Small-amplitude protein conformational dynamics: Second-order analytic relation between cartesian coordinates and dihedral angles

An analytic expression for protein atomic displacements in Cartesian coordinate space (CCS) against small changes in dihedral angles is derived. To study time-dependent dynamics of a native protein molecule in CCS from dynamics in the internal coordinate space (ICS), it is necessary to convert small changes of internal coordinate variables to Cartesian coordinate variables. When we are interested in molecular motion, six degrees of freedom for translational and rotational motion of the molecule must be eliminated in this conversion, and this conversion is achieved by requiring the Eckart condition to hold. In this article, only dihedral angles are treated as independent internal variables (i.e., bond angles and bond lengths are fixed), and Cartesian coordinates of atoms are given analytically by a second-order Taylor expansion in terms of small deviations of variable dihedral angles. Coefficients of the first-order terms are collected in the K matrix obtained previously by Noguti and Go (1983) (see ref. 2). Coefficients of the second-order terms, which are for the first time derived here, are associated with the (newly termed) L matrix. The effect of including the resulting quadratic terms is compared against the precise numerical treatment using the Eckart condition. A normal mode analysis (NMA) in the dihedral angle space (DAS) of the protein bovine pancreatic trypsin inhibitor (BPTI) has been performed to calculate shift of mean atomic positions and mean square fluctuations around the mean positions. The analysis shows that the second-order terms involving the L matrix have significant contributions to atomic fluctuations at room temperature. This indicates that NMA in CCS involves significant errors when applied for such large molecules as proteins. These errors can be avoided by carrying out NMA in DAS and by considering terms up to second order in the conversion of atomic motion from DAS to CCS. © 1995 by John Wiley & Sons, Inc.     

Geometry optimization of periodic systems using internal coordinates

An algorithm is proposed for the structural optimization of periodic systems in internal (chemical) coordinates. Internal coordinates may include in addition to the usual bond lengths, bond angles, out-of-plane and dihedral angles, various "lattice internal coordinates" such as cell edge lengths, cell angles, cell volume, etc. The coordinate transformations between Cartesian (or fractional) and internal coordinates are performed by a generalized Wilson B-matrix, which in contrast to the previous formulation by Kudin et al. [J. Chem. Phys. 114, 2919 (2001)] includes the explicit dependence of the lattice parameters on the positions of all unit cell atoms. The performance of the method, including constrained optimizations, is demonstrated on several examples, such as layered and microporous materials (gibbsite and chabazite) as well as the urea molecular crystal. The calculations used energies and forces from the ab initio density functional theory plane wave method in the projector-augmented wave formalism.     

Transient molecular electro-optics Cartesian rotation vector versus Eulerian angles

Comparing the Euler angles, the classical choice of generalized coordinates describing the three rotational degrees of freedom of a rigid body, and the Cartesian rotation vector, we show that they both have their advantages and disadvantages in kinetic theory and Brownian dynamics analysis of molecular electro-optics. The Eulerian angles often yield relatively simple, yet singular, equations of motion, while their counterparts expressed in terms of Cartesian rotation vector are non-singular but more complex. In a special case, we show that the generalized force associated with the Cartesian rotation vector equals the torque. In addition, we introduce a new graphical approach to qualitatively track how changes in the Eulerian angles affect the Cartesian rotation vector.     

Ferreting out correlations from trajectory data

Thermally driven materials characterized by complex energy landscapes, such as proteins, exhibit motions on a broad range of space and time scales. Principal component analysis (PCA) is often used to extract modes of motion from protein trajectory data that correspond to coherent, functional motions. In this work, two other methods, maximum covariance analysis (MCA) and canonical correlation analysis (CCA) are formulated in a way appropriate to analyze protein trajectory data. Both methods partition the coordinates used to describe the system into two sets (two measurement domains) and inquire as to the correlations that may exist between them. MCA and CCA provide rotations of the original coordinate system that successively maximize the covariance (MCA) or correlation (CCA) between modes of each measurement domain under suitable constraint conditions. We provide a common framework based on the singular value decomposition of appropriate matrices to derive MCA and CCA. The differences between and strengths and weaknesses of MCA and CCA are discussed and illustrated. The application presented here examines the correlation between the backbone and side chain of the peptide met-enkephalin as it fluctuates between open conformations, found in solution, to closed conformations appropriate to when it is bound to its receptor. Difficulties with PCA carried out in Cartesian coordinates are found and motivate a formulation in terms of dihedral angles for the backbone atoms and selected atom distances for the side chains. These internal coordinates are a more reliable basis for all the methods explored here. MCA uncovers a correlation between combinations of several backbone dihedral angles and selected side chain atom distances of met-enkephalin. It could be used to suggest residues and dihedral angles to focus on to favor specific side chain conformers. These methods could be applied to proteins with domains that, when they rearrange upon ligand binding, may have correlated functional motions or, for multi-subunit proteins, may exhibit correlated subunit motions.     

DiTe2: Calculating the diffusion tensor for flexible molecules

We report on an extended hydrodynamic modeling of the friction tensorial properties of flexible molecules including all types of natural, Z-Matrix like, internal coordinates. We implement the new methodology by extending and updating the software DiTe [Barone et al. J. Comput. Chem. 30, 2 (2009)]. DiTe (DIffusion TEnsor) implements a hydrodynamic modeling of the generalized translational, rotational, and configurational friction and diffusion tensors of flexible molecules in which flexibility is described in terms of dihedral angles. The new tool, DiTe2, has been renewed to include also stretching and bending types of internal mobility. Furthermore, DiTe2 is able to calculate the friction and diffusion tensors along collective (or reaction) coordinates defined as linear combinations of the internal natural ones. A number of tests are reported to show the new features of DiTe2. As leitmotiv for the tests, the calmodulin protein is taken into consideration, described both at all-atom and coarse-grained levels. © 2018 Wiley Periodicals, Inc.     

Extension of QM/MM docking and its applications to metalloproteins

To overcome the limitation of conventional docking methods which assume fixed charge model from force field parameters, combined quantum mechanics/molecular mechanics (QM/MM) method has been applied to docking as a variable charge model and shown to exhibit improvement on the docking accuracy over fixed charge based methods. However, it has also been shown that there are a number of examples for which adoption of variable‐charge model fails to reproduce the native binding modes. In particular, for metalloproteins, previously implemented method of QM/MM docking failed most often. This class of proteins has highly polarized binding sites at which high‐coordinate‐numbered metal ions reside. We extend the QM/MM docking method so that protein atoms surrounding the binding site along with metal ions are included as quantum region, as opposed to only ligand atoms. This extension facilitates the required scaling of partial charges on metal ions leading to prediction of correct binding modes in metalloproteins. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Evaluation of the searching abilities of HBOP and HBSITE for binding pocket detection

We evaluated the pocket‐searching abilities of the computer programs HBOP and HBSITE, in which hydrophobic potentials calculated on grid points generated around a protein function as an indicator of the pocket‐like property‐using a test set of 458 experimentally observed structures of protein–ligand complexes. The results were compared with those obtained using the alternative algorithms PASS and SiteID, which only consider geometric properties, and Q‐SiteFinder, which considers not only geometry but also physicochemical properties. The comparison revealed that HBOP and HBSITE could detect experimentally observed ligand‐binding pockets for more test complexes than PASS and SiteID. In addition, the success rate of HBOP for detecting binding pockets was higher than that of Q‐SiteFinder, and that of HBSITE was comparable with that of Q‐SiteFinder. Results of tests for ligand‐unbound state proteins indicated that HBSITE was more appropriate than Q‐SiteFinder for pocket searches of ligand‐unbound proteins, and HBSITE was more robust than Q‐SiteFinder against structural differences between ligand‐bound and ‐unbound state proteins. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

Modeling and predicting binding affinity of phencyclidine‐like compounds using machine learning methods

Machine learning methods have always been promising in the science and engineering fields, and the use of these methods in chemistry and drug design has advanced especially since the 1990s. In this study, molecular electrostatic potential (MEP) surfaces of phencyclidine‐like (PCP‐like) compounds are modeled and visualized in order to extract features that are useful in predicting binding affinities. In modeling, the Cartesian coordinates of MEP surface points are mapped onto a spherical self‐organizing map (SSOM). The resulting maps are visualized using electrostatic potential (ESP) values. These values also provide features for a prediction system. Support vector machines and partial least‐squares method are used for predicting binding affinities of compounds. Copyright © 2009 John Wiley & Sons, Ltd.     

Organic Ligand Binding by a Hydrophobic Cavity in a Designed Tetrameric Coiled‐Coil Protein

The design and characterization of a hydrophobic cavity in de novo designed proteins provides a wide range of information about the functions of de novo proteins. We designed a de novo tetrameric coiled‐coil protein with a hydrophobic pocketlike cavity. Tetrameric coiled coils with hydrophobic cavities have previously been reported. By replacing one Leu residue at the a position with Ala, hydrophobic cavities that did not flatten out due to loose peptide chains were reliably created. To perform a detailed examination of the ligand‐binding characteristics of the cavities, we originally designed two other coiled‐coil proteins: AM2, with eight Ala substitutions at the adjacent a and d positions at the center of a bundled structure, and AM2W, with one Trp and seven Ala substitutions at the same positions. To increase the association of the helical peptides, each helical peptide was connected with flexible linkers, which resulted in a single peptide chain. These proteins exhibited CD spectra corresponding to superhelical structures, despite weakened hydrophobic packing. AM2W exhibited binding affinity for size‐complementary organic compounds. The dissociation constants, K_d, of AM2W were 220 nM for adamantane, 81 μM for 1‐adamantanol, and 294 μM for 1‐adamantaneacetic acid, as measured by fluorescence titration analyses. Although it was contrary to expectations, AM2 did not exhibit any binding affinity, probably due to structural defects around the designed hydrophobic cavity. Interestingly, AM2W exhibited incremental structure stability through ligand binding. Plugging of structural defects with organic ligands would be expected to facilitate protein folding.     

Dihedral angle principal component analysis of molecular dynamics simulations

It has recently been suggested by Mu et al. [Proteins 58, 45 (2005)] to use backbone dihedral angles instead of Cartesian coordinates in a principal component analysis of molecular dynamics simulations. Dihedral angles may be advantageous because internal coordinates naturally provide a correct separation of internal and overall motion, which was found to be essential for the construction and interpretation of the free energy landscape of a biomolecule undergoing large structural rearrangements. To account for the circular statistics of angular variables, a transformation from the space of dihedral angles {phi(n)} to the metric coordinate space {x(n)=cos phi(n),y(n)=sin phi(n)} was employed. To study the validity and the applicability of the approach, in this work the theoretical foundations underlying the dihedral angle principal component analysis (dPCA) are discussed. It is shown that the dPCA amounts to a one-to-one representation of the original angle distribution and that its principal components can readily be characterized by the corresponding conformational changes of the peptide. Furthermore, a complex version of the dPCA is introduced, in which N angular variables naturally lead to N eigenvalues and eigenvectors. Applying the methodology to the construction of the free energy landscape of decaalanine from a 300 ns molecular dynamics simulation, a critical comparison of the various methods is given.     

Carbohydrate‐binding protein identification by coupling structural similarity searching with binding affinity prediction

Carbohydrate‐binding proteins (CBPs) are potential biomarkers and drug targets. However, the interactions between carbohydrates and proteins are challenging to study experimentally and computationally because of their low binding affinity, high flexibility, and the lack of a linear sequence in carbohydrates as exists in RNA, DNA, and proteins. Here, we describe a structure‐based function‐prediction technique called SPOT‐Struc that identifies carbohydrate‐recognizing proteins and their binding amino acid residues by structural alignment program SPalign and binding affinity scoring according to a knowledge‐based statistical potential based on the distance‐scaled finite‐ideal gas reference state (DFIRE). The leave‐one‐out cross‐validation of the method on 113 carbohydrate‐binding domains and 3442 noncarbohydrate binding proteins yields a Matthews correlation coefficient of 0.56 for SPalign alone and 0.63 for SPOT‐Struc (SPalign + binding affinity scoring) for CBP prediction. SPOT‐Struc is a technique with high positive predictive value (79% correct predictions in all positive CBP predictions) with a reasonable sensitivity (52% positive predictions in all CBPs). The sensitivity of the method was changed slightly when applied to 31 APO (unbound) structures found in the protein databank (14/31 for APO versus 15/31 for HOLO). The result of SPOT‐Struc will not change significantly if highly homologous templates were used. SPOT‐Struc predicted 19 out of 2076 structural genome targets as CBPs. In particular, one uncharacterized protein in Bacillus subtilis (1oq1A) was matched to galectin‐9 from Mus musculus. Thus, SPOT‐Struc is useful for uncovering novel carbohydrate‐binding proteins. SPOT‐Struc is available at http://sparks‐lab.org . © 2014 Wiley Periodicals, Inc.     

Enzyme Selectivity Fine‐Tuned through Dynamic Control of a Loop

Allostery has been revealed as an essential property of all proteins. For enzymes, shifting of the structural equilibrium distribution at one site can have substantial impacts on protein dynamics and selectivity. Promising sites of remotely shifting such a distribution by changing the dynamics would be at flexible loops because relatively large changes may be achieved with minimal modification of the protein. A ligand‐selective change of binding affinity to the active site of cyclophilin is presented involving tuning of the dynamics of a highly flexible loop. Binding affinity is increased upon substitution of double Gly to Ala at the hinge regions of the loop. Quenching of the motional amplitudes of the loop slightly rearranges the active site. In particular, key residues for binding Phe60 and His126 adopt a more fixed orientation in the bound protein. Our system may serve as a model system for studying the effects of various time scales of loop motion on protein function tuned by mutations.     

Realistic sampling of amino acid geometries for a multipolar polarizable force field

The Quantum Chemical Topological Force Field (QCTFF) uses the machine learning method kriging to map atomic multipole moments to the coordinates of all atoms in the molecular system. It is important that kriging operates on relevant and realistic training sets of molecular geometries. Therefore, we sampled single amino acid geometries directly from protein crystal structures stored in the Protein Databank (PDB). This sampling enhances the conformational realism (in terms of dihedral angles) of the training geometries. However, these geometries can be fraught with inaccurate bond lengths and valence angles due to artefacts of the refinement process of the X‐ray diffraction patterns, combined with experimentally invisible hydrogen atoms. This is why we developed a hybrid PDB/nonstationary normal modes (NM) sampling approach called PDB/NM. This method is superior over standard NM sampling, which captures only geometries optimized from the stationary points of single amino acids in the gas phase. Indeed, PDB/NM combines the sampling of relevant dihedral angles with chemically correct local geometries. Geometries sampled using PDB/NM were used to build kriging models for alanine and lysine, and their prediction accuracy was compared to models built from geometries sampled from three other sampling approaches. Bond length variation, as opposed to variation in dihedral angles, puts pressure on prediction accuracy, potentially lowering it. Hence, the larger coverage of dihedral angles of the PDB/NM method does not deteriorate the predictive accuracy of kriging models, compared to the NM sampling around local energetic minima used so far in the development of QCTFF. © 2015 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Self-consistent Karplus parametrization of 3J couplings depending on the polypeptide side-chain torsion chi1.

Recently proposed self-consistent 3J coupling analysis (Schmidt, J. M.; Blümel, M.; L?hr, F.; Rüterjans, H. J. Biomol. NMR 1999, 14, 1-12) has been carried out to calibrate Karplus parameters constituting the empirical dependence of 3J coupling constants on the chi1 dihedral angle in amino acid side chains. The procedure involves simultaneous least-squares optimization of six sets of three Karplus coefficients related to all six 3J coupling types accessible in 15N,13C-labeled proteins. A simple concept of fundamental and incremental component couplings is proposed to account for substituent effects, eventually yielding amino acid topology-specific Karplus parameters. The method is exemplified with recombinant Desulfovibrio vulgaris flavodoxin (147 amino acids, 16 kDa) with reference to a total of 749 experimental 3JHalpha,Hbeta, 3JN',Hbeta, 3JC',Hbeta, 3JHalpha,Cgamma, 3JN',Cgamma, and 3JC',Cgamma coupling constants. Unlike other parametrizations, the present method does not make reference to X-ray coordinates, so that the Karplus coefficients obtained are not influenced by differences between solution and crystal states. Cross validation using X-ray torsion angles demonstrates the improvement relative to previous parametrizations. The Karplus coefficients derived are applicable to other proteins, too. Parameter refinement also yields a series of chi1 torsion angles, providing valuable constraints for protein structure determination, as well as optional parameters of local angular mobility in the contexts of Gaussian random fluctuation or a three-site jump model. The procedure permits automatic stereospecific assignments of Hbeta and Cgamma chemical shifts. The majority of the flavodoxin side-chain conformations agrees with high-resolution X-ray structures of the protein. Marked deviations between NMR and X-ray datasets are attributed to different rotameric states due to crystal-packing effects and to conformational equilibria between multiple chi1 rotamers.     

Q‐DockLHM: Low‐resolution refinement for ligand comparative modeling

The success of ligand docking calculations typically depends on the quality of the receptor structure. Given improvements in protein structure prediction approaches, approximate protein models now can be routinely obtained for the majority of gene products in a given proteome. Structure‐based virtual screening of large combinatorial libraries of lead candidates against theoretically modeled receptor structures requires fast and reliable docking techniques capable of dealing with structural inaccuracies in protein models. Here, we present Q‐Dock^LHM, a method for low‐resolution refinement of binding poses provided by FINDSITE^LHM, a ligand homology modeling approach. We compare its performance to that of classical ligand docking approaches in ligand docking against a representative set of experimental (both holo and apo) as well as theoretically modeled receptor structures. Docking benchmarks reveal that unlike all‐atom docking, Q‐Dock^LHM exhibits the desired tolerance to the receptor's structure deformation. Our results suggest that the use of an evolution‐based approach to ligand homology modeling followed by fast low‐resolution refinement is capable of achieving satisfactory performance in ligand‐binding pose prediction with promising applicability to proteome‐scale applications. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

The effect of water displacement on binding thermodynamics: concanavalin A

Interactions at the binding interface of biomolecular complexes are often mediated by ordered water molecules. In this work, we considered two concanavalin A-carbohydrate complexes. In the first, a water molecule is buried at the binding interface. In the second, this water molecule is displaced by a modification of the ligand (Clarke, C.; Woods, R. J.; Gluska, J.; Cooper, A.; Nutley, M. A.; Boons, G. J. J. Am. Chem. Soc. 2001, 123, 12238-12247). We computed the contribution of this water molecule to the thermodynamic properties using statistical mechanical formulas for the energy and entropy and molecular dynamics simulations. Other contributions to the binding affinity, including desolvation, entropy of conformational restriction, and interaction between the ligand and protein, were also computed. The thermodynamic consequences of displacement of the ordered water molecule by ligand modification were in qualitative agreement with experimental data. The free energy contribution of the water molecule (-17.2 kcal/mol; -19.2 enthalpic and +2 entropic) was nearly equivalent to the additional protein-ligand interactions in trimannoside 2 (-18.9 kcal/mol). The two structural ions interact more strongly with the water than with the hydroxyl of trimannoside 2, thus favoring trimannoside 1. The contributions from desolvation and conformational entropy are much smaller but significant, compared to the binding free energy difference. The picture that emerges is that the final outcome of water displacement is sensitive to the details of the binding site and cannot be predicted by simple empirical rules.     

Computational study of ligand binding in lipid transfer proteins: Structures,interfaces, and free energies of protein‐lipid complexes

Plant nonspecific lipid transfer proteins (nsLTPs) bind a wide variety of lipids, which allows them to perform disparate functions. Recent reports on their multifunctionality in plant growth processes have posed new questions on the versatile binding abilities of these proteins. The lack of binding specificity has been customarily explained in qualitative terms on the basis of a supposed structural flexibility and nonspecificity of hydrophobic protein‐ligand interactions. We present here a computational study of protein‐ligand complexes formed between five nsLTPs and seven lipids bound in two different ways in every receptor protein. After optimizing geometries in molecular dynamics calculations, we computed Poisson‐Boltzmann electrostatic potentials, solvation energies, properties of the protein‐ligand interfaces, and estimates of binding free energies of the resulting complexes. Our results provide the first quantitative information on the ligand abilities of nsLTPs, shed new light into protein‐lipid interactions, and reveal new features which supplement commonly held assumptions on their lack of binding specificity. © 2012 Wiley Periodicals, Inc.     

Ligand‐Induced Conformational Changes in HSP90 Monitored Time Resolved and Label Free—Towards a Conformational Activity Screening for Drug Discovery

Investigation of protein–ligand interactions is crucial during early drug‐discovery processes. ATR‐FTIR spectroscopy can detect label‐free protein–ligand interactions with high spatiotemporal resolution. Here we immobilized, as an example, the heat shock protein HSP90 on an ATR crystal. This protein is an important molecular target for drugs against several diseases including cancer. With our novel approach we investigated a ligand‐induced secondary structural change. Two specific binding modes of 19 drug‐like compounds were analyzed. Different binding modes can lead to different efficacy and specificity of different drugs. In addition, the k_obs values of ligand dissociation were obtained. The results were validated by X‐ray crystallography for the structural change and by SPR experiments for the dissociation kinetics, but our method yields all data in a single and simple experiment.     

Fast and accurate predictions of binding free energies using MM‐PBSA and MM‐GBSA

In the drug discovery process, accurate methods of computing the affinity of small molecules with a biological target are strongly needed. This is particularly true for molecular docking and virtual screening methods, which use approximated scoring functions and struggle in estimating binding energies in correlation with experimental values. Among the various methods, MM‐PBSA and MM‐GBSA are emerging as useful and effective approaches. Although these methods are typically applied to large collections of equilibrated structures of protein‐ligand complexes sampled during molecular dynamics in water, the possibility to reliably estimate ligand affinity using a single energy‐minimized structure and implicit solvation models has not been explored in sufficient detail. Herein, we thoroughly investigate this hypothesis by comparing different methods for the generation of protein‐ligand complexes and diverse methods for free energy prediction for their ability to correlate with experimental values. The methods were tested on a series of structurally diverse inhibitors of Plasmodium falciparum DHFR with known binding mode and measured affinities. The results showed that correlations between MM‐PBSA or MM‐GBSA binding free energies with experimental affinities were in most cases excellent. Importantly, we found that correlations obtained with the use of a single protein‐ligand minimized structure and with implicit solvation models were similar to those obtained after averaging over multiple MD snapshots with explicit water molecules, with consequent save of computing time without loss of accuracy. When applied to a virtual screening experiment, such an approach proved to discriminate between true binders and decoy molecules and yielded significantly better enrichment curves. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010