Thermodynamic geometry of minimum-dissipation driven barrier crossing

We explore the thermodynamic geometry of a simple system that models the bistable dynamics of nucleic acid hairpins in single molecule force-extension experiments. Near equilibrium, optimal (minimum-dissipation) driving protocols are governed by a generalized linear response friction coefficient. Our analysis demonstrates that the friction coefficient of the driving protocols is sharply peaked at the interface between metastable regions, which leads to minimum-dissipation protocols that drive rapidly within a metastable basin, but then linger longest at the interface, giving thermal fluctuations maximal time to kick the system over the barrier. Intuitively, the same principle applies generically in free energy estimation (both in steered molecular dynamics simulations and in single-molecule experiments), provides a design principle for the construction of thermodynamically efficient coupling between stochastic objects, and makes a prediction regarding the construction of evolved biomolecular motors.

Figure 1

The total potential $E(x,x_{\mathrm{s}})$ is the sum of a time-independent bistable potential $E_{\mathrm{m}}\left(x\right)$ (solid curve) and a harmonic potential $E_{\mathrm{t}}(x,x_{\mathrm{s}})$ (dashed curve) whose location depends on time through the control parameter $\lambda =x_{\mathrm{s}}$, the trap minimum.

Figure 2

Total potential energy landscape $E(x,x_{\mathrm{s}})$ [bistable potential $E_{\mathrm{m}}\left(x\right)$ plus quadratic potential $E_{\mathrm{s}}(x,x_{\mathrm{s}})]$ as a function of particle position $x$. Different curves within a subplot have different trap minima $x_{\mathrm{s}}$. Ascending rows have higher molecular barrier heights $\beta \mathrm{\Delta }{E}_{\mathrm{m}}^{‡}$. Columns to the right have higher trap strengths $k_{\mathrm{s}}$.

Figure 3

Equilibrium force autocorrelation function ${\langle \delta f\left(0\right)\delta f\left(t\right)\rangle }_{x_{\mathrm{s}}}$ for trap minimum $x_{\mathrm{s}}$, with same variation of $x_{\mathrm{s}},k_{\mathrm{s}}$, and $\beta \mathrm{\Delta }{E}_{\mathrm{m}}^{‡}$ as in Fig. 2.

Figure 4

(a) Force variance ${\langle \delta f^2\rangle }_{x_{\mathrm{s}}}$, (b) relaxation time $\tau \left(x_{\mathrm{s}}\right)$, and (c) their product, the generalized friction coefficient $\zeta \left(x_{\mathrm{s}}\right)$, across a range of trap minima $x_{\mathrm{s}}$ (within a curve), for varying molecular barrier heights $\beta \mathrm{\Delta }{E}_{\mathrm{m}}^{‡}$ (different curves within given sub-plot) and varying trap stiffness $k_{\mathrm{s}}$ (left to right subplots).

Figure 5

Excess power ${\mathcal{P}}_{\mathrm{ex}}$ as a function of control parameter $x_{\mathrm{s}}$, calculated directly via numerical simulation (dashed blue curves) or estimated using the control parameter velocity and the cumulative distribution function form of the friction coefficient (solid black curves). Dotted black horizontal lines show asymptotic excess powers at $x_{\mathrm{s}}=\pm \infty$. Same variation of $k_{\mathrm{s}}$ and $\beta \mathrm{\Delta }{E}_{\mathrm{m}}^{‡}$ as in Fig. 2.

Figure 6

Control parameter velocity $d{x}_{\mathrm{s}}/dt$ (in arbitrary units) as a function of control parameter $x_{\mathrm{s}}$, for naive constant-velocity protocols (red dashed lines) and optimal minimum-dissipation protocols under the linear response approximation (solid black curves). Same variation of $k_{\mathrm{s}}$ and $\beta \mathrm{\Delta }{E}_{\mathrm{m}}^{‡}$ as in Fig. 2.

Figure 7

Control parameter position $x_{\mathrm{s}}$ as a function of time (in arbitrary units), for naive constant-velocity protocols (red dashed lines) and optimal minimum-dissipation protocols under the linear response approximation (solid black curves). Same variation of $k_{\mathrm{s}}$ and $\beta \mathrm{\Delta }{E}_{\mathrm{m}}^{‡}$ as in Fig. 2.

Figure 8

(a) Naive excess work, calculated from numerical simulations. (b) Ratio of excess works [Eq. (17)] for the naive (constant-velocity) and optimal (minimum-dissipation) protocols, estimated from the linear response approximation. $k_{\mathrm{s}}$ varies within a curve and $\beta \mathrm{\Delta }{E}_{\mathrm{m}}^{‡}$ varies across curves.