Interpreting NMR data for beta-peptides using molecular dynamics simulations

NMR is one of the most used techniques to resolve structure of proteins and peptides in solution. However, inconsistencies may occur due to the fact that a polypeptide may adopt more than one conformation. Since the NOE distance bounds and (3)J-values used in such structure determination represent a nonlinear average over the total ensemble of conformers, imposition of NOE or (3)J-value restraints to obtain one unique conformation is not an appropriate procedure in such cases. Here, we show that unrestrained MD simulation of a solute in solution using a high-quality force field yields a conformational ensemble that is largely compatible with the experimental NMR data on the solute. Four 100 ns MD simulations of two forms of a nine-residue beta-peptide in methanol at two temperatures produced conformational ensembles that were used to interpret the NMR data on this molecule and resolve inconsistencies between the experimental NOEs. The protected and unprotected forms of the beta-peptide adopt predominantly a 12/10-helix in agreement with the qualitative interpretation of the NMR data. However, a particular NOE was not compatible with this helix indicating the presence of other conformations. The simulations showed that 3(14)()-helical structures were present in the ensemble of the unprotected form and that their presence correlates with the fulfillment of the particular NOE. Additionally, all inter-hydrogen distances were calculated to compare NOEs predicted by the simulations to the ones observed experimentally. The MD conformational ensembles allowed for a detailed and consistent interpretation of the experimental data and showed the small but specific conformational differences between the protected and unprotected forms of the peptide.     

Novel inhibitor discovery through virtual screening against multiple protein conformations generated via ligand-directed modeling: a maternal embryonic leucine zipper kinase example

Kinase targets have been demonstrated to undergo major conformational reorganization upon ligand binding. Such protein conformational plasticity remains a significant challenge in structure-based virtual screening methodology and may be approximated by screening against an ensemble of diverse protein conformations. Maternal embryonic leucine zipper kinase (MELK), a member of serine-threonine kinase family, has been recently found to be involved in the tumerogenic state of glioblastoma, breast, ovarian, and colon cancers. We therefore modeled several conformers of MELK utilizing the available chemogenomic and crystallographic data of homologous kinases. We carried out docking pose prediction and virtual screening enrichment studies with these conformers. The performances of the ensembles were evaluated by their ability to reproduce known inhibitor bioactive conformations and to efficiently recover known active compounds early in the virtual screen when seeded with decoy sets. A few of the individual MELK conformers performed satisfactorily in reproducing the native protein-ligand pharmacophoric interactions up to 50% of the cases. By selecting an ensemble of a few representative conformational states, most of the known inhibitor binding poses could be rationalized. For example, a four conformer ensemble is able to recover 95% of the studied actives, especially with imperfect scoring function(s). The virtual screening enrichment varied considerably among different MELK conformers. Enrichment appears to improve by selection of a proper protein conformation. For example, several holo and unliganded active conformations are better to accommodate diverse chemotypes than ATP-bound conformer. These results prove that using an ensemble of diverse conformations could give a better performance. Applying this approach, we were able to screen a commercially available library of half a million compounds against three conformers to discover three novel inhibitors of MELK, one from each template. Among the three compounds validated via experimental enzyme inhibition assays, one is relatively potent (15; K(d) = 0.37 μM), one moderately active (12; K(d) = 3.2 μM), and one weak but very selective (9; K(d) = 18 μM). These novel hits may be utilized to assist in the development of small molecule therapeutic agents useful in diseases caused by deregulated MELK, and perhaps more importantly, the approach demonstrates the advantages of choosing an appropriate ensemble of a few conformers in pursuing compound potency, selectivity, and novel chemotypes over using single target conformation for structure-based drug design in general.     

Conformational Ensemble of Disordered Proteins Probed by Solvent Paramagnetic Relaxation Enhancement (sPRE)

Characterization of the conformational ensemble of disordered proteins is highly important for understanding protein folding and aggregation mechanisms, but remains a computational and experimental challenge owing to the dynamic nature of these proteins. New observables that can provide unique insights into transient residual structures in disordered proteins are needed. Here using denatured ubiquitin as a model system, NMR solvent paramagnetic relaxation enhancement (sPRE) measurements provide an accurate and highly sensitive probe for detecting low populations of residual structure in a disordered protein. Furthermore, a new ensemble calculation approach based on sPRE restraints in conjunction with residual dipolar couplings (RDCs) and small‐angle X‐ray scattering (SAXS) is used to define the conformational ensemble of disordered proteins at atomic resolution. The approach presented should be applicable to a wide range of dynamic macromolecules.     

Molecular properties from conformational ensembles. 1. Dipole moments of molecules with multiple internal rotations

We present a novel method for constructing the stable conformational space of small molecules with many rotatable bonds that uses our iterative stochastic elimination (ISE) algorithm, a robust stochastic search method capable of finding ensembles of best solutions for large combinatorial problems. To validate the method, we show that ISE reproduces the best conformers found in a fully exhaustive search, as well as compare computed dipole moments to experimental values, based on molecular ensembles and their Boltzmann distributions. Results were also compared to the alternative molecular dynamics and simulated annealing methods. Our results clarify that many low energy conformations may be required to reproduce molecular properties, while single low energy conformers or ensembles of low energy conformers cannot account for the experimental properties of flexible molecules. Whereas ISE well reproduces conformations that are not separated by very large energy barriers, it has not been successful in reproducing conformations of strained molecules.     

Protein structure along the order-disorder continuum

Thermal fluctuations cause proteins to adopt an ensemble of conformations wherein the relative stability of the different ensemble members is determined by the topography of the underlying energy landscape. "Folded" proteins have relatively homogeneous ensembles, while "unfolded" proteins have heterogeneous ensembles. Hence, the labels "folded" and "unfolded" represent attempts to provide a qualitative characterization of the extent of structural heterogeneity within the underlying ensemble. In this work, we introduce an information-theoretic order parameter to quantify this conformational heterogeneity. We demonstrate that this order parameter can be estimated in a straightforward manner from an ensemble and is applicable to both unfolded and folded proteins. In addition, a simple formula for approximating the order parameter directly from crystallographic B factors is presented. By applying these metrics to a large sample of proteins, we show that proteins span the full range of the order-disorder axis.     

Searching and optimizing structure ensembles for complex flexible sugars

NMR restrictions are suitable to specify the geometry of a molecule when a single well-defined global free energy minimum exists that is significantly lower than other local minima. Carbohydrates are quite flexible, and therefore, NMR observables do not always correlate with a single conformer but instead with an ensemble of low free energy conformers that can be accessed by thermal fluctuations. In this communication, we describe a novel procedure to identify and weight the contribution to the ensemble of local minima conformers based on comparison to residual dipolar couplings (RDCs) or other NMR observables, such as scalar couplings. A genetic algorithm is implemented to globally minimize the R factor comparing calculated RDCs to experiment. This is done by optimizing the weights of different conformers derived from the exhaustive local minima conformational search program, fast sugar structure prediction software (FSPS). We apply this framework to six human milk sugars, LND-1, LNF-1, LNF-2, LNF-3, LNnT, and LNT, and are able to determine corresponding population weights for the ensemble of conformers. Interestingly, our results indicate that in all cases the RDCs can be well represented by only a few most important conformers. This confirms that several, but not all of the glycosidic linkages in histo-blood group "epitopes" are quite rigid.     

Quantifying polypeptide conformational space: sensitivity to conformation and ensemble definition

Quantifying the density of conformations over phase space (the conformational distribution) is needed to model important macromolecular processes such as protein folding. In this work, we quantify the conformational distribution for a simple polypeptide (N-mer polyalanine) using the cumulative distribution function (CDF), which gives the probability that two randomly selected conformations are separated by less than a "conformational" distance and whose inverse gives conformation counts as a function of conformational radius. An important finding is that the conformation counts obtained by the CDF inverse depend critically on the assignment of a conformation's distance span and the ensemble (e.g., unfolded state model): varying ensemble and conformation definition (1 --> 2 A) varies the CDF-based conformation counts for Ala(50) from 10(11) to 10(69). In particular, relatively short molecular dynamics (MD) relaxation of Ala(50)'s random-walk ensemble reduces the number of conformers from 10(55) to 10(14) (using a 1 A root-mean-square-deviation radius conformation definition) pointing to potential disconnections in comparing the results from simplified models of unfolded proteins with those from all-atom MD simulations. Explicit waters are found to roughen the landscape considerably. Under some common conformation definitions, the results herein provide (i) an upper limit to the number of accessible conformations that compose unfolded states of proteins, (ii) the optimal clustering radius/conformation radius for counting conformations for a given energy and solvent model, (iii) a means of comparing various studies, and (iv) an assessment of the applicability of random search in protein folding.     

Simulation studies of the fidelity of biomolecular structure ensemble recreation

We examine the ability of Bayesian methods to recreate structural ensembles for partially folded molecules from averaged data. Specifically we test the ability of various algorithms to recreate different transition state ensembles for folding proteins using a multiple replica simulation algorithm using input from "gold standard" reference ensembles that were first generated with a Go-like Hamiltonian having nonpairwise additive terms. A set of low resolution data, which function as the "experimental" phi values, were first constructed from this reference ensemble. The resulting phi values were then treated as one would treat laboratory experimental data and were used as input in the replica reconstruction algorithm. The resulting ensembles of structures obtained by the replica algorithm were compared to the gold standard reference ensemble, from which those "data" were, in fact, obtained. It is found that for a unimodal transition state ensemble with a low barrier, the multiple replica algorithm does recreate the reference ensemble fairly successfully when no experimental error is assumed. The Kolmogorov-Smirnov test as well as principal component analysis show that the overlap of the recovered and reference ensembles is significantly enhanced when multiple replicas are used. Reduction of the multiple replica ensembles by clustering successfully yields subensembles with close similarity to the reference ensembles. On the other hand, for a high barrier transition state with two distinct transition state ensembles, the single replica algorithm only samples a few structures of one of the reference ensemble basins. This is due to the fact that the phi values are intrinsically ensemble averaged quantities. The replica algorithm with multiple copies does sample both reference ensemble basins. In contrast to the single replica case, the multiple replicas are constrained to reproduce the average phi values, but allow fluctuations in phi for each individual copy. These fluctuations facilitate a more faithful sampling of the reference ensemble basins. Finally, we test how robustly the reconstruction algorithm can function by introducing errors in phi comparable in magnitude to those suggested by some authors. In this circumstance we observe that the chances of ensemble recovery with the replica algorithm are poor using a single replica, but are improved when multiple copies are used. A multimodal transition state ensemble, however, turns out to be more sensitive to large errors in phi (if appropriately gauged) and attempts at successful recreation of the reference ensemble with simple replica algorithms can fall short.     

Generation of multiple pharmacophore hypotheses using multiobjective optimisation techniques

Summary Pharmacophore methods provide a way of establishing a structure--activity relationship for a series of known active ligands. Often, there are several plausible hypotheses that could explain the same set of ligands and, in such cases, it is important that the chemist is presented with alternatives that can be tested with different synthetic compounds. Existing pharmacophore methods involve either generating an ensemble of conformers and considering each conformer of each ligand in turn or exploring conformational space on-the-fly. The ensemble methods tend to produce a large number of hypotheses and require considerable effort to analyse the results, whereas methods that vary conformation on-the-fly typically generate a single solution that represents one possible hypothesis, even though several might exist. We describe a new method for generating multiple pharmacophore hypotheses with full conformational flexibility being explored on-the-fly. The method is based on multiobjective evolutionary algorithm techniques and is designed to search for an ensemble of diverse yet plausible overlays which can then be presented to the chemist for further investigation.     

Role of backbone-solvent interactions in determining conformational equilibria of intrinsically disordered proteins

Intrinsically disordered proteins (IDPs) are functional proteins that do not fold into well-defined three-dimensional structures under physiological conditions. IDP sequences have low hydrophobicity, and hence, recent experiments have focused on quantitative studies of conformational ensembles of archetypal IDP sequences such as polyglutamine and glycine-serine block copolypeptides. Results from these experiments show that, despite the absence of hydrophobic residues, polar IDPs prefer ensembles of collapsed structures in aqueous milieus. Do these preferences originate in interactions that are unique to polar sidechains? The current study addresses this issue by analyzing conformational equilibria for polyglycine and a glycine-serine block copolypeptide in two environments, namely, water and 8 M urea. Polyglycine, a poly secondary-amide, has no sidechains and is a useful model system for generic polypeptide backbones. Results based on large-scale molecular dynamics simulations show that polyglycine forms compact, albeit disordered, globules in water and swollen, disordered coils in 8 M urea. There is minimal overlap between conformational ensembles in the two environments. Analysis of order parameters derived from theories for flexible polymers show that water at ambient temperatures is a poor solvent for generic polypeptide backbones. Therefore, the experimentally observed preferences for polyglutamine and glycine-serine block copolypeptides must originate, at least partially, in polypeptide backbones. A preliminary analysis of the driving forces that lead to distinct conformational preferences for polyglycine in two different environments is presented. Implications for describing conformational ensembles of generic IDP sequences are also discussed.     

Disorder-to-order transition of an intrinsically disordered region of sortase revealed by multiscale enhanced sampling

Molecular functions of intrinsically disordered proteins (IDPs) or intrinsically disordered regions (IDRs), such as molecular recognition and cellular signaling, are ascribed to dynamic changes in the conformational space in response to binding of target molecules. Sortase, a transpeptitase in Gram-positive bacteria, has an IDR in a loop which undergoes a disordered-to-ordered transition (called "disordered loop"), accompanying a tilt of another loop ("dynamic loop"), upon binding of a signal peptide and a calcium ion. In this study, all-atom conformational ensembles of sortase were calculated for the four different binding states (with/without the peptide and with/without a calcium ion) by the multiscale enhanced sampling (MSES) simulation to examine how the binding of the peptide and/or calcium influences the conformational ensemble. The MSES is a multiscale and multicopy simulation method that allows an enhanced sampling of the all-atom model of large proteins including explicit solvent. A 100 ns MSES simulation of the ligand-free sortase using 20 replicas (in total 2 μs) demonstrated large flexibility in both the disordered and dynamic loops; however, their distributions were not random but had a clear preference which populates the N-terminal part of the disordered loop near the bound form. The MSES simulations of the three binding states clarified the allosteric mechanism of sortase: the N- and C-terminal parts of the disordered loop undergo a disorder-to-order transition independently of each other upon binding of the peptide and a calcium ion, respectively; however, upon binding of both ligands, the two parts work cooperatively to stabilize the bound peptide.     

Replica exchange with guided annealing for accelerated sampling of disordered protein conformations

We critically examine a recently proposed convective replica exchange (cRE) method for enhanced sampling of protein conformation based on theoretical and numerical analysis. The results demonstrate that cRE and related replica exchange with guided annealing (RE‐GA) schemes lead to unbalanced exchange attempt probabilities and break detailed balance whenever the system undergoes slow conformational transitions (relative to the temperature diffusion timescale). Nonetheless, numerical simulations suggest that approximate canonical ensembles can be generated for systems with small conformational transition barriers. This suggests that RE‐GA maybe suitable for simulating intrinsically disordered proteins, an important class of newly recognized functional proteins. The efficacy of RE‐GA is demonstrated by calculating the conformational ensembles of intrinsically disordered kinase inducible domain protein. The results show that RE‐GA helps the protein to escape nonspecific compact states more efficiently and provides several fold speedups in generating converged and largely correct ensembles compared to the standard temperature RE. © 2014 Wiley Periodicals, Inc.     

Assessment of conformational ensemble sizes necessary for specific resolutions of coverage of conformational space

The size of conformational ensembles required for regular coverage of the conformational space of druglike molecules was examined. Using the conformer generation program Omega, the number of regularly distributed conformers (NRC) of flexible compounds was determined as a function of the root-mean-square deviation (RMSD) resolution of coverage. A regression equation was developed predicting the NRC of a molecule as a function of RMSD. The model yielded R(2) of 0.91 for both training and test sets, which consisted of 3414 and 3352 compounds, respectively. Utilizing 14 504 ligands from the Protein Data Bank with experimentally determined 3-D conformations, the regression equation was applied to the estimation of the NRC and the success rate of reproduction of experimental conformations from a theoretical conformation ensemble as a function of RMSD and flexibility was explored.     

Structure and dynamics of the Abeta(21-30) peptide from the interplay of NMR experiments and molecular simulations

We combine molecular dynamics simulations and new high-field NMR experiments to describe the solution structure of the Abeta(21-30) peptide fragment that may be relevant for understanding structural mechanisms related to Alzheimer's disease. By using two different empirical force-field combinations, we provide predictions of the three-bond scalar coupling constants ((3)J(H(N)H(alpha))), chemical-shift values, (13)C relaxation parameters, and rotating-frame nuclear Overhauser effect spectroscopy (ROESY) crosspeaks that can then be compared directly to the same observables measured in the corresponding NMR experiment of Abeta(21-30). We find robust prediction of the (13)C relaxation parameters and medium-range ROESY crosspeaks by using new generation TIP4P-Ew water and Amber ff99SB protein force fields, in which the NMR validates that the simulation yields both a structurally and dynamically correct ensemble over the entire Abeta(21-30) peptide. Analysis of the simulated ensemble shows that all medium-range ROE restraints are not satisfied simultaneously and demonstrates the structural diversity of the Abeta(21-30) conformations more completely than when determined from the experimental medium-range ROE restraints alone. We find that the structural ensemble of the Abeta(21-30) peptide involves a majority population (approximately 60%) of unstructured conformers, lacking any secondary structure or persistent hydrogen-bonding networks. However, the remaining minority population contains a substantial percentage of conformers with a beta-turn centered at Val24 and Gly25, as well as evidence of the Asp23 to Lys28 salt bridge important to the fibril structure. This study sets the stage for robust theoretical work on Abeta(1-40) and Abeta(1-42), for which collection of detailed NMR data on the monomer will be more challenging because of aggregation and fibril formation on experimental timescales at physiological conditions. In addition, we believe that the interplay of modern molecular simulation and high-quality NMR experiments has reached a fruitful stage for characterizing structural ensembles of disordered peptides and proteins in general.     

Relationship among ligand conformations in solution, in the solid state, and at the Hsp90 binding site: geldanamycin and radicicol

The unknown effects of a receptor's environment on a ligand's conformation presents a difficult challenge in predicting feasible bioactive conformations, particularly if the receptor is ill-defined. The primary hypothesis of this work is that a structure's conformational ensemble in solution presents viable candidates for protein binding. The experimental solution profile can be achieved with the NAMFIS (NMR analysis of molecular flexibility in solution) method, which deconvolutes the average NMR spectrum of small flexible molecules into individual contributing conformations with varying populations. Geldanamycin and radicicol are structurally different macrocycles determined by X-ray crystallography to bind to a common site on the cellular chaperone heat shock protein 90 (Hsp90). Without benefit of a receptor structure, NAMFIS has identified the bioactive conformers of geldanamycin and radicicol in CDCl3 solution with populations of 4% and 21%, respectively. Conversely, docking the set of NAMFIS conformers into the unliganded proteins with GLIDE followed by MM-GBSA scoring reproduces the experimental crystallographic binding poses.     

A Method to Explore Protein Side Chain Conformational Variability Using Experimental Data

Experimentally measured values of molecular properties or observables of biomolecules such as proteins are generally averages over time and space, which do not contain su?cient information to determine the underlying conformational distribution of the molecules in solution. The relationship between experimentally measured NMR ³J‐coupling values and the corresponding dihedral angle values is a particularly complicated case due to its nonlinear, multiple‐valued nature. Molecular dynamics (MD) simulations at constant temperature can generate Boltzmann ensembles of molecular structures that are free from a priori assumptions about the nature of the underlying conformational distribution. They suffer, however, from limited sampling with respect to time and conformational space. Moreover, the quality of the obtained structures is dependent on the choice of force ?eld and solvation model. A recently proposed method that uses time‐averaging with local‐elevation (LE) biasing of the conformational search provides an elegant means of overcoming these three problems. Using a set of side chain ³J‐coupling values for the FK506 binding protein (FKBP), we ?rst investigate the uncertainty in the angle values predicted theoretically. We then propose a simple MD‐based technique to detect inconsistencies within an experimental data set and identify degrees of freedom for which conformational averaging takes place or for which force ?eld parameters may be de?cient. Finally, we show that LE MD is the best method for producing ensembles of structures that, on average, ?t the experimental data.     

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.     

Motion of a disordered polypeptide chain as studied by paramagnetic relaxation enhancements, 15N relaxation, and molecular dynamics simulations: how fast is segmental diffusion in denatured ubiquitin?

Molecular dynamics (MD) simulations have been widely used to analyze dynamic conformational equilibria of folded proteins, especially in relation to NMR observables. However, this approach found little use in the studies of disordered proteins, where the sampling of vast conformational space presents a serious problem. In this paper, we demonstrate that the latest advances in computation technology make it possible to overcome this limitation. The experimentally validated (calibrated) MD models allow for new insights into structure/dynamics of disordered proteins. As a test system, we have chosen denatured ubiquitin in solution with 8 M urea at pH 2. High-temperature MD simulations in implicit solvent have been carried out for the wild-type ubiquitin as well as MTSL-tagged Q2C, D32C, and R74C mutants. To recalibrate the MD data (500 K) in relation to the experimental conditions (278 K, 8 M urea), the time axes of the MD trajectories were rescaled. The scaling factor was adjusted such as to maximize the agreement between the simulated and experimental (15)N relaxation rates. The resulting effective length of the trajectories, 311 μs, ensures good convergence properties of the MD model. The constructed MD model was validated against the array of experimental data, including additional (15)N relaxation parameters, multiple sets of paramagnetic relaxation enhancements (PREs), and the radius of gyration. In each case, a near-quantitative agreement has been obtained, suggesting that the model is successful. Of note, the MD-based approach rigorously predicts the quantities that are inherently dynamic, i.e., dependent on the motional correlation times. This cannot be accomplished, other than in empirical fashion, on the basis of static structural models (conformational ensembles). The MD model was further used to investigate the relative translational motion of the MTSL label and the individual H(N) atoms. The derived segmental diffusion coefficients proved to be nearly uniform along the peptide chain, averaging to D = 0.49-0.55 × 10(-6) cm(2)/s. This result was verified by direct analysis of the experimental PRE data using the recently proposed Ullman-Podkorytov model. In this model, MTSL and H(N) moieties are treated as two tethered spheres undergoing mutual diffusion in a harmonic potential. The fitting of the experimental data involving D as a single adjustable parameter leads to D = 0.45 × 10(-6) cm(2)/s, in good agreement with the MD-based analyses. This result can be compared with the range of estimates obtained from the resonance energy transfer experiments, D = 0.2-6.0 × 10(-6) cm(2)/s.     

Evaluation of the intramolecular basis set superposition error in the calculations of larger molecules: [n]helicenes and Phe-Gly-Phe tripeptide

Correlated ab initio calculations on large systems, such as the popular MP2 (or RI-MP2) method, suffer from the intramolecular basis set superposition error (BSSE). This error is typically manifested in molecules with folded structures, characterized by intramolecular dispersion interactions. It can dramatically affect the energy differences between various conformers as well as intramolecular stabilities, and it can even impair the accuracy of the predictions of the equilibrium molecular structures. In this study, we will present two extreme cases of intramolecular BSSE, the internal stability of [n]helicene molecules and the relative energies of various conformers of phenylalanyl-glycyl-phenylalanine tripeptide (Phe-Gly-Phe), and compare the calculated data with benchmark values (experimental or high-level theoretical data). As a practical and cheap solution to the accurate treatment of the systems with large anticipated value of intramolecular BSSE, the recently developed density functional method augmented with an empirical dispersion term (DFT-D) is proposed and shown to provide very good results in both of the above described representative cases.     

Root mean square deviation probability analysis of molecular dynamics trajectories on DNA

The comparison and detection of the commonalities and differences in multiple structural ensembles is an important step in the use of molecular simulations to gain insight into the conformation and dynamics of complex biomacromolecules. While the average structure is often employed as the representative of an ensemble of structures in such comparisons, dynamic molecular systems with multiple conformational substates call for a more accurate representation that captures the complete dynamical range of the ensemble. We present a probability analysis procedure based on the root-mean-square differences among the structural ensembles that efficiently and accurately performs the relevant comparison.