Improving free-energy estimates from unidirectional work measurements: theory and experiment

Free-energy measuring nanopore device

Free energies (FEs) in molecular sciences can be used to quantify the stability of folded molecules. In this article, we introduce nanopores for measuring FEs. We pull DNA hairpin-forming molecules through a nanopore, measure work, and estimate the FE change in the slow limit, and with the Jarzynski fluctuation theorem (FT) at fast pulling times. We also pull our molecule with optical tweezers, compare it to nanopores, and explore how sampling single molecules from equilibrium or a folded ensemble affects the FE estimate via the FT. The nanopore experiment helps us address and overcome the conceptual problem of equilibrium sampling in single-molecule pulling experiments. Only when molecules are sampled from an equilibrium ensemble do nanopore and tweezer FE estimates mutually agree. We demonstrate that nanopores are very useful tools for comparing FEs of two molecules at finite times and we propose future applications.

Folding Free Energy Determination of an RNA Three-Way Junction Using Fluctuation Theorems

Nonequilibrium work relations and fluctuation theorems permit us to extract equilibrium information from nonequilibrium measurements. They find application in single-molecule pulling experiments where molecular free energies can be determined from irreversible work measurements by using unidirectional (e.g., Jarzynski’s equality) and bidirectional (e.g., Crooks fluctuation theorem and Bennet’s acceptance ratio (BAR)) methods. However, irreversibility and the finite number of pulls limit their applicability: the higher the dissipation, the larger the number of pulls necessary to estimate ΔG within a few kBT. Here, we revisit pulling experiments on an RNA three-way junction (3WJ) that exhibits significant dissipation and work-distribution long tails upon mechanical unfolding. While bidirectional methods are more predictive, unidirectional methods are strongly biased. We also consider a cyclic protocol that combines the forward and reverse work values to increase the statistics of the measurements. For a fixed total experimental time, faster pulling rates permit us to efficiently sample rare events and reduce the bias, compensating for the increased dissipation. This analysis provides a more stringent test of the fluctuation theorem in the large irreversibility regime.

Molten globule-like transition state of protein barnase measured with calorimetric force spectroscopy

Understanding the molecular forces driving the unfolded polypeptide chain to self-assemble into a functional native structure remains an open question. However, identifying the states visited during protein folding (e.g., the transition state between the unfolded and native states) is tricky due to their transient nature. Here, we introduce calorimetric force spectroscopy in a temperature jump optical trap to determine the enthalpy, entropy, and heat capacity of the transition state of protein barnase. We find that the transition state has the properties of a dry molten globule, that is, high free energy and low configurational entropy, being structurally similar to the native state. This experimental single-molecule study characterizes the thermodynamic properties of the transition state in funneled energy landscapes.

Understanding how proteins fold into their native structure is a fundamental problem in biophysics, crucial for protein design. It has been hypothesized that the formation of a molten globule intermediate precedes folding to the native conformation of globular proteins; however, its thermodynamic properties are poorly known. We perform single-molecule pulling experiments of protein barnase in the range of 7 ^∘C to 37 ^∘C using a temperature-jump optical trap. We derive the folding free energy, entropy and enthalpy, and heat capacity change (ΔCp = 1,050 ± 50 cal/mol·K) at low ionic strength conditions. From the measured unfolding and folding kinetic rates, we also determine the thermodynamic properties of the transition state, finding a significant change in ΔCp (∼90%) between the unfolded and the transition states. In contrast, the major change in enthalpy (∼80%) occurs between the transition and native states. These results highlight a transition state of high energy and low configurational entropy structurally similar to the native state, in agreement with the molten globule hypothesis.

Force Dependence of Proteins' Transition State Position and the Bell-Evans Model

Single-molecule force spectroscopy has opened a new field of research in molecular biophysics and biochemistry. Pulling experiments on individual proteins permit us to monitor conformational transitions with high temporal resolution and measure their free energy landscape. The force–extension curves of single proteins often present large hysteresis, with unfolding forces that are higher than refolding ones. Therefore, the high energy of the transition state (TS) in these molecules precludes kinetic rates measurements in equilibrium hopping experiments. In irreversible pulling experiments, force-dependent kinetic rates measurements show a systematic discrepancy between the sum of the folding and unfolding TS distances derived by the kinetic Bell–Evans model and the full molecular extension predicted by elastic models. Here, we show that this discrepancy originates from the force-induced movement of TS. Specifically, we investigate the highly kinetically stable protein barnase, using pulling experiments and the Bell–Evans model to characterize the position of its kinetic barrier. Experimental results show that while the TS stays at a roughly constant distance relative to the native state, it shifts with force relative to the unfolded state. Interestingly, a conversion of the protein extension into amino acid units shows that the TS position follows the Leffler–Hammond postulate: the higher the force, the lower the number of unzipped amino acids relative to the native state. The results are compared with the quasi-reversible unfolding–folding of a short DNA hairpin.

Free energy measurements by the generalized fluctuation theorems: Theory and numerical study of a model filament

We measure the free energy of a model filament, which undergoes deformations and structural transitions, as a function of its extension, in silico. We perform Brownian Dynamics (BD) simulations of pulling experiments at various speeds, following a protocol close to experimental ones. The results from the fluctuation theorems are compared with the estimates from Monte Carlo (MC) simulation, where the rugged free energy landscape is produced by the density of states method. The fluctuation theorems (FT) give accurate estimates of the free energy up to moderate pulling speeds. At higher pulling speeds, the work distributions do not efficiently sample the domain of small work and FT slightly overestimates free energy. In order to comprehend the differences, we analyze the work distributions from the BD simulations in the framework of trajectory thermodynamics and propose the generalized fluctuation theorems that take into account the information (relative entropy) evaluated in the expanded phase space. The measured work - free energy relation is consistent with the results obtained from the generalized fluctuation theorems. We discuss operational methods to improve the estimates at high pulling speed.

Entropic bonding of the type 1 pilus from experiment and simulation

The type 1 pilus is a bacterial filament consisting of a long coiled proteic chain of subunits joined together by non-covalent bonding between complementing β -strands. Its strength and structural stability are critical for its anchoring function in uropathogenic Escherichia coli bacteria. The pulling and unravelling of the FimG subunit of the pilus was recently studied by atomic force microscopy experiments and steered molecular dynamics simulations (Alonso-Caballero et al. 2018 Nat. Commun. 9, 2758. (doi:10.1038/s41467-018-05107-6)). In this work, we perform a quantitative comparison between experiment and simulation, showing a good agreement in the underlying work values for the unfolding. The simulation results are then used to estimate the free energy difference for the detachment of FimG from the complementing strand of the neighbouring subunit in the chain, FimF. Finally, we show that the large free energy difference for the unravelling and detachment of the subunits which leads to the high stability of the chain is entirely entropic in nature.

Two-State Folding Energy Determination Based on Transition Points in Nonequilibrium Single-Molecule Experiments

Many small protein domains or nucleic acid structures undergo two-state unfolding-refolding transitions during mechanical stretching using single-molecule techniques. Here, by applying the Jarzynski equality (JE), we analytically express the folding energy of a molecule as a function of the experimentally measured transition points ξ* obtained with two typical time-varying mechanical constraints: the force constraints F(t) and the position constraints R(t) of a Hookian spring attached to one end of the molecule. Compared to previous applications of JE based on the integration of accurately measured force-extension curves of a tether that typically contains the molecule of interest and handles, our approach just needs to accurately measure a single data point. In the case of the F(t) process, the calculation is handle-independent. The broad applications of the theory are demonstrated by measuring the folding energies of a DNA hairpin, a DNA G-quadruplex, and the titin I27 domain based on transition forces using magnetic tweezers.

Quantum work fluctuations in connection with the Jarzynski equality

A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change ΔF of a system at inverse temperature β from an ensemble average of nonequilibrium exponential work, i.e., 〈e^{-βW}〉=e^{-βΔF}. The number of experimental work values needed to reach a given accuracy of ΔF is determined by the variance of e^{-βW}, denoted var(e^{-βW}). We discover in this work that var(e^{-βW}) in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging var(e^{-βW}) at sufficiently low temperatures, markedly different from the classical behavior. The divergence of var(e^{-βW}) indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.

A physical picture for mechanical dissociation of biological complexes: from forces to free energies

Single-molecule force spectroscopy is a powerful technique based on the application of controlled forces to macromolecules. In order to relate the measured response of the molecule to its equilibrium and dynamic properties, a suitable physical picture of the involved process is necessary. In this work, we introduce a plausible model for mechanical unbinding of some molecular complexes, based on a novel free energy profile. We combine two standard theoretical frameworks for analyzing force spectroscopy experiments on two protein:protein complexes, obtaining key magnitudes of the underlying free energy profile, which are only understood within the mentioned model. Additionally, we carry out detailed stochastic dynamics simulations to prove the validity of the analysis protocol and the reliability of the free energy profile. Remarkably, we can compare directly the obtained unbinding free energies with the previously known bulk binding free energies, bridging the gap between bulk and single molecule techniques.

Mechanical Folding and Unfolding of Protein Barnase at the Single-Molecule Level

The unfolding and folding of protein barnase has been extensively investigated in bulk conditions under the effect of denaturant and temperature. These experiments provided information about structural and kinetic features of both the native and the unfolded states of the protein, and debates about the possible existence of an intermediate state in the folding pathway have arisen. Here, we investigate the folding/unfolding reaction of protein barnase under the action of mechanical force at the single-molecule level using optical tweezers. We measure unfolding and folding force-dependent kinetic rates from pulling and passive experiments, respectively, and using Kramers-based theories (e.g., Bell-Evans and Dudko-Hummer-Szabo models), we extract the position of the transition state and the height of the kinetic barrier mediating unfolding and folding transitions, finding good agreement with previous bulk measurements. Measurements of the force-dependent kinetic barrier using the continuous effective barrier analysis show that protein barnase verifies the Leffler-Hammond postulate under applied force and allow us to extract its free energy of folding, ΔG0. The estimated value of ΔG0 is in agreement with our predictions obtained using fluctuation relations and previous bulk studies. To address the possible existence of an intermediate state on the folding pathway, we measure the power spectrum of force fluctuations at high temporal resolution (50 kHz) when the protein is either folded or unfolded and, additionally, we study the folding transition-path time at different forces. The finite bandwidth of our experimental setup sets the lifetime of potential intermediate states upon barnase folding/unfolding in the submillisecond timescale.

Resolving Non-Specific and Specific Adhesive Interactions of Catechols at Solid/Liquid Interfaces at the Molecular Scale

The adhesive system of mussels evolved into a powerful and adaptive system with affinity to a wide range of surfaces. It is widely known that thereby 3,4-dihydroxyphenylalanine (Dopa) plays a central role. However underlying binding energies remain unknown at the single molecular scale. Here, we use single-molecule force spectroscopy to estimate binding energies of single catechols with a large range of opposing chemical functionalities. Our data demonstrate significant interactions of Dopa with all functionalities, yet most interactions fall within the medium-strong range of 10-20 kB T. Only bidentate binding to TiO2 surfaces exhibits a higher binding energy of 29 kB T. Our data also demonstrate at the single-molecule level that oxidized Dopa and amines exhibit interaction energies in the range of covalent bonds, confirming the important role of Dopa for cross-linking in the bulk mussel adhesive. We anticipate that our approach and data will further advance the understanding of biologic and technologic adhesives.

Scaling Law for Irreversible Entropy Production in Critical Systems

We examine the Jarzynski equality for a quenching process across the critical point of second-order phase transitions, where absolute irreversibility and the effect of finite-sampling of the initial equilibrium distribution arise in a single setup with equal significance. We consider the Ising model as a prototypical example for spontaneous symmetry breaking and take into account the finite sampling issue by introducing a tolerance parameter. The initially ordered spins become disordered by quenching the ferromagnetic coupling constant. For a sudden quench, the deviation from the Jarzynski equality evaluated from the ideal ensemble average could, in principle, depend on the reduced coupling constant ε₀ of the initial state and the system size L. We find that, instead of depending on ε₀ and L separately, this deviation exhibits a scaling behavior through a universal combination of ε₀ and L for a given tolerance parameter, inherited from the critical scaling laws of second-order phase transitions. A similar scaling law can be obtained for the finite-speed quench as well within the Kibble-Zurek mechanism.

Convergence of large-deviation estimators

We study the convergence of statistical estimators used in the estimation of large-deviation functions describing the fluctuations of equilibrium, nonequilibrium, and manmade stochastic systems. We give conditions for the convergence of these estimators with sample size, based on the boundedness or unboundedness of the quantity sampled, and discuss how statistical errors should be defined in different parts of the convergence region. Our results shed light on previous reports of "phase transitions" in the statistics of free energy estimators and establish a general framework for reliably estimating large-deviation functions from simulation and experimental data and identifying parameter regions where this estimation converges.

Scaling from single molecule to macroscopic adhesion at polymer/metal interfaces

Understanding the evolution of macroscopic adhesion based on fundamental molecular interactions is crucial to designing strong and smart polymer/metal interfaces that play an important role in many industrial and biomedical applications. Here we show how macroscopic adhesion can be predicted on the basis of single molecular interactions. In particular, we carry out dynamic single molecule-force spectroscopy (SM-AFM) in the framework of Bell-Evans' theory to gain information about the energy barrier between the bound and unbound states of an amine/gold junction. Furthermore, we use Jarzynski's equality to obtain the equilibrium ground-state energy difference of the amine/gold bond from these nonequilibrium force measurements. In addition, we perform surface forces apparatus (SFA) experiments to measure macroscopic adhesion forces at contacts where approximately 10(7) amine/gold bonds are formed simultaneously. The SFA approach provides an amine/gold interaction energy (normalized by the number of interacting molecules) of (36 ± 1)k(B)T, which is in excellent agreement with the interaction free energy of (35 ± 3)k(B)T calculated using Jarzynski's equality and single-molecule AFM experiments. Our results validate Jarzynski's equality for the field of polymer/metal interactions by measuring both sides of the equation. Furthermore, the comparison of SFA and AFM shows how macroscopic interaction energies can be predicted on the basis of single molecular interactions, providing a new strategy to potentially predict adhesive properties of novel glues or coatings as well as bio- and wet adhesion.

Direct quantification of loop interaction and π-π stacking for G-quadruplex stability at the submolecular level

The well-demonstrated biological functions of DNA G-quadruplex inside cells call for small molecules that can modulate these activities by interacting with G-quadruplexes. However, the paucity of the understanding of the G-quadruplex stability contributed from submolecular elements, such as loops and tetraguanine (G) planes (or G-quartets), has hindered the development of small-molecule binders. Assisted by click chemistry, herein, we attached pulling handles via two modified guanines in each of the three G-quartets in human telomeric G-quadruplex. Mechanical unfolding using these handles revealed that the loop interaction contributed more to the G-quadruplex stability than the stacking of G-quartets. This result was further confirmed by the binding of stacking ligands, such as telomestatin derivatives, which led to similar mechanical stability for all three G-quartets by significant reduction of loop interactions for the top and bottom G-quartets. The direct comparison of loop interaction and G-quartet stacking in G-quadruplex provides unprecedented insights for the design of more efficient G-quadruplex-interacting molecules. Compared to traditional experiments, in which mutations are employed to elucidate the roles of specific residues in a biological molecule, our submolecular dissection offers a complementary approach to evaluate individual domains inside a molecule with fewer disturbances to the native structure.

On the improvement of free-energy calculation from steered molecular dynamics simulations using adaptive stochastic perturbation protocols

The potential of mean force (PMF) calculation in single molecule manipulation experiments performed via the steered molecular dynamics (SMD) technique is a computationally very demanding task because the analyzed system has to be perturbed very slowly to be kept close to equilibrium. Faster perturbations, far from equilibrium, increase dissipation and move the average work away from the underlying free energy profile, and thus introduce a bias into the PMF estimate. The Jarzynski equality offers a way to overcome the bias problem by being able to produce an exact estimate of the free energy difference, regardless of the perturbation regime. However, with a limited number of samples and high dissipation the Jarzynski equality also introduces a bias. In our previous work, based on the Brownian motion formalism, we introduced three stochastic perturbation protocols aimed at improving the PMF calculation with the Jarzynski equality in single molecule manipulation experiments and analogous computer simulations. This paper describes the PMF reconstruction results based on full-atom molecular dynamics simulations, obtained with those three protocols. We also want to show that the protocols are applicable with the second-order cumulant expansion formula. Our protocols offer a very noticeable improvement over the simple constant velocity pulling. They are able to produce an acceptable estimate of PMF with a significantly reduced bias, even with very fast perturbation regimes. Therefore, the protocols can be adopted as practical and efficient tools for the analysis of mechanical properties of biological molecules.

Free-energy inference from partial work measurements in small systems

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 dual-trap 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.

Unfolding kinetics of periodic DNA hairpins

DNA hairpin molecules with periodic base sequences can be expected to exhibit a regular coarse-grained free energy landscape (FEL) as a function of the number of open base pairs and applied mechanical force. Using a commonly employed model, we first analyze for which types of sequences a particularly simple landscape structure is predicted, where forward and backward energy barriers between partly unfolded states are decreasing linearly with force. Stochastic unfolding trajectories for such molecules with simple FEL are subsequently generated by kinetic Monte Carlo simulations. Introducing probabilities that can be sampled from these trajectories, it is shown how the parameters characterizing the FEL can be estimated. Already 300 trajectories, as typically generated in experiments, provide faithful results for the FEL parameters.

Nascent RNA transcripts facilitate the formation of G-quadruplexes

Recent discovery of the RNA/DNA hybrid G-quadruplexes (HQs) and their potential wide-spread occurrence in human genome during transcription have suggested a new and generic transcriptional control mechanism. The G-rich sequence in which HQ may form can coincide with that for DNA G-quadruplexes (GQs), which are well known to modulate transcriptions. Understanding the molecular interaction between HQ and GQ is, therefore, of pivotal importance to dissect the new mechanism for transcriptional regulation. Using a T7 transcription model, herein we found that GQ and HQ form in a natural sequence, (GGGGA)4, downstream of many transcription start sites. Using a newly-developed single-molecular stalled-transcription assay, we revealed that RNA transcripts helped to populate quadruplexes at the expense of duplexes. Among quadruplexes, HQ predominates GQ in population and mechanical stabilities, suggesting HQ may serve as a better mechanical block during transcription. The fact that HQ and GQ folded within tens of milliseconds in the presence of RNA transcripts provided justification for the co-transcriptional folding of these species. The catalytic role of RNA transcripts in the GQ formation was strongly suggested as the GQ folded >7 times slower without transcription. These results shed light on the possible synergistic effect of GQs and HQs on transcriptional controls.

Nonequilibrium work by charge control in a Josephson junction

We consider a single Josephson junction in the presence of time varying gate charge, and examine the nonequilibrium work done by the charge control in the framework of fluctuation theorems. Assuming first a high quality junction with negligible Ohmic current, we obtain the probability distribution functions of the work and confirm the Crooks relation to give the estimation of the free energy changes ΔF=0. The reliability of ΔF estimated from the Jarzynksi equality is crucially dependent on protocol parameters, while the Bennett's acceptance ratio method yields consistently ΔF=0. We examine the behaviors of the work average and point out its relation to heat and entropy production associated with the circuit control. Finally considering finite tunnel resistance we discuss dissipation effects on the work statistics.

Probing small-scale intermittency with a fluctuation theorem

We characterize statistical properties of the flow field in developed turbulence using concepts from stochastic thermodynamics. On the basis of data from a free air-jet experiment, we demonstrate how the dynamic fluctuations induced by small-scale intermittency generate analogs of entropy-consuming trajectories with sufficient weight to make fluctuation theorems observable at the macroscopic scale. We propose an integral fluctuation theorem for the entropy production associated with the stochastic evolution of velocity increments along the eddy hierarchy and demonstrate its extreme sensitivity to the accurate description of the tails of the velocity distributions.

Structural and mechanical properties of individual human telomeric G-quadruplexes in molecularly crowded solutions

Recent experiments provided controversial observations that either parallel or non-parallel G-quadruplex exists in molecularly crowded buffers that mimic cellular environment. Here, we used laser tweezers to mechanically unfold structures in a human telomeric DNA fragment, 5'-(TTAGGG)4TTA, along three different trajectories. After the end-to-end distance of each unfolding geometry was measured, it was compared with PDB structures to identify the best-matching G-quadruplex conformation. This method is well-suited to identify biomolecular structures in complex settings not amenable to conventional approaches, such as in a solution with mixed species or at physiologically significant concentrations. With this approach, we found that parallel G-quadruplex coexists with non-parallel species (1:1 ratio) in crowded buffers with dehydrating cosolutes [40% w/v dimethyl sulfoxide (DMSO) or acetonitrile (ACN)]. In crowded solutions with steric cosolutes [40% w/v bovine serum albumin (BSA)], the parallel G-quadruplex constitutes only 10% of the population. This difference unequivocally supports the notion that dehydration promotes the formation of parallel G-quadruplexes. Compared with DNA hairpins that have decreased unfolding forces in crowded (9 pN) versus diluted (15 pN) buffers, those of G-quadruplexes remain the same (20 pN). Such a result implies that in a cellular environment, DNA G-quadruplexes, instead of hairpins, can stop DNA/RNA polymerases with stall forces often <20 pN.

Stochastic thermodynamics, fluctuation theorems and molecular machines

Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.

Comparison of free-energy estimators and their dependence on dissipated work

The estimate of free energy changes based on Bennett's acceptance ratio method is examined in several limiting cases and compared with other estimates based on the Jarzynski equality and on the Crooks relation. While the absolute amount of the dissipated work, defined as the surplus of the average work over the free energy difference, limits the practical applicability of Jarzynski's and Crooks' methods, the reliability of Bennett's approach is restricted by the difference of the dissipated works in the forward and the backward processes. We illustrate these points by considering a Gaussian chain and a hairpin chain which both are extended during the forward and accordingly compressed during the backward protocols. The reliability of the Crooks relation predominantly depends on the sample size; for the Jarzynski estimator the slowness of the work protocol is crucial, and the Bennett method is shown to give precise estimates irrespective of the pulling speed and sample size as long as the dissipated works are the same for the forward and the backward processes as it is the case for Gaussian work distributions. With an increasing dissipated work difference the Bennett estimator also acquires a bias which increases roughly in proportion to this difference. A substantial simplification of the Bennett estimator is provided by the 1/2 formula which expresses the free energy difference by the algebraic average of the Jarzynski estimates for the forward and the backward processes. It agrees with the Bennett estimate in all cases when the Jarzynski and the Crooks estimates fail to give reliable results.

Click chemistry assisted single-molecule fingerprinting reveals a 3D biomolecular folding funnel

A 3D folding funnel was proposed in the 1990s to explain the fast kinetics exhibited by a biomacromolecule in presence of seemingly unlimited folding pathways. Over the years, numerous simulations have been performed with this concept; however, experimental verification is yet to be attained even for the simplest proteins. Here, we have used a click chemistry based strategy to introduce six pairs of handles in a human telomeric DNA sequence. A laser-tweezers-based, single-molecule structural fingerprinting on the six inter-handle distances reveals the formation of a hybrid-1 G-quadruplex in the sequence. Kinetic and thermodynamic fingerprinting on the six trajectories defined by each handle-pair depict a 3D folding funnel and a kinetic topology in which the kinetics pertaining to each handle residue is annotated for this G-quadruplex. We anticipate the methods and the concepts developed here are well applicable to other biomacromolecules, including RNA and proteins.

G-quadruplex and i-motif are mutually exclusive in ILPR double-stranded DNA

G-quadruplex has demonstrated its biological functions in vivo. Although G-quadruplex in single-stranded DNA (ssDNA) has been well characterized, investigation of this species in double-stranded DNA (dsDNA) lags behind. Here we use chemical footprinting and laser-tweezers-based single-molecule approaches to demonstrate that a dsDNA fragment found in the insulin-linked polymorphic region (ILPR), 5'-(ACA GGGG TGT GGGG)2 TGT, can fold into a G-quadruplex at pH 7.4 with 100 mM K+, and an i-motif at pH 5.5 with 100 mM Li+. Surprisingly, under a condition that favors the formation of both G-quadruplex and i-motif (pH 5.5, 100 mM K+), a unique determination of change in the free energy of unfolding (ΔGunfold) by laser-tweezers experiments provides compelling evidence that only one species is present in each dsDNA. Under this condition, molecules containing G-quadruplex are more stable than those with i-motif. These two species have mechanical stabilities (rupture force≥17 pN) comparable to the stall force of RNA polymerases, which, from a mechanical perspective alone, could justify a regulatory mechanism for tetraplex structures in the expression of human insulin.

Single-molecule measurements of the binding between small molecules and DNA aptamers

Aptamers that bind small molecules can serve as basic biosensing platforms. Evaluation of the binding constant between an aptamer and a small molecule helps to determine the effectiveness of the aptamer-based sensors. Binding constants are often measured by a series of experiments with varying ligand or aptamer concentrations. Such experiments are time-consuming, material nonprudent, and prone to low reproducibility. Here, we use laser tweezers to determine the dissociation constant for aptamer-ligand interactions at the single-molecule level from only one ligand concentration. Using an adenosine 5'-triphosphate disodium salt (ATP) binding aptamer as an example, we have observed that the mechanical stabilities of aptamers bound with ATP are higher than those without a ligand. Comparison of the change in free energy of unfolding (ΔG(unfold)) between these two aptamers yields a ΔG of 33 ± 4 kJ/mol for the binding. By applying a Hess-like cycle at room temperature, we obtained a dissociation constant (K(d)) of 2.0 ± 0.2 μM, a value consistent with the K(d) obtained from our equilibrated capillary electrophoresis (CE) (2.4 ± 0.4 μM) and close to that determined by affinity chromatography in the literature (6 ± 3 μM). We anticipate that our laser tweezers and CE methodologies may be used to more conveniently evaluate the binding between receptors and ligands and also serve as analytical tools for force-based biosensing.

Phase transition in the Jarzynski estimator of free energy differences

The transition between a regime in which thermodynamic relations apply only to ensembles of small systems coupled to a large environment and a regime in which they can be used to characterize individual macroscopic systems is analyzed in terms of the change in behavior of the Jarzynski estimator of equilibrium free energy differences from nonequilibrium work measurements. Given a fixed number of measurements, the Jarzynski estimator is unbiased for sufficiently small systems. In these systems the directionality of time is poorly defined and the configurations that dominate the empirical average, but which are in fact typical of the reverse process, are sufficiently well sampled. As the system size increases the arrow of time becomes better defined. The dominant atypical fluctuations become rare and eventually cannot be sampled with the limited resources that are available. Asymptotically, only typical work values are measured. The Jarzynski estimator becomes maximally biased and approaches the exponential of minus the average work, which is the result that is expected from standard macroscopic thermodynamics. In the proper scaling limit, this regime change has been recently described in terms of a phase transition in variants of the random energy model. In this paper this correspondence is further demonstrated in two examples of physical interest: the sudden compression of an ideal gas and adiabatic quasistatic volume changes in a dilute real gas.