Information-to-work conversion in single-molecule experiments: From discrete to continuous feedback

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Information-to-work conversion in single-molecule experiments: From discrete to continuous feedback

Regina K. Schmitt, Patrick P. Potts, Heiner Linke, Jonas Johansson, Peter Samuelsson, Marc Rico-Pasto, and Felix Ritort

Phys. Rev. E 107, L052104 – Published 25 May 2023

Abstract

We theoretically investigate the extractable work in single molecule unfolding-folding experiments with applied feedback. Using a simple two-state model, we obtain a description of the full work distribution from discrete to continuous feedback. The effect of the feedback is captured by a detailed fluctuation theorem, accounting for the information aquired. We find analytical expressions for the average work extraction as well as an experimentally measurable bound thereof, which becomes tight in the continuous feedback limit. We further determine the parameters for maximal power or rate of work extraction. Although our two-state model only depends on a single effective transition rate, we find qualitative agreement with Monte Carlo simulations of DNA hairpin unfolding-folding dynamics.

Received 10 October 2021
Accepted 10 April 2023

DOI:https://doi.org/10.1103/PhysRevE.107.L052104

Physics Subject Headings (PhySH)

DNA folding Fluctuation theorems Nonequilibrium & irreversible thermodynamics Thermodynamics

Statistical Physics & Thermodynamics

Authors & Affiliations

Regina K. Schmitt, Patrick P. Potts , Heiner Linke, Jonas Johansson, and Peter Samuelsson

Department of Physics and NanoLund, Lund University, Box 188, SE-221 00 Lund, Sweden

Marc Rico-Pasto and Felix Ritort

Department of Condensed Matter Physics, Small Biosystems Laboratory, Universitat de Barcelona, C/Marti i Franques 1, 08028 Barcelona, Spain

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Issue

Vol. 107, Iss. 5 — May 2023

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Images

Figure 1
(a) Force-position trace for single DNA hairpin experiment, with force jumps showing folding/unfolding events (see the text). The inset: DNA molecule held between a micropipette and movable optical tweezers, with a force applied. (b) Schematics of the molecule free energy landscape for initial (solid) and final (dashed) times of the protocol. The two molecular states $U$ (unfolded) and $F$ (folded) shown at their corresponding local energy minima. (c)–(e) Energy-time trajectories in state space with transitions at times $τ_{n}$ , denoted by vertical arrows. The dissipated (dimensionless) heat at the transitions is $q_{n} = \pm 2 τ_{n}$ . Regions in time where work is extracted are shaded blue. In (d) measurements and FB are performed at times $τ_{n}^{*}$ in intervals $δ τ$ . At the measurement detecting the first transition $F \to U$ the states are shifted with infinite drive speed $ζ^{'} = 0$ to their final energies. In (e), measurement and FB is continuous ( $δ τ = 0$ ) with nonero $ζ^{'}$ : Directly after a first transition, the drive speed is increased from $ζ$ to $ζ^{'} < ζ$ . Red and blue lines in (c)–(e) stand for the folded and unfolded energy branches.
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Figure 2
(a) Probability distribution of work $w$ for protocols with no FB, (dashed lines) and continuous FB (solid lines) for three different effective rates $ζ$ . Continuous FB shifts the distribution towards negative $w$ , most notably in the quasistatic regime $ζ ≫ 1$ . The inset shows the fraction $p_{TV}$ of TV contributions without feedback (dashed line) and continuous feedback (solid line) as a function of $ζ$ . (b) Average work from dynamical simulations of continuous FB with a finite rate $ζ^{'}$ after the first transition (see the text) for different sets of $ζ, ζ^{'}$ (empty symbols). The work without FB, Eq. (5), (dashed line) and continuous FB for $ζ^{'} \to 0$ , Eq. (7), (solid line) and $ζ^{'} = 0.02$ (dashed-dot line) for the two-state system are shown for reference.
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Figure 3
Average work ${〈 w 〉}_{δ τ}$ for the discrete FB protocol as a function of the time between the measurements $δ τ$ for $τ_{0} = - 10$ . Note that the time axis goes from large to small $δ τ$ . The work (black line) decreases with decreasing $δ τ$ before it becomes negative and eventually saturates at the continuous measurement result ${〈 w 〉}_{c}$ (cyan line). Three different entropies (see the text) $〈 I 〉$ ( transfer entropy, red), $ln (γ)$ (logarithmic efficacy, yellow) and $〈 Φ_{s} 〉$ (blue) are plotted, showing that whereas $H$ and $- ln (γ)$ diverge in the continuous FB limit, $〈 Φ_{s} 〉$ is finite, constituting a tight bound on the extractable work.
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Figure 4
Contour plot of average work ${〈 w 〉}_{c}^{ζ^{'}}$ as a function of $ζ, ζ^{'}$ . Negative work values are found for small $ζ^{'}$ and large $ζ$ . Side panel: Work probability distributions for three sets of $ζ, ζ^{'}$ , marked in the main panel. Trajectories with refolding events (RE) contribute to a shoulder at high work values with height increasing with increasing $ζ$ . For two uppermost panels, the corresponding work probability distributions obtained from the dynamical simulations are shown (black, thin lines). For the lowest panel, the dynamical simulation were found to be unfeasible, i.e., too time consuming for the parameters chosen.
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