Free-energy inference from partial work measurements in small systems

Abstract

Fluctuation relations (FRs) are among the few existing general results in nonequilibrium systems. Their verification requires the measurement of the total work performed on a system. Nevertheless in many cases only a partial measurement of the work is possible. Here we consider FRs in dual-trap optical tweezers where two different forces (one per trap) are measured. With this setup we perform pulling experiments on single molecules by moving one trap relative to the other. We demonstrate that work should be measured using the force exerted by the trap that is moved. The force that is measured in the trap at rest fails to provide the full dissipation in the system, leading to a (incorrect) work definition that does not satisfy the FR. The implications to single-molecule experiments and free-energy measurements are discussed. In the case of symmetric setups a second work definition, based on differential force measurements, is introduced. This definition is best suited to measure free energies as it shows faster convergence of estimators. We discuss measurements using the (incorrect) work definition as an example of partial work measurement. We show how to infer the full work distribution from the partial one via the FR. The inference process does also yield quantitative information, e.g., the hydrodynamic drag on the dumbbell. Results are also obtained for asymmetric dualtrap setups. We suggest that this kind of inference could represent a previously unidentified and general application of FRs to extract information about irreversible processes in small systems.statistical mechanics | biophysics | fluctuation theorems F luctuation relations (FRs) are mathematical equations connecting non equilibrium work measurements to equilibrium free-energy differences. FRs, such as the Jarzynski equality (JE) or the Crooks fluctuation relation (CFR), have become a valuable tool in single-molecule biophysics where they are used to measure folding free energies from irreversible pulling experiments (1, 2). Such measurements have been carried out with laser optical tweezers on different nucleic acid structures such as hairpins (3-6), G quadruplexes (7,8), and proteins (9-12) and with atomic force microscopes on proteins (13) and bimolecular complexes (14). An important issue regarding FRs is the correct definition of work, which rests on the correct identification of configurational variables and control parameters. In the singletrap optical tweezers configuration this issue has been thoroughly discussed (17)(18)(19).The situation of how to correctly measure work in small systems becomes subtle when there are different forces applied to the system. In this case theory gives the prescription to correctly define the work (W Γ ) for a given trajectory (Γ): Integrate the generalized force (f λ ) (conjugated to the control parameter, λ) over λ along Γ, W Γ = R Γ f λ dλ. However, in some cases one cannot measure the proper generalized force or has limited experimental access to partial sources of the entropy production, leading to wha...

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From free energy measurements to thermodynamic inference in nonequilibrium small systems

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