Experimental validation of free-energy-landscape reconstruction from non-equilibrium single-molecule force spectroscopy measurements (original) (raw)
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Biophysical Journal, 2011
Free-energy-landscape formalisms provide the fundamental conceptual framework for physical descriptions of how proteins and nucleic acids fold into specific three-dimensional structures 1,2 . Although folding landscapes are difficult to measure experimentally, recent theoretical work by Hummer and Szabo 3 has shown that landscape profiles can be reconstructed from non-equilibrium single-molecule force spectroscopy measurements using an extension of the Jarzynski equality 4 . This method has been applied to simulations 5,6 and experiments 7,8 but never validated experimentally. We tested it using force-extension measurements on DNA hairpins with distinct, sequence-dependent folding landscapes. Quantitative agreement was found between the landscape profiles obtained from the non-equilibrium reconstruction and those from equilibrium probability distributions 9 . We also tested the method on a riboswitch aptamer with three partially folded intermediate states, successfully reconstructing the landscape but finding some states difficult to resolve owing to low occupancy or overlap of the potential wells. These measurements validate the landscape-reconstruction method and provide a new test of non-equilibrium work relations.
Probing Position-Dependent Diffusion in Folding Reactions Using Single-Molecule Force Spectroscopy
Biophysical journal, 2018
Folding of proteins and nucleic acids involves a diffusive search over a multidimensional conformational energy landscape for the minimal-energy structure. When examining the projection of conformational motions onto a one-dimensional reaction coordinate, as done in most experiments, the diffusion coefficient D is generally position dependent. However, it has proven challenging to measure such position-dependence experimentally. We investigated the position-dependence of D in the folding of DNA hairpins as a simple model system in two ways: first, by analyzing the round-trip time to return to a given extension in constant-force extension trajectories measured by force spectroscopy, and second, by analyzing the fall time required to reach a given extension in force jump measurements. These methods yielded conflicting results: the fall time implied a fairly constant D, but the round-trip time implied variations of over an order of magnitude. Comparison of experiments with computationa...
Biophysical Journal, 2006
We present, to our knowledge, a new theory that takes internal dynamics of proteins into account to describe forced-unfolding and force-quench refolding in single molecule experiments. In the current experimental setup (using either atomic force microscopy or laser optical tweezers) the distribution of unfolding times, P(t), is measured by applying a constant stretching force f S from which the apparent f S -dependent unfolding rate is obtained. To describe the complexity of the underlying energy landscape requires additional probes that can incorporate the dynamics of tension propagation and relaxation of the polypeptide chain upon force quench. We introduce a theory of force correlation spectroscopy to map the parameters of the energy landscape of proteins. In force correlation spectroscopy, the joint distribution P(T, t) of folding and unfolding times is constructed by repeated application of cycles of stretching at constant f S separated by release periods T during which the force is quenched to f Q , f S . During the release period, the protein can collapse to a manifold of compact states or refold. We show that P(T, t) at various f S and f Q values can be used to resolve the kinetics of unfolding as well as formation of native contacts. We also present methods to extract the parameters of the energy landscape using chain extension as the reaction coordinate and P(T, t). The theory and a wormlike chain model for the unfolded states allows us to obtain the persistence length l p and the f Q -dependent relaxation time, giving us an estimate of collapse timescale at the single molecular level, in the coil states of the polypeptide chain. Thus, a more complete description of landscape of protein native interactions can be mapped out if unfolding time data are collected at several values of f S and f Q . We illustrate the utility of the proposed formalism by analyzing simulations of unfolding-refolding trajectories of a coarse-grained protein (S1) with b-sheet architecture for several values of f S , T, and f Q ¼ 0. The simulations of stretch-relax trajectories are used to map many of the parameters that characterize the energy landscape of S1.
Biophysical Journal
Force spectroscopy is commonly used to measure the kinetics of processes occurring in single biological molecules. These measurements involve attaching the molecule of interest to micron-sized or larger force probes via compliant linkers. Recent theoretical work has described how the properties of the probes and linkers can alter the observed kinetics from the intrinsic behavior of the molecule in isolation. We applied this theory to estimate the errors in measurements of folding made using optical tweezers. Errors in the folding rates arising from instrument artifacts were only~20% for constant-force measurements of DNA hairpins with typical choices of linker length and probe size. Measurements of transition paths using a constant trap position at high trap stiffness were also found to be in the low-artifact limit. These results indicate that typical optical trap measurements of kinetics reflect the dynamics of the molecule fairly well, and suggest practical limitations on experimental design to ensure reliable kinetic measurements.
Transition Path Times for DNA and RNA Folding from Force Spectroscopy
Physics, 2012
The duration of structural transitions in biopolymers is only a fraction of the time spent searching diffusively over the configurational energy landscape. We found the transition time, TP , and the diffusion constant, D, for DNA and RNA folding using energy landscapes obtained from single-molecule trajectories under tension in optical traps. DNA hairpins, RNA pseudoknots, and a riboswitch all had TP $ 10 s and D $ 10 À13-14 m 2 =s, despite widely differing unfolding rates. These results show how energy-landscape analysis can be harnessed to characterize brief but critical events during folding reactions.
Dynamic force spectroscopy of DNA hairpins: I. Force kinetics and free energy landscapes
Journal of Statistical Mechanics-theory and Experiment, 2009
We investigate the thermodynamics and kinetics of DNA hairpins that fold/unfold under the action of applied mechanical force. We introduce the concept of the molecular free energy landscape and derive simplified expressions for the force dependent Kramers-Bell rates. To test the theory we have designed a specific DNA hairpin sequence that shows two-state cooperative folding under mechanical tension and carried out pulling experiments using optical tweezers. We show how we can determine the parameters that characterize the molecular free energy landscape of such sequence from rupture force kinetic studies. Finally we combine such kinetic studies with experimental investigations of the Crooks fluctuation relation to derive the free energy of formation of the hairpin at zero force.
Efficient methods for determining folding free energies in single-molecule pulling experiments
Journal of Statistical Mechanics: Theory and Experiment, 2019
The remarkable accuracy and versatility of single-molecule techniques make possible new measurements that are not feasible in bulk assays. Among these, the precise estimation of folding free energies using fluctuation theorems in nonequilibrium pulling experiments has become a benchmark in modern biophysics. In practice, the use of fluctuation relations to determine free energies requires a thorough evaluation of the usually large energetic contributions caused by the elastic deformation of the different elements of the experimental setup (such as the optical trap, the molecular linkers and the stretched-unfolded polymer). We review and describe how to optimally estimate such elastic energy contributions to extract folding free energies, using DNA and RNA hairpins as model systems pulled by laser optical tweezers. The methodology is generally applicable to other force-spectroscopy techniques and molecular systems.
The journal of physical chemistry letters, 2017
Biomolecules diffusively explore their energy landscape overcoming energy barriers via thermally activated processes to reach the biologically relevant conformation. Mechanically induced unfolding and folding reactions offer an excellent playground to feature these processes at the single-molecule level by monitoring changes in the molecular extension. Here we investigate two-state DNA hairpins designed to have the transition states at different locations. We use optical tweezers to characterize the force-dependent behavior of the kinetic barrier from nonequilibrium pulling experiments by using the continuous effective barrier approach (CEBA). We introduce the mechanical fragility and the molecular transition-state susceptibility, both useful quantities to characterize the response of the transition state to an applied force. Our results demonstrate the validity of the Leffler-Hammond postulate where the transition state approaches the folded state as force increases, implying monot...
Physical Review Letters, 2012
The duration of structural transitions in biopolymers is only a fraction of the time spent searching diffusively over the configurational energy landscape. We found the transition time, TP , and the diffusion constant, D, for DNA and RNA folding using energy landscapes obtained from single-molecule trajectories under tension in optical traps. DNA hairpins, RNA pseudoknots, and a riboswitch all had TP $ 10 s and D $ 10 À13-14 m 2 =s, despite widely differing unfolding rates. These results show how energy-landscape analysis can be harnessed to characterize brief but critical events during folding reactions.
2011
The work presented in this dissertation focuses on the kinetics of biomolecular reactions under mechanical force, including protein unfolding and disulfide-bond reduction, probed at the single-molecule level. The advent of single-molecule force spectroscopy has allowed the direct measure of force-dependent reaction rates, providing a powerful approach to extract the kinetic information and to characterize the underlying energy landscape that governs the reaction. The widely accepted two-state kinetic model for protein unfolding describes that the protein unfolds by crossing over a single energy barrier, with the implicit assumption of a single transition state and a well-defined activation energy barrier. Based on this assump-tion, the ensemble-averaged survival probability is expected to follow single exponential time dependence. However, it has become increasingly clear that the saddle point of the free-energy surface in most reactions is populated by ensembles of conformations, l...