Experimentally accessible measurement of irreversibility in stochastic systems by categorizing single-molecule displacements

Technical Summary: Experimentally Accessible Measurement of Irreversibility in Stochastic Systems

Problem Statement
Quantifying the irreversibility and dissipation of non-equilibrium processes is fundamental to understanding the behavior, capabilities, and efficiency of stochastic systems, particularly in biological contexts like protein folding. While stochastic thermodynamics provides a rigorous theoretical framework (e.g., fluctuation theorems, Kullback-Leibler divergences), applying these concepts to experimental data is challenging. Existing methods often require:

1. Complete knowledge of the forces acting on the system and the diffusion coefficient.
2. Access to full trajectory data, which is often impossible due to finite temporal resolution.
3. Specific conditions (e.g., stationarity) or complex data processing (e.g., machine learning hyperparameter tuning).
4. Both forward and backward protocols for certain estimators (e.g., Crooks' relation).

Consequently, there is a need for a model-free, experimentally accessible method to measure irreversibility in systems with limited information, such as single-molecule tracking where forces and diffusion coefficients may be unknown or difficult to estimate.

Methodology
The authors introduce a technique to quantify irreversibility in overdamped Langevin systems based on the categorization of single-molecule displacements over finite time intervals ($\Delta t$).

1. Displacement Categorization: Instead of analyzing full trajectories, the method observes the initial position ($x'$) and final position ($x$) of a particle after a time $\Delta t$, defining the displacement $\ell = x - x'$. These displacements are categorized into classes ($C$) based on their spatial location relative to defined boundaries (e.g., crossing a boundary $k$ from left to right, right to left, or remaining in the bulk).
2. Irreversibility Measure ($\Sigma_{\Delta t}$): The authors define a measure of irreversibility as the Kullback-Leibler (KL) divergence between the joint probability distributions of forward displacements and their time-reversed counterparts across these classes:
   $$\Sigma_{\Delta t}(C_n) \equiv k_B \sum_{C \in C_n} \int d\ell \, q_{\Delta t}(\ell, C) \ln \frac{q_{\Delta t}(\ell, C)}{q_{\Delta t}(-\ell, \tilde{C})}$$
   where $q_{\Delta t}(\ell, C)$ is the probability density of a displacement $\ell$ belonging to class $C$, and $\tilde{C}$ is the time-reversed class.
3. Theoretical Bounds:
   • Stationary Systems: For stationary dynamics, $\Sigma_{\Delta t}$ provides a rigorous lower bound to the average entropy production $\langle \Delta S_{tot} \rangle$. This bound tightens as the number of displacement classes increases (a hierarchy of bounds).
   • Time-Dependent Systems: For non-stationary protocols, the measure provides a lower bound in the limit of vanishing $\Delta t$.
   • Local Dissipation: By focusing on displacements crossing a specific boundary, the method can estimate the local rate of dissipation.
4. Experimental Implementation: The method is applied to single-molecule magnetic tweezer experiments on the protein talin R3 (wild-type and mutant). The system is subjected to cyclic linear force ramps (unfolding and refolding). The irreversibility is calculated by binning displacements into four classes relative to a boundary between the folded and unfolded states.

Key Contributions and Results

• Model-Free Estimation: The method requires no prior knowledge of the deterministic forces, the diffusion coefficient, or the underlying energy landscape. It relies solely on the statistics of observed displacements.
• Validation on Simulated Systems:
  • In a driven particle on a ring with a square potential, the method successfully captured the entropy production. The authors demonstrated that increasing the number of displacement classes (partitioning space more finely) significantly improved the estimate, especially for systems near equilibrium where the Thermodynamic Uncertainty Relation (TUR) alone provided a loose bound.
  • The method was shown to accurately map the spatial profile of local dissipation.
• Application to Protein Folding (Talin R3):
  • Wild-Type vs. Mutant: The method distinguished between the wild-type talin R3 and a more mechanically stable mutant (R3IVVI). It correctly identified that the mutant exhibits higher total irreversibility.
  • Asymmetry in Folding Dynamics: A key finding was the detection of asymmetry in the mutant's dynamics: unfolding was found to be systematically more irreversible than refolding. This was not observed in the wild-type.
  • Landscape Inference: By comparing experimental results with simulations of asymmetric double-well potentials, the authors linked the observed asymmetry in irreversibility to the underlying free-energy landscape. Specifically, the mutant's behavior suggested an asymmetric landscape where the unfolded well is broader and further from the barrier than the folded well.
• Robustness: The method proved robust against measurement noise (simulated Gaussian noise) and limited data sets, maintaining the ability to rank systems by dissipation and identify the timing of irreversible events (peaks in irreversibility rate corresponding to transition events).

Significance and Claims
The paper claims to provide a "rigorous tool for rationalizing the function of mechanically active proteins" by offering a quantitative framework for non-equilibrium thermodynamics in biomolecular transitions.

• Experimental Accessibility: The primary significance lies in the method's feasibility for experimental systems where full trajectory reconstruction or force knowledge is unavailable. It bypasses the need to sample the exponentially large space of trajectories by focusing on coarse-grained initial and final states.
• Sensitivity to Landscape Features: The authors demonstrate that quantifying irreversibility can unveil subtle properties of the energy landscape (e.g., asymmetries in barrier distances) that are not easily accessible through standard equilibrium measurements or hysteresis loop areas alone.
• Comparison to Existing Methods: The authors position their method as superior to the Thermodynamic Uncertainty Relation (TUR) for spatially heterogeneous systems and more robust/interpretable than machine-learning approaches or methods requiring both forward and backward protocols (like Crooks' relation or specific binning methods). They note that while their measure is a lower bound, it captures the relative trends and differences between systems (e.g., wild-type vs. mutant, different pulling rates) with high fidelity.

The paper concludes that this approach allows researchers to assess how far from equilibrium a system operates and to characterize trade-offs between dissipation and performance, specifically highlighting its utility in dissecting the thermodynamic signatures of protein folding mechanisms.