Reconstructing folding energy landscape profiles from nonequilibrium pulling curves with an inverse Weierstrass integral transform

Appearance of de Gennes length in force-induced transitions

Using Langevin dynamic simulations, a simple coarse-grained model of a DNA protein construct is used to study the DNA rupture and the protein unfolding. We identify three distinct states: (i) zipped DNA and collapsed protein, (ii) unzipped DNA and stretched protein, and (iii) unzipped DNA and collapsed protein. Here, we find a phase diagram that shows these states depending on the size of the DNA handle and the protein. For a less stable protein, unfolding is solely governed by the size of the linker DNA, whereas if the protein's stability increases, complete unfolding becomes impossible because the rupture force for DNA has reached a saturation regime influenced by the de Gennes length. We show that unfolding occurs via a few intermediate states by monitoring the force-extension curve of the entire protein. We extend our study to a heterogeneous protein system, where similar intermediate states in two systems can lead to different protein unfolding paths.

Connections between efficient control and spontaneous transitions in an Ising model

A system can be driven between metastable configurations by a time-dependent driving protocol, which uses external control parameters to change the potential energy of the system. Here we investigate the correspondence between driving protocols that are designed to minimize work and the spontaneous transition paths of the system in the absence of driving. We study the spin-inversion reaction in a 2D Ising model, quantifying the timing of each spin flip and heat flow to the system during both a minimum-work protocol and a spontaneous transition. The general order of spin flips during the transition mechanism is preserved between the processes, despite the coarseness of control parameters that are unable to reproduce more detailed features of the spontaneous mechanism. Additionally, external control parameters provide energy to each system component to compensate changes in internal energy, showing how control parameters are tuned during a minimum-work protocol to counteract underlying energetic features. This paper supports a correspondence between minimum-work protocols and spontaneous transition mechanisms.

Multidimensional minimum-work control of a 2D Ising model

A system's configurational state can be manipulated using dynamic variation of control parameters, such as temperature, pressure, or magnetic field; for finite-duration driving, excess work is required above the equilibrium free-energy change. Minimum-work protocols in multidimensional control-parameter space have potential to significantly reduce work relative to one-dimensional control. By numerically minimizing a linear-response approximation to the excess work, we design protocols in control-parameter spaces of a 2D Ising model that efficiently drive the system from the all-down to all-up configuration. We find that such designed multidimensional protocols take advantage of more flexible control to avoid control-parameter regions of high system resistance, heterogeneously input and extract work to make use of system relaxation, and flatten the energy landscape, making accessible many configurations that would otherwise have prohibitively high energy and thus decreasing spin correlations. Relative to one-dimensional protocols, this speeds up the rate-limiting spin-inversion reaction, thereby keeping the system significantly closer to equilibrium for a wide range of protocol durations, and significantly reducing resistance and hence work.

Single-Molecule Force Spectroscopy of Protein Folding

The use of force probes to induce unfolding and refolding of single molecules through the application of mechanical tension, known as single-molecule force spectroscopy (SMFS), has proven to be a powerful tool for studying the dynamics of protein folding. Here we provide an overview of what has been learned about protein folding using SMFS, from small, single-domain proteins to large, multi-domain proteins. We highlight the ability of SMFS to measure the energy landscapes underlying folding, to map complex pathways for native and non-native folding, to probe the mechanisms of chaperones that assist with native folding, to elucidate the effects of the ribosome on co-translational folding, and to monitor the folding of membrane proteins.

Effect of Memory and Inertial Contribution on Transition-Time Distributions: Theory and Simulations

Transition paths refer to the time taken by molecules to cross a barrier separating two molecular conformations. In this work, we study how memory, as well as inertial contribution in the dynamics along a reaction coordinate, can affect the distribution of the transition-path time. We use a simple model of dynamics governed by a generalized Langevin equation with a power-law memory along with the inertial term, which was neglected in previous studies, where memory effects were explored only in the overdamped limit. We derive an approximate expression for the transit-time distribution and discuss our results for the short- and long-time limits and also compare it with known results in the high friction (overdamped) limit as well as in the Markovian limit. We have developed a numerical algorithm to test our theoretical results against extensive numerical simulations.

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.

Errors in Energy Landscapes Measured with Particle Tracking

Tracking Brownian particles is often employed to map the energy landscape they explore. Such measurements have been exploited to study many biological processes and interactions in soft materials. Yet video tracking is irremediably contaminated by localization errors originating from two imaging artifacts: the "static" errors come from signal noise, and the "dynamic" errors arise from the motion blur due to finite frame-acquisition time. We show that these errors result in systematic and nontrivial biases in the measured energy landscapes. We derive a relationship between the true and the measured potential that elucidates, among other aberrations, the presence of false double-well minima in the apparent potentials reported in recent studies. We further assess several canonical trapping and pair-interaction potentials by using our analytically derived results and Brownian dynamics simulations. In particular, we show that the apparent spring stiffness of harmonic potentials (such as optical traps) is increased by dynamic errors but decreased by static errors. Our formula allows for the development of efficient corrections schemes, and we also present in this work a provisional method for reconstructing true potentials from the measured ones.

High-Precision Single-Molecule Characterization of the Folding of an HIV RNA Hairpin by Atomic Force Microscopy

The folding of RNA into a wide range of structures is essential for its diverse biological functions from enzymatic catalysis to ligand binding and gene regulation. The unfolding and refolding of individual RNA molecules can be probed by single-molecule force spectroscopy (SMFS), enabling detailed characterization of the conformational dynamics of the molecule as well as the free-energy landscape underlying folding. Historically, high-precision SMFS studies of RNA have been limited to custom-built optical traps. Although commercial atomic force microscopes (AFMs) are widely deployed and offer significant advantages in ease-of-use over custom-built optical traps, traditional AFM-based SMFS lacks the sensitivity and stability to characterize individual RNA molecules precisely. Here, we developed a high-precision SMFS assay to study RNA folding using a commercial AFM and applied it to characterize a small RNA hairpin from HIV that plays a key role in stimulating programmed ribosomal frameshifting. We achieved rapid data acquisition in a dynamic assay, unfolding and then refolding the same individual hairpin more than 1,100 times in 15 min. In comparison to measurements using optical traps, our AFM-based assay featured a stiffer force probe and a less compliant construct, providing a complementary measurement regime that dramatically accelerated equilibrium folding dynamics. Not only did kinetic analysis of equilibrium trajectories of the HIV RNA hairpin yield the traditional parameters used to characterize folding by SMFS (zero-force rate constants and distances to the transition state), but we also reconstructed the full 1D projection of the folding free-energy landscape comparable to state-of-the-art studies using dual-beam optical traps, a first for this RNA hairpin and AFM studies of nucleic acids in general. Looking forward, we anticipate that the ease-of-use of our high-precision assay implemented on a commercial AFM will accelerate studying folding of diverse nucleic acid structures.

Improved free-energy landscape reconstruction of bacteriorhodopsin highlights local variations in unfolding energy

Precisely quantifying the energetics that drive the folding of membrane proteins into a lipid bilayer remains challenging. More than 15 years ago, atomic force microscopy (AFM) emerged as a powerful tool to mechanically extract individual membrane proteins from a lipid bilayer. Concurrently, fluctuation theorems, such as the Jarzynski equality, were applied to deduce equilibrium free energies (ΔG0) from non-equilibrium single-molecule force spectroscopy records. The combination of these two advances in single-molecule studies deduced the free-energy of the model membrane protein bacteriorhodopsin in its native lipid bilayer. To elucidate this free-energy landscape at a higher resolution, we applied two recent developments. First, as an input to the reconstruction, we used force-extension curves acquired with a 100-fold higher time resolution and 10-fold higher force precision than traditional AFM studies of membrane proteins. Next, by using an inverse Weierstrass transform and the Jarzynski equality, we removed the free energy associated with the force probe and determined the molecular free-energy landscape of the molecule under study, bacteriorhodopsin. The resulting landscape yielded an average unfolding free energy per amino acid (aa) of 1.0 ± 0.1 kcal/mol, in agreement with past single-molecule studies. Moreover, on a smaller spatial scale, this high-resolution landscape also agreed with an equilibrium measurement of a particular three-aa transition in bacteriorhodopsin that yielded 2.7 kcal/mol/aa, an unexpectedly high value. Hence, while average unfolding ΔG0 per aa is a useful metric, the derived high-resolution landscape details significant local variation from the mean. More generally, we demonstrated that, as anticipated, the inverse Weierstrass transform is an efficient means to reconstruct free-energy landscapes from AFM data.

Exploring biomolecular energy landscapes

The potential energy landscape perspective provides both a conceptual and a computational framework for predicting, understanding and designing molecular properties. In this Feature Article, we highlight some recent advances that greatly facilitate structure prediction and analysis of global thermodynamics and kinetics in proteins and nucleic acids. The geometry optimisation procedures, on which these calculations are based, can be accelerated significantly using local rigidification of selected degrees of freedom, and through implementations on graphics processing units. Results of progressive local rigidification are first summarised for trpzip1, including a systematic analysis of the heat capacity and rearrangement rates. Benchmarks for all the essential optimisation procedures are then provided for a variety of proteins. Applications are then illustrated from a study of how mutation affects the energy landscape for a coiled-coil protein, and for transitions in helix morphology for a DNA duplex. Both systems exhibit an intrinsically multifunnel landscape, with the potential to act as biomolecular switches.

The importance of thermodynamics for molecular systems, and the importance of molecular systems for thermodynamics

Improved understanding of molecular systems has only emphasised the sophistication of networks within the cell. Simultaneously, the advance of nucleic acid nanotechnology, a platform within which reactions can be exquisitely controlled, has made the development of artificial architectures and devices possible. Vital to this progress has been a solid foundation in the thermodynamics of molecular systems. In this pedagogical review and perspective, we discuss how thermodynamics determines both the overall potential of molecular networks, and the minute details of design. We then argue that, in turn, the need to understand molecular systems is helping to drive the development of theories of thermodynamics at the microscopic scale.

Comparing the energy landscapes for native folding and aggregation of PrP

Protein sequences are evolved to encode generally one folded structure, out of a nearly infinite array of possible folds. Underlying this code is a funneled free energy landscape that guides folding to the native conformation. Protein misfolding and aggregation are also a manifestation of free-energy landscapes. The detailed mechanisms of these processes are poorly understood, but often involve rare, transient species and a variety of different pathways. The inherent complexity of misfolding has hampered efforts to measure aggregation pathways and the underlying energy landscape, especially using traditional methods where ensemble averaging obscures important rare and transient events. We recently studied the misfolding and aggregation of prion protein by examining 2 monomers tethered in close proximity as a dimer, showing how the steps leading to the formation of a stable aggregated state can be resolved in the single-molecule limit and the underlying energy landscape thereby reconstructed. This approach allows a more quantitative comparison of native folding versus misfolding, including fundamental differences in the dynamics for misfolding. By identifying key steps and interactions leading to misfolding, it should help to identify potential drug targets. Here we describe the importance of characterizing free-energy landscapes for aggregation and the challenges involved in doing so, and we discuss how single-molecule studies can help test proposed structural models for PrP aggregates.

Transition Path Times Measured by Single-Molecule Spectroscopy

The transition path is a tiny fraction of a molecular trajectory during which the free-energy barrier is crossed. It is a single-molecule property and contains all mechanistic information of folding processes of biomolecules such as proteins and nucleic acids. However, the transition path has been difficult to probe because it is short and rarely visited when transitions actually occur. Recent technical advances in single-molecule spectroscopy have made it possible to directly probe transition paths, which has opened up new theoretical and experimental approaches to investigating folding mechanisms. This article reviews recent single-molecule fluorescence and force spectroscopic measurements of transition path times and their connection to both theory and simulations.

Direct measurement of sequence-dependent transition path times and conformational diffusion in DNA duplex formation

The conformational diffusion coefficient, D, sets the timescale for microscopic structural changes during folding transitions in biomolecules like nucleic acids and proteins. D encodes significant information about the folding dynamics such as the roughness of the energy landscape governing the folding and the level of internal friction in the molecule, but it is challenging to measure. The most sensitive measure of D is the time required to cross the energy barrier that dominates folding kinetics, known as the transition path time. To investigate the sequence dependence of D in DNA duplex formation, we measured individual transition paths from equilibrium folding trajectories of single DNA hairpins held under tension in high-resolution optical tweezers. Studying hairpins with the same helix length but with G:C base-pair content varying from 0 to 100%, we determined both the average time to cross the transition paths, τ_tp, and the distribution of individual transit times, P_TP(t). We then estimated D from both τ_tp and P_TP(t) from theories assuming one-dimensional diffusive motion over a harmonic barrier. τ_tp decreased roughly linearly with the G:C content of the hairpin helix, being 50% longer for hairpins with only A:T base pairs than for those with only G:C base pairs. Conversely, D increased linearly with helix G:C content, roughly doubling as the G:C content increased from 0 to 100%. These results reveal that G:C base pairs form faster than A:T base pairs because of faster conformational diffusion, possibly reflecting lower torsional barriers, and demonstrate the power of transition path measurements for elucidating the microscopic determinants of folding.

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.

Pseudoknots in RNA folding landscapes

MOTIVATION

The function of an RNA molecule is not only linked to its native structure, which is usually taken to be the ground state of its folding landscape, but also in many cases crucially depends on the details of the folding pathways such as stable folding intermediates or the timing of the folding process itself. To model and understand these processes, it is necessary to go beyond ground state structures. The study of rugged RNA folding landscapes holds the key to answer these questions. Efficient coarse-graining methods are required to reduce the intractably vast energy landscapes into condensed representations such as barrier trees or basin hopping graphs : BHG) that convey an approximate but comprehensive picture of the folding kinetics. So far, exact and heuristic coarse-graining methods have been mostly restricted to the pseudoknot-free secondary structures. Pseudoknots, which are common motifs and have been repeatedly hypothesized to play an important role in guiding folding trajectories, were usually excluded.

RESULTS

We generalize the BHG framework to include pseudoknotted RNA structures and systematically study the differences in predicted folding behavior depending on whether pseudoknotted structures are allowed to occur as folding intermediates or not. We observe that RNAs with pseudoknotted ground state structures tend to have more pseudoknotted folding intermediates than RNAs with pseudoknot-free ground state structures. The occurrence and influence of pseudoknotted intermediates on the folding pathway, however, appear to depend very strongly on the individual RNAs so that no general rule can be inferred.

AVAILABILITY AND IMPLEMENTATION

The algorithms described here are implemented in C++ as standalone programs. Its source code and Supplemental material can be freely downloaded from http://www.tbi.univie.ac.at/bhg.html.

CONTACT

qin@bioinf.uni-leipzig.de

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

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.

Probing the structural dynamics of proteins and nucleic acids with optical tweezers

Conformational changes are an essential feature of most molecular processes in biology. Optical tweezers have emerged as a powerful tool for probing conformational dynamics at the single-molecule level because of their high resolution and sensitivity, opening new windows on phenomena ranging from folding and ligand binding to enzyme function, molecular machines, and protein aggregation. By measuring conformational changes induced in a molecule by forces applied by optical tweezers, new insight has been gained into the relationship between dynamics and function. We discuss recent advances from studies of how structure forms in proteins and RNA, including non-native structures, fluctuations in disordered proteins, and interactions with chaperones assisting native folding. We also review the development of assays probing the dynamics of complex protein-nucleic acid and protein-protein assemblies that reveal the dynamic interactions between biomolecular machines and their substrates.

Testing Landscape Theory for Biomolecular Processes with Single Molecule Fluorescence Spectroscopy

Although Kramers' theory for diffusive barrier crossing on a 1D free energy profile plays a central role in landscape theory for complex biomolecular processes, it has not yet been rigorously tested by experiment. Here we test this 1D diffusion scenario with single molecule fluorescence measurements of DNA hairpin folding. We find an upper bound of 2.5  μs for the average transition path time, consistent with the predictions by theory with parameters determined from optical tweezer measurements.