Force-dependent hopping rates of RNA hairpins can be estimated from accurate measurement of the folding landscapes

Quantitative Elasticity of Flexible Polymer Chains Using Interferometer-Based AFM

We estimate the elasticity of single polymer chains using atomic force microscope (AFM)-based oscillatory experiments. An accurate estimate of elasticity using AFM is limited by assumptions in describing the dynamics of an oscillating cantilever. Here, we use a home-built fiber-interferometry-based detection system that allows a simple and universal point-mass description of cantilever oscillations. By oscillating the cantilever base and detecting changes in cantilever oscillations with an interferometer, we extracted stiffness versus extension profiles for polymers. For polyethylene glycol (PEG) in a good solvent, stiffness-extension data showed significant deviation from conventional force-extension curves (FECs) measured in constant velocity pulling experiments. Furthermore, modeling stiffness data with an entropic worm-like chain (WLC) model yielded a persistence length of (0.5 ± 0.2 nm) compared to anomaly low value (0.12 nm ± 0.01) in conventional pulling experiments. This value also matched well with equilibrium measurements performed using magnetic tweezers. In contrast, polystyrene (PS) in a poor solvent, like water, showed no deviation between the two experiments. However, the stiffness profile for PS in good solvent (8M Urea) showed significant deviation from conventional force-extension curves. We obtained a persistence length of (0.8 ± 0.2 nm) compared to (0.22 nm ± 0.01) in pulling experiments. Our unambiguous measurements using interferometer yield physically acceptable values of persistence length. It validates the WLC model in good solvents but suggests caution for its use in poor solvents.

Modulation of a protein-folding landscape revealed by AFM-based force spectroscopy notwithstanding instrumental limitations

Single-molecule force spectroscopy is a powerful tool for studying protein folding. Over the last decade, a key question has emerged: how are changes in intrinsic biomolecular dynamics altered by attachment to μm-scale force probes via flexible linkers? Here, we studied the folding/unfolding of α₃D using atomic force microscopy (AFM)-based force spectroscopy. α₃D offers an unusual opportunity as a prior single-molecule fluorescence resonance energy transfer (smFRET) study showed α₃D's configurational diffusion constant within the context of Kramers theory varies with pH. The resulting pH dependence provides a test for AFM-based force spectroscopy's ability to track intrinsic changes in protein folding dynamics. Experimentally, however, α₃D is challenging. It unfolds at low force (<15 pN) and exhibits fast-folding kinetics. We therefore used focused ion beam-modified cantilevers that combine exceptional force precision, stability, and temporal resolution to detect state occupancies as brief as 1 ms. Notably, equilibrium and nonequilibrium force spectroscopy data recapitulated the pH dependence measured using smFRET, despite differences in destabilization mechanism. We reconstructed a one-dimensional free-energy landscape from dynamic data via an inverse Weierstrass transform. At both neutral and low pH, the resulting constant-force landscapes showed minimal differences (∼0.2 to 0.5 k _B T) in transition state height. These landscapes were essentially equal to the predicted entropic barrier and symmetric. In contrast, force-dependent rates showed that the distance to the unfolding transition state increased as pH decreased and thereby contributed to the accelerated kinetics at low pH. More broadly, this precise characterization of a fast-folding, mechanically labile protein enables future AFM-based studies of subtle transitions in mechanoresponsive proteins.

Desmosome architecture derived from molecular dynamics simulations and cryo-electron tomography

The desmosome is a major cell–cell junction connecting cells in tissues under high mechanical load. Currently, while structures of the constituent cadherins are known, the desmosome architecture has remained elusive. The primary reason is the high plasticity of the cadherins. As many other cellular structures, their high flexibility cannot be easily addressed by conventional structural techniques that rely on averaging many identical structures. For this, we combine high-end cryo-electron tomography with large-scale molecular dynamics simulations to produce a molecular model of the desmosome that integrates new with decades-old observations, accounts for the remarkable biophysical properties, and maps the intermolecular interactions.

Desmosomes are cell–cell junctions that link tissue cells experiencing intense mechanical stress. Although the structure of the desmosomal cadherins is known, the desmosome architecture—which is essential for mediating numerous functions—remains elusive. Here, we recorded cryo-electron tomograms (cryo-ET) in which individual cadherins can be discerned; they appear variable in shape, spacing, and tilt with respect to the membrane. The resulting sub-tomogram average reaches a resolution of ∼26 Å, limited by the inherent flexibility of desmosomes. To address this challenge typical of dynamic biological assemblies, we combine sub-tomogram averaging with atomistic molecular dynamics (MD) simulations. We generate models of possible cadherin arrangements and perform an in silico screening according to biophysical and structural properties extracted from MD simulation trajectories. We find a truss-like arrangement of cadherins that resembles the characteristic footprint seen in the electron micrograph. The resulting model of the desmosomal architecture explains their unique biophysical properties and strength.

Molecular free energy profiles from force spectroscopy experiments by inversion of observed committors

In single-molecule force spectroscopy experiments, a biomolecule is attached to a force probe via polymer linkers and the total extension of the molecule plus apparatus is monitored as a function of time. In a typical unfolding experiment at constant force, the total extension jumps between two values that correspond to the folded and unfolded states of the molecule. For several biomolecular systems, the committor, which is the probability to fold starting from a given extension, has been used to extract the molecular activation barrier (a technique known as "committor inversion"). In this work, we study the influence of the force probe, which is much larger than the molecule being measured, on the activation barrier obtained by committor inversion. We use a two-dimensional framework in which the diffusion coefficient of the molecule and of the pulling device can differ. We systematically study the free energy profile along the total extension obtained from the committor by numerically solving the Onsager equation and using Brownian dynamics simulations. We analyze the dependence of the extracted barrier on the linker stiffness, molecular barrier height, and diffusion anisotropy and, thus, establish the range of validity of committor inversion. Along the way, we showcase the committor of 2-dimensional diffusive models and illustrate how it is affected by barrier asymmetry and diffusion anisotropy.

Conformational heterogeneity in human interphase chromosome organization reconciles the FISH and Hi-C paradox

Hi-C experiments are used to infer the contact probabilities between loci separated by varying genome lengths. Contact probability should decrease as the spatial distance between two loci increases. However, studies comparing Hi-C and FISH data show that in some cases the distance between one pair of loci, with larger Hi-C readout, is paradoxically larger compared to another pair with a smaller value of the contact probability. Here, we show that the FISH-Hi-C paradox can be resolved using a theory based on a Generalized Rouse Model for Chromosomes (GRMC). The FISH-Hi-C paradox arises because the cell population is highly heterogeneous, which means that a given contact is present in only a fraction of cells. Insights from the GRMC is used to construct a theory, without any adjustable parameters, to extract the distribution of subpopulations from the FISH data, which quantitatively reproduces the Hi-C data. Our results show that heterogeneity is pervasive in genome organization at all length scales, reflecting large cell-to-cell variations.

Studies comparing Hi-C and FISH data show that in some cases the distance between one pair of loci is paradoxically larger compared to another pair with a smaller value of the contact probability. Here the authors use a theory based on a Generalized Rouse Model for Chromosomes to resolve this paradox.

Mechanical and Structural Tuning of Reversible Hydrogen Bonding in Interlocked Calixarene Nanocapsules

We present force probe molecular dynamics simulations of dimers of interlocked calixarene nanocapsules and study the impact of structural details and solvent properties on the mechanical unfolding pathways. The system consists of two calixarene "cups" that form a catenane structure via interlocked aliphatic loops of tunable length. The dimer shows reversible rebinding, and the kinetics of the system can be understood in terms of a two-state model for shorter loops (≤14 CH₂ units) and a three-state model for longer loops (≥15 CH₂ units). The various conformational states of the dimer are stabilized by networks of hydrogen bonds, the mechanical susceptibility of which can be altered by changing the polarity and proticity of the solvent. The variation of the loop length and the solvent properties in combination with changes in the pulling protocol allows to tune the reversibility of the conformational transitions.

Generalized Langevin Equation as a Model for Barrier Crossing Dynamics in Biomolecular Folding

Conformational memory in single-molecule dynamics has attracted recent attention and, in particular, has been invoked as a possible explanation of some of the intriguing properties of transition paths observed in single-molecule force spectroscopy (SMFS) studies. Here we study one candidate for a non-Markovian model that can account for conformational memory, the generalized Langevin equation with a friction force that depends not only on the instantaneous velocity but also on the velocities in the past. The memory in this model is determined by a time-dependent friction memory kernel. We propose a method for extracting this kernel directly from an experimental signal and illustrate its feasibility by applying it to a generalized Rouse model of a SMFS experiment, where the memory kernel is known exactly. Using the same model, we further study how memory affects various statistical properties of transition paths observed in SMFS experiments and evaluate the performance of recent approximate analytical theories of non-Markovian dynamics of barrier crossing. We argue that the same type of analysis can be applied to recent single-molecule observations of transition paths in protein and DNA folding.

Computational Methods for Modeling Aptamers and Designing Riboswitches

Riboswitches, which are located within certain noncoding RNA region perform functions as genetic "switches", regulating when and where genes are expressed in response to certain ligands. Understanding the numerous functions of riboswitches requires computation models to predict structures and structural changes of the aptamer domains. Although aptamers often form a complex structure, computational approaches, such as RNAComposer and Rosetta, have already been applied to model the tertiary (three-dimensional (3D)) structure for several aptamers. As structural changes in aptamers must be achieved within the certain time window for effective regulation, kinetics is another key point for understanding aptamer function in riboswitch-mediated gene regulation. The coarse-grained self-organized polymer (SOP) model using Langevin dynamics simulation has been successfully developed to investigate folding kinetics of aptamers, while their co-transcriptional folding kinetics can be modeled by the helix-based computational method and BarMap approach. Based on the known aptamers, the web server Riboswitch Calculator and other theoretical methods provide a new tool to design synthetic riboswitches. This review will represent an overview of these computational methods for modeling structure and kinetics of riboswitch aptamers and for designing riboswitches.

Advances in Quantum Mechanochemistry: Electronic Structure Methods and Force Analysis

In quantum mechanochemistry, quantum chemical methods are used to describe molecules under the influence of an external force. The calculation of geometries, energies, transition states, reaction rates, and spectroscopic properties of molecules on the force-modified potential energy surfaces is the key to gain an in-depth understanding of mechanochemical processes at the molecular level. In this review, we present recent advances in the field of quantum mechanochemistry and introduce the quantum chemical methods used to calculate the properties of molecules under an external force. We place special emphasis on quantum chemical force analysis tools, which can be used to identify the mechanochemically relevant degrees of freedom in a deformed molecule, and spotlight selected applications of quantum mechanochemical methods to point out their synergistic relationship with experiments.

Structure-Based Derivation of Protein Folding Intermediates and Energies from Optical Tweezers

Optical tweezers (OTs) measure the force-dependent time-resolved extension of a single macromolecule tethered between two trapped beads. From this measurement, it is possible to determine the folding intermediates, energies, and kinetics of the macromolecule. Previous data analysis generally has used the extension as a reaction coordinate to characterize the observed folding transitions. Despite its convenience, the extension poorly describes folding in the absence of force. Here, we chose the contour length of the unfolded polypeptide as a reaction coordinate and modeled the extensions of protein structures along their predicted folding pathways based on high-resolution structures of the proteins in their native states. We included the extension in our model to calculate the total extensions, energies, and transition rates of the proteins as a function of force. We fit these calculations to the corresponding experimental measurements and obtained the best-fit conformations and energies of proteins in different folding states. We applied our method to analyze single-molecule trajectories of two representative protein complexes responsible for membrane fusion, the HIV-1 glycoprotein 41 and the synaptic SNARE proteins, which involved transitions between two and five states, respectively. Nonlinear fitting of the model to the experimental data revealed the structures of folding intermediates and transition states and their associated energies. Our results demonstrate that the contour length is a useful reaction coordinate to characterize protein folding and that intrinsic extensions of protein structures should be taken into account to properly derive the conformations and energies of protein folding intermediates from single-molecule manipulation experiments.

The Power of Force: Insights into the Protein Folding Process Using Single-Molecule Force Spectroscopy

One of the major challenges in modern biophysics is observing and understanding conformational changes during complex molecular processes, from the fundamental protein folding to the function of molecular machines. Single-molecule techniques have been one of the major driving forces of the huge progress attained in the last few years. Recent advances in resolution of the experimental setups, aided by theoretical developments and molecular dynamics simulations, have revealed a much higher degree of complexity inside these molecular processes than previously reported using traditional ensemble measurements. This review sums up the evolution of these developments and gives an outlook on prospective discoveries.

Quantifying Instrumental Artifacts in Folding Kinetics Measured by Single-Molecule Force Spectroscopy

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.

Effect of elastic colored noise in the hopping dynamics of single molecules in stretching experiments

The influence of colored noise induced by elastic fluctuations in single-molecule stretching experiments is theoretically and numerically studied. Unlike in the thermal white noise case currently considered in the literature, elastically induced hopping dynamics between folded and unfolded states is manifested through critical oscillations showing smaller end-to-end distance fluctuations (δx∼1.25nm) within the free energy wells corresponding to both states. Our results are derived by analyzing the elastic coupling between the Handle-Molecule-Handle system and the laser optical tweezers (LOT) array. It is shown that an Ornstein-Uhlenbeck process related to this elastic coupling may trigger the hopping transitions via a colored noise with an intensity proportional to the elastic constant of the LOT array. Evolution equations of the variables of the system were derived by using the irreversible thermodynamics of small systems recently proposed. Theoretical expressions for the corresponding stationary probability densities are provided and the viability of inferring the shape of the free energy from direct measurements is discussed.

Communication: Does force spectroscopy of biomolecules probe their intrinsic dynamic properties?

In single-molecule pulling experiments, the molecule of interest is attached to a much larger object such as an atomic force microscope tip or a micrometer sized bead. The measured dynamics of molecular transitions is therefore affected by the hydrodynamic drag on the pulling instrument itself. By considering the transitions within the combined system (the molecule and the instrument), it is shown here that two distinct physical regimes exist: when the intrinsic stiffness of the molecule is greater than that of the linker connecting the molecule to the pulling setup then the pulling experiment probes the intrinsic dynamics of the molecule with only relatively small (and quantifiable) corrections resulting from the pulling setup. In contrast, when the stiffness of the linker exceeds that of the molecule, the molecular transition in question involves concerted motion of the molecule and the pulling setup and the hydrodynamic drag on the pulling instrument becomes the dominant source of friction along the molecular reaction coordinate. An analytical formula interpolating between these two cases is further derived. These results explain recent conflicting observations where some single-molecule pulling measurements report anomalously low diffusion coefficients along molecular reaction coordinates while others do not.

On artifacts in single-molecule force spectroscopy

In typical force spectroscopy experiments, a small biomolecule is attached to a soft polymer linker that is pulled with a relatively large bead or cantilever. At constant force, the total extension stochastically changes between two (or more) values, indicating that the biomolecule undergoes transitions between two (or several) conformational states. In this paper, we consider the influence of the dynamics of the linker and mesoscopic pulling device on the force-dependent rate of the conformational transition extracted from the time dependence of the total extension, and the distribution of rupture forces in force-clamp and force-ramp experiments, respectively. For these different experiments, we derive analytic expressions for the observables that account for the mechanical response and dynamics of the pulling device and linker. Possible artifacts arise when the characteristic times of the pulling device and linker become comparable to, or slower than, the lifetimes of the metastable conformational states, and when the highly anharmonic regime of stretched linkers is probed at high forces. We also revisit the problem of relating force-clamp and force-ramp experiments, and identify a linker and loading rate-dependent correction to the rates extracted from the latter. The theory provides a framework for both the design and the quantitative analysis of force spectroscopy experiments by highlighting, and correcting for, factors that complicate their interpretation.

Extracting intrinsic dynamic parameters of biomolecular folding from single-molecule force spectroscopy experiments

Single-molecule studies in which a mechanical force is transmitted to the molecule of interest and the molecular extension or position is monitored as a function of time are versatile tools for probing the dynamics of protein folding, stepping of molecular motors, and other biomolecular processes involving activated barrier crossing. One complication in interpreting such studies, however, is the fact that the typical size of a force probe (e.g., a dielectric bead in optical tweezers or the atomic force microscope tip/cantilever assembly) is much larger than the molecule itself, and so the observed molecular motion is affected by the hydrodynamic drag on the probe. This presents the experimenter with a nontrivial task of deconvolving the intrinsic molecular parameters, such as the intrinsic free energy barrier and the effective diffusion coefficient exhibited while crossing the barrier from the experimental signal. Here we focus on the dynamical aspect of this task and show how the intrinsic diffusion coefficient along the molecular reaction coordinate can be inferred from single-molecule measurements of the rates of biomolecular folding and unfolding. We show that the feasibility of accomplishing this task is strongly dependent on the relationship between the intrinsic molecular elasticity and that of the linker connecting the molecule to the force probe and identify the optimal range of instrumental parameters allowing determination of instrument-free molecular dynamics.

Nonlinear irreversible thermodynamics of single-molecule experiments

Irreversible thermodynamics of single-molecule experiments subject to external constraining forces of a mechanical nature is presented. Extending Onsager's formalism to the nonlinear case of systems under nonequilibrium external constraints, we are able to calculate the entropy production and the general nonlinear kinetic equations for the variables involved. In particular, we analyze the case of RNA stretching protocols obtaining critical oscillations between different configurational states when forced by external means to remain in the unstable region of its free-energy landscape, as observed in experiments. We also calculate the entropy produced during these hopping events and show how resonant phenomena in stretching experiments of single RNA macromolecules may arise. We also calculate the hopping rates using Kramer's approach obtaining a good comparison with experiments.

The role of binding site on the mechanical unfolding mechanism of ubiquitin

We apply novel atomistic simulations based on potential energy surface exploration to investigate the constant force-induced unfolding of ubiquitin. At the experimentally-studied force clamping level of 100 pN, we find a new unfolding mechanism starting with the detachment between β₅ and β₃ involving the binding site of ubiquitin, the Ile44 residue. This new unfolding pathway leads to the discovery of new intermediate configurations, which correspond to the end-to-end extensions previously seen experimentally. More importantly, it demonstrates the novel finding that the binding site of ubiquitin can be responsible not only for its biological functions, but also its unfolding dynamics. We also report in contrast to previous single molecule constant force experiments that when the clamping force becomes smaller than about 300 pN, the number of intermediate configurations increases dramatically, where almost all unfolding events at 100 pN involve an intermediate configuration. By directly calculating the life times of the intermediate configurations from the height of the barriers that were crossed on the potential energy surface, we demonstrate that these intermediate states were likely not observed experimentally due to their lifetimes typically being about two orders of magnitude smaller than the experimental temporal resolution.

Using simulations and kinetic network models to reveal the dynamics and functions of riboswitches

Riboswitches, RNA elements found in the untranslated region, regulate gene expression by binding to target metaboloites with exquisite specificity. Binding of metabolites to the conserved aptamer domain allosterically alters the conformation in the downstream expression platform. The fate of gene expression is determined by the changes in the downstream RNA sequence. As the metabolite-dependent cotranscriptional folding and unfolding dynamics of riboswitches are the key determinant of gene expression, it is important to investigate both the thermodynamics and kinetics of riboswitches both in the presence and absence of metabolite. Single molecule force experiments that decipher the free energy landscape of riboswitches from their mechanical responses, theoretical and computational studies have recently shed light on the distinct mechanism of folding dynamics in different classes of riboswitches. Here, we first discuss the dynamics of water around riboswitch, highlighting that water dynamics can enhance the fluctuation of nucleic acid structure. To go beyond native state fluctuations, we used the Self-Organized Polymer model to predict the dynamics of add adenine riboswitch under mechanical forces. In addition to quantitatively predicting the folding landscape of add-riboswitch, our simulations also explain the difference in the dynamics between pbuE adenine- and add adenine-riboswitches. In order to probe the function in vivo, we use the folding landscape to propose a system level kinetic network model to quantitatively predict how gene expression is regulated for riboswitches that are under kinetic control.

Exploring the topography of the stress-modified energy landscapes of mechanosensitive molecules

We propose a method for computing the activation barrier for chemical reactions involving molecules subjected to mechanical stress. The method avoids reactant and transition-state saddle optimizations at every force by, instead, solving the differential equations governing the force dependence of the critical points (i.e., minima and saddles) on the system's potential energy surface (PES). As a result, only zero-force geometry optimization (or, more generally, optimization performed at a single force value) is required by the method. In many cases, minima and transition-state saddles only exist within a range of forces and disappear beyond a certain critical point. Our method identifies such force-induced instabilities as points at which one of the Hessian eigenvalues vanishes. We elucidate the nature of those instabilities as fold and cusp catastrophes, where two or three critical points on the force-modified PES coalesce, and provide a classification of various physically distinct instability scenarios, each illustrated with a concrete chemical example.

Reconstructing folding energy landscapes by single-molecule force spectroscopy

Folding may be described conceptually in terms of trajectories over a landscape of free energies corresponding to different molecular configurations. In practice, energy landscapes can be difficult to measure. Single-molecule force spectroscopy (SMFS), whereby structural changes are monitored in molecules subjected to controlled forces, has emerged as a powerful tool for probing energy landscapes. We summarize methods for reconstructing landscapes from force spectroscopy measurements under both equilibrium and nonequilibrium conditions. Other complementary, but technically less demanding, methods provide a model-dependent characterization of key features of the landscape. Once reconstructed, energy landscapes can be used to study critical folding parameters, such as the characteristic transition times required for structural changes and the effective diffusion coefficient setting the timescale for motions over the landscape. We also discuss issues that complicate measurement and interpretation, including the possibility of multiple states or pathways and the effects of projecting multiple dimensions onto a single coordinate.

Stretching of single poly-ubiquitin molecules revisited: dynamic disorder in the non-exponential unfolding kinetics

A theoretical framework based on a generalized Langevin equation (GLE) with fractional Gaussian noise (fGn) and a power-law memory kernel is presented to describe the non-exponential kinetics of the unfolding of a single poly-ubiquitin molecule under a constant force [T.-L. Kuo, S. Garcia-Manyes, J. Li, I. Barel, H. Lu, B. J. Berne, M. Urbakh, J. Klafter, and J. M. Fernández, Proc. Natl. Acad. Sci. U.S.A. 107, 11336 (2010)]. Such a GLE-fGn strategy is made on the basis that the pulling coordinate variable x undergoes subdiffusion, usually resulting from conformational fluctuations, over a one-dimensional force-modified free-energy surface U(x, F). By using the Kramers' rate theory, we have obtained analytical formulae for the time-dependent rate coefficient k(t, F), the survival probability S(t, F) as well as the waiting time distribution function f(t, F) as functions of time t and force F. We find that our results can fit the experimental data of f(t, F) perfectly in the whole time range with a power-law exponent γ = 1/2, the characteristic of typical anomalous subdiffusion. In addition, the fitting of the survival probabilities for different forces facilitates us to reach rather reasonable estimations for intrinsic properties of the system, such as the free-energy barrier and the distance between the native conformation and the transition state conformation along the reaction coordinate, which are in good agreements with molecular dynamics simulations in the literatures. Although static disorder has been implicated in the original work of Kuo et al., our work suggests a sound and plausible alternative interpretation for the non-exponential kinetics in the stretching of poly-ubiquitin molecules, associated with dynamic disorder.

Kinetics of allosteric transitions in S-adenosylmethionine riboswitch are accurately predicted from the folding landscape

Riboswitches are RNA elements that allosterically regulate gene expression by binding cellular metabolites. The SAM-III riboswitch, one of several classes that binds S-adenosylmethionine (SAM), represses translation upon binding SAM (OFF state) by encrypting the ribosome binding sequence. We have carried out simulations of the RNA by applying mechanical force (f) to the ends of SAM-III, with and without SAM, to get quantitative insights into the f-dependent structural changes. Force-extension (z) curves (FECs) for the apo (ON) state, obtained in simulations in which f is increased at a constant loading rate, show three intermediates, with the first one being the rupture of SAM binding region, which is greatly stabilized in the OFF state. Force-dependent free energy profiles, G(z,f), as a function of z, obtained in equilibrium constant force simulations, reveal the intermediates observed in FECs. The predicted stability difference between the ON and OFF states using G(z,f) is in excellent agreement with experiments. Remarkably, using G(z,f)s and estimate of an effective diffusion constant at a single value of f allows us to predict the f-dependent transition rates using theory of first passage times for both the apo and holo states. To resolve the kinetics of assembly of SAM-III riboswitch in structural terms, we use force stretch-quench pulse sequences in which the force on RNA is maintained at a low (fq) value starting from a high value for a time period tq. Variation of tq over a wide range results in resolution of elusive states involved in the SAM binding pocket and leads to accurate determination of folding times down to fq = 0. Quantitative measure of the folding kinetics, obtained from the folding landscape, allows us to propose that, in contrast to riboswitches regulating transcription, SAM-III functions under thermodynamic control provided the basal concentration of SAM exceeds a small critical value. All of the predictions are amenable to tests in single molecule pulling experiments.

Direct measurement of sequential folding pathway and energy landscape of human telomeric G-quadruplex structures

Single-stranded guanine-rich sequences fold into compact G-quadruplexes. Although G-triplexes have been proposed and demonstrated as intermediates in the folding of G-quadruplexes, there is still a debate on their folding pathways. In this work, we employed magnetic tweezers to investigate the folding kinetics of single human telomeric G-quadruplexes in 100 mM Na(+) buffer. The results are consistent with a model in which the G-triplex is an in-pathway intermediate in the folding of the G-quadruplex. By finely tuning the force exerted on the G-quadruplex, we observed reversible transitions from the G-quadruplex to the G-triplex as well as from the G-triplex to the unfolded coil when the force was increased from 26 to 39 pN. The energy landscape derived from the probability distribution shows clearly that the G-quadruplex goes through an intermediate when it is unfolded, and vice versa.

From mechanical folding trajectories to intrinsic energy landscapes of biopolymers

In single-molecule laser optical tweezer (LOT) pulling experiments, a protein or RNA is juxtaposed between DNA handles that are attached to beads in optical traps. The LOT generates folding trajectories under force in terms of time-dependent changes in the distance between the beads. How to construct the full intrinsic folding landscape (without the handles and beads) from the measured time series is a major unsolved problem. By using rigorous theoretical methods--which account for fluctuations of the DNA handles, rotation of the optical beads, variations in applied tension due to finite trap stiffness, as well as environmental noise and limited bandwidth of the apparatus--we provide a tractable method to derive intrinsic free-energy profiles. We validate the method by showing that the exactly calculable intrinsic free-energy profile for a generalized Rouse model, which mimics the two-state behavior in nucleic acid hairpins, can be accurately extracted from simulated time series in a LOT setup regardless of the stiffness of the handles. We next apply the approach to trajectories from coarse-grained LOT molecular simulations of a coiled-coil protein based on the GCN4 leucine zipper and obtain a free-energy landscape that is in quantitative agreement with simulations performed without the beads and handles. Finally, we extract the intrinsic free-energy landscape from experimental LOT measurements for the leucine zipper.

Effect of handle length and microsphere size on transition kinetics in single-molecule experiments

When subject to constant tension, a DNA or RNA hairpin will typically make abrupt transitions between the open and closed state. Although the transition kinetics are an intrinsic property of the molecule, the transition rates measured in single-molecule experiments can be influenced by the configuration of the measurement system. We investigate the transition kinetics for a DNA hairpin held under constant force by an optical trap as a function of microsphere size and double-stranded DNA handle length. We find the apparent transition lifetime cannot be expressed as a function of the drag coefficient of the microsphere alone or as a function of time scales relevant to the optical trap. The apparent transition lifetime is found to be a linear function of the factor β(eff)·α(handle), where β(eff) is the effective drag coefficient of the microsphere near the surface and α(handle) is the stiffness of the DNA tether. The results provide insight into the perturbation to the hairpin transition kinetics due to experimental configuration and guidance for designing single-molecule experiments which determine the intrinsic molecular kinetics.

Force as a single molecule probe of multidimensional protein energy landscapes

Force spectroscopy has developed into an indispensable tool for studying folding and binding of proteins on a single molecule level in real time. Design of the pulling geometry allows tuning the reaction coordinate in a very precise manner. Many recent experiments have taken advantage of this possibility and have provided detailed insight the folding pathways on the complex high dimensional energy landscape. Beyond its potential to provide control over the reaction coordinate, force is also an important physiological parameter that affects protein conformation under in vivo conditions. Single molecule force spectroscopy studies have started to unravel the response and adaptation of force bearing protein structures to mechanical loads.

RNA under tension: Folding Landscapes, Kinetic partitioning Mechanism, and Molecular Tensegrity

Non-coding RNA sequences play a great role in controlling a number of cellular functions, thus raising the need to understand their complex conformational dynamics in quantitative detail. In this perspective, we first show that single molecule pulling when combined with with theory and simulations can be used to quantitatively explore the folding landscape of nucleic acid hairpins, and riboswitches with tertiary interactions. Applications to riboswitches, which are non-coding RNA elements that control gene expression by undergoing dynamical conformational changes in response to binding of metabolites, lead to an organization principle that assembly of RNA is determined by the stability of isolated helices. We also point out the limitations of single molecule pulling experiments, with molecular extension as the only accessible parameter, in extracting key parameters of the folding landscapes of RNA molecules.

Weak temporal signals can synchronize and accelerate the transition dynamics of biopolymers under tension

In addition to thermal noise, which is essential to promote conformational transitions in biopolymers, the cellular environment is replete with a spectrum of athermal fluctuations that are produced from a plethora of active processes. To understand the effect of athermal noise on biological processes, we studied how a small oscillatory force affects the thermally induced folding and unfolding transition of an RNA hairpin, whose response to constant tension had been investigated extensively in both theory and experiments. Strikingly, our molecular simulations performed under overdamped condition show that even at a high (low) tension that renders the hairpin (un)folding improbable, a weak external oscillatory force at a certain frequency can synchronously enhance the transition dynamics of RNA hairpin and increase the mean transition rate. Furthermore, the RNA dynamics can still discriminate a signal with resonance frequency even when the signal is mixed among other signals with nonresonant frequencies. In fact, our computational demonstration of thermally induced resonance in RNA hairpin dynamics is a direct realization of the phenomena called stochastic resonance and resonant activation. Our study, amenable to experimental tests using optical tweezers, is of great significance to the folding of biopolymers in vivo that are subject to the broad spectrum of cellular noises.

Kinetic mechanism of conformational switch between bistable RNA hairpins

Transitions between the different conformational states play a critical role in many RNA catalytic and regulatory functions. In this study, we use the Kinetic Monte Carlo method to investigate the kinetic mechanism for the conformational switches between bistable RNA hairpins. We find three types of conformational switch pathways for RNA hairpins: refolding after complete unfolding, folding through basepair-exchange pathways and through pseudoknot-assisted pathways, respectively. The result of the competition between the three types of pathways depends mainly on the location of the rate-limiting base stacks (such as the GC base stacks) in the structures. Depending on the structural relationships between the two bistable hairpins, the conformational switch can follow single or multiple dominant pathways. The predicted folding pathways are supported by the activation energy results derived from the Arrhenius plot as well as the NMR spectroscopy data.

Quantifying internal friction in unfolded and intrinsically disordered proteins with single-molecule spectroscopy

Internal friction, which reflects the "roughness" of the energy landscape, plays an important role for proteins by modulating the dynamics of their folding and other conformational changes. However, the experimental quantification of internal friction and its contribution to folding dynamics has remained challenging. Here we use the combination of single-molecule Förster resonance energy transfer, nanosecond fluorescence correlation spectroscopy, and microfluidic mixing to determine the reconfiguration times of unfolded proteins and investigate the mechanisms of internal friction contributing to their dynamics. Using concepts from polymer dynamics, we determine internal friction with three complementary, largely independent, and consistent approaches as an additive contribution to the reconfiguration time of the unfolded state. We find that the magnitude of internal friction correlates with the compactness of the unfolded protein: its contribution dominates the reconfiguration time of approximately 100 ns of the compact unfolded state of a small cold shock protein under native conditions, but decreases for more expanded chains, and approaches zero both at high denaturant concentrations and in intrinsically disordered proteins that are expanded due to intramolecular charge repulsion. Our results suggest that internal friction in the unfolded state will be particularly relevant for the kinetics of proteins that fold in the microsecond range or faster. The low internal friction in expanded intrinsically disordered proteins may have implications for the dynamics of their interactions with cellular binding partners.

Computational investigation of the effect of thermal perturbation on the mechanical unfolding of titin I27

The emergence of single-molecule force measurement experiments has facilitated a better understanding of protein folding pathways and the thermodynamics involved. Computational methods such as steered molecular dynamics (SMD) simulations are helpful in providing atomistic level information on the unfolding pathways. Recent experimental studies have showed that combinations of single-molecule experiments with traditional methods such as chemical and/or thermal denaturation yield additional insights into the folding phenomenon. In this study, we report results from extensive computations (a total of about 60 SMD simulations with a total length of about 0.4 μs) that address the effect of thermal perturbation on the mechanical stability of the I27 domain of the protein titin. A wide range of temperatures (280-340 K) were considered for the pulling, which was done at both constant velocity and constant force using SMD simulations. Good agreement with experimental data, such as for the trends in changes in average force and the maximum force with respect to the temperature, was obtained. This study identifies two competing pathways for the mechanical unfolding of I27, and illustrates the significance of combining various techniques to examine protein folding.

Measuring the folding landscape of a harmonically constrained biopolymer

Pioneering studies have shown that the probability distribution of opening length for a DNA hairpin, recorded under constant force using an optical trap, can be used to reconstruct the energy landscape of the transition. However, measurements made under constant force are subject to some limitations. Under constant force a system with a sufficiently high energy barrier spends most of its time in the closed or open conformation, with relatively few statistics collected in the transition state region. We describe a measurement scheme in which the system is driven progressively through the transition by an optical trap and an algorithm is used to extract the energy landscape of the transition from the fluctuations recorded during this process. We illustrate this technique in simulations and demonstrate its effectiveness in experiments on a DNA hairpin. We find that the combination of this technique with the use of short DNA handles facilitates a high-resolution measurement of the hairpin's folding landscape with a very short measurement time.

Direction-dependent mechanical unfolding and green fluorescent protein as a force sensor

An Ising-like model of proteins is used to investigate the mechanical unfolding of the green fluorescent protein along different directions. When the protein is pulled from its ends, we recover the major and minor unfolding pathways observed in experiments. Upon varying the pulling direction, we find the correct order of magnitude and ranking of the unfolding forces. Exploiting the direction dependence of the unfolding force at equilibrium, we propose a force sensor whose luminescence depends on the applied force.

Improving signal/noise resolution in single-molecule experiments using molecular constructs with short handles

We investigate unfolding/folding force kinetics in DNA hairpins exhibiting two and three states with newly designed short dsDNA handles (29 bp) using optical tweezers. We show how the higher stiffness of the molecular setup moderately enhances the signal/noise ratio (SNR) in hopping experiments as compared to conventional long-handled constructs (≅700 bp). The shorter construct results in a signal of higher SNR and slower folding/unfolding kinetics, thereby facilitating the detection of otherwise fast structural transitions. A novel analysis, as far as we are aware, of the elastic properties of the molecular setup, based on high-bandwidth measurements of force fluctuations along the folded branch, reveals that the highest SNR that can be achieved with short handles is potentially limited by the marked reduction of the effective persistence length and stretch modulus of the short linker complex.

Free-energy landscapes and thermodynamic parameters of complex molecules from nonequilibrium simulation trajectories

Thermodynamic parameters such as free energies and heat capacities are important quantities for understanding processes involving structural transitions in complex molecules such as proteins. Computational investigations provide simulated data that can be used for calculating thermodynamic parameters. However, calculations give accurate results only if the simulations sample all of configuration space with the appropriate temperature-dependent Boltzmann equilibrium probabilities. For many systems, truly comprehensive sampling of configuration space is not computationally feasible. We present an approximation technique for the calculations that will give accurate values for thermodynamic parameters when the data is incomplete. Our work is applicable to systems in which there are two distinct, important regions of configuration space that must be sampled. Importantly, the results are also valid when the system is more complex than two-state systems. Transition pathways that involve intermediate configurations between two stable regions are allowed in this treatment, and therefore the results are valid for multistate systems.

Survey on indirect optical manipulation of cells, nucleic acids, and motor proteins

Optical tweezers have emerged as a promising technique for manipulating biological objects. Instead of direct laser exposure, more often than not, optically-trapped beads are attached to the ends or boundaries of the objects for translation, rotation, and stretching. This is referred to as indirect optical manipulation. In this paper, we utilize the concept of robotic gripping to explain the different experimental setups which are commonly used for indirect manipulation of cells, nucleic acids, and motor proteins. We also give an overview of the kind of biological insights provided by this technique. We conclude by highlighting the trends across the experimental studies, and discuss challenges and promising directions in this domain of active current research.

Compaction and tensile forces determine the accuracy of folding landscape parameters from single molecule pulling experiments

We establish a framework for assessing whether the transition state location of a biopolymer, which can be inferred from single molecule pulling experiments, corresponds to the ensemble of structures that have equal probability of reaching either the folded or unfolded states (P(fold)=0.5). Using results for the forced unfolding of a RNA hairpin, an exactly soluble model, and an analytic theory, we show that P(fold) is solely determined by s, an experimentally measurable molecular tensegrity parameter, which is a ratio of the tensile force and a compaction force that stabilizes the folded state. Applications to folding landscapes of DNA hairpins and a leucine zipper with two barriers provide a structural interpretation of single molecule experimental data. Our theory can be used to assess whether molecular extension is a good reaction coordinate using measured free energy profiles.