A two-state kinetic model for the unfolding of single molecules by mechanical force

Nucleosome Dynamics Derived at the Single-Molecule Level Bridges Its Structures and Functions

Nucleosome, the building block of chromatin, plays pivotal roles in all DNA-related processes. While cryogenic-electron microscopy (cryo-EM) has significantly advanced our understanding of nucleosome structures, the emerging field of single-molecule force spectroscopy is illuminating their dynamic properties. This technique is crucial for revealing how nucleosome behavior is influenced by chaperones, remodelers, histone variants, and post-translational modifications, particularly in their folding and unfolding mechanisms under tension. Such insights are vital for deciphering the complex interplay in nucleosome assembly and structural regulation, highlighting the nucleosome's versatility in response to DNA activities. In this Perspective, we aim to consolidate the latest advancements in nucleosome dynamics, with a special focus on the revelations brought forth by single-molecule manipulation. Our objective is to highlight the insights gained from studying nucleosome dynamics through this innovative approach, emphasizing the transformative impact of single-molecule manipulation techniques in the field of chromatin research.

Information-to-work conversion in single-molecule experiments: From discrete to continuous feedback

We theoretically investigate the extractable work in single molecule unfolding-folding experiments with applied feedback. Using a simple two-state model, we obtain a description of the full work distribution from discrete to continuous feedback. The effect of the feedback is captured by a detailed fluctuation theorem, accounting for the information aquired. We find analytical expressions for the average work extraction as well as an experimentally measurable bound thereof, which becomes tight in the continuous feedback limit. We further determine the parameters for maximal power or rate of work extraction. Although our two-state model only depends on a single effective transition rate, we find qualitative agreement with Monte Carlo simulations of DNA hairpin unfolding-folding dynamics.

Steps minimize dissipation in rapidly driven stochastic systems

Micro- and nanoscale systems driven by rapid changes in control parameters (control protocols) dissipate significant energy. In the fast-protocol limit, we find that protocols that minimize dissipation at fixed duration are universally given by a two-step process, jumping to and from a point that balances jump size with fast relaxation. Jump protocols could be exploited by molecular machines or thermodynamic computing to improve energetic efficiency, and implemented in nonequilibrium free-energy estimation to improve accuracy.

Recovery of Equilibrium Free Energy from Nonequilibrium Thermodynamics with Mechanosensitive Ion Channels in E. coli

In situ measurements of the free energy difference between the open and closed states of ion channels are challenging due to hysteresis effects and inactivation. Exploiting recent developments in statistical physics, we present a general formalism to extract the free energy difference ΔF between the closed and open states of mechanosensitive ion channels from nonequilibrium work distributions associated with the opening and closing of the channels (gating) in response to ramp stimulation protocols recorded in native patches. We show that the work distributions obtained from the gating of MscS channels in E. coli membrane satisfy the strong symmetry relation predicted by the Crooks fluctuation theorem. Our approach enables the determination of ΔF using patch-clamp experiments, which are often inherently restricted to the nonequilibrium regime.

Apparent violations of the second law in two-level systems

We examine two-level systems which undergo an instantaneous change in the energy level differences due to the application of an external protocol. In particular the behavior of the work distribution as well as the probability of second-law violations as the thermodynamic limit, and separately the quasistatic limit, is approached are investigated.

Nonequilibrium free-energy estimation conditioned on measurement outcomes

The Jarzynski equality is one of the most influential results in the field of nonequilibrium statistical mechanics. This celebrated equality allow the calculation of equilibrium free-energy differences from work distributions of nonequilibrium processes. In practice, such calculations often suffer from poor convergence due to the need to sample rare events. Here we examine if the inclusion of measurement and feedback can improve the convergence of nonequilibrium free-energy calculations. A modified version of the Jarzynski equality in which realizations with a given outcome are kept, while others are discarded, is used. We find that discarding realizations with unwanted outcomes can result in improved convergence compared to calculations based on the Jarzynski equality. We argue that the observed improved convergence is closely related to Bennett's acceptance ratio method, which was developed without any reference to measurements or feedback.

Direct Observation of Folding Energy Landscape of RNA Hairpin at Mechanical Loading Rates

By applying a controlled mechanical load using optical tweezers, we measured the diffusive barrier crossing in a 49 nt long P5ab RNA hairpin. We find that in the free-energy landscape the barrier height (G^‡) and transition distance (x^‡) are dependent on the loading rate (r) along the pulling direction, x, as predicted by Bell. The barrier shifted toward the initial state, whereas ΔG^‡ reduced significantly from 50 to 5 kT, as r increased from 0 to 32 pN/s. However, the equilibrium work (ΔG) during strand separation, as estimated by Crook's fluctuation theorem, remained unchanged at different rates. Previously, helix formation and denaturation have been described as two-state (F ↔ U) transitions for P5ab. Herein, we report three intermediate states I₁, I, and I₂ located at 4, 11, and 16 nm respectively, from the folded conformation. The intermediates were observed only when the hairpin was subjected to an optimal r, 7.6 pN/s. The results indicate that the complementary strands in P5ab can zip and unzip through complex routes, whereby mismatches act as checkpoints and often impose barriers. The study highlights the significance of loading rates in force-spectroscopy experiments that are increasingly being used to measure the folding properties of biomolecules.

Analytic descriptions of stochastic bistable systems under force ramp

Solving the two-state master equation with time-dependent rates, the ubiquitous driven bistable system, is a long-standing problem that does not permit a complete solution for all driving rates. Here we show an accurate approximation to this problem by considering the system in the control parameter regime. The results are immediately applicable to a diverse range of bistable systems including single-molecule mechanics.

Thermodynamics of force-dependent folding and unfolding of small protein and nucleic acid structures

In this paper, we outline the theoretical framework for understanding the equilibrium force-dependent folding and unfolding transitions of protein domains and small nucleic acid structures, both having small rigid folded structures and highly flexible unfolded polymeric chain conformations. A complete statistical description of the state described by the probability function ρ(ξ)(n,x), is obtained, where n is an index denoting the structural state, and x is the extension of the molecule. ξ denotes an external constraint applied to the molecule, which is either a constant force or a harmonic spring attached to one end of the molecule. The extension probability distribution regardless of the structural state: , the free energy landscape: -kBT ln(ρ(ξ)(x)), and the probability of the states regardless of the extension: , are analyzed using the force-dependent structural transitions of the classic titin I27 domain as an example. The impact of different external constraints is also discussed.

Free energies of molecular clusters determined by guided mechanical disassembly

The excess free energy of a molecular cluster is a key quantity in models of the nucleation of droplets from a metastable vapor phase; it is often viewed as the free energy arising from the presence of an interface between the two phases. We show how this quantity can be extracted from simulations of the mechanical disassembly of a cluster using guide particles in molecular dynamics. We disassemble clusters ranging in size from 5 to 27 argonlike Lennard-Jones atoms, thermalized at 60 K, and obtain excess free energies, by means of the Jarzynski equality, that are consistent with previous studies. We only simulate the cluster of interest, in contrast to approaches that require a series of comparisons to be made between clusters differing in size by one molecule. We discuss the advantages and disadvantages of the scheme and how it might be applied to more complex systems.

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.

Microcanonical work and fluctuation relations for an open system: An exactly solvable model

We calculate the probability distribution of work for an exactly solvable model of a system interacting with its environment. The system of interest is a harmonic oscillator with a time-dependent control parameter, the environment is modeled by N-independent harmonic oscillators with arbitrary frequencies, and the system-environment coupling is bilinear and not necessarily weak. The initial conditions of the combined system and environment are sampled from a microcanonical distribution and the system is driven out of equilibrium by changing the control parameter according to a prescribed protocol. In the limit of infinitely large environment, i.e., N→∞, we recover the nonequilibrium work relation and Crooks's fluctuation relation. Moreover, the microcanonical Crooks relation is verified for finite environments. Finally, we show the equivalence of multitime correlation functions of the system in the infinite environment limit for canonical and microcanonical ensembles.

An energetic model for macromolecules unfolding in stretching experiments

We propose a simple approach, based on the minimization of the total (entropic plus unfolding) energy of a two-state system, to describe the unfolding of multi-domain macromolecules (proteins, silks, polysaccharides, nanopolymers). The model is fully analytical and enlightens the role of the different energetic components regulating the unfolding evolution. As an explicit example, we compare the analytical results with a titin atomic force microscopy stretch-induced unfolding experiment showing the ability of the model to quantitatively reproduce the experimental behaviour. In the thermodynamic limit, the sawtooth force-elongation unfolding curve degenerates to a constant force unfolding plateau.

Analysis of diffuse K+ and Mg2+ ion binding to a two-base-pair kissing complex by single-molecule mechanical unfolding

The folding and stability of RNA tertiary interactions depend critically on cationic conditions. It is usually difficult, however, to isolate such effects on tertiary interactions from those on the entire RNA. By manipulating conformations of single RNA molecules using optical tweezers, we distinguished individual steps of breaking and forming of a two-base-pair kissing interaction from those of secondary folding. The binding of metal ions to the small tertiary structure appeared to be saturable with an apparent Kd of 160 mM for K(+) and 1.5 mM for Mg(2+). The kissing formation was estimated to be associated with binding of ~2-3 diffuse K(+) or Mg(2+) ions. At their saturated binding, Mg(2+) provided ~3 kcal/mol more stabilizing energy to the structure than K(+). Furthermore, the cations change the unkissing forces significantly more than the kissing ones. For example, the presence of Mg(2+) ions increased the average unkissing force from 21 pN to 44 pN, surprisingly high for breaking merely two base pairs; in contrast, the mean kissing force was changed by only 4.5 pN. Interestingly, the differential salt effects on the transition forces were not caused by different changes in the height of the kinetic barriers but were instead attributed to how different molecular structures respond to the applied force. Our results showed the importance of diffuse cation binding to the stability of tertiary interaction and demonstrated the utility of mechanical unfolding in studying tertiary interactions.

Global conformational selection and local induced fit for the recognition between intrinsic disordered p53 and CBP

The transactivation domain (TAD) of tumor suppressor p53 can bind with the nuclear coactivator binding domain (NCBD) of cyclic-AMP response element binding protein (CBP) and activate transcription. NMR experiments demonstrate that both apo-NCBD and TAD are intrinsic disordered and bound NCBD/TAD undergoes a transition to well folded. The recognition mechanism between intrinsic disordered proteins is still hotly debated. Molecular dynamics (MD) simulations in explicit solvent are used to study the recognition mechanism between intrinsic disordered TAD and NCBD. The average RMSD values between bound and corresponding apo states and Kolmogorov-Smirnov P test analysis indicate that TAD and NCBD may follow an induced fit mechanism. Quantitative analysis indicates there is also a global conformational selection. In summary, the recognition of TAD and NCBD might obey a local induced fit and global conformational selection. These conclusions are further supported by high-temperature unbinding kinetics and room temperature landscape analysis. These methods can be used to study the recognition mechanism of other intrinsic disordered proteins.

Biological mechanisms, one molecule at a time

The last 15 years have witnessed the development of tools that allow the observation and manipulation of single molecules. The rapidly expanding application of these technologies for investigating biological systems of ever-increasing complexity is revolutionizing our ability to probe the mechanisms of biological reactions. Here, we compare the mechanistic information available from single-molecule experiments with the information typically obtained from ensemble studies and show how these two experimental approaches interface with each other. We next present a basic overview of the toolkit for observing and manipulating biology one molecule at a time. We close by presenting a case study demonstrating the impact that single-molecule approaches have had on our understanding of one of life's most fundamental biochemical reactions: the translation of a messenger RNA into its encoded protein by the ribosome.

Challenges in thermodynamics: Irreversible processes, nonextensive entropies, and systems without equilibrium states

Recent works on evaporation and condensation demonstrate that even these simplest irreversible processes, studied for over 100 years, are not well understood. In the case of a liquid evaporating into its vapor, the liquid temperature is constant during evaporation and the evaporation flux is governed by the heat transfer from the hotter vapor into the colder liquid. Whether liquid evaporates into its own vapor or into the vacuum, the irreversible pathway in the process goes through a number of steps which quickly lead to the steady-state conditions with mechanical equilibrium in most parts of the system—the fact overlooked in all previous studies. Even less is known about general rules which govern systems far from equilibrium. Recently, it has been demonstrated that a work done in an irreversible process can be related to the free energy difference between equilibrium states joined by the process. Finally, a real challenge in thermodynamics is a description of living systems since they do not have equilibrium states, are nonextensive, (i.e., they cannot be divided into subsystems), and cannot be isolated. Thus, their proper description requires new paradigms in thermodynamics.

Statistics of trajectories in two-state master equations

We derive a simple expression for the probability of trajectories of a master equation. The expression is particularly useful when the number of states is small and permits the calculation of observables that can be defined as functionals of whole trajectories. We illustrate the method with a two-state master equation, for which we calculate the distribution of the time spent in one state and the distribution of the number of transitions, each in a given time interval. These two expressions are obtained analytically in terms of modified Bessel functions.

Work distribution for the adiabatic compression of a dilute and interacting classical gas

We consider a simple, physically motivated model of a dilute classical gas of interacting particles, initially equilibrated with a heat bath, undergoing adiabatic and quasistatic compression or expansion. This provides an example of a thermodynamic process for which non-Gaussian work fluctuations can be computed exactly from microscopic principles. We find that the work performed during this process is described statistically by a gamma distribution, and we use this result to show that the model satisfies the nonequilibrium work and fluctuation theorems, but not a prediction based on linear response theory.

Determination of thermodynamics and kinetics of RNA reactions by force

Single-molecule methods have made it possible to apply force to an individual RNA molecule. Two beads are attached to the RNA; one is on a micropipette, the other is in a laser trap. The force on the RNA and the distance between the beads are measured. Force can change the equilibrium and the rate of any reaction in which the product has a different extension from the reactant. This review describes use of laser tweezers to measure thermodynamics and kinetics of unfolding/refolding RNA. For a reversible reaction the work directly provides the free energy; for irreversible reactions the free energy is obtained from the distribution of work values. The rate constants for the folding and unfolding reactions can be measured by several methods. The effect of pulling rate on the distribution of force-unfolding values leads to rate constants for unfolding. Hopping of the RNA between folded and unfolded states at constant force provides both unfolding and folding rates. Force-jumps and force-drops, similar to the temperature jump method, provide direct measurement of reaction rates over a wide range of forces. The advantages of applying force and using single-molecule methods are discussed. These methods, for example, allow reactions to be studied in non-denaturing solvents at physiological temperatures; they also simplify analysis of kinetic mechanisms because only one intermediate at a time is present. Unfolding of RNA in biological cells by helicases, or ribosomes, has similarities to unfolding by force.

Evidence for non-two-state kinetics in the nucleocapsid protein chaperoned opening of DNA hairpins

In HIV-1 reverse transcription, the nucleocapsid protein, NC, induces secondary structure fluctuations in specific DNA and RNA hairpins. Time-resolved single-molecule fluorescence resonance energy transfer was used to study NC chaperoned opening of DNA hairpins over a broader range of conditions and in more depth than in previous studies. The experiments reveal a complex mechanism for secondary structure fluctuations with dynamic processes occurring over a wide time range, i.e., approximately 5 to >250 ms and with the involvement of long-lived intermediates. The dynamic role of DNA loop regions and NC binding/dissociation events are discussed.

Single-molecule experiments in biological physics: methods and applications

I review single-molecule experiments (SMEs) in biological physics. Recent technological developments have provided the tools to design and build scientific instruments of high enough sensitivity and precision to manipulate and visualize individual molecules and measure microscopic forces. Using SMEs it is possible to manipulate molecules one at a time and measure distributions describing molecular properties, characterize the kinetics of biomolecular reactions and detect molecular intermediates. SMEs provide additional information about thermodynamics and kinetics of biomolecular processes. This complements information obtained in traditional bulk assays. In SMEs it is also possible to measure small energies and detect large Brownian deviations in biomolecular reactions, thereby offering new methods and systems to scrutinize the basic foundations of statistical mechanics. This review is written at a very introductory level, emphasizing the importance of SMEs to scientists interested in knowing the common playground of ideas and the interdisciplinary topics accessible by these techniques. The review discusses SMEs from an experimental perspective, first exposing the most common experimental methodologies and later presenting various molecular systems where such techniques have been applied. I briefly discuss experimental techniques such as atomic-force microscopy (AFM), laser optical tweezers (LOTs), magnetic tweezers (MTs), biomembrane force probes (BFPs) and single-molecule fluorescence (SMF). I then present several applications of SME to the study of nucleic acids (DNA, RNA and DNA condensation) and proteins (protein-protein interactions, protein folding and molecular motors). Finally, I discuss applications of SMEs to the study of the nonequilibrium thermodynamics of small systems and the experimental verification of fluctuation theorems. I conclude with a discussion of open questions and future perspectives.

Energy dissipation and fluctuation response for particles in fluids

An equality was recently proved relating energy dissipation to the difference of the response and velocity correlation functions for a class of Langevin equations. We generalize this for the physically important case of particles in a fluid, where bath fluctuations are nonlocal in time due to hydrodynamic modes. We also show that the inclusion of a mass term does not alter the result and provide a simple physical interpretation of the original equality.

Force-dependent fragility in RNA hairpins

We apply Kramers theory to investigate the dissociation of multiple bonds under mechanical force and interpret experimental results for the unfolding and refolding force distributions of an RNA hairpin pulled at different loading rates using laser tweezers. We identify two different kinetic regimes depending on the range of forces explored during the unfolding and refolding process. The present approach extends the range of validity of the two-states approximation by providing a theoretical framework to reconstruct free-energy landscapes and identify force-induced structural changes in molecular transition states using single molecule pulling experiments. The method should be applicable to RNA hairpins with multiple kinetic barriers.

Mechanism of titin unfolding by force: insight from quasi-equilibrium molecular dynamics calculations

We have studied the unfolding by force of one of the immunoglobulin domains of the muscle protein titin using molecular dynamics simulations at 300 K. Previous studies, done at constant pulling rates, showed that under the effect of the force two strands connected to each other by six backbone H-bonds are pulled apart. No details about the mechanism of H-bond breaking were provided. Our simulation protocol "pull and wait" was designed to correspond to very slow pulling, more similar to the rates used in experiments than are the protocols used in previous computational studies. Under these conditions interstrand backbone H-bonds are not "ripped apart" by the application of the force. Instead, small elongations produced by the force weaken specific backbone H-bonds with respect to water-backbone H-bonds. These weakened bonds allow a single water molecule to make H-bonds to the CO and the NH of the same backbone H-bond while they are still bound to each other. The backbone H-bond then breaks (distance > 3.6 A), but its donor and acceptor atoms remain bound to the same water molecule. Further separation of the chains takes place when a second water molecule makes an H-bond with either the protein backbone donor or acceptor atom. Thus, the force does not directly break the main chain H-bonds: it destabilizes them in such a way that they are replaced by H-bonds to water. With this mechanism, the force necessary to break all the H-bonds required to separate the two strands will be strongly dependent on the pulling speed. Further simulations carried out at low forces but long waiting times (> or = 500 ps, < or = 10 ns) show that, given enough time, even a very small pulling force (< 400 pN) is sufficient to destabilize the interstrand H-bonds and allow them to be replaced by H-bonds to two water molecules. As expected, increasing the temperature to 350 K allows the interstrand H-bonds to break at lower forces than those required at 300 K.

Sub-angstrom conformational changes of a single molecule captured by AFM variance analysis

A system's equilibrium variance can be analyzed to probe its underlying dynamics at higher resolution. Here, using single-molecule atomic-force microscope techniques, we show how the variance in the length of a single dextran molecule can be used to establish thermodynamic equilibrium and to detect conformational changes not directly observable with other methods. Dextran is comprised of a chain of pyranose rings that each undergoes an Angstrom-scale transition from a chair to boat conformation under a stretching force. Our analysis of the variance of the molecule's fluctuations verifies equilibrium throughout the force-extension curve, consistent with the expected thermodynamic ensemble. This validates further analysis of the variance in the transition region, which reveals an intermediate conformation between the chair and the boat on the sub-Angstrom scale. Our test of thermal equilibrium as well as our variance analysis can be readily extended to a wide variety of molecules, including proteins.

Calculating potentials of mean force and diffusion coefficients from nonequilibrium processes without Jarzynski’s equality

In general, the direct application of the Jarzynski equality (JE) to reconstruct potentials of mean force (PMFs) from a small number of nonequilibrium unidirectional steered molecular-dynamics (SMD) paths is hindered by the lack of sampling of extremely rare paths with negative dissipative work. Such trajectories that transiently violate the second law of thermodynamics are crucial for the validity of JE. As a solution to this daunting problem, we propose a simple and efficient method, referred to as the FR method, for calculating simultaneously both the PMF U(z) and the corresponding diffusion coefficient D(z) along a reaction coordinate z for a classical many-particle system by employing a small number of fast SMD pullings in both forward (F) and time reverse (R) directions, without invoking JE. By employing Crooks [Phys. Rev. E 61, 2361 (2000)] transient fluctuation theorem (that is more general than JE) and the stiff-spring approximation, we show that (i) the mean dissipative work W(d) in the F and R pullings is the same, (ii) both U(z) and W(d) can be expressed in terms of the easily calculable mean work of the F and R processes, and (iii) D(z) can be expressed in terms of the slope of W(d). To test its viability, the FR method is applied to determine U(z) and D(z) of single-file water molecules in single-walled carbon nanotubes (SWNTs). The obtained U(z) is found to be in very good agreement with the results from other PMF calculation methods, e.g., umbrella sampling. Finally, U(z) and D(z) are used as input in a stochastic model, based on the Fokker-Planck equation, for describing water transport through SWNTs on a mesoscopic time scale that in general is inaccessible to MD simulations.

A coarse-grained model for force-induced protein deformation and kinetics

Force-induced changes in protein conformation are thought to be responsible for certain cellular responses to mechanical force. Changes in conformation subsequently initiate a biochemical response by alterations in, for example, binding affinity to another protein or enzymatic activity. Here, a model of protein extension under external forcing is created inspired by Kramers' theory for reaction rate kinetics in liquids. The protein is assumed to have two distinct conformational states: a relaxed state, C(1), preferred in the absence of external force, and an extended state, C(2), favored under force application. In the context of mechanotransduction, the extended state is a conformation from which the protein can initiate signaling. Appearance and persistence of C(2) are assumed to lead to transduction of the mechanical signal into a chemical one. The protein energy landscape is represented by two harmonic wells of stiffness kappa(1) and kappa(2), whose minima correspond to conformations C(1) and C(2). First passage time t(f) from C(1) to C(2) is determined from the Fokker-Plank equation employing several different approaches found in the literature. These various approaches exhibit significant differences in behavior as force increases. Although the level of applied force and the energy difference between states largely determine equilibrium, the dominant influence on t(f) is the height of the transition state. Distortions in the energy landscape due to force can also have a significant influence, however, exhibiting a weaker force dependence than exponential as previously reported, approaching a nearly constant value at a level of force that depends on the ratio kappa(1)/kappa(2). Two model systems are used to demonstrate the utility of this approach: a short alpha-helix undergoing a transition between two well-defined states and a simple molecular motor.

Force unfolding single RNAs

We develop a continue time Monte Carlo algorithm to simulate single RNAs unfolded by a time-dependent external force on the secondary structure level. Two recent unfolding RNA experiments carried out by Bustamante group are mainly investigated. We find that, in contrast to popular two-state assumption about the RNAs free energy landscape along the molecular extension, the molecules used in the experiments do not present apparent energy barriers. The strong cooperative folding and unfolding transitions of the RNAs observed in the experiments and in our simulations arise from the interaction of the molecules and the light trap. In addition, we also investigate the properties of Jarzynski's remarkable equality, whose experimental test has received considerable attention.

Structure and energy of a DNA dodecamer under tensile load

In the last decade, methods to study single DNA molecules under tensile load have been developed. These experiments measure the force required to stretch and melt the double helix and provide insights into the structural stability of DNA. However, it is not easy to directly relate the shape of the force curve to the structural changes that occur in the double helix under tensile load. Here, state-of-the-art computer simulations of short DNA sequences are preformed to provide an atomistic description of the stretching of the DNA double helix. These calculations show that for extensions larger that ∼25% the DNA undergoes a structural transformation and a few base pairs are lost from both the terminal and central part of the helix. This locally melted DNA duplex is stable and can be extended up to ∼50–60% of the equilibrium length at a constant force. It is concluded that melting under tension cannot be modeled as a simple two-state process. Finally, the important role of the cantilever stiffness in determining the shape of the force–extension curve and the most probable rupture force is discussed.

Work distribution functions for hysteresis loops in a single-spin system

We compute the distribution of the work done in driving a single Ising spin with a time-dependent magnetic field. Using Glauber dynamics we perform Monte Carlo simulations to find the work distributions at different driving rates. We find that in general the work distributions are broad with a significant probability for processes with negative dissipated work. The special cases of slow and fast driving rates are studied analytically. We verify that various work fluctuation theorems corresponding to equilibrium initial states are satisfied while a steady state version is not.

Thermodynamic calculations in biological systems

The ability to compute intra- and inter-molecular interactions provides the opportunity to gain a deeper understanding of previously intractable problems in biochemistry and biophysics. This review presents three examples in which molecular dynamics calculations were used to gain insight into the atomic detail underlying important experimental observations. The three examples are the following: (1) Entropic contribution to rate acceleration that results from conformational constraints imposed on the reactants; (2) Mechanism of force unfolding of a small protein molecule by the application of a force that separates its N- and C-terminals; and (3) Loss of translational entropy experienced by small molecules when they bind to proteins.

Work-probability distribution in systems driven out of equilibrium

We derive the differential equation describing the time evolution of the work probability distribution function of a stochastic system which is driven out of equilibrium by the manipulation of a parameter. We consider both systems described by their microscopic state or by a collective variable which identifies a quasiequilibrium state. We show that the work probability distribution can be represented by a path integral, which is dominated by "classical" paths in the large system size limit. We compare these results with simulated manipulation of mean-field systems. We discuss the range of applicability of the Jarzynski equality for evaluating the system free energy using these out-of-equilibrium manipulations. Large fluctuations in the work and the shape of the work distribution tails are also discussed.

Equality connecting energy dissipation with a violation of the fluctuation-response relation

In systems driven away from equilibrium, the velocity correlation function and the linear-response function to a small perturbation force do not satisfy the fluctuation-response relation (FRR) due to the lack of detailed balance in contrast to equilibrium systems. In this Letter, an equality between an extent of the FRR violation and the rate of energy dissipation is proved for Langevin systems under nonequilibrium conditions. This equality enables us to calculate the rate of energy dissipation by quantifying the extent of the FRR violation, which can be measured experimentally.

Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies

Atomic force microscopes and optical tweezers are widely used to probe the mechanical properties of individual molecules and molecular interactions, by exerting mechanical forces that induce transitions such as unfolding or dissociation. These transitions often occur under nonequilibrium conditions and are associated with hysteresis effects-features usually taken to preclude the extraction of equilibrium information from the experimental data. But fluctuation theorems allow us to relate the work along nonequilibrium trajectories to thermodynamic free-energy differences. They have been shown to be applicable to single-molecule force measurements and have already provided information on the folding free energy of a RNA hairpin. Here we show that the Crooks fluctuation theorem can be used to determine folding free energies for folding and unfolding processes occurring in weak as well as strong nonequilibrium regimes, thereby providing a test of its validity under such conditions. We use optical tweezers to measure repeatedly the mechanical work associated with the unfolding and refolding of a small RNA hairpin and an RNA three-helix junction. The resultant work distributions are then analysed according to the theorem and allow us to determine the difference in folding free energy between an RNA molecule and a mutant differing only by one base pair, and the thermodynamic stabilizing effect of magnesium ions on the RNA structure.

Thermodynamic and kinetic aspects of RNA pulling experiments

Recent single-molecule pulling experiments have shown how it is possible to manipulate RNA molecules using laser tweezers. In this article we investigate a minimal model for the experimental setup which includes an RNA molecule connected to two polymers (handles) and a bead trapped in the optical potential and attached to one of the handles. We start by considering the case of small single-domain RNA molecules, which unfold in a cooperative way. The model qualitatively reproduces the experimental results and allows us to investigate the influence of the bead and handles on the unfolding reaction. A main ingredient of the model is to consider the appropriate statistical ensemble and the corresponding thermodynamic potential describing thermal fluctuations in the system. We then investigate several questions relevant to extract thermodynamic information from experimental data. The kinetics of unfolding is also studied by introducing a dynamical model. Finally, we apply the model to the more general problem of a multidomain RNA molecule with Mg(2+) tertiary contacts that unfolds in a sequential way.