hummer, and A. Szabo, Intrinsic rates and activation free energies from single-molecule pulling experiments, Phys. Rev. Lett. 96, 108101 (2006).
- O. K. Dudko, G. Hummer, and A. Szabo, Theory, analysis, and interpretation of single-molecule force spectroscopy experiments, Proc. Natl. Acad. Sci. U.S.A. 105, 15755 (2008).
- G. Hummer and A. Szabo, Free energy profiles from single-molecule pulling experiments, Proc. Natl. Acad. Sci. U.S.A. 107, 21441 (2010).
- M. T. Woodside and S. M. Block, Reconstructing folding energy landscapes by single-molecule force spectroscopy, Annu. Rev. Biophys. 43, 19 (2014).
- J. C. M. Gebhardt, T. Bornschlögl, and M. Rief, Full distance-resolved folding energy landscape of one single protein molecule, Proc. Natl. Acad. Sci. U.S.A. 107, 2013 (2010).
- R. Walder, W. J. Van Patten, D. B. Ritchie, R. K. Montange, T. W. Miller, M. T. Woodside, and T. T. Perkins, High-precision single-molecule characterization of the folding of an HIV RNA hairpin by atomic force microscopy, Nano Lett. 18, 6318 (2018).
- A. N. Gupta, A. Vincent, K. Neupane, H. Yu, F. Wang, and M. T. Woodside, Experimental validation of free-energy-landscape reconstruction from non-equilibrium single-molecule force spectroscopy measurements, Nat. Phys. 7, 631 (2011).
- P. Cossio, G. Hummer, and A. Szabo, On artifacts in single-molecule force spectroscopy, Proc. Natl. Acad. Sci. U.S.A. 112, 14248 (2015).
- K. Neupane and M. T. Woodside, Quantifying instrumental artifacts in folding kinetics measured by single-molecule force spectroscopy, Biophys. J. 111, 283 (2016).
- M. Hinczewski, C. M. Gebhardt, M. Rief, and D. Thirumalai, From mechanical folding trajectories to intrinsic energy landscapes of biopolymers, Proc. Natl. Acad. Sci. U.S.A. 110, 4500 (2013).
- L. Dingeldein, P. Cossio, and R. Covino, Simulation-based inference of single-molecule force spectroscopy, Mach. Learn. 4, 025009 (2022).
- R. Covino, M. T. Woodside, G. Hummer, A. Szabo, and P. Cossio, Molecular free energy profiles from force spectroscopy experiments by inversion of observed committors, J. Chem. Phys. 151, 154115 (2019).
- L. Dingeldein, P. Cossio, and R. Covino, Simulation-based inference of single-molecule experiments, Curr. Opin. Struct. Biol. 91, 102988 (2025).
- K. Cranmer, J. Brehmer, and G. Louppe, The frontier of simulation-based inference, Proc. Natl. Acad. Sci. U.S.A. 117, 30055 (2020).
- G. Papamakarios and I. Murray, Fast -free inference of simulation models with Bayesian conditional density estimation, Adv. Neural Inf. Process. Syst. 29 (2016).
- G. Papamakarios, D. Sterratt, and I. Murray, Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows, in The 22nd International Conference on Artificial Intelligence and Statistics (2019), pp. 837–848.
- C. Durkan, I. Murray, and G. Papamakarios, On contrastive learning for likelihood-free inference, in International Conference on Machine Learning (2020), pp. 2771–2781.
- M. Dax, S. R. Green, J. Gair, J. H. Macke, A. Buonanno, and B. Schölkopf, Real-time gravitational wave science with neural posterior estimation, Phys. Rev. Lett. 127, 241103 (2021).
- B. Régaldo-Saint Blancard, C. H. Hahn, S. Ho, J. Hou, P. Lemos, E. Massara, C. Modi, A. M. Dizgah, L. Parker, Y. Yao, and M. Eickenberg, Galaxy clustering analysis with simbig and the wavelet scattering transform, Phys. Rev. D 109, 083535 (2024).
- R. Gao, M. Deistler, A. Schulz, P. J. Gonçalves, and J. H. Macke, Deep inverse modeling reveals dynamic-dependent invariances in neural circuit mechanisms, Biorxiv (2024).
- J.-M. Lueckmann, P. J. Goncalves, G. Bassetto, K. Öcal, M. Nonnenmacher, and J. H. Macke, Flexible statistical inference for mechanistic models of neural dynamics, Adv. Neural Inf. Process. Syst. 30, 1289 (2017).
- J.-M. Lueckmann, J. Boelts, D. Greenberg, P. Goncalves, and J. Macke, Benchmarking simulation-based inference, in International Conference on Artificial Intelligence and Statistics (PMLR, San Diego, California, USA, 2021), pp. 343–351.
- K. Neupane, A. P. Manuel, J. Lambert, and M. T. Woodside, Transition-path probability as a test of reaction-coordinate quality reveals DNA hairpin folding is a one-dimensional diffusive process, J. Phys. Chem. Lett. 6, 1005 (2015).
- K. Neupane, D. B. Ritchie, H. Yu, Daniel A. Foster, F. Wang, and M. T. Woodside, Transition path times for nucleic acid folding determined from energy-landscape analysis of single-molecule trajectories, Phys. Rev. Lett. 109, 068102 (2012).
- A. P. Manuel, J. Lambert, and M. T. Woodside, Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories, Proc. Natl. Acad. Sci. U.S.A. 112, 7183 (2015).
- J. Liphardt, B. Onoa, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, Reversible unfolding of single RNA molecules by mechanical force, Science 292, 733 (2001).
- G. Varani, Exceptionally stable nucleic acid hairpins, Annu. Rev. Biophys. Biomol. Struct. 24, 379 (1995).
- E. K. Davydova, T. J. Santangelo, and L. B. Rothman-Denes, Bacteriophage N4 virion RNA polymerase interaction with its promoter DNA hairpin, Proc. Natl. Acad. Sci. U.S.A. 104, 7033 (2007).
- K. Neupane, A. P. Manuel, and M. T. Woodside, Protein folding trajectories can be described quantitatively by one-dimensional diffusion over measured energy landscapes, Nat. Phys. 12, 700 (2016).
- M. C. Engel, D. B. Ritchie, Daniel A. N. Foster, Kevin S. D. Beach, and M. T. Woodside, Reconstructing folding energy landscape profiles from nonequilibrium pulling curves with an inverse Weierstrass integral transform, Phys. Rev. Lett. 113, 238104 (2014).
- A. Lyons, A. Devi, N. Q. Hoffer, and M. T. Woodside, Quantifying the properties of nonproductive attempts at thermally activated energy-barrier crossing through direct observation, Phys. Rev. X 14, 011017 (2024).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/yltb-6jkj for details of the simulator, the simulation-based inference framework, and experimental setup, which includes Refs. [40–45].
- C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios, Neural spline flows, Adv. Neural Inf. Process. Syst. 32, 7509 (2019).
- G. Papamakarios, T. Pavlakou, and I. Murray, Masked autoregressive flow for density estimation, Adv. Neural Inf. Process. Syst. 30, 2338 (2017).
- D. Greenberg, M. Nonnenmacher, and J. Macke, Automatic posterior transformation for likelihood-free inference, in International Conference on Machine Learning (2019), pp. 2404–2414.
- F. Nielsen, On a generalization of the Jensen–Shannon divergence and the Jensen–Shannon centroid, Entropy 22, 221 (2020).
- E. Englesson and H. Azizpour, Generalized Jensen-Shannon divergence loss for learning with noisy labels, Adv. Neural Inf. Process. Syst. 34, 30284 (2021).
- W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, Passive all-optical force clamp for high-resolution laser trapping, Phys. Rev. Lett. 95, 208102 (2005).
- A. G. T. Pyo and M. T. Woodside, Memory effects in single-molecule force spectroscopy measurements of biomolecular folding, Phys. Chem. Chem. Phys. 21, 24527 (2019).
- R. Satija, A. Das, and D. E. Makarov, Transition path times reveal memory effects and anomalous diffusion in the dynamics of protein folding, J. Chem. Phys. 147, 152707 (2017).
- R. Satija and D. E. Makarov, Generalized Langevin equation as a model for barrier crossing dynamics in biomolecular folding, J. Phys. Chem. B 123, 802 (2019).
- C. A. Pierse and O. K. Dudko, Distinguishing signatures of multipathway conformational transitions, Phys. Rev. Lett. 118, 088101 (2017).
- R. Satija, A. M. Berezhkovskii, and D. E. Makarov, Broad distributions of transition-path times are fingerprints of multidimensionality of the underlying free energy landscapes, Proc. Natl. Acad. Sci. U.S.A. 117, 27116 (2020).
- K. Neupane, H. Yu, D. A. Foster, F. Wang, and M. T. Woodside, Single-molecule force spectroscopy of the add adenine riboswitch relates folding to regulatory mechanism, Nucleic Acids Res. 39, 7677 (2011).
- N. Q. Hoffer, K. Neupane, and M. T. Woodside, Observing the base-by-base search for native structure along transition paths during the folding of single nucleic acid hairpins, Proc. Natl. Acad. Sci. U.S.A. 118, e2101006118 (2021).
- L. Dingeldein, A. Lyons, P. Cossio, M. Woodside, and R. Covino, Code and data: “Quantitative and predictive folding models from limited single-molecule data using simulation-based inference”, Zenodo, 10.5281/zenodo.19881932 (2026).
- L. Dingeldein, A. Lyons, P. Cossio, M. Woodside, and R. Covino, sbismfs: Simulation-based inference for single-molecule force spectroscopy, GitHub repository, https://github.com/covinolab/SBIsmfs (2026).
- J. Boelts et al., SBI reloaded: A toolkit for simulation-based inference workflows, J. Open Source Software 10, 7754 (2025).