Nonlinear irreversible thermodynamics of single-molecule experiments

Irreversible thermodynamics of single-molecule experiments subject to external constraining forces of a mechanical nature is presented. Extending Onsager's formalism to the nonlinear case of systems under nonequilibrium external constraints, we are able to calculate the entropy production and the general nonlinear kinetic equations for the variables involved. In particular, we analyze the case of RNA stretching protocols obtaining critical oscillations between different configurational states when forced by external means to remain in the unstable region of its free-energy landscape, as observed in experiments. We also calculate the entropy produced during these hopping events and show how resonant phenomena in stretching experiments of single RNA macromolecules may arise. We also calculate the hopping rates using Kramer's approach obtaining a good comparison with experiments.

Figure 1

Elongation of RNA molecules subject to tension near the folding and unfolding states. The corresponding shape of the free energy as obtained by fitting simulation data from Ref. [6] is also shown. (a) For low external tensions applied on the RNA molecule the fitting of the free energy implies that the second term at the right-hand side of Eq. (8) is ${\tau }_{\text{int}}^{\left(c\right)}-\tau =-4.72$ pN, where the absolute minimum of the free energy corresponds to the folded configuration. (b) For external tensions in the range of the unstable region of the force-extension curve, the free energy is bistable and the effective tension takes the value ${\tau }_{\text{int}}^{\left(c\right)}-\tau =-5.24$ pN. (c) For large enough tensions, ${\tau }_{\text{int}}^{\left(c\right)}-\tau =-6.42$ pN, the absolute minimum of the free energy corresponds to the unfolded configuration. (d) Effect of a periodically driving force $\tau \left(t\right)$ on the hopping dynamics of the molecule. The parameters of the oscillation are $\omega =\gamma /10$ and ${\tau }_{\text{int}}^{\left(c\right)}-\tau \simeq -5.32$ pN.

Figure 2

The entropy produced as a function of time (lines) by integrating Eq. (4) for the single noise realization shown. Different intensities of the noise term $\sigma =k_{B}T\gamma$ were considered. Increasing noise intensity (larger $\sigma$) implies larger entropy produced because hopping events are promoted. $l_p$ is the persistence length of the handles, $L$ the length of the handles, and $x_{\text{sys}}$ the dimensions of the hairpin according to Ref. [5].

Figure 3

Three probability distributions as a function of elongation for different times obtained numerically from Eq. (13). (a) Two probability distributions (modal non-Gaussian and bimodal) from the initial stage to the stationary regime in the case of a constant external tension. (b) Two oscillatory behaviors of the bimodal distribution in the presence of a periodic external force, the alternative increase of probability in folded and unfolded states, and the oscillation of the maximum corresponding to the unfolded state.