Conformational ensembles and sampled energy landscapes: Analysis and comparison

We present novel algorithms and software addressing four core problems in computational structural biology, namely analyzing a conformational ensemble, comparing two conformational ensembles, analyzing a sampled energy landscape, and comparing two sampled energy landscapes. Using recent developments in computational topology, graph theory, and combinatorial optimization, we make two notable contributions. First, we present a generic algorithm analyzing height fields. We then use this algorithm to perform density‐based clustering of conformations, and to analyze a sampled energy landscape in terms of basins and transitions between them. In both cases, topological persistence is used to manage (geometric) frustration. Second, we introduce two algorithms to compare transition graphs. The first is the classical earth mover distance metric which depends only on local minimum energy configurations along with their statistical weights, while the second incorporates topological constraints inherent to conformational transitions. Illustrations are provided on a simplified protein model (BLN69), whose frustrated potential energy landscape has been thoroughly studied. The software implementing our tools is also made available, and should prove valuable wherever conformational ensembles and energy landscapes are used. © 2015 Wiley Periodicals, Inc.     

Detection-dependent kinetics as a probe of folding landscape microstructure

The folding landscapes of polypeptides and proteins exhibit a hierarchy of local minima. The causes range from proline isomerization all the way down to microstructure in the free energy caused by residual frustration inherent in even the best 20 amino acid design. The corresponding time scales range from hours to submicroseconds. The smallest microstructures are difficult to detect. We have measured the folding/unfolding kinetics of the engineered trpzip2 peptide at different tryptophan fluorescence wavelengths, each yielding a different rate. Wavelength-dependent folding kinetics on 0.1-2 mus time scales show that different microstructures with a range of solvent exposure and local dynamics are populated. We estimate a lower limit for the roughness of the free energy surface based on the range of rates observed.     

From Valleys to Ridges: Exploring the Dynamic Energy Landscape of Single Membrane Proteins

Membrane proteins are involved in essential biological processes such as energy conversion, signal transduction, solute transport and secretion. All biological processes, also those involving membrane proteins, are steered by molecular interactions. Molecular interactions guide the folding and stability of membrane proteins, determine their assembly, switch their functional states or mediate signal transduction. The sequential steps of molecular interactions driving these processes can be described by dynamic energy landscapes. The conceptual energy landscape allows to follow the complex reaction pathways of membrane proteins while its modifications describe why and how pathways are changed. Single‐molecule force spectroscopy (SMFS) detects, quantifies and locates interactions within and between membrane proteins. SMFS helps to determine how these interactions change with temperature, point mutations, oligomerization and the functional states of membrane proteins. Applied in different modes, SMFS explores the co‐existence and population of reaction pathways in the energy landscape of the protein and thus reveals detailed insights into local mechanisms, determining its structural and functional relationships. Here we review how SMFS extracts the defining parameters of an energy landscape such as the barrier position, reaction kinetics and roughness with high precision.     

Amyloid Polymorphism in the Protein Folding and Aggregation Energy Landscape

Protein folding involves a large number of steps and conformations in which the folding protein samples different thermodynamic states characterized by local minima. Kinetically trapped on‐ or off‐pathway intermediates are metastable folding intermediates towards the lowest absolute energy minima, which have been postulated to be the natively folded state where intramolecular interactions dominate, and the amyloid state where intermolecular interactions dominate. However, this view largely neglects the rich polymorphism found within amyloid species. We review the protein folding energy landscape in view of recent findings identifying specific transition routes among different amyloid polymorphs. Observed transitions such as twisted ribbon→crystal or helical ribbon→nanotube, and forbidden transitions such helical ribbon?crystal, are discussed and positioned within the protein folding and aggregation energy landscape. Finally, amyloid crystals are identified as the ground state of the protein folding and aggregation energy landscape.     

Geometrical features of the protein folding mechanism are a robust property of the energy landscape: a detailed investigation of several reduced models

The concept of a funneled energy landscape and the principle of minimal frustration are the theoretical foundation justifying the applicability of structure-based models. In simulations, a protein is commonly reduced to a C(alpha)-bead representation. These simulations are sufficient to predict the geometrical features of the folding mechanism observed experimentally utilizing a concise formulation of the Hamiltonian with low computational costs. Toward a better understanding of the interplay between energetic and geometrical features in folding, the side chain is now explicitly included in the simulations. The simplest choice is the addition of C(beta)-beads at the center-of-mass position of the side chains. While one varies the energetic parameters of the model, the geometric aspects of the folding mechanism remain robust for a broad range of parameters. Energetic properties like folding barriers and protein stability are sensitive to the details of simulations. This robustness to geometry and sensitivity to energetic properties provide flexibility in choosing different parameters to represent changes in sequences, environments, stability or folding rate effects. Therefore, minimal frustration and the funnel concept guarantee that the geometrical features are robust properties of the folding landscape, while mutations and/or changes in the environment easily influence energy-dependent properties like folding rates or stability.     

Multidomain protein solves the folding problem by multifunnel combined landscape: theoretical investigation of a y-family DNA polymerase

Approximately three-fourths of eukaryotic proteins are composed of multiple independently folded domains. However, much of our understanding is based on single domain proteins or isolated domains whose studies directly lead to well-known energy landscape theory in which proteins fold by navigating through a funneled energy landscape toward native structure ensembles. The degrees of freedom for proteins with multiple domains are many orders of magnitude larger than that for single domain proteins. Now, the question arises: How do the multidomain proteins solve the "protein folding problem"? Here, we specifically address this issue by exploring the structure folding relationship of Sulfolobus solfataricus DNA polymerase IV (DPO4), a prototype Y-family DNA polymerase which contains a polymerase core consisting of a palm (P domain), a finger (F domain), and a thumb domain (T domain) in addition to a little finger domain (LF domain). The theoretical results are in good agreement with the experimental data and lead to several theoretical predictions. Finally, we propose that for rapid folding into well-defined conformations which carry out the biological functions, four-domain DPO4 employs a divide-and-conquer strategy, that is, combining multiple individual folding funnels into a single funnel (domains fold independently and then coalesce). In this way, the degrees of freedom for multidomain proteins are polynomial rather than exponential, and the conformational search process can be reduced effectively from a large to a smaller time scale.     

Frustration and hydrophobicity interplay in protein folding and protein evolution

A lattice model is used to study mutations and compacting effects on protein folding rates and folding temperature. In the context of protein evolution, we address the question regarding the best scenario for a polypeptide chain to fold: either a fast nonspecific collapse followed by a slow rearrangement to form the native structure or a specific collapse from the unfolded state with the simultaneous formation of the native state. This question is investigated for optimized sequences, whose native state has no frustrated contacts between monomers, and also for mutated sequences, whose native state has some degree of frustration. It is found that the best scenario for folding may depend on the amount of frustration of the native structure. The implication of this result on protein evolution is discussed.     

Crystallographic B-factors highlight energetic frustration in aldolase folding

Kinetic simulations of the folding and unfolding of the mammalian TIM barrel protein aldolase were conducted to determine if a minimalist monomeric Gō model, using the native structure to determine attractive energies in the protein chain, could capture the experimentally determined folding pathway. The folding order, that is, the order in which different secondary structures fold, between the Gō model simulations and that from hydrogen-deuterium exchange experiments, did not agree. To explain this discrepancy, two alternate variant of the basic Gō model were simulated: (1) a monomer Gō model with native contact energies weighted by a statistical potential (SP model) and (2) a monomer Gō model with native contact energies inversely weighted by crystallographic B factors (B model). The B model demonstrated the best agreement between simulation and experiments. The success of the B model is attributed to the ability of B factors to highlight local energetic frustration in the aldolase structure which results in weaker native contacts in these frustrated regions. The predictive success of the B model also reveals the potential use of B factor information for energetic weighting in general protein modeling.     

Energy Landscape Exploration of Sub‐Nanometre Copper–Silver Clusters

The energy landscapes of sub‐nanometre bimetallic coinage metal clusters are explored with the Threshold Algorithm coupled with the Birmingham Cluster Genetic Algorithm. Global and energetically low‐lying minima along with their permutational isomers are located for the Cu${_4 }$ Ag${_4 }$ cluster with the Gupta potential and density functional theory (DFT). Statistical tools are employed to map the connectivity of the energy landscape and the growth of structural basins, while the thermodynamics of interconversion are probed, based on probability flows between minima. Asymmetric statistical weights are found for pathways across dividing states between stable geometries, while basin volumes are observed to grow independently of the depth of the minimum. The DFT landscape is found to exhibit significantly more frustration than that of the Gupta potential, including several open, pseudo‐planar geometries which are energetically competitive with the global minimum. The differences in local minima and their transition barriers between the two levels of theory indicate the importance of explicit electronic structure for even simple, closed shell clusters.     

Inherent structure analysis of protein folding

An analysis in terms of the inherent structures (IS, local minima) of the multidimensional potential energy landscape is applied to proteins. Detailed calculations are performed for the 46 bead BLN model, which folds into a four-stranded beta-barrel. Enhanced sampling has allowed determination of 239 199 IS states, believed to encompass nearly all the compact, low-energy states, and of well-averaged thermodynamic quantities at low temperature. The density of states shows distinct lobes for compact and extended states, and entropic barriers for the collapse and local ordering transitions. A two-dimensional scatterplot or density of states clearly shows the multifunnel structure of the energy landscape. The anharmonic vibrational free energy is found to play a crucial role in protein folding. The problem of determining the folding transition in a multifunnel system is discussed, and novel indicators of folding are introduced. A particularly clear picture is obtained through the occupation probabilities, pi, of individual low-lying IS, which become finite below the collapse temperature; it is suggested that poor foldability corresponds to a large "misfolding interval" where the excited state pi>0 exceeds that of the native state p0.     

Folding behavior of model proteins with weak energetic frustration

The native structure of fast-folding proteins, albeit a deep local free-energy minimum, may involve a relatively small energetic penalty due to nonoptimal, though favorable, contacts between amino acid residues. The weak energetic frustration that such contacts represent varies among different proteins and may account for folding behavior not seen in unfrustrated models. Minimalist model proteins with heterogeneous contacts--as represented by lattice heteropolymers consisting of three types of monomers--also give rise to weak energetic frustration in their corresponding native structures, and the present study of their equilibrium and nonequilibrium properties reveals some of the breadth in their behavior. In order to capture this range within a detailed study of only a few proteins, four candidate protein structures (with their cognate sequences) have been selected according to a figure of merit called the winding index--a characteristic of the number of turns the protein winds about an axis. The temperature-dependent heat capacities reveal a high-temperature collapse transition, and an infrequently observed low-temperature rearrangement transition that arises because of the presence of weak energetic frustration. Simulation results motivate the definition of a new measure of folding affinity as a sequence-dependent free energy--a function of both a reduced stability gap and high accessibility to non-native structures--that correlates strongly with folding rates.     

Rerouting the folding pathway of the Notch ankyrin domain by reshaping the energy landscape

The modular nature of repeat proteins has made them a successful target for protein design. Ankyrin repeat, TPR, and leucine rich repeat domains that have been designed solely on consensus information have been shown to have higher thermostability than their biological counterparts. We have previously shown that we can reshape the energy landscape of a repeat protein by adding multiple C-terminal consensus ankyrin repeats to the five N-terminal repeats of the Notch ankyrin domain. Here we explore how the folding mechanism responds to reshaping of the energy landscape. We have used analogous substitutions of a conserved alanine with glycine in each repeat to determine the distribution of structure in the transition state ensembles of constructs containing one (Nank1-5C1) and two consensus (Nank1-5C2) ankyrin repeats. Whereas folding of the wild-type Notch ankyrin domain is slowed by substitutions in its central repeats, (1) folding of Nank1-5C1 and Nank1-5C2 is slowed by substitutions in the C-terminal repeats. Thus, the addition of C-terminal stabilizing repeats shifts the transition state ensemble toward the C-terminal repeats, rerouting the folding pathway of the ankyrin repeat domain. These findings indicate that, for the Notch ankyrin domain, folding pathways are selected based on local energetics.     

Statistical theory for protein ensembles with designed energy landscapes

Combinatorial protein libraries provide a promising route to investigate the determinants and features of protein folding and to identify novel folding amino acid sequences. A library of sequences based on a pool of different monomer types are screened for folding molecules, consistent with a particular foldability criterion. The number of sequences grows exponentially with the length of the polymer, making both experimental and computational tabulations of sequences infeasible. Herein a statistical theory is extended to specify the properties of sequences having particular values of global energetic quantities that specify their energy landscape. The theory yields the site-specific monomer probabilities. A foldability criterion is derived that characterizes the properties of sequences by quantifying the energetic separation of the target state from low-energy states in the unfolded ensemble and the fluctuations of the energies in the unfolded state ensemble. For a simple lattice model of proteins, excellent agreement is observed between the theory and the results of exact enumeration. The theory may be used to provide a quantitative framework for the design and interpretation of combinatorial experiments.     

Accelerated molecular dynamics simulations of protein folding

Folding of four fast‐folding proteins, including chignolin, Trp‐cage, villin headpiece and WW domain, was simulated via accelerated molecular dynamics (aMD). In comparison with hundred‐of‐microsecond timescale conventional molecular dynamics (cMD) simulations performed on the Anton supercomputer, aMD captured complete folding of the four proteins in significantly shorter simulation time. The folded protein conformations were found within 0.2–2.1 Å of the native NMR or X‐ray crystal structures. Free energy profiles calculated through improved reweighting of the aMD simulations using cumulant expansion to the second‐order are in good agreement with those obtained from cMD simulations. This allows us to identify distinct conformational states (e.g., unfolded and intermediate) other than the native structure and the protein folding energy barriers. Detailed analysis of protein secondary structures and local key residue interactions provided important insights into the protein folding pathways. Furthermore, the selections of force fields and aMD simulation parameters are discussed in detail. Our work shows usefulness and accuracy of aMD in studying protein folding, providing basic references in using aMD in future protein‐folding studies. © 2015 Wiley Periodicals, Inc.     

Site‐Resolved Observation of Vibrational Energy Transfer Using a Genetically Encoded Ultrafast Heater

Allosteric information transfer in proteins has been linked to distinct vibrational energy transfer (VET) pathways in a number of theoretical studies. Experimental evidence for such pathways, however, is sparse because site‐selective injection of vibrational energy into a protein, that is, localized heating, is required for their investigation. Here, we solved this problem by the site‐specific incorporation of the non‐canonical amino acid β‐(1‐azulenyl)‐l ‐alanine (AzAla) through genetic code expansion. As an exception to Kasha's rule, AzAla undergoes ultrafast internal conversion and heating after S₁ excitation while upon S₂ excitation, it serves as a fluorescent label. We equipped PDZ3, a protein interaction domain of postsynaptic density protein 95, with this ultrafast heater at two distinct positions. We indeed observed VET from the incorporation sites in the protein to a bound peptide ligand on the picosecond timescale by ultrafast IR spectroscopy. This approach based on genetically encoded AzAla paves the way for detailed studies of VET and its role in a wide range of proteins.     

Forced Folding of a Disordered Protein Accesses an Alternative Folding Landscape

Intrinsically disordered proteins (IDPs) are involved in diverse cellular functions. Many IDPs can interact with multiple binding partners, resulting in their folding into alternative ligand‐specific functional structures. For such multi‐structural IDPs, a key question is whether these multiple structures are fully encoded in the protein sequence, as is the case in many globular proteins. To answer this question, here we employed a combination of single‐molecule and ensemble techniques to compare ligand‐induced and osmolyte‐forced folding of α‐synuclein. Our results reveal context‐dependent modulation of the protein′s folding landscape, suggesting that the codes for the protein′s native folds are partially encoded in its primary sequence, and are completed only upon interaction with binding partners. Our findings suggest a critical role for cellular interactions in expanding the repertoire of folds and functions available to disordered proteins.