Roughness Of Energy Landscape Research Articles

Observing the base-by-base search for native structure along transition paths during the folding of single nucleic acid hairpins

Biomolecular folding involves searching among myriad possibilities for the native conformation, but the elementary steps expected from theory for this search have never been detected directly. We probed the dynamics of folding at high resolution using optical tweezers, measuring individual trajectories as nucleic acid hairpins passed through the high-energy transition states that dominate kinetics and define folding mechanisms. We observed brief but ubiquitous pauses in the transition states, with a dwell time distribution that matched microscopic theories of folding quantitatively. The sequence dependence suggested that pauses were dominated by microbarriers from nonnative conformations during the search by each nucleotide residue for the native base-pairing conformation. Furthermore, the pauses were position dependent, revealing subtle local variations in energy-landscape roughness and allowing the diffusion coefficient describing the microscopic dynamics within the barrier to be found without reconstructing the shape of the energy landscape. These results show how high-resolution measurements can elucidate key microscopic events during folding to test fundamental theories of folding.

The role of solvent quality and chain stiffness on the end-to-end contact kinetics of semiflexible polymers.

Motivated by loop closure during protein folding and DNA packing, we systemically studied the effects of the solvent quality and chain stiffness on the thermodynamics and kinetics of the end-to-end contact formation for semiflexible polymer chains with reactive ends by Langevin dynamics simulations. In thermodynamics, a rich variety of products of the end-to-end contact have been discovered, such as loop, hairpin, toroid, and rodlike bundle, the populations of which are dependent on the solvent quality and chain stiffness. In kinetics, the overall pathways to form the end-to-end contact have been identified. The change of solvent quality and chain stiffness can tune the roughness of energy landscape and modulate the kinetic partitioning of the end-to-end contact formation pathways, leading to differing kinetic behaviors. In good or poor solvents, the first end-to-end contact rate k c decreases with increasing the strength of bending stiffness k θ monotonically. In very poor solvents, however, the dependence of the logarithm of the first end-to-end contact rate ln k c on k θ exhibits erratic behavior, which stems from more rugged energy landscape due to the polymer chain getting trapped into the intermediate state composed of the rodlike bundles with two ends in separation. For semiflexible chains, with increasing chain length N, the rate k c increases initially and then decreases: in good solvents, the rate k c exhibits a power-law relationship to chain length N with an exponent of ∼-1.50 in the region of long chains, which is in good agreement with the value derived from the experiment in the asymptotic limit of large N; and in poor solvents, the rate k c exhibits a significantly stronger chain length dependence than those observed in good solvents in the region of long chains due to frustration to form the end-to-end contact along a specific path, especially the scaling exponent between the rate k c and chain length N is ∼-3.62 for the case of polymer chains with k θ = 4 at the solvent quality ε ij = 1, in accord with the value obtained from the experiments.

Conformational changes during permeation of Na+ through a modified cyclic peptide nanotube promote energy landscape roughness.

Using a metadynamics approach, we investigate the potential of mean force for Na+ permeation inside a cyclic peptide nanotube (CPN) with modified interior as a function of ion position, coordination number, and lumen chemistry. We show that functionalizing the lumen of a CPN with a methyl-benzoic acid group introduces non-periodic variations in the internal energy of the nanotube, which dictate the overall free energy roughness during the permeation of Na+. These non-periodic variations arise from the structural dynamics of the functional group, where changes in the dihedral angles induced by the proximity of the ion give rise to conformational changes that increase landscape roughness and thereby decrease transport rate. Our computational framework emphasizes the advantages of using the coordination number as a collective variable to investigate the available conformations during ion permeation through CPNs, and reveals new structure-function relations for chemically tunable CPNs, paving the way for rational design of nano-porous systems with tunable selectivity and flux.

The roughness of the protein energy landscape results in anomalous diffusion of the polypeptide backbone.

Although protein folding is often described by motion on a funnel-shaped overall topology of the energy landscape, the many local interactions that can occur result in considerable landscape roughness which slows folding by increasing internal friction. Recent experimental results have brought to light that this roughness also causes unusual diffusional behaviour of the backbone of an unfolded protein, i.e. the relative motion of protein sections cannot be described by the normal diffusion equation, but shows strongly subdiffusional behaviour with a nonlinear time dependence of the mean square displacement, 〈r(2)(t)〉∝t(α) (α≪ 1). This results in significantly slower configurational equilibration than had been assumed hitherto. Analysis of the results also allows quantification of the energy landscape roughness, i.e. the root-mean-squared depth of local minima, yielding a value of 4-5kBT for a typical small protein.

Measurement of energy landscape roughness of folded and unfolded proteins.

The dynamics of protein conformational changes, from protein folding to smaller changes, such as those involved in ligand binding, are governed by the properties of the conformational energy landscape. Different techniques have been used to follow the motion of a protein over this landscape and thus quantify its properties. However, these techniques often are limited to short timescales and low-energy conformations. Here, we describe a general approach that overcomes these limitations. Starting from a nonnative conformation held by an aromatic disulfide bond, we use time-resolved spectroscopy to observe nonequilibrium backbone dynamics over nine orders of magnitude in time, from picoseconds to milliseconds, after photolysis of the disulfide bond. We find that the reencounter probability of residues that initially are in close contact decreases with time following an unusual power law that persists over the full time range and is independent of the primary sequence. Model simulations show that this power law arises from subdiffusional motion, indicating a wide distribution of trapping times in local minima of the energy landscape, and enable us to quantify the roughness of the energy landscape (4-5 k(B)T). Surprisingly, even under denaturing conditions, the energy landscape remains highly rugged with deep traps (>20 k(B)T) that result from multiple nonnative interactions and are sufficient for trapping on the millisecond timescale. Finally, we suggest that the subdiffusional motion of the protein backbone found here may promote rapid folding of proteins with low contact order by enhancing contact formation between nearby residues.

A Model with Two Elastically Coupled Reaction Coordinates for the Internal Friction of Proteins

The rates of protein conformational changes are usually not only limited by external but also internal friction, however, the origin and significance of this latter phenomenon is poorly understood. It is often found experimentally that a linear fit to the reciprocal of the reaction rate as a function of the viscosity of the external medium has a non-zero value at zero viscosity, signifying the presence of internal friction. Furthermore, some of the experiments performed at different temperatures indicate that the internal friction follows an Arrhenius-like temperature dependence. To explain these phenomena we suggest a simple model for protein conformational changes in terms of two elastically coupled reaction coordinates, one of which being in contact with the external medium, and the other one experiencing the rough energy landscape of the protein. Our analytical calculations, supplemented with numerical simulations demonstrate that depending on the coupling strength (which is related to the flexibility of the protein) the short-wavelength components of the energy landscape roughness can be observed as an increased apparent internal friction with an apparent activation energy.

Protein Flexibility Partitions the Effects of Energy Landscape Roughness Between Activation Energy and Internal Friction

The rate of protein conformational changes are usually not only limited by external but also internal friction, however, the origin and significance of this latter phenomenon is poorly understood. By investigating the internal friction during the activation of two trypsin mutants at various temperatures and external viscosities we have discovered that the temperature dependence of the internal friction shows an Arrhenius-like behavior. The characteristic energy of the Arrhenius formula, however, can change dramatically upon the replacement of a single amino acid at a hinge position (thereby affecting the flexibility of the protein), or by crossing a critical temperature. At the same time, the activation energy of the conformational transition also changes with a similar magnitude, but in the opposite direction. These observations shed light on the intricate interplay between the apparent internal friction and activation energy. Moreover, we have found that the more flexible a protein is the greater proportion of its activation energy is partitioned into internal friction. All these results have allowed us to come to the general conclusion that the different hierarchical levels of the roughness of the energy landscape along a conformational transition can be observed as either activation energy or internal friction depending on the degree of flexibility of the protein.

The Plastic Energy Landscape of Protein Folding: A TRIANGULAR FOLDING MECHANISM WITH AN EQUILIBRIUM INTERMEDIATE FOR A SMALL PROTEIN DOMAIN

Protein domains usually fold without or with only transiently populated intermediates, possibly to avoid misfolding, which could result in amyloidogenic disease. Whether observed intermediates are productive and obligatory species on the folding reaction pathway or dispensable by-products is a matter of debate. Here, we solved the crystal structure of a small protein domain, SAP97 PDZ2 I342W C378A, and determined its folding pathway. The presence of a folding intermediate was demonstrated both by single and double-mixing kinetic experiments using urea-induced (un)folding as well as ligand-induced folding. This protein domain was found to fold via a triangular scheme, where the folding intermediate could be either on- or off-pathway, depending on the experimental conditions. Furthermore, we found that the intermediate was present at equilibrium, which is rarely seen in folding reactions of small protein domains. The folding mechanism observed here illustrates the roughness and plasticity of the protein folding energy landscape, where several routes may be employed to reach the native state. The results also reconcile the folding mechanisms of topological variants within the PDZ domain family.

The transition state transit time of WW domain folding is controlled by energy landscape roughness

Protein folding barriers can be so low that a substantial protein population diffusing in the transition state region can be detected. The very fast kinetic phase contributed by transition state transit is the molecular phase. We detect the molecular phase of the beta-sheet protein FiP35 from 60 to 83 degrees C by T-jump relaxation experiments. The molecular phase actually slows down slightly with increasing temperature. Thus the friction that controls the prefactor in Kramers' transition state model does not scale with solvent viscosity. Instead, we postulate that an increase in the energy landscape roughness as the hydrophobic effect strengthens with increasing temperature explains the slowing of the molecular phase. We measured that the duration tau(m) of the molecular phase depends slightly on the size of the T-jump, in agreement with this explanation. The tau(m) measured here provides the best current estimate for the transit time from folded to unfolded state of a single protein molecule. We confirm this by directly comparing relaxation and single molecule signals computed by using Langevin trajectory models on a realistic FiP35 free energy surface.

How to mark off paths on the protein energy landscape

Protein folding has run the gamut from complexity, to simplicity, and in a way, back to complexity. In the beginning, there were the slow and complex folding mechanisms of multidomain proteins involving multiple parallel or sequential intermediates en route to the native state (1). Then followed small proteins that can fold by a rapid two-state mechanism (2). Finally, when the single leftover activation barrier was reduced as much as possible through protein engineering, energy landscape roughness was revealed in folding kinetics (3). Landscape roughness temporarily traps very fast folders as they navigate many pathways toward the native state (4). The article by Noe et al. (5) in this issue of PNAS shows how molecular dynamics simulations can be used very efficiently to create a roadmap of the pathways on which a protein explores the energy landscape.

Protein energy landscape roughness

The 'new view' of proteins sees protein reactions as parallel processes occurring along funnelled energy landscapes. These landscapes are generally not smooth, but are superimposed by hills and valleys of different heights and widths leading to roughness on the energy surface. In the present paper, we describe the origins of protein energy landscape roughness, measurements of its scale and its implications.

Beyond Induced-Fit Receptor-Ligand Interactions: Structural Changes that Can Significantly Extend Bond Lifetimes

While the lifetime of conventional receptor-ligand interactions is shortened by tensile mechanical force, some recently discovered interactions, termed catch bonds, can be strengthened by force. Motivated by the search for the underpinning structural mechanisms, we here explore the structural dynamics of the binding site of the bacterial adhesive protein FimH by molecular dynamics and steered molecular dynamics. While the crystal structure of only one FimH conformation has been reported so far, we describe two distinctively different conformations of the mannose-bound FimH binding site. Force-induced dissociation was slowed when the mannose ring rotated such that additional force-bearing hydrogen bonds formed with the base of the FimH binding pocket. The lifetime of the complex was further enhanced significantly by rigidifying this base. We finally show how even sub-angstrom spatial alterations of the hydrogen bonding pattern within the base can lead to significantly decreased bond lifetimes.

Studies of molecular docking between fibroblast growth factor and heparin using generalized simulated annealing

AbstractSince the middle 70s, the main molecular docking problem consists in limitations to treat adequately the degrees of freedom of protein (or a receptor) due to the energy landscape roughness and the high computational cost. Until recently, only few algorithms considering flexible simultaneously both ligand and receptor at low computational cost were developed. As a recent proposed Statistical Mechanics, generalized simulated annealing (GSA) has been employed at diverse works concerning global optimization problems. In this work, we used this method exploring the molecular docking problem taking into account the FGF‐2 and heparin complex. Since the requirements of an efficient docking algorithm are accuracy and velocity, we tested the influence of GSA parameters qA (new configuration acceptance index), qV (energy surface visiting index), and qT (temperature decreasing control) on the performance of GSADOCK program. Our simulations showed that as temperature parameter qT increases, qA parameter follows this behavior in the interval ranging from 1.1 to 2.3. We found that the GSA parameters have the best performance for the qA values ranging from 1.1 to 1.3, qV values from 1.3 to 1.5, and qT values from 1.1 to 1.7. Most of good qV values were equal or next the good qT values. Finally, the implemented algorithm is trustworthy and can be employed as a tool of molecular modeling methods. The final version of the program will be free of charge and will be accessible at our home‐page or could be requested to the authors for e‐mail. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Energy landscape roughness of the streptavidin–biotin interaction

Molecular interactions between receptors and ligands can be characterized by their free energy landscape. In its simplest representation, the energy landscape is described by a barrier of certain height and width that determines the dissociation rate of the complex, as well as its dynamic strength. Some interactions, however, require a more complex landscape with additional barriers and roughness along the reaction coordinate. This roughness slows down the dissociation kinetics of the interaction and contributes to its dynamic strength. The streptavidin-biotin complex has been extensively studied due to its remarkably low dissociation kinetics. However, single molecule measurements from independent experiments showed scattered and disparate results. In this work, the energy landscape roughness of the streptavidin-biotin interaction was estimated to be in the range of 5-8kBT using dynamic force spectroscopy (DFS) measurements at three different temperatures. These results can be used to explain both its slow dissociation kinetics and the discrepancies in the reported force measurements.

Direct measurement of protein energy landscape roughness

The energy landscape of proteins is thought to have an intricate, corrugated structure. Such roughness should have important consequences on the folding and binding kinetics of proteins, as well as on their equilibrium fluctuations. So far, no direct measurement of protein energy landscape roughness has been made. Here, we combined a recent theory with single-molecule dynamic force spectroscopy experiments to extract the overall energy scale of roughness epsilon for a complex consisting of the small GTPase Ran and the nuclear transport receptor importin-beta. The results gave epsilon > 5k(B)T, indicating a bumpy energy surface, which is consistent with the ability of importin-beta to accommodate multiple conformations and to interact with different, structurally distinct ligands.

Effects of Confinement in Chaperonin Assisted Protein Folding: Rate Enhancement by Decreasing the Roughness of the Folding Energy Landscape

Chaperonins, such as the GroE complex of the bacteria Escherichia coli, assist the folding of proteins under non-permissive folding conditions by providing a cavity in which the newly translated or translocated protein can be encapsulated. Whether the chaperonin cage plays a passive role in protecting the protein from aggregation, or an active role in accelerating folding rates, remains a matter of debate. Here, we investigate the role of confinement in chaperonin mediated folding through molecular dynamics simulations. We designed a substrate protein with an α/β sandwich fold, a common structural motif found in GroE substrate proteins and confined it to a spherical hydrophilic cage which mimicked the interior of the GroEL/ES cavity. The thermodynamics and kinetics of folding were studied over a wide range of temperature and cage radii. Confinement was seen to significantly raise the collapse temperature, Tc, as a result of the associated entropy loss of the unfolded state. The folding temperature, Tf, on the other hand, remained unaffected by encapsulation, a consequence of the folding mechanism of this protein that involves an initial collapse to a compact misfolded state prior to rearranging to the native state. Folding rates were observed to be either accelerated or retarded compared to bulk folding rates, depending on the temperature of the simulation. Rate enhancements due to confinement were observed only at temperatures above the temperature Tm, which corresponds to the temperature at which the protein folds fastest. For this protein, Tm lies above the folding temperature, Tf, implying that encapsulation alone will not lead to a rate enhancement under conditions where the native state is stable (T<Tf). For confinement to positively impact folding rates under physiological conditions, it is hence necessary for the protein to exhibit a folding transition above the temperature at which it exhibits its fastest folding rate (Tm<Tf). We designed a protein with this property by reducing the energetic frustration in the original α/β sandwich substrate protein. The modified protein exhibited a twofold acceleration in folding rates upon encapsulation. This rate enhancement is due to a mechanistic change in folding involving the elimination, upon encapsulation, of accessible local energy minima corresponding to structures with large radii of gyration. For this protein, confinement hence plays more than the role of a passive cage, but rather adopts an active role, accelerating folding rates by decreasing the roughness of the energy landscape of the protein.

Can energy landscape roughness of proteins and RNA be measured by using mechanical unfolding experiments?

By considering temperature effects on the mechanical unfolding rates of proteins and RNA, whose energy landscape is rugged, the question posed in the title is answered in the affirmative. Adopting a theory by Zwanzig [Zwanzig, R. (1988) Proc. Natl. Acad. Sci. USA 85, 2029-2030], we show that, because of roughness characterized by an energy scale epsilon, the unfolding rate at constant force is retarded. Similarly, in nonequilibrium experiments done at constant loading rates, the most probable unfolding force increases because of energy landscape roughness. The effects are dramatic at low temperatures. Our analysis suggests that, by using temperature as a variable in mechanical unfolding experiments of proteins and RNA, the ruggedness energy scale epsilon, can be directly measured.

Smoothing the landscapes of protein folding: Insights from a minimal model

This work addresses the consideration of the energy landscape roughness in protein sequence design. The proteins are modeled by 2D lattice chains, initially designed to maximize the energy gap between the folded and unfolded states. Additional optimization and control of the folding properties is achieved by specific sequence mutations that alter the energetic and geometric roughness of the landscape. It is found that mutations that reduce the energetic roughness at the expense of increasing the native-state energy generally lead to a fast folding and stable protein at lower temperatures. Such mutations are also found to modify the geometric roughness (related to nucleation effects) creating variations in the folding time that depends specifically on each sequence and can lead in many cases to a reduction of the total landscape roughness. An additional reduction of the geometric roughness is achieved by adding local bond-angle propensities to selected sequence sites.

Relaxation in a perfect funnel

We have exactly solved the relaxational dynamics of a model protein which possesses a kinetically perfect funnel-like energy landscape. We find that the dependence of the relaxation time, $\tau$, on the density of states (DOS) and the energy level spacing distributions of the model displays several main types of behavior depending on the temperature $T$. This allows us to identify possible generic features of the relaxation. For some ranges of $T$, $\tau$ is insensitive to the density of states; for intermediate values of $T$ it depends on the energy level spacing distribution rather than on the DOS directly, and it becomes gradually more dependent on DOS with increasing temperature; finally, the relaxation can also be determined exclusively by the presence of a deep gap in the energy spectrum rather than by the detailed features of the density of states. We found that the behavior of $\tau$ crucially depends on the degeneracy of the energy spectrum. For the special case of exponentially increasing degeneracy, we were able to identify a characteristic temperature which roughly separates the relaxational regimes controlled by energetics and by entropy, respectively. Finally, the validity of our theory is discussed when roughness of energy landscape is added.