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Using a system’s equilibrium behavior to reduce its energy dissipation in nonequilibrium processes
Edited by Attila Szabo, NIH, Bethesda, MD, and approved February 12, 2019 (received for review October 21, 2018)

Significance
Biomolecular machines implement many vital activities in cells and must operate quickly and in functional directions, requiring energy dissipation. Recent experiments reveal that some evolved machines are quite energetically efficient, engendering interest in design principles that achieve such high efficiency. Recent theory predicts how to use equilibrium measurements (in the absence of driving) to design ways to drive a system to minimize energy dissipation. Here, we experimentally demonstrate the utility of this theory for designing efficient driving processes (“protocols”) by rapidly unfolding and folding single DNA hairpins. We show that such designed protocols systematically and significantly reduce energy dissipation over large variations of driving speed and DNA hairpin friction. Similar protocols may underlie the high efficiency observed in molecular machines.
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.
Reversible heat engines operating infinitely slowly according to the Carnot cycle do not dissipate energy; their energetic efficiency is limited only by the entropy increase of the surroundings associated with the transfer of heat from a hot to a cold reservoir. In contrast, for engines operating irreversibly, the extra nonequilibrium energy cost associated with carrying out a process at a finite rate further reduces their efficiency (1). This is the case of biological machines (2) that must operate under signaling, transport, and cell cycle time constraints. For instance, FoF1-ATP synthase, the primary machine responsible for ATP synthesis, can rotate up to ∼350 revolutions per second (3); the bacteriophage φ29 packaging motor internalizes the 19.3-kbp viral genome into a small capsid at rates of 100 bp/s—faster than the relaxation rate of the confined DNA (4); and during sporulation, the Bacillus subtilis DNA translocase, SpoIIIE, transfers two-thirds of its
Recently, a generalized friction coefficient—which can be obtained from equilibrium measurements—was shown to be the parameter that governs the near-equilibrium energy dissipation during a finite rate process (8). Here, we demonstrate experimentally the utility of this theoretical framework for designing energetically efficient nonequilibrium processes and propose that similar operation protocols may underlie the high efficiency observed in molecular machines. To this end, we subject single DNA hairpins to mechanical unfolding and refolding using protocols dictated by this theory; we show that these protocols systematically and significantly reduce energy dissipation during the process. DNA hairpins are ideally suited for this test, as the magnitude of the friction coefficient can be tuned by changing the molecule’s length, the free energy difference, the free energy barrier, and the transition rates between its folded and unfolded states (9).
According to this near-equilibrium linear response theory, the excess power dissipated by a system taken from an initial to a final state by varying a control parameter λ according to a protocol (time schedule) Λ is proportional to a generalized friction coefficient ζ (8):
It can be shown (12) that, near equilibrium, the driving protocol that minimizes the dissipation for a given total duration,
To obtain the generalized friction coefficient of the DNA hairpin, we monitored the equilibrium force fluctuations of molecules tethered between two optical traps at various fixed trap separations, X. For very small or very large trap separations, the force fluctuates around a single mean value corresponding to the folded or unfolded conformation, respectively (Fig. 1A); for intermediate trap separations, the force fluctuates between two different values, reflecting the hopping dynamics of the DNA hairpin sampling the folded and unfolded conformations (Fig. 1A). For each separation X, we calculated the force autocorrelation function,
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 τrelax(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.
Next, we calculated the force variance (Fig. 1D) and the force relaxation time (Fig. 1E) from the force autocorrelation function. The force variance peaks at an intermediate trap separation,
As mentioned above, the theory predicts that (near equilibrium) the minimum dissipation protocol proceeds with a pulling speed—or velocity of the steering trap—that scales as the inverse square root of the friction coefficient (8): pulling fast at extreme separations, where the friction coefficient is small, and slow around
Next, we measured force as a function of trap separation during designed and naive protocols with total durations ranging from 3.7 to 0.13 s. These force separation curves of naive and designed protocols display significant differences in the force at which the DNA hairpins unfold/refold (Fig. 2A). Fig. 2B shows the distributions of unfolding force differences,
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 Fnaive – Fdesigned 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.
Analogous to Fig. 2A, Fig. 3A depicts the cycle work for a typical realization of an unfolding/refolding cycle. According to Eq. 1, when driving a system at a constant velocity, more work is dissipated at trap separations where the friction coefficient is larger. Consistently, the constant velocity protocols produce higher dissipation around
Designed protocols consistently require less work than corresponding naive protocols. (A) Example force separation curves showing the cycle work WU + WR 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 〈𝒫ex(X)〉/〈𝒫ex〉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 WU + WR for naive (yellow) and designed (green) protocols for protocols ranging from slow (Top) to fast (Middle and Bottom). (D) Mean cycle work 〈WU + WR〉 during naive (green) and designed (orange) protocols as a function of protocol duration.
The data presented here correspond to a DNA hairpin that allowed relatively rapid folded–unfolded equilibration such that transitions to the folded or unfolded conformations occurred even for 0.13-s protocols. This feature allowed us to interrogate the hairpin’s nonequilibrium response over a broad range of protocol durations. In SI Appendix, we show that these results also hold for a different DNA hairpin sequence with significantly (∼100 times) slower equilibration.
In summary, we have sampled the equilibrium force fluctuations in DNA hairpins, displaying the dynamics of a two-state system (Fig. 1). We showed that the generalized friction coefficient—determined from such equilibrium fluctuations—can be used to design driving schedules (Fig. 1G) that significantly reduce the excess work dissipated compared with constant velocity schedules (naive protocols) completed in the same total time (Fig. 3D). This result held for protocol durations that vary by a factor of ∼30 (Fig. 3D), even when driven far from equilibrium (dissipating up to ∼10 kBT, which greatly exceeds the ∼1-kBT energy fluctuations at equilibrium). These observations indicate that this near-equilibrium theory is still able to reduce dissipation even beyond the regime of the theory’s strict validity.
This experiment represents the design and implementation of a single-molecule protocol that systematically reduces the nonequilibrium energy dissipation in a process constrained to a finite duration.
These results have immediate applications in the streamlining of single-molecule experiments and steered molecular dynamics simulations (17). For instance, when using the Jarzynski equality or Crooks fluctuation theorem to infer the free energy difference in a given process (such as protein unfolding), the farther the system is from equilibrium during experiment or simulation, the slower the rate of convergence and accuracy of the free energy estimator, which depends inversely on the energy dissipated (18). Therefore, by sampling the equilibrium fluctuations of a biomolecular process, it should be possible to estimate the generalized friction coefficient across the control parameter landscape; next, it would be possible to craft nonequilibrium protocols that dissipate significantly less energy, thereby speeding up the convergence and increasing the accuracy of any given free energy estimator.
There are tantalizing hints of molecular machines conserving energy while operating out of equilibrium (4, 19): the φ29 DNA packaging motor is more likely to slow down and pause at high packaging fractions, where the storing of additional DNA involves significantly higher dissipation, and translating ribosomes facing RNA hairpins—that impose a large barrier to translation—change “gear,” operating slower while crossing the barrier (20). Based on the theoretical framework presented here, both cases can be seen as examples in which the molecular machines implement driving protocols that proceed slower where the friction coefficient is higher, thereby reducing dissipation and increasing their efficiency. We hypothesize that a molecular biophysical system can waste less energy through naturally evolved dynamics that is rationalizable in terms of the generalized friction coefficient; specifically, such molecular motors may have evolved to slow down their operation in regions of their control parameter space corresponding to high values of the friction coefficient as a way to harness fluctuations from the thermal bath, thus improving their operation efficiency.
The agreement of theory (8) and our experiments suggests extensions to more complex contexts. In particular, we conjecture that molecular machines may have evolved to slow down in regions of large friction and speed up in regions of small friction. The rotary motor F1-ATP synthase is known to be a remarkably efficient machine (21), where the Fo subunit―powered by proton flow down a concentration gradient―forces rotation of the γ-subunit, a molecular crankshaft that drives synthesis of ATP by F1 (6). After attaching a magnetic bead to the crankshaft of F1 (22), one could―analogous to the procedure described in this study―use a magnetic tweezers instrument to hold the bead at various angles so as to extract the equilibrium torque fluctuations of the rotary crankshaft, from which one could extract the friction coefficient at each position (in this experiment, the angle corresponds to the control parameter for driving F1 in analogy to the trap separation for driving the unfolding of the DNA hairpin). One could then estimate the minimum dissipation protocol and determine the ratio of energy input (work done to rotate the crankshaft) to energy output (ATP molecules synthesized) (22, 23) for designed and naive protocols. These ratios quantify the energetic efficiency with which the respective protocols induce F1 to synthesize ATP, and their difference determines the energetic savings.
We have seen here that the linear response theory provides a useful qualitative guide to design protocols that systematically require less work than naive ones. Moreover, since this theoretical framework naturally generalizes to stochastic protocols (24), future experiments could be designed to more closely match autonomous machines driven by fluctuating forces. Insights from the experiments designed with this framework should provide a deeper understanding of the nonequilibrium energetic efficiency of biomolecular machines and ultimately, guide the operation of efficient synthetic nanomachines.
Materials and Methods
Basic Optical Trap Setup.
High-resolution force separation measurements were conducted on a dual-trap instrument using a solid-state 1,064-nm laser as described previously (25). Traps were calibrated as previously described (26). DNA tethers were formed between a 0.90-μm-diameter streptavidin-coated bead and a 1-μm-diameter antidigoxigenin-coated bead (Spherotech) held in separate optical traps. An oxygen scavenging system [100 μg mL−1 glucose oxidase, 5 mg mL−1 dextrose (Sigma-Aldrich), 20 μg mL−1 catalase (Calbiochem)] was included in the buffer to prevent the formation of reactive singlet oxygen, thus increasing the lifetime of the DNA tethers.
DNA Molecules.
Hairpin DNA sequences were selected to display hopping dynamics such that determining
Bead size variation, small differences in chemical attachments, and nonspecific interactions with the bead surface can lead to molecule to molecule variation. We minimized the contribution of trap distance variation by subtracting the value of
Equilibrium Sampling.
Each of 20 molecules is initially probed to find
For each molecule, each separation is sampled for 30 s in order from smallest (
Equilibrium force fluctuations at each of several fixed separations were measured independently in each of 20 different molecules. From these fluctuations, the generalized friction coefficient was estimated using Eq. 2. At each separation, we jackknife resampled from the set of 20 friction estimates to calculate the mean generalized friction and SE (29).
We fit several piecewise-constant acceleration profiles of protocol velocity to the minimum dissipation one (
Naive and Designed Protocols.
We estimate the work W during a trajectory of forces
There are 14, 9, 8, 8, 10, and 9 separate molecules sampled with 888 (444), 590 (295), 396 (198), 590 (295), 592 (296), and 472 (236) individual realizations (full cycles) of protocols for durations of 0.13, 0.24, 0.48, 0.93, 1.8, and 3.7 s, respectively. The cycle work (hysteresis)
We investigate six different protocol durations ranging from 0.13 to 3.7 s. For each protocol duration, we calculate the work along ∼1,200 individual realizations, ∼300 of each of the four protocol types: designed or naive and unfolding or refolding.
To estimate the unfolding (refolding) force in a given force separation curve, we first smooth the force trace using a second-order Savitsky–Golay filter with window width of ∼0.4 ms. We report the unfolding (refolding) force as the maximum (minimum) force before the final unfolding (refolding) event takes place. We control for intermolecular variation by analyzing the difference between unfolding/refolding forces along naive and designed protocols for a given molecule instead of raw unfolding/refolding forces.
The excess power in a protocol interval (Fig. 3B) is calculated by adding the total unfolding work in an interval
Acknowledgments
We thank Nancy Forde, John Bechhoefer, Aidan Brown, and Kamdin Mirsanaye (Simon Fraser University); Michael Woodside (University of Alberta); and Ronen Gabizon and Antony Lee (University of California, Berkeley) for useful discussions. This work is supported in part by the Nanomachines program (KC1203) funded by the office of Basic Energy Sciences of the US Department of Energy [contract no. DE-AC02-05CH11231 (to C.B.)]; the University of California Mexus graduate fellowship (to S.T.); Natural Sciences and Engineering Research Council of Canada (NSERC) Canada Graduate Scholarships-Master's and Alexander Graham Bell Canada Graduate Scholarships-Doctoral (to S.J.L.); the Howard Hughes Medical Institute (C.B.); an NSERC Discovery Grant (to D.A.S.); the Faculty of Science, Simon Fraser University through President’s Research Startup Grant (to D.A.S.); and a Tier-II Canada Research Chair (D.A.S.).
Footnotes
↵1S.T. and S.J.L. contributed equally to this work.
↵2Present address: Scientific Application Development Group, Lumicks, Cambridge, MA 02139.
- ↵3To whom correspondence may be addressed. Email: carlosb{at}berkeley.edu or dsivak{at}sfu.ca.
Author contributions: S.T., S.J.L., S.L., C.B., and D.A.S. designed research; S.T. performed research; S.J.L. and D.A.S. analyzed data; and S.T., S.J.L., S.L., C.B., and D.A.S. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1817778116/-/DCSupplemental.
Published under the PNAS license.
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