Statistical and conformational analysis of the electron density of protein side chains

Protein side chains make most of the specific contacts between proteins and other molecules, and their conformational properties have been studied for many years. These properties have been analyzed primarily in the form of rotamer libraries, which cluster the observed conformations into groups and provide frequencies and average dihedral angles for these groups. In recent years, these libraries have improved with higher resolution structures and using various criteria such as high thermal factors to eliminate side chains that may be misplaced within the crystallographic model coordinates. Many of these side chains have highly non-rotameric dihedral angles. The origin of side chains with high B-factors and/or with non-rotameric dihedral angles is of interest in the determination of protein structures and in assessing the prediction of side chain conformations. In this paper, using a statistical analysis of the electron density of a large set of proteins, it is shown that: (1) most non-rotameric side chains have low electron density compared to rotameric side chains; (2) up to 15% of chi1 non-rotameric side chains in PDB models can clearly be fit to density at a single rotameric conformation and in some cases multiple rotameric conformations; (3) a further 47% of non-rotameric side chains have highly dispersed electron density, indicating potentially interconverting rotameric conformations; (4) the entropy of these side chains is close to that of side chains annotated as having more than one chi(1) rotamer in the crystallographic model; (5) many rotameric side chains with high entropy clearly show multiple conformations that are not annotated in the crystallographic model. These results indicate that modeling of side chains alternating between rotamers in the electron density is important and needs further improvement, both in structure determination and in structure prediction.     

Structure and dynamics of an amphiphilic peptide in a lipid bilayer: a molecular dynamics study

A molecular dynamics simulation of a simple model membrane system composed of a single amphiphilic helical peptide (ace-K2GL16K2A-amide) in a fully hydrated 1,2-dimyristoyl-sn-glycero-3-phosphocholine bilayer was performed for a total of 1060 ps. The secondary structure of the peptide and its stability were described in terms of average dihedral angles, phi and psi, and the C alpha torsion angles formed by backbone atoms; by the average translation per residue along the helix axis; and by the intramolecular peptide hydrogen bonds. The results indicated that residues 6 through 15 remain in a stable right-handed alpha-helical conformation, whereas both termini exhibit substantial fluctuations. A change in the backbone dihedral angles for residues 16 and 17 is accompanied by the loss of two intramolecular hydrogen bonds, leading to a local but long-lived disruption of the helix. The dynamics of the peptide was characterized in terms of local and global helix motions. The local motions of the N-H bond angles were described in terms of the autocorrelation functions of P2[cos thetaNH(t, t + tau)] and reflected the different degrees of local peptide order as well as a variation in time scale for local motions. The chi1 and chi2 dihedral angles of the leucine side chains underwent frequent transitions between potential minima. No connection between the side-chain positions and their mobility was observed, however. In contrast, the lysine side chains displayed little mobility during the simulation. The global peptide motions were characterized by the tilting and bending motions of the helix. Although the peptide was initially aligned parallel to the bilayer normal, during the simulation it was observed to tilt away from the normal, reaching an angle of approximately 25 degrees by the end of the simulation. In addition, a slight bend of the helix was detected. Finally, the solvation of the peptide backbone and side-chain atoms was also investigated.     

New Insights into the Interdependence between Amino Acid Stereochemistry and Protein Structure

To successfully design new proteins and understand the effects of mutations in natural proteins, we must understand the geometric and physicochemical principles underlying protein structure. The side chains of amino acids in peptides and proteins adopt specific dihedral angle combinations; however, we still do not have a fundamental quantitative understanding of why some side-chain dihedral angle combinations are highly populated and others are not. Here we employ a hard-sphere plus stereochemical constraint model of dipeptide mimetics to enumerate the side-chain dihedral angles of leucine (Leu) and isoleucine (Ile), and identify those conformations that are sterically allowed versus those that are not as a function of the backbone dihedral angles ? and ψ. We compare our results with the observed distributions of side-chain dihedral angles in proteins of known structure. With the hard-sphere plus stereochemical constraint model, we obtain agreement between the model predictions and the observed side-chain dihedral angle distributions for Leu and Ile. These results quantify the extent to which local, geometrical constraints determine protein side-chain conformations.     

New Insights into the Interdependence between Amino Acid Stereochemistry and Protein Structure

To successfully design new proteins and understand the effects of mutations in natural proteins, we must understand the geometric and physicochemical principles underlying protein structure. The side chains of amino acids in peptides and proteins adopt specific dihedral angle combinations; however, we still do not have a fundamental quantitative understanding of why some side-chain dihedral angle combinations are highly populated and others are not. Here we employ a hard-sphere plus stereochemical constraint model of dipeptide mimetics to enumerate the side-chain dihedral angles of leucine (Leu) and isoleucine (Ile), and identify those conformations that are sterically allowed versus those that are not as a function of the backbone dihedral angles ϕ and ψ. We compare our results with the observed distributions of side-chain dihedral angles in proteins of known structure. With the hard-sphere plus stereochemical constraint model, we obtain agreement between the model predictions and the observed side-chain dihedral angle distributions for Leu and Ile. These results quantify the extent to which local, geometrical constraints determine protein side-chain conformations.     

Quantification of thermal ring flexibilities of aromatic and heteroaromatic compounds

The consequences of thermal fluctuations occurring at room temperatures on the aromatic character of a broad group of compounds were analyzed in three distinct ways. First of all, the ring deformations were modeled along normal coordinates coming from quantum thermo-chemistry computations. The amplitudes of vibrations were estimated according to absorbed energies at room temperature. Alternatively, in-plane and out-of-plane ring deformations were modeled via scanning procedure with partial relaxation of the molecular geometry. The influence of ring deformations on π–electron delocalization was expressed in terms of HOMA values. Besides, the ring deformability was defined as the averaged change of bond angles or dihedral angles constituting the ring that was associated with 1.5 kcal mol^-1 increase of the system energy. The molecules structures adopted during vibrations at room temperature can lead to significant heterogeneity of structural index of aromaticity. The broad span of HOMA values was obtained for analyzed five- or six-membered aromatic and heteroaromatic rings. However, the averaged values obtained for such fluctuations almost perfectly match HOMA values of molecule in the ground state. It has been demonstrated that the ring deformability imposed by bond angle changes is much smaller than for dihedral angles with the same rise of system energy. Interestingly in the case of out-of-plane vibrations modeled by scanning procedure there is observed linear correlation between ring deformability and HOMA values. Proposed method for inclusion of thermal vibrations in the framework of π–electron delocalization provides natural shift of the way of thinking about aromaticity from a static quantity to a dynamic and heterogeneous one due to inclusion of a more realistic object of analysis – thermally deformed structures. From this perspective the thermal fluctuations are supposed to be non-negligible contributions to aromaticity phenomenon.     

A method for characterizing transition concertedness from polymer dynamics computer simulations

A statistical method based on classifying the transitions among a set of dihedral angles within an “energy transfer window” is developed, and used to analyze Brownian (BD) and molecular dynamics (MD) simulations of the acyl chains in a lipid bilayer, and MD of neat hexadecane. It is shown for the BD simulation that when a transition of the dihedral angle in the center of the chain occurs, a transition of a particular next nearest neighbor (or angle 2-apart) will follow concertedly with a probability of approximately 0.10 within a lime window of approximately 3 ps. The MD bilayer simulations, which are based on a more flexible model of the hydrocarbon chains, yield corresponding concerted transition probabilities of approximately 0.083 and window sizes of 1–2 ps. An analysis of angles 4-apart yields concerted transition probabilities of 0.03 and 0.04 for the BD and MD bilayer simulations, respectively, and window sizes close to those of the corresponding 2-apart cases. Statistical hypothesis testing very strongly rejects the assertion that these follower transitions are occurring at random. Similar analysis reveals marginal or no evidence of concertedness between 1-apart (nearest neighbor) and between 3-apart dihedral angle transitions. The pattern of concertedness for hexadecane is qualitatively similar to that of the lipid chains, although concertedness is somewhat stronger for the 3-apart transitions and somewhat weaker for those 4-apart. Finally, it is suggested that the diffusion of small solute molecules in membranes is better facilitated by non concerted transitions, which are associated with relatively large displacements of the chains, than by concerted transitions, which do little to change the chain shape. © 1995 John Wiley & Sons, Inc.     

The geometry of alpha-sheet: Implications for its possible function as amyloid precursor in proteins

alpha-sheet has been proposed as the main constituent of the prefibrillar intermediate during amyloid formation. Here the helical parameters of the alpha-sheet strand are calculated from average main-chain dihedral angles reported from molecular dynamics simulations. It is an almost linear polypeptide that forms a right-handed helix of about 100 A diameter, with 100 residues and a rise of 30 A per turn. The strands are curved but untwisted, which implies that neighboring strands need not coil to make interstrand hydrogen bonds. This suggests that compared to beta-sheets in native folded proteins, alpha-sheets can be larger and stack more easily to create extensive 3D blocks. It is shown that alpha-sheet is related to a category of structures termed "mirror" structures. Mirror structures have repetitive pairs of main-chain dihedral angles at residues i and i+1 that satisfy the condition phi(i) (+1) = -psi(i), psi(i) (+1) = -phi(i). They are uniquely identified by the two orientations of their peptide planes, specified by phi(i) and psi(i). Their side chains point alternately in opposite directions. Interestingly, their conformations are insensitive to phi(i) and psi(i) in that the pseudo dihedral angle formed by four consecutive C(alpha) atoms is always close to 180 degrees . There are two types: "beta-mirror" and "alpha-mirror" structure; beta-mirror structures relate to beta-sheet by small peptide plane rotations, of less than 90 degrees , while alpha-mirror structures are close to alpha-sheet and relate to beta-sheet by approximately 180 degrees peptide plane flips. Most mirror structures, and in particular the alpha-mirror, form wide helices with diameters 50-70 A. Their gentle curvature, and therefore that of the alpha-sheet, arises from the orientation of successive peptide units causing the difference in the bond angles at the C and N atoms of the peptide unit to gradually change the direction of the chain.     

Representation of a protein molecule as a tree and application to modular computer programs which calculate and modify atomic coordinates

The concept and representation of a logical tree as defined in computer science is applied to obtain a suitable representation of protein molecules in computer programs which handle or calculate atomic coordinates of protein molecules. On the basis of this analysis and of the analysis of the calculation and modification of the structure of a protein from bond lengths, bond angles, and dihedral angles, which is reproduced in an appendix, program modules which accomplish the various required computations are described. Three such modules are given in Iverson notation; in fact, it is hoped that this article will serve as a reasonably complete basis for the preparation of machine programs by moderately proficient programmers.     

Equilibrium transitions between side‐chain conformations in leucine and isoleucine

Despite recent improvements in computational methods for protein design, we still lack a quantitative, predictive understanding of the intrinsic probabilities for amino acids to adopt particular side‐chain conformations. Surprisingly, this question has remained unsettled for many years, in part because of inconsistent results from different experimental approaches. To explicitly determine the relative populations of different side‐chain dihedral angles, we performed all‐atom hard‐sphere Langevin Dynamics simulations of leucine (Leu) and isoleucine (Ile) dipeptide mimetics with stereo‐chemical constraints and repulsive‐only steric interactions between non‐bonded atoms. We determine the relative populations of the different χ₁ and χ₂ dihedral angle combinations as a function of the backbone dihedral angles ? and ψ. We also propose, and test, a mechanism for inter‐conversion between the different side‐chain conformations. Specifically, we discover that some of the transitions between side‐chain dihedral angle combinations are very frequent, whereas others are orders of magnitude less frequent, because they require rare coordinated motions to avoid steric clashes. For example, to transition between different values of χ₂, the Leu side‐chain bond angles κ₁ and κ₂ must increase, whereas to transition in χ₁, the Ile bond angles λ₁ and λ₂ must increase. These results emphasize the importance of computational approaches in stimulating further experimental studies of the conformations of side‐chains in proteins. Moreover, our studies emphasize the power of simple steric models to inform our understanding of protein structure, dynamics, and design. Proteins 2015; 83:1488–1499. © 2015 Wiley Periodicals, Inc.     

Deuterium isotope effects on <Superscript>15</Superscript>N backbone chemical shifts in proteins

Quantum mechanical calculations are presented that predict that one-bond deuterium isotope effects on the ¹⁵N chemical shift of backbone amides of proteins, ¹Δ¹⁵N(D), are sensitive to backbone conformation and hydrogen bonding. A quantitative empirical model for ¹Δ¹⁵N(D) including the backbone dihedral angles, Φ and Ψ, and the hydrogen bonding geometry is presented for glycine and amino acid residues with aliphatic side chains. The effect of hydrogen bonding is rationalized in part as an electric-field effect on the first derivative of the nuclear shielding with respect to N–H bond length. Another contributing factor is the effect of increased anharmonicity of the N–H stretching vibrational state upon hydrogen bonding, which results in an altered N–H/N–D equilibrium bond length ratio. The N–H stretching anharmonicity contribution falls off with the cosine of the N–H···O bond angle. For residues with uncharged side chains a very good prediction of isotope effects can be made. Thus, for proteins with known secondary structures, ¹Δ¹⁵N(D) can provide insights into hydrogen bonding geometries. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Simulation of conformational possibilities of DNA via calculation of nonbonded interactions of complementary dinucleoside phosphate complexes

Using classical potential functions, we carried out potential-energy calculations on the complementary deoxydinucleoside phosphate complexes dApdA:dUpdU, dUpdA:dUpdA, and dApdU:dApdU. All dihedral and bond angles, except those of the nitrogen bases, were varied. The resulting minimum-energy conformations of the complexes are close to DNA A- and B-family conformations, with a typical arrangement of the nitrogen bases. The dihedral and bond angles of one of the molecules forming the complex can thereby differ by several degrees from those of the other molecule. For different base sequences, some dihedral and bond angles may vary over a range of several degrees without appreciably changing the total energy of the complex. Some low-energy conformations of the complexes corresponding to other regions of the conformational space are also found. The biological consequences of possible changes in dihedral and bond angles, occurring on interaction with other molecules, are discussed.     

Local deformation studies of chain molecules: differential conditions for changes of dihedral angles

New first and second-order differential equations for changes of dihedral angles characterizing local deformations of chain molecules with fixed bond lengths and bond angles are derived. Two methods for integrating the differential relations are given. The proposed method is used to generate a path of locally deformed conformations around a β-turn region of a small protein, bovine pancreatic trypsin inhibitor. The variable regions change their conformations by more than 3 Å root-mean-square distance value whereas the fixed regions stay within 0.02 Å. Possible applications of this method are in the field of computer graphics, Monte Carlo simulations, and energy minimization calculations of chain molecules.     

Identification of folding preferences of cleavage junctions of HIV-1 precursor proteins for regulation of cleavability

Human immunodeficiency virus type 1 protease (HIV-1 PR) cleaves two viral precursor proteins, Gag and Gag-Pol, at multiple sites. Although the processing proceeds in the rank order to assure effective viral replication, the molecular mechanisms by which the order is regulated are not fully understood. In this study, we used bioinformatics approaches to examine whether the folding preferences of the cleavage junctions influence their cleavabilities by HIV-1 PR. The folding of the eight-amino-acid peptides corresponding to the seven cleavage junctions of the HIV-1_HXB2 Gag and Gag-Pol precursors were simulated in the PR-free and PR-bound states with molecular dynamics and homology modeling methods, and the relationships between the folding parameters and the reported kinetic parameters of the HIV-1_HXB2 peptides were analyzed. We found that a folding preference for forming a dihedral angle of Cβ (P1)-Cα (P1)- Cα (P1’)-Cβ (P1’) in the range of 150 to 180 degrees in the PR-free state was positively correlated with the 1/K_m (R = 0.95, P = 0.0008) and that the dihedral angle of the O (P2)-C (P2)- C (P1)- O (P1) of the main chains in the PR-bound state was negatively correlated with k_cat (R = 0.94, P = 0.001). We further found that these two folding properties influenced the overall cleavability of the precursor protein when the sizes of the side chains at the P1 site were similar. These data suggest that the dihedral angles at the specific positions around the cleavage junctions before and after binding to PR are both critical for regulating the cleavability of precursor proteins by HIV-1 PR.     

A statistically derived parameterization for the collagen triple-helix

The triple-helix is a unique secondary structural motif found primarily within the collagens. In collagen, it is a homo- or hetero-tripeptide with a repeating primary sequence of (Gly-X-Y)(n), displaying characteristic peptide backbone dihedral angles. Studies of bulk collagen fibrils indicate that the triple-helix must be a highly repetitive secondary structure, with very specific constraints. Primary sequence analysis shows that most collagen molecules are primarily triple-helical; however, no high-resolution structure of any entire protein is yet available. Given the drastic morphological differences in self-assembled collagen structures with subtle changes in assembly conditions, a detailed knowledge of the relative locations of charged and sterically bulky residues in collagen is desirable. Its repetitive primary sequence and highly conserved secondary structure make collagen, and the triple-helix in general, an ideal candidate for a general parameterization for prediction of residue locations and for the use of a helical wheel in the prediction of residue orientation. Herein, a statistical analysis of the currently available high-resolution X-ray crystal structures of model triple-helical peptides is performed to produce an experimentally based parameter set for predicting peptide backbone and C(beta) atom locations for the triple-helix. Unlike existing homology models, this allows easy prediction of an entire triple-helix structure based on all existing high-resolution triple-helix structures, rather than only on a single structure or on idealized parameters. Furthermore, regional differences based on the helical propensity of residues may be readily incorporated. The parameter set is validated in terms of the predicted bond lengths, backbone dihedral angles, and interchain hydrogen bonding.     

Vibrational approach to the dynamics of an α-helix

The dynamics of a finite α-helix have been studied in the harmonic approximation by a vibrational analysis of the atomic motions about their equilibrium positions. The system were represented by an empirical potential energy function, and all degrees of freedom (bond lengths, bond angles, and torsional angles) were allowed to vary. The complete results were compared with a more restrictive model in which the peptide dihedral angle was kept rigid; also, a model potential excluding hydrogen bonds was examined. Thermal fluctuations in the backbone dihedral angles ? and ψ are 12° to 15°. The fluctuations of adjacent dihedral angles are highly correlated, and the correlation pattern is affected by the flexibility of the peptide dihedral angle. Time-dependent autocorrelations in the motion of ? and ψ appear to decay due to dephasing in less than 1 psec, while the motions of the carbonyl oxygen and amide hydrogens out of the peptide plane are more harmonic. Length fluctuations have been evaluated and exhibit a strong end effect; the calculated elastic modulus is in agreement with other values. Rigid and adiabatic total energy surfaces corresponding to dihedral angle rotations in the middle of the helix have been obtained and compared with the quadratic approximation to those surfaces. The magnitudes and correlations between the fluctuations obtained by averaging over the adiabatic energy surface most closely resemble the vibrational results. Of particular interest is the fact that hydrogen bonds play a relatively small role in the local dihedral angle fluctuations, though the hydrogen bonds are important in the energy of overall length changes.     

On the rotational operators in protein structure simulations

The reduction of the computational complexity of the algorithms dealing with protein structure analysis and conformation predictions is of prime importance. One common element in most of these algorithms is the process of transforming geometrical information between dihedral angles and Cartesian coordinates of the atoms in the protein using rotational operators. In the literature, the operators used in protein structures are rotation matrices, quaternions in vector and matrix forms and the Rodrigues-Gibbs formula. In the protein structure-related literature, the most widely promoted rotational operator is the quaternions operator. In this work, we studied the computational efficiency of the mathematical operations of the above rotational operators applied to protein structures. A similar study applied to protein structures has not been reported previously. We concluded that the computational efficiency of these rotational operators applied to protein chains is different from those reported for other applications (such as mechanical machinery) and the conclusions are not analogous. Rotation matrices are the most efficient mathematical operators in the protein chains. We examined our findings in two protein molecules: Ab1 tyrosine kinase and heparin-binding growth factor 2. We found that the rotation matrix operator has between 2 and 187% fewer mathematical operations than the other rotational operators.     

Fast flexible modeling of RNA structure using internal coordinates

Modeling the structure and dynamics of large macromolecules remains a critical challenge. Molecular dynamics (MD) simulations are expensive because they model every atom independently, and are difficult to combine with experimentally derived knowledge. Assembly of molecules using fragments from libraries relies on the database of known structures and thus may not work for novel motifs. Coarse-grained modeling methods have yielded good results on large molecules but can suffer from difficulties in creating more detailed full atomic realizations. There is therefore a need for molecular modeling algorithms that remain chemically accurate and economical for large molecules, do not rely on fragment libraries, and can incorporate experimental information. RNABuilder works in the internal coordinate space of dihedral angles and thus has time requirements proportional to the number of moving parts rather than the number of atoms. It provides accurate physics-based response to applied forces, but also allows user-specified forces for incorporating experimental information. A particular strength of RNABuilder is that all Leontis-Westhof basepairs can be specified as primitives by the user to be satisfied during model construction. We apply RNABuilder to predict the structure of an RNA molecule with 160 bases from its secondary structure, as well as experimental information. Our model matches the known structure to 10.2 Angstroms RMSD and has low computational expense.     

Determination of the three-dimensional structure of oligosaccharides in the solid state from experimental 13C NMR data and ab initio chemical shift surfaces

A novel method for the determination of the three-dimensional (3D) structure of oligosaccharides in the solid state using experimental 13C NMR data is presented. The approach employs this information, combined with 13C chemical shift surfaces (CSSs) for the glycosidic bond carbons in the generation of NMR pseudopotential energy functions suitable for use as constraints in molecular modeling simulations. Application of the method to trehalose, cellobiose, and cellotetraose produces 3D models that agree remarkably well with the reported X-ray structures, with phi and psi dihedral angles that are within 10 degrees from the ones observed in the crystals. The usefulness of the approach is further demonstrated in the determination of the 3D structure of the cellohexaose, an hexasaccharide for which no X-ray data has been reported, as well as in the generation of accurate structural models for cellulose II and amylose V6.     

BuildBeta—A system for automatically constructing beta sheets

We describe a method that can thoroughly sample a protein conformational space given the protein primary sequence of amino acids and secondary structure predictions. Specifically, we target proteins with β‐sheets because they are particularly challenging for ab initio protein structure prediction because of the complexity of sampling long‐range strand pairings. Using some basic packing principles, inverse kinematics (IK), and β‐pairing scores, this method creates all possible β‐sheet arrangements including those that have the correct packing of β‐strands. It uses the IK algorithms of ProteinShop to move α‐helices and β‐strands as rigid bodies by rotating the dihedral angles in the coil regions. Our results show that our approach produces structures that are within 4–6 Å RMSD of the native one regardless of the protein size and β‐sheet topology although this number may increase if the protein has long loops or complex α‐helical regions. Proteins 2010. © Published 2009 Wiley‐Liss, Inc.     

Pressure-dependent <Superscript>13</Superscript>C chemical shifts in proteins: origins and applications

Pressure-dependent ¹³C chemical shifts have been measured for aliphatic carbons in barnase and Protein G. Up to 200 MPa (2 kbar), most shift changes are linear, demonstrating pressure-independent compressibilities. CH₃, CH₂ and CH carbon shifts change on average by +0.23, −0.09 and −0.18 ppm, respectively, due to a combination of bond shortening and changes in bond angles, the latter matching one explanation for the γ-gauche effect. In addition, there is a residue-specific component, arising from both local compression and conformational change. To assess the relative magnitudes of these effects, residue-specific shift changes for protein G were converted into structural restraints and used to calculate the change in structure with pressure, using a genetic algorithm to convert shift changes into dihedral angle restraints. The results demonstrate that residual ¹³Cα shifts are dominated by dihedral angle changes and can be used to calculate structural change, whereas ¹³Cβ shifts retain significant dependence on local compression, making them less useful as structural restraints.