Measuring irreversibility in stochastic systems by categorizing single-molecule displacements

TL;DR

The paper addresses how to quantify irreversibility in small stochastic systems from single-molecule trajectories without detailed force information. It introduces a model-free measure $\Sigma_{\Delta t}$ based on KL divergences between forward and time-reversed displacement classes, connected to entropy production via a conditional fluctuation theorem. A hierarchy of bounds is developed by refining displacement partitions, and a local dissipation proxy $\Sigma_{\Delta t}^{trans}$ captures spatial dissipation near chosen boundaries. The method is validated theoretically on Langevin models, demonstrated on nonequilibrium steady states and driven double-well systems, and applied to experimental force ramp data for talin, revealing landscape-dependent asymmetries in folding dynamics. The approach offers a practical, robust tool for mapping dissipation and understanding biomolecular mechanochemistry under non-equilibrium conditions, with potential extensions to time-dependent driving and higher-dimensional systems.

Abstract

Quantifying the irreversibility and dissipation of non-equilibrium processes is crucial to understanding their behavior, assessing their possible capabilities, and characterizing their efficiency. We introduce a physical quantity that quantifies the irreversibility of stochastic Langevin systems from the observation of individual molecules' displacements. Categorizing these displacements into a few groups based on their initial and final position allows us to measure irreversibility precisely without the need to know the forces and magnitude of the fluctuations acting on the system. Our model-free estimate of irreversibility is related to entropy production by a conditional fluctuation theorem and provides a lower bound to the average entropy production. We validate the method on single-molecule force spectroscopy experiments of proteins subject to force ramps. We show that irreversibility is sensitive to detailed features of the energy landscape underlying the protein folding dynamics and suggest how our methods can be employed to unveil key properties of protein folding processes.

Figures (19)

• Figure 1: (a) Illustration of displacements $\ell$ of three trajectories in the time window $\Delta t$, categorized into displacement classes with respect to the boundary at $k$. Trajectories of class $k^\rightarrow$ cross $k$ left to right, $k^\leftarrow$ right to left and $B$ remain on the same side. (b) Sample displacement distributions at $k=0$ for the square potential on ring [shown in Fig. \ref{['fig: fig2']}(a)], at different driving speeds with $v_0=0.5~µm\per s$. At equilibrium $q(\ell,k^\rightarrow)=q(-\ell,k^\leftarrow)$.
• Figure 2: (a) Profile of square potential $W(x)$ and different classes of displacements: top $\mathcal{C}_3(k_1)$ and bottom $\mathcal{C}_6(k_1,k_2)$. (b) Steady-state probability distributions for a particle in a square potential on a periodic topology, subject to different constant drivings $v$ with $v_0=0.5~µm\per s$. For the equilibrium (eq.) distribution, $v=0$. (c) Local dissipation: the solid lines give the density of entropy production rate $\dot{\sigma}$ defined in Eq. \ref{['eq: sigmadot']} for near ($v=0.5$) and far ($v=2.5$) from equilibrium driving. The dashed curves are the local irreversibility estimates obtained from $\dot{\Sigma}_{\Delta t}^{trans}=\Sigma_{\Delta t}^{trans}/\Delta t$ [see Eq. \ref{['eq: dkltrans']} and \ref{['eq: transExpand']}], as a function of the position of the boundary $k_1=x$. (d) Fraction of irreversibility captured ($\Sigma_{\Delta t}/\langle\Delta S_{tot}\rangle$) for three types of displacement partitions $\mathcal{I}$ (purple dotted line), $\mathcal{C}_3(k_1=-0.1)$ (purple dashed line), $\mathcal{C}_6(k_1=-0.1,k_2=\pm5)$ (purple solid line) and TUR (orange line) as a function of the nonequilibrium driving $v$. (e) $\Sigma_{\Delta t}/\langle\Delta S_{tot}\rangle$ for different partitions as a function of the boundary location $k_1$ for far and near equilibrium driving (blue and red lines, respectively). For all drivings considered in the plots above, $D = 1$ µms, $\gamma=1\, k_BT$ sµm, $\Delta W = \ln{8}\; k_BT$, and irreversibilities are measured with $\Delta t=10$ m s.
• Figure 3: (a) Symmetric double-well potential with a high ($\Delta U=4k_BT$) and low ($\Delta U=2k_BT$) barrier and $x_M=L=10nm$. Arrows indicate the types of displacements measured in the $\mathcal{C}_4$ classification used to calculate $\Sigma_{\Delta t}(\mathcal{C}_4)$. The boundary used to define the displacements is located at $k$. (b) Influence of displacement time $\Delta t$ on the total measured irreversibility $\Delta \Sigma_{t_f}(\mathcal{C}_4)$ in each force ramp, for the high barrier case with measurement noise. (c) Entropy production rate $\langle\dot S_{tot}\rangle$ (dotted lines), computed via Eq. \ref{['eq: thermoEnt']}, and irreversibility rate $\dot\Sigma_{\Delta t}$ for different pulling rates for low and high potential barriers, respectively. Solid curves: running average of the irreversibility rate estimate $\dot\Sigma_{\Delta t}(\mathcal{C}_4)$, measured every millisecond with $\Delta t=5~m s$, using 5000 trajectories and adding measurement noise. The running averages are taken over 70, 35, 18 points for the $|r|=5,\,10,\,20$ curves, respectively, while the shaded error regions are taken from the running standard deviation over the same windows. Dashed curves: $\dot\Sigma_{\Delta t}(\mathcal{C}_4)$ but estimated from 50000 trajectories without measurement noise. Inset: Cumulative time integral of measured irreversibility $\Delta\Sigma_t(\mathcal{C}_4)$ and entropy $\langle \Delta S_{tot}\rangle$. Bottom panel: trace of pulling force curves with time, for the different ramp rates as color-coded above. $k_BT=4.11~pNnm$ and $D=3000~nm\squared\per s$.
• Figure 4: (a) Schematic of experimental magnetic tweezer setup. (b) Thin traces: example trajectories of the Talin R3 extension as the pulling force is linearly increased (blue) and decreased (orange) at $r=\pm10$ pNs, for the wild-type R3$^{\rm{WT}}$ (top) and mutant R3$^{\rm{IVVI}}$ (bottom). The thick curves show the respective mean extension over all of the experiments. (c) Irreversibility rate $\dot\Sigma_{\Delta t}(\mathcal{C}_4)$ measurements from experiments on the R3$^{\rm{WT}}$ (top) and mutant R3$^{\rm{IVVI}}$ (bottom) protein domains at different pulling rates. Measuring every millisecond with $\Delta t=5~ms$, we show the running average over 70, 35, 18 points for the $|r|=5,\,10,\,20$ curves, respectively. Inset: Cumulative time integral $\Delta\Sigma_t(\mathcal{C}_4)$ of measured irreversibility. Lower panels: pulling forces for each ramp rate as color-coded above. (d) Total irreversibility at the end of each protocol as a function of unfolding (positive) and refolding (negative) pulling rates. The shown data is obtained from 1682, 5000, 1441 trajectories for $|r|=5,\,10,\,20$ on the R3$^{\rm{WT}}$ and 3933, 3500, 5322 on the R3$^{\rm{IVVI}}$, respectively. (e) Simulations are performed on the shown asymmetric potentials, with $D=3000$ nms, $k_BT=4.11~pNnm$. The potential with the lower energy barrier mimics the R3$^{\rm{WT}}$ protein and the high barrier the mutant R3$^{\rm{IVVI}}$ construct. (f) As in (c) for 5000 simulated trajectories, with added noise on the recorded positions. (g) As in (d) for the simulated data.
• Figure S1: Displacement distributions $q_{\Delta t}(\ell,C)$ (orange) of the $\mathcal{C}_6(k_1,k_2)=\{k_1^\rightarrow,k_1^\leftarrow,B_L,B_R,k_2^\rightarrow,k_2^\leftarrow\}$ classification, for the near-equilibrium (a) and far-from-equilibrium (b) driving in the periodic square potential analyzed in Fig. \ref{['fig: fig2']}. Their respective distributions under time-reversal $q_{\Delta t}(-\ell,\tilde{C})$ are shown in blue. $k_1=0.11$ for $v=0.5~µm\per s$ and $k_1=-0.13$ for $v=2.5~µm\per s$, the placements offering the tightest bound $\Sigma_{\Delta t}(\mathcal{C}_6)$ to $\langle\Delta S_{tot}\rangle$.