Challenges in Inferring the Directionality of Active Molecular Processes from Single-Molecule Fluorescence Resonance Energy Transfer Trajectories

The significance of fuzzy boundaries of the barrier regions in single-molecule measurements of failed barrier crossing attempts

A recent ground-breaking experimental study [Lyons et al., Phys. Rev. X 14(1), 011017 (2024)] reports on measuring the temporal duration and the spatial extent of failed attempts to cross an activation barrier (i.e., "loops") for a folding transition in a single molecule and for a Brownian particle trapped within a bistable potential. Within the model of diffusive dynamics, however, both of these quantities are, on average, exactly zero because of the recrossings of the barrier region boundary. That is, an observer endowed with infinite spatial and temporal resolution would find that finite loops do not exist (or, more precisely, form a set of measure zero). Here we develop a description of the experiment that takes the "fuzziness" of the boundaries caused by finite experimental resolution into account and show how the experimental uncertainty of localizing the point, in time and space, where the barrier is crossed leads to observable distributions of loop times and sizes. Although these distributions generally depend on the experimental resolution, this dependence, in certain cases, may amount to a simple resolution-dependent factor and, therefore, the experiments do probe inherent properties of barrier crossing dynamics.

General theory for localizing the where and when of entropy production meets single-molecule experiments

The laws of thermodynamics apply to biophysical systems on the nanoscale as described by the framework of stochastic thermodynamics. This theory provides universal, exact relations for quantities like work, which have been verified in experiments where a fully resolved description allows direct access to such quantities. Complementary studies consider partially hidden, coarse-grained descriptions, in which the mean entropy production typically is not directly accessible but can be bounded in terms of observable quantities. Going beyond the mean, we introduce a fluctuating entropy production that applies to individual trajectories in a coarse-grained description under time-dependent driving. Thus, this concept is applicable to the broad and experimentally significant class of driven systems in which not all relevant states can be resolved. We provide a paradigmatic example by studying an experimentally verified protein unfolding process. As a consequence, the entire distribution of the coarse-grained entropy production rather than merely its mean retains spatial and temporal information about the microscopic process. In particular, we obtain a bound on the distribution of the physical entropy production of individual unfolding events.

Information-theoretical limit on the estimates of dissipation by molecular machines using single-molecule fluorescence resonance energy transfer experiments

Single-molecule fluorescence resonance energy transfer (FRET) experiments are commonly used to study the dynamics of molecular machines. While in vivo molecular processes often break time-reversal symmetry, the temporal directionality of cyclically operating molecular machines is often not evident from single-molecule FRET trajectories, especially in the most common two-color FRET studies. Solving a more quantitative problem of estimating the energy dissipation/entropy production by a molecular machine from single-molecule data is even more challenging. Here, we present a critical assessment of several practical methods of doing so, including Markov-model-based methods and a model-free approach based on an information-theoretical measure of entropy production that quantifies how (statistically) dissimilar observed photon sequences are from their time reverses. The Markov model approach is computationally feasible and may outperform model free approaches, but its performance strongly depends on how well the assumed model approximates the true microscopic dynamics. Markov models are also not guaranteed to give a lower bound on dissipation. Meanwhile, model-free, information-theoretical methods systematically underestimate entropy production at low photoemission rates, and long memory effects in the photon sequences make these methods demanding computationally. There is no clear winner among the approaches studied here, and all methods deserve to belong to a comprehensive data analysis toolkit.

Estimator of entropy production for partially accessible Markov networks based on the observation of blurred transitions

A central task in stochastic thermodynamics is the estimation of entropy production for partially accessible Markov networks. We establish an effective transition-based description for such networks with transitions that are not distinguishable and therefore blurred for an external observer. We demonstrate that, in contrast to a description based on fully resolved transitions, this effective description is typically non-Markovian at any point in time. Starting from an information-theoretic bound, we derive an operationally accessible entropy estimator for this observation scenario. We illustrate the operational relevance and the quality of this entropy estimator with a numerical analysis of various representative examples.

Milestoning estimators of dissipation in systems observed at a coarse resolution

Many nonequilibrium, active processes are observed at a coarse-grained level, where different microscopic configurations are projected onto the same observable state. Such "lumped" observables display memory, and in many cases, the irreversible character of the underlying microscopic dynamics becomes blurred, e.g., when the projection hides dissipative cycles. As a result, the observations appear less irreversible, and it is very challenging to infer the degree of broken time-reversal symmetry. Here we show, contrary to intuition, that by ignoring parts of the already coarse-grained state space we may-via a process called milestoning-improve entropy-production estimates. We present diverse examples where milestoning systematically renders observations "closer to underlying microscopic dynamics" and thereby improves thermodynamic inference from lumped data assuming a given range of memory, and we hypothesize that this effect is quite general. Moreover, whereas the correct general physical definition of time reversal in the presence of memory remains unknown, we here show by means of physically relevant examples that at least for semi-Markov processes of first and second order, waiting-time contributions arising from adopting a naive Markovian definition of time reversal generally must be discarded.

Emergence of Memory in Equilibrium versus Nonequilibrium Systems

Experiments often probe observables that correspond to low-dimensional projections of high-dimensional dynamics. In such situations distinct microscopic configurations become lumped into the same observable state. It is well known that correlations between the observable and the hidden degrees of freedom give rise to memory effects. However, how and under which conditions these correlations emerge remain poorly understood. Here we shed light on two fundamentally different scenarios of the emergence of memory in minimal stationary systems, where observed and hidden degrees of freedom either evolve cooperatively or are coupled by a hidden nonequilibrium current. In the reversible setting the strongest memory manifests when the timescales of hidden and observed dynamics overlap, whereas, strikingly, in the driven setting maximal memory emerges under a clear timescale separation. Our results hint at the possibility of fundamental differences in the way memory emerges in equilibrium versus driven systems that may be utilized as a "diagnostic" of the underlying hidden transport mechanism.

Cochaperones convey the energy of ATP hydrolysis for directional action of Hsp90

The molecular chaperone and heat shock protein Hsp90 is part of many protein complexes in eukaryotic cells. Together with its cochaperones, Hsp90 is responsible for the maturation of hundreds of clients. Although having been investigated for decades, it still is largely unknown which components are necessary for a functional complex and how the energy of ATP hydrolysis is used to enable cyclic operation. Here we use single-molecule FRET to show how cochaperones introduce directionality into Hsp90's conformational changes during its interaction with the client kinase Ste11. Three cochaperones are needed to couple ATP turnover to these conformational changes. All three are therefore essential for a functional cyclic operation, which requires coupling to an energy source. Finally, our findings show how the formation of sub-complexes in equilibrium followed by a directed selection of the functional complex can be the most energy efficient pathway for kinase maturation.

Waiting Time Distributions in Hybrid Models of Motor-Bead Assays: A Concept and Tool for Inference

In single-molecule experiments, the dynamics of molecular motors are often observed indirectly by measuring the trajectory of an attached bead in a motor-bead assay. In this work, we propose a method to extract the step size and stalling force for a molecular motor without relying on external control parameters. We discuss this method for a generic hybrid model that describes bead and motor via continuous and discrete degrees of freedom, respectively. Our deductions are solely based on the observation of waiting times and transition statistics of the observable bead trajectory. Thus, the method is non-invasive, operationally accessible in experiments and can, in principle, be applied to any model describing the dynamics of molecular motors. We briefly discuss the relation of our results to recent advances in stochastic thermodynamics on inference from observable transitions. Our results are confirmed by extensive numerical simulations for parameters values of an experimentally realized F1-ATPase assay.