Using a system's equilibrium behavior to reduce its energy dissipation in nonequilibrium processes

Abstract

Cells must operate far from equilibrium, utilizing and dissipating energy continuously to maintain their organization and to avoid stasis and death. However, they must also avoid unnecessary waste of energy. Recent studies have revealed that molecular machines are extremely efficient thermodynamically compared with their macroscopic counterparts. However, the principles governing the efficient out-of-equilibrium operation of molecular machines remain a mystery. A theoretical framework has been recently formulated in which a generalized friction coefficient quantifies the energetic efficiency in nonequilibrium processes. Moreover, it posits that, to minimize energy dissipation, external control should drive the system along the reaction coordinate with a speed inversely proportional to the square root of that friction coefficient. Here, we demonstrate the utility of this theory for designing and understanding energetically efficient nonequilibrium processes through the unfolding and folding of single DNA hairpins.

Keywords: DNA hairpins; dissipation; energetic efficiency; nonequilibrium; single molecule.

Fig. 1.

Equilibrium sampling reveals that the friction coefficient peaks strongly at the hopping regime. (A) Sample force traces as a function of time for folded hairpin (Left; red), hopping hairpin (Center; purple), and unfolded hairpin (Right; blue). (B) Equilibrium force distributions and (C) force correlation as a function of lag time for corresponding fixed optical trap separations. (D) Force variance 〈δF²〉_x, (E) force relaxation time τ_relax(X), and their product (F) the generalized friction coefficient ζ(X) as a function of fixed optical trap separation. (G) For a 0.13-s protocol duration, the designed velocity dX/dt ∝ ζ^−1/2 (green points) with best-fit model (green curve) minimizes Akaike information criterion (30) compared with naive velocity (orange line). (H) Designed and naive velocities scale inversely with protocol duration τ, and designed (green) and naive (yellow) protocols are plotted as functions of t/τ.

Fig. 2.

Designed protocols consistently unfold at lower force and refold at higher force. (A) Example force separation curves from a sample molecule for protocol duration τ = 0.13 s, highlighting the unfolding (Upper) and refolding (Lower) events (black dots) and the corresponding forces (dashed lines) for designed (dark blue and dark red) and naive (light blue and pink) protocols. The raw data (thin lines) are Savitsky–Golay filtered to obtain a smoothed force separation curve (thick lines). (B) Distributions of differences F_naive – F_designed between naive and designed unfolding (blue) and refolding (red) forces. (C) Mean and SE for unfolding and refolding force differences as a function of protocol duration. On average, the designed protocol unfolds at a lower force and refolds as a higher force than the corresponding naive protocol.

Fig. 3.

Designed protocols consistently require less work than corresponding naive protocols. (A) Example force separation curves showing the cycle work W^U + W^R for naive (Left; orange) and designed (Right; green) protocols. The raw force separation curve (thin) is smoothed by a Savitsky–Golay filter (thick). (B) Excess power 〈𝒫_ex(X)〉/〈𝒫_ex〉_naive normalized by average naive excess power as a function of trap separation for naive (yellow) and designed (green) protocols. (C) Distributions of cycle work W^U + W^R for naive (yellow) and designed (green) protocols for protocols ranging from slow (Top) to fast (Middle and Bottom). (D) Mean cycle work 〈W^U + W^R〉 during naive (green) and designed (orange) protocols as a function of protocol duration.