Single-protein conformational dynamics was reported by Hu, Hong, Smith, Neusius, Cheng and Smith (Nature Physics 12, 171–174, 2016) to be non-equilibrium, self-similar and ageing over at least thirteen decades in time (10^-12 s to ~10^2 s), with diffusion over the energy landscape described by a non-ergodic continuous time random walk on a fractal conformational transition network. The follow-up literature (2016–2026) leaves a precise open core: (1) the timescale at which an individual folded protein finally equilibrates and becomes ergodic is undetermined, and even the existence of ageing and non-ergodicity is under published dispute; and (2) the microscopic mechanism of the observed subdiffusion is contested among at least three incompatible pictures, and no quantitative predictive mapping from energy-barrier statistics or network geometry to the anomalous exponent exists. Two further literature gaps motivate the same objective: no continuous single-protocol measurement validates the uninterrupted thirteen-decade scaling, and the magnitude of the effect of non-equilibrium conformational fluctuations on catalytic rate-constant distributions remains unquantified. The problem is to convert this phenomenological description into a predictive, mechanistically grounded theory of single-protein dynamics.

Background

Proteins are not rigid objects: their atoms undergo continual internal fluctuations that are essential for function, from enzyme catalysis to molecular-machine cycles. Molecular dynamics (MD) simulations, which numerically integrate the equations of motion of all atoms, together with spectroscopy experiments, show that these fluctuations are subdiffusive — the mean-square displacement of a structural coordinate grows slower than linearly with time, roughly as tαt^{\alpha} with anomalous exponent α<1\alpha < 1 — and that relaxation is non-exponential, with effective relaxation times spread over many decades, from picoseconds up to about 10210^{2} s. A system is ergodic when the time average of an observable along a single long trajectory equals its ensemble average over many copies; non-ergodicity means a single molecule can persistently explore only part of the accessible conformational space, so its long-time behavior differs from the ensemble's. A continuous time random walk (CTRW) is a random process in which a particle waits a random waiting time between jumps; when the waiting-time distribution is heavy-tailed (decaying as a power law with no characteristic scale), its mean diverges, the walk is subdiffusive, non-ergodic, and exhibits ageing — its statistical properties depend on the age of the process, i.e., on how long the system has evolved before observation begins, so measured relaxation times grow systematically with the observation time itself.

Building on an earlier proposal that peptide subdiffusion originates from the fractal-like (scale-free, self-similar) structure of the configuration space (Neusius, Daidone, Sokolov and Smith, Physical Review Letters 100, 188103, 2008), Hu et al. (Nature Physics 12, 171–174, 2016) combined MD simulations on three globular proteins (timescales 10−1210^{-12} to 10−510^{-5} s) with published single-molecule experiments and concluded that the dynamics of individual protein molecules is non-equilibrium, self-similar over thirteen decades in time, exhibits ageing, and is described by a non-ergodic CTRW whose fractal character arises from the topology and geometry of the underlying energy landscape. Comparing with single-molecule data, they suggested this non-equilibrium behavior persists up to timescales approaching the in-vivo lifetime of an individual protein molecule. This is a striking claim: if a single folded protein never (or only extremely slowly) equilibrates during its biological life, the standard ergodic equilibrium-statistical-mechanics treatment of protein kinetics — the basis for interpreting rate constants, NMR relaxation, and MD-derived equilibrium observables — requires re-examination for individual molecules.

The claim immediately drew scrutiny and spawned an active literature. Goychuk and Pöschel (Nature Physics 17, 2021) published a Comment arguing that the evidence for ageing 'is based on a fallacy'; the original authors, joined by colleagues, replied that ageing is present under the critics' own criteria (Li et al., Nature Physics 17, 775–776, 2021), so the non-equilibrium interpretation remains under published dispute rather than settled. On longer timescales, plasmon rulers — pairs of gold nanospheres linked by a single protein whose distance is read out optically — allowed Ye et al. (Nano Letters 18, 4666, 2018) to monitor a single heat-shock protein 90 molecule continuously for 24 hours at video rate, extending the single-protein observation bandwidth by three to four orders of magnitude beyond single-molecule Förster resonance energy transfer (smFRET, a technique that reports inter-dye distances in single molecules); they found previously unseen conformational dynamics on the scale of minutes to hours, indicating slow dynamics persist to at least 10410^{4} s, but did not determine an equilibration crossover. On the mechanism side, three incompatible explanations of protein subdiffusion now coexist: the original non-ergodic CTRW on a fractal energy landscape; Markovian (memory-less) diffusion over a fractal network of metastable conformational states, which Maggi and Orozco (Physical Review E 109, 034402, 2024) argued explains subdiffusion while ruling out the CTRW — a conclusion contested by Neusius, Sokolov and Smith (Physical Review E 111, 056401, 2025) and disputed in Maggi and Orozco's reply (Physical Review E 111, 056402, 2025); and hierarchical friction memory, in which a non-Markovian generalized Langevin equation with a multi-scale memory kernel — the friction force at a given time depending on the system's earlier configurational history — causes the subdiffusion of fast-folding proteins, as Klimek, Dalton, Tepper and Netz (PNAS 123, e2516506123, 2026) concluded, explicitly finding that memory effects, not the free-energy landscape, dominate the scaling. Separately, Maggi (Physical Chemistry Chemical Physics 28, 6447, 2026) re-analyzed MD trajectories and claimed that apparent deviations from ergodicity are artifacts of insufficient convergence of the measured quantities on picosecond–nanosecond windows, while noting that longer windows were not directly investigated; and Ormeño and General (Communications Chemistry 7, 26, 2024) analyzed when MD trajectories reach thermodynamic equilibrium, providing methodology relevant to, but not a determination of, the single-protein crossover. For the functional implications, single-molecule enzymology has long observed 'dynamic disorder' — fluctuating catalytic rates caused by slow conformational changes — and Grossman-Haham, Rosenblum, Namani and Hofmann (PNAS 115, 513–518, 2018) showed that slow domain reconfiguration produces power-law kinetics in a two-state enzyme; but no quantitative prediction connects the thirteen-decade self-similar framework to catalytic barrier heights or measured rate-constant distributions.

Problem Statement

Determine whether, and on what timescale, the internal dynamics of an individual folded protein molecule reaches dynamical equilibrium and ergodicity — or establishes that the dynamics remains non-equilibrium and self-similar (with power-law, observation-time-dependent relaxation times and ageing) over the protein's entire biological lifetime — and give the phenomenon a quantitative, predictive microscopic foundation. Concretely, with respect to the framework of Hu et al. (Nature Physics 12, 171–174, 2016): (1) measure or predict the crossover timescale at which time-averaged and ensemble-averaged observables of a single protein coincide (single-molecule data bound it from below by roughly 10210^{2} s; the observed crossover must be determined, with genuine non-equilibrium behavior cleanly separated from finite-sampling convergence artifacts); and (2) establish the microscopic mechanism of the observed subdiffusion — currently the subject of a published controversy among the non-ergodic continuous time random walk on a fractal energy landscape, Markovian diffusion over a fractal network of metastable conformational states, and hierarchical friction-memory (generalized Langevin) dynamics — and derive explicit quantitative relations connecting the microscopic statistics that control the scaling under the established mechanism (energy-barrier-height statistics and transition-network geometric measures, such as fractal dimensions, or the structure of the multiscale friction-memory kernel) to the anomalous exponent α\alpha and waiting-time statistics, testable against simulation and experiment. Both deliverables serve one objective: converting the phenomenological thirteen-decade self-similar description of single-protein dynamics into a predictive theory whose transport law and crossover behavior follow from the underlying microscopic mechanism. Continuous long-time validation of the uninterrupted scaling, and the quantification of how non-equilibrium conformational fluctuations modulate catalytic barrier heights and the statistical distributions of measured rate constants in enzymes, remain open related directions and motivating context for this objective.

References

  1. Xiaohu Hu, Liang Hong, Micholas Dean Smith, Thomas Neusius, Xiaolin Cheng, Jeremy C. Smith, 'The dynamics of single protein molecules is non-equilibrium and self-similar over thirteen decades in time', Nature Physics 12, 171–174 (2016). DOI: 10.1038/nphys3553, https://doi.org/10.1038/nphys3553
  2. Igor Goychuk, Thorsten Pöschel, 'Insufficient evidence for ageing in protein dynamics', Nature Physics 17 (2021). DOI: 10.1038/s41567-021-01269-1, https://doi.org/10.1038/s41567-021-01269-1
  3. Jun Li, Xiaohu Hu, Thomas Neusius, Xiaolin Cheng, Micholas Dean Smith, Aljaž Godec, Liang Hong, Ralf Metzler, Jeremy C. Smith, 'Reply to: Insufficient evidence for ageing in protein dynamics', Nature Physics 17, 775–776 (2021). DOI: 10.1038/s41567-021-01270-8, https://doi.org/10.1038/s41567-021-01270-8
  4. Weixiang Ye, Markus Götz, Sirin Celiksoy, Laura Tüting, Christoph Ratzke, Janak Prasad, Julia Ricken, Seraphine V. Wegner, Rubén Ahijado-Guzmán, Thorsten Hugel, Carsten Sönnichsen, 'Conformational Dynamics of a Single Protein Monitored for 24 h at Video Rate', Nano Letters 18 (2018). DOI: 10.1021/acs.nanolett.8b03342, https://doi.org/10.1021/acs.nanolett.8b03342
  5. Thomas Neusius, Isabella Daidone, Igor M. Sokolov, Jeremy C. Smith, 'Subdiffusion in Peptides Originates from the Fractal-Like Structure of Configuration Space', Physical Review Letters 100, 188103 (2008). DOI: 10.1103/PhysRevLett.100.188103, https://doi.org/10.1103/PhysRevLett.100.188103
  6. Luca Maggi, Modesto Orozco, 'Main role of fractal-like nature of conformational space in subdiffusion in proteins', Physical Review E 109, 034402 (2024). DOI: 10.1103/physreve.109.034402, https://doi.org/10.1103/physreve.109.034402
  7. Thomas Neusius, Igor M. Sokolov, Jeremy C. Smith, 'Comment on "Main role of fractal-like nature of conformational space in subdiffusion in proteins"', Physical Review E 111, 056401 (2025). DOI: 10.1103/physreve.111.056401, https://doi.org/10.1103/physreve.111.056401
  8. Luca Maggi, Modesto Orozco, 'Reply to "Comment on Main role of fractal-like nature of conformational space in subdiffusion in proteins"', Physical Review E 111, 056402 (2025). DOI: 10.1103/physreve.111.056402, https://doi.org/10.1103/physreve.111.056402
  9. Luca Maggi, 'Investigation of the non-ergodicity of protein dynamics', Physical Chemistry Chemical Physics 28, 6447–6456 (2026). DOI: 10.1039/d5cp03703d, https://doi.org/10.1039/d5cp03703d
  10. Anton Klimek, Benjamin A. Dalton, Lucas Tepper, Roland R. Netz, 'Hierarchical friction memory leads to subdiffusive configurational dynamics of fast-folding proteins', PNAS 123, e2516506123 (2026). DOI: 10.1073/pnas.2516506123, https://doi.org/10.1073/pnas.2516506123 (preprint arXiv:2506.18823)
  11. Franco Ormeño, Ignacio J. General, 'Convergence and equilibrium in molecular dynamics simulations', Communications Chemistry 7, 26 (2024). DOI: 10.1038/s42004-024-01114-5, https://doi.org/10.1038/s42004-024-01114-5
  12. Iris Grossman-Haham, Gabriel Rosenblum, Trishool Namani, Hagen Hofmann, 'Slow domain reconfiguration causes power-law kinetics in a two-state enzyme', PNAS 115, 513–518 (2018). DOI: 10.1073/pnas.1714401115, https://doi.org/10.1073/pnas.1714401115
  13. Gilad Haran, Hisham Mazal, 'How fast are the motions of tertiary-structure elements in proteins?', The Journal of Chemical Physics 153, 085101 (2020). DOI: 10.1063/5.0024972, https://doi.org/10.1063/5.0024972 (preprint arXiv:2009.06439)
  14. Benedikt Sohmen, Christian Beck, Veronika Frank, Tilo Seydel, Ingo Hoffmann, Bianca Hermann, Mark Nüesch, Marco Grimaldo, Frank Schreiber, Steffen Wolf, Felix Roosen-Runge, Thorsten Hugel, 'The Onset of Molecule-Spanning Dynamics in Heat Shock Protein Hsp90', Advanced Science 10, 2304262 (2023). DOI: 10.1002/advs.202304262, https://doi.org/10.1002/advs.202304262