Energy Landscape of All-Atom Protein-Protein Interactions Revealed by Multiscale Enhanced Sampling

Protein-protein interactions are regulated by a subtle balance of complicated atomic interactions and solvation at the interface. To understand such an elusive phenomenon, it is necessary to thoroughly survey the large configurational space from the stable complex structure to the dissociated states using the all-atom model in explicit solvent and to delineate the energy landscape of protein-protein interactions. In this study, we carried out a multiscale enhanced sampling (MSES) simulation of the formation of a barnase-barstar complex, which is a protein complex characterized by an extraordinary tight and fast binding, to determine the energy landscape of atomistic protein-protein interactions. The MSES adopts a multicopy and multiscale scheme to enable for the enhanced sampling of the all-atom model of large proteins including explicit solvent. During the 100-ns MSES simulation of the barnase-barstar system, we observed the association-dissociation processes of the atomistic protein complex in solution several times, which contained not only the native complex structure but also fully non-native configurations. The sampled distributions suggest that a large variety of non-native states went downhill to the stable complex structure, like a fast folding on a funnel-like potential. This funnel landscape is attributed to dominant configurations in the early stage of the association process characterized by near-native orientations, which will accelerate the native inter-molecular interactions. These configurations are guided mostly by the shape complementarity between barnase and barstar, and lead to the fast formation of the final complex structure along the downhill energy landscape.     

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.     

The Role of Solvent in Protein Folding and in Aggregation

We discuss features of the effect of solvent on protein folding andaggregation, highlighting the physics related to the particulate nature and the peculiar structure of the aqueous solvent, and the biological significance of interactions between solvent and proteins. To this purpose we use a generalized energy landscape of extended dimensionality. A closer look at the properties of solvent induced interactions and forces proves useful for understanding the physical grounds of `ad hoc' interactions and for devising realistic ways of accounting for solvent effects. The solvent has long been known to be a crucially important part of biological systems, and times appear mature for it to be adequately accounted for in the protein folding problem. Use of the extended dimensionality energy landscape helpseliciting the possibility of coupling among conformational changes and aggregation, such as proved by experimental data in the literature.     

Effects of multiple-bond ruptures on kinetic parameters extracted from force spectroscopy measurements: revisiting biotin-streptavidin interactions

Force spectroscopy measurements of the rupture of the molecular bond between biotin and streptavidin often results in a wide distribution of rupture forces. We attribute the long tail of high rupture forces to the nearly simultaneous rupture of more than one molecular bond. To decrease the number of possible bonds, we employed hydrophilic polymeric tethers to attach biotin molecules to the atomic force microscope probe. It is shown that the measured distributions of rupture forces still contain high forces that cannot be described by the forced dissociation from a deep potential well. We employed a recently developed analytical model of simultaneous rupture of two bonds connected by polymer tethers with uneven length to fit the measured distributions. The resulting kinetic parameters agree with the energy landscape predicted by molecular dynamics simulations. It is demonstrated that when more than one molecular bond might rupture during the pulling measurements there is a noise-limited range of probe velocities where the kinetic parameters measured by force spectroscopy correspond to the true energy landscape. Outside this range of velocities, the kinetic parameters extracted by using the standard most probable force approach might be interpreted as artificial energy barriers that are not present in the actual energy landscape. Factors that affect the range of useful velocities are discussed.     

The effects of nonnative interactions on protein folding rates: theory and simulation

Proteins are minimally frustrated polymers. However, for realistic protein models, nonnative interactions must be taken into account. In this paper, we analyze the effect of nonnative interactions on the folding rate and on the folding free energy barrier. We present an analytic theory to account for the modification on the free energy landscape upon introduction of nonnative contacts, added as a perturbation to the strong native interactions driving folding. Our theory predicts a rate-enhancement regime at fixed temperature, under the introduction of weak, nonnative interactions. We have thoroughly tested this theoretical prediction with simulations of a coarse-grained protein model, by using an off-lattice C(alpha)model of the src-SH3 domain. The strong agreement between results from simulations and theory confirm the nontrivial result that a relatively small amount of nonnative interaction energy can actually assist the folding to the native structure.     

Stretching the immunoglobulin 27 domain of the titin protein: the dynamic energy landscape

Molecular simulations are carried out on the Immunoglobulin 27 domain of the titin protein. The energy landscape is mapped out using an implicit solvent model, and molecular dynamics simulations are run with the solvent explicitly modeled. Stretching a protein is shown to produce a dynamic energy landscape in which the energy minima move in configuration space, change in depth, and are created and destroyed. The connections of these landscape changes to the mechanical unfolding of the Immunoglobulin 27 domain are addressed. Hydrogen bonds break upon stretching by either intrabasin processes associated with the movement of energy minima, or interbasin processes associated with transitions between energy minima. Intrabasin changes are reversible and dominate for flexible interactions, whereas interbasin changes are irreversible and dominate for stiff interactions. The most flexible interactions are Glu-Lys salt bridges, which can act like tethers to bind strands even after all backbone interactions between the strands have been broken. As the protein is stretched, different types of structures become the lowest energy structures, including structures that incorporate nonnative hydrogen bonds. Structures that have flat energy versus elongation profiles become the lowest energy structures at elongations of several Angstroms, and are associated with the unfolding intermediate state observed experimentally.     

Characterizing protein energy landscape by self-learning multiscale simulations: application to a designed β-hairpin

Characterizing the energy landscape of proteins at atomic resolution is still a very challenging problem, since it simultaneously requires high accuracy in estimating specific interactions and high efficiency in conformational sampling. Here, for these two requirements to meet, we extended the self-learning multiscale simulation (SLMS) method developed recently and applied it to the designed β-hairpin CLN025. The SLMS integrates all-atom and coarse-grained (CG) models in an iterative way such that the conformational sampling is performed by the CG model, the AA energy is used to calibrate the energy landscape, and the CG model is improved by the calibrated energy landscape. We extended the SLMS in two aspects, use of the energy decomposition for self-learning of the CG potential and a two-bead/residue CG model. The results show that the self-learning greatly improved the CG potential, and with the derived CG potential, the β-hairpin CLN025 robustly folded to the native structure. The self-learning iteration progressively enhanced the context dependence in the CG potential and increased the energy gap between the native and the denatured states of the CG model, leading to a funnel-like energy landscape. By using the SLMS method, without prior knowledge of the native structure but with the help of the AA energy, we can obtain a tailor-made CG potential specific to the target protein. The method can be useful for de novo structure prediction as well.     

Landscape effects on gene flow for a climate‐sensitive montane species,the American pika

Climate change is arguably the greatest challenge to conservation of our time. Most vulnerability assessments rely on past and current species distributions to predict future persistence but ignore species' abilities to disperse through landscapes, which may be particularly important in fragmented habitats and crucial for long‐term persistence in changing environments. Landscape genetic approaches explore the interactions between landscape features and gene flow and can clarify how organisms move among suitable habitats, but have suffered from methodological uncertainties. We used a landscape genetic approach to determine how landscape and climate‐related features influence gene flow for American pikas (Ochotona princeps) in Crater Lake National Park. Pikas are heat intolerant and restricted to cool microclimates; thus, range contractions have been predicted as climate changes. We evaluated the correlation between landscape variables and genetic distance using partial Mantel tests in a causal modelling framework, and used spatially explicit simulations to evaluate methods of model optimization including a novel approach based on relative support and reciprocal causal modelling. We found that gene flow was primarily restricted by topographic relief, water and west‐facing aspects, suggesting that physical restrictions related to small body size and mode of locomotion, as well as exposure to relatively high temperatures, limit pika dispersal in this alpine habitat. Our model optimization successfully identified landscape features influencing resistance in the simulated data for this landscape, but underestimated the magnitude of resistance. This is the first landscape genetic study to address the fundamental question of what limits dispersal and gene flow in the American pika.     

Solvent-induced free energy landscape and solute-solvent dynamic coupling in a multielement solute

Molecular dynamics simulations using a simple multielement model solute with internal degrees of freedom and accounting for solvent-induced interactions to all orders in explicit water are reported. The potential energy landscape of the solute is flat in vacuo. However, the sole untruncated solvent-induced interactions between apolar (hydrophobic) and charged elements generate a rich landscape of potential of mean force exhibiting typical features of protein landscapes. Despite the simplicity of our solute, the depth of minima in this landscape is not far in size from free energies that stabilize protein conformations. Dynamical coupling between configurational switching of the system and hydration reconfiguration is also elicited. Switching is seen to occur on a time scale two orders of magnitude longer than that of the reconfiguration time of the solute taken alone, or that of the unperturbed solvent. Qualitatively, these results are unaffected by a different choice of the water-water interaction potential. They show that already at an elementary level, solvent-induced interactions alone, when fully accounted for, can be responsible for configurational and dynamical features essential to protein folding and function.     

Salt-Dependent Folding Energy Landscape of RNA Three-Way Junction

RNAs are highly negatively charged chain molecules. Metal ions play a crucial role in RNA folding stability and conformational changes. In this work, we employ the recently developed tightly bound ion (TBI) model, which accounts for the correlation between ions and the fluctuation of ion distributions, to investigate the ion-dependent free energy landscape for the three-way RNA junction in a 16S rRNA domain. The predicted electrostatic free energy landscape suggests that 1), ion-mediated electrostatic interactions cause an ensemble of unfolded conformations narrowly populated around the maximally extended structure; and 2), Mg²⁺ ion-induced correlation effects help bring the helices to the folded state. Nonelectrostatic interactions, such as noncanonical interactions within the junctions and between junctions and helix stems, might further limit the conformational diversity of the unfolded state, resulting in a more ordered unfolded state than the one predicted from the electrostatic effect. Moreover, the folded state is predominantly stabilized by the coaxial stacking force. The TBI-predicted folding stability agrees well with the experimental measurements for the different Na⁺ and Mg²⁺ ion concentrations. For Mg²⁺ solutions, the TBI model, which accounts for the Mg²⁺ ion correlation effect, gives more improved predictions than the Poisson-Boltzmann theory, which tends to underestimate the role of Mg²⁺ in stabilizing the folded structure. Detailed control tests indicate that the dominant ion correlation effect comes from the charge-charge Coulombic correlation rather than the size (excluded volume) correlation between the ions. Furthermore, the model gives quantitative predictions for the ion size effect in the folding energy landscape and folding cooperativity.     

Is landscape fragmentation always detrimental for species conservation? The case of the Iberian lynx in central Spain

The patch-corridor-matrix is the most commonly used model when dealing with landscape characterization studies, but it shows relevant limitations to detect landscape heterogeneity. Other authors have used a functional approach, since it is well known that nutrient, mineral and energy flows exist among ecosystems. These flows can be perceived in boundaries between different landcovers, making possible the identification of spatial units sharing a common pattern of ecological interactions known as mosaics. While the influence of each mosaic over a certain species has been previously addressed, no attention has been given to the intra-mosaic variation. The aim of this research is to assess the influence of functional diversity and connectivity, on the habitat suitability of the Iberian lynx (Lynx pardinus). For this, we built two GLMs to test if these features show a differential effect on lynxes’ habitat suitability depending on the mosaic. Both GLMs built show that the influence of these landscape features on lynxes’ habitat suitability depends on the landscape spatial organization and landcover composition, suggesting that there is no unique response of a species to changes in landscape diversity and/or connectivity. Thus, we conclude that considering both landscape and species features would allow to a better integration of land management strategies and conservation actions, which could favor species adaptation to highly human-modified landscapes.     

Interaction cutoff effect on ruggedness of protein-protein energy landscape

The concept of the energy landscape is important for better understanding of protein-protein interactions and for designing adequate docking procedures. The intermolecular landscape has a rugged terrain that impedes search procedures. Its inherent ruggedness is related to the conformational characteristics of the molecules and to the form of the potential function--more rugged for short-range potentials and less rugged for "soft," typically long-range potentials. Our study determined that the landscape ruggedness is further substantially exacerbated by truncation of the potentials. This additional ruggedness appears below certain critical interaction ranges that depend on the form of the potential. The theoretical model describing the cutoff effect on the landscape ruggedness is confirmed by the energy calculation on a dataset of protein-protein complexes. The negative effect of the potentials cutoff is well known. However, revealing its physical basis in terms of the energy landscape is important for better understanding of intermolecular interactions.     

Collective properties of hydration: long range and specificity of hydrophobic interactions.

We report results of molecular dynamics (MD) simulations of composite model solutes in explicit molecular water solvent, eliciting novel aspects of the recently demonstrated, strong many-body character of hydration. Our solutes consist of identical apolar (hydrophobic) elements in fixed configurations. Results show that the many-body character of PMF is sufficiently strong to cause 1) a remarkable extension of the range of hydrophobic interactions between pairs of solute elements, up to distances large enough to rule out pairwise interactions of any type, and 2) a SIF that drives one of the hydrophobic solute elements toward the solvent rather than away from it. These findings complement recent data concerning SIFs on a protein at single-residue resolution and on model systems. They illustrate new important consequences of the collective character of hydration and of PMF and reveal new aspects of hydrophobic interactions and, in general, of SIFs. Their relevance to protein recognition, conformation, function, and folding and to the observed slight yet significant nonadditivity of functional effects of distant point mutations in proteins is discussed. These results point out the functional role of the configurational and dynamical states (and related statistical weights) corresponding to the complex configurational energy landscape of the two interacting systems: biomolecule + water.     

A small tripeptide AFA undergoes two state cooperative conformational transitions: implications for conformational biases in unfolded states

It is important to understand the conformational biases that are present in unfolded states to understand protein folding. In this context, it is surprising that even a short tripeptide like AFA samples folded/ordered conformation as demonstrated recently by NMR experiments of the peptide in aqueous solution at 280 K. In this paper, we present molecular dynamics simulation of the peptide in explicit water using OPLS-AA/L all-atom force field. The results are in overall agreement with NMR results and provide some further insights. The peptide samples turn and extended conformational forms corresponding to minima in free energy landscape. Frequent transitions between the minima are observed due to modest free energy barriers. The turn conformation seems to be stabilized by hydrophobic interactions and possibly by bridging water molecules between backbone donors and acceptors. Thus the peptide does not sample conformations randomly, but samples well defined conformations. The peptide served as a model for folding-unfolding equilibrium in the context of peptide folding. Further, implications for drug design are also discussed.     

Theoretical probes of conformational fluctuations in S-peptide and RNase A/3′–UMP enzyme product complex

The dynamic properties of the RNase A/3′–UMP enzyme/product complex and the S-peptide of RNase A have been investigated by molecular dynamics simulations using suitable generalization of ideas introduced to probe the energy landscape in structural glasses. We introduce two measures, namely, the kinetic energy fluctuation metric and the force metric, both of which are used to calculate the time needed for sampling the conformation space of the molecules. The calculation of the fluctuation metric requires a single trajectory whereas the force metric is computed using two independent trajectories. The vacuum MD simulations show that for both systems the time required for kinetic energy equipartitioning is surprisingly long even at high temperatures. We show that the force metric is a powerful means of probing the nature and relative importance of conformational substates which determine the dynamics at low temperatures. In particular the time dependence of the non-bonded force metric is used to demonstrate that at low temperatures the system is predominantly localized hi a single cluster of conformational substates. The force metric is used to show that relaxation of long range (in sequence space) interactions must be mediated by a sequence of local dihedral angle transitions. We also argue that the time needed for compact structure formation is intimately related to the time needed for the relaxation of the dihedral angle degrees of freedom. The tame for non-bonded interactions, which drive protein molecules to fold under appropriate conditions, to relax becomes extremely long as the temperature is lowered suggesting that the formation of maximally compact structure hi proteins must be a very slow process. © 1993 Wiley-Liss, Inc.     

Protein-protein recognition: exploring the energy funnels near the binding sites

We present a rapidly executable minimal binding energy model for molecular docking and use it to explore the energy landscape in the vicinity of the binding sites of four different enzyme inhibitor complexes. The structures of the complexes are calculated starting with the crystal structures of the free monomers, using DOCK 4.0 to generate a large number of potential configurations, and screening with the binding energy target function. In order to investigate possible correlations between energy and variation from the native structure, we introduce a new measure of similarity, which removes many of the difficulties associated with root mean square deviation. The analysis uncovers energy gradients, or funnels, near the binding site, with decreasing energy as the degree of similarity between the native and docked structures increases. Such energy funnels can increase the number of random collisions that may evolve into productive stable complex, and indicate that short-range interactions in the precomplexes can contribute to the association rate. The finding could provide an explanation for the relatively rapid association rates that are observed even in the absence of long-range electrostatic steering.     

Exploiting Human Resource Requirements to Infer Human Movement Patterns for Use in Modelling Disease Transmission Systems: An Example from Eastern Province,Zambia

In this research, an agent-based model (ABM) was developed to generate human movement routes between homes and water resources in a rural setting, given commonly available geospatial datasets on population distribution, land cover and landscape resources. ABMs are an object-oriented computational approach to modelling a system, focusing on the interactions of autonomous agents, and aiming to assess the impact of these agents and their interactions on the system as a whole. An A* pathfinding algorithm was implemented to produce walking routes, given data on the terrain in the area. A* is an extension of Dijkstra’s algorithm with an enhanced time performance through the use of heuristics. In this example, it was possible to impute daily activity movement patterns to the water resource for all villages in a 75 km long study transect across the Luangwa Valley, Zambia, and the simulated human movements were statistically similar to empirical observations on travel times to the water resource (Chi-squared, 95% confidence interval). This indicates that it is possible to produce realistic data regarding human movements without costly measurement as is commonly achieved, for example, through GPS, or retrospective or real-time diaries. The approach is transferable between different geographical locations, and the product can be useful in providing an insight into human movement patterns, and therefore has use in many human exposure-related applications, specifically epidemiological research in rural areas, where spatial heterogeneity in the disease landscape, and space-time proximity of individuals, can play a crucial role in disease spread.     

Fitting protein-folding free energy landscape for a certain conformation to an NK fitness landscape

The NK fitness landscape is a mathematical landscape model with a parameter k that governs the degree of ruggedness of the landscape. We presented a procedure to fit a given landscape to the NK fitness landscape by introducing the “apparent k-value” k_app. In this paper, we defined the protein free energy (ΔG) landscape in amino acid sequence space, where ΔG is the folding free energy from a random coil to a “certain conformation”. Applying this landscape to our fitting procedure, we examined the statistical properties of the landscape. For calculation of a conformation energy, amino acid residues are represented by points, and interaction energies among amino acid residues are given as (1+K)-body interactions, that is, an unit of interacting (1+K) residues cooperatively contribute a single energy value to the conformational energy. Our results suggest that the apparent k-value of the free energy landscape is k_app≈K, and that the number of possible interactions among residues is unrelated to the k_app value. This leads to the inference that k_app takes values about 1-3 in real landscapes, if nature adopts two-body ∼four-body interaction energies.     

Folding a protein with equal probability of being helix or hairpin

We explore the possibility for the native structure of a protein being inherently multiconformational in an ab initio coarse-grained model. Based on the Wang-Landau algorithm, the complete free energy landscape for the designed sequence 2DX4: INYWLAHAKAGYIVHWTA is constructed. It is shown that 2DX4 possesses two nearly degenerate native structures: one is a helix structure with the other a hairpinlike structure, and their free energy difference is <2% of that of local minima. Two degenerate native structures are stabilized by an energy barrier of ～10 kcal/mol. Furthermore, the hydrogen-bond and dipole-dipole interactions are found to be two major competing interactions in transforming one conformation into the other. Our results indicate that two degenerate native structures are stabilized by subtle balance between different interactions in proteins. In particular, for small proteins, balance between the hydrogen-bond and dipole-dipole interactions happens for proteins of sizes being ～18 amino acids and is shown to the main driving mechanism for the occurrence of degeneracy. These results provide important clues to the study of native structures of proteins.