Construction of the free energy landscape of biomolecules via dihedral angle principal component analysis

A systematic approach to construct a low-dimensional free energy landscape from a classical molecular dynamics (MD) simulation is presented. The approach is based on the recently proposed dihedral angle principal component analysis (dPCA), which avoids artifacts due to the mixing of internal and overall motions in Cartesian coordinates and circumvents problems associated with the circularity of angular variables. Requiring that the energy landscape reproduces the correct number, energy, and location of the system's metastable states and barriers, the dimensionality of the free energy landscape (i.e., the number of essential components) is obtained. This dimensionality can be determined from the distribution and autocorrelation of the principal components. By performing an 800 ns MD simulation of the folding of hepta-alanine in explicit water and using geometric and kinetic clustering techniques, it is shown that a five-dimensional dPCA energy landscape is a suitable and accurate representation of the full-dimensional landscape. In the second step, the dPCA energy landscape can be employed (e.g., in a Langevin simulation) to facilitate a detailed investigation of biomolecular dynamics in low dimensions. Finally, several ways to visualize the multidimensional energy landscape are discussed.     

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.     

Algorithm for normal mode analysis with general internal coordinates

A technique for performing normal vibrational analysis for biological macromolecules using general internal coordinates is proposed. The technique is based on the conventional algorithm for calculating the second derivatives of potential and kinetic energies using intramolecular dihedral angles, intermolecular translation, and rotation as variables [Braun, W. et al., J Phys Soc Jpn 1984, 53, 3269]. We extend the algorithm to include more general internal coordinates, bond stretching, angle bending, and so forth, without assuming two-body interactions. The essential point is the separation of the variables for potential functions and vibrational analysis. With our technique, we can arbitrarily choose any combination of internal coordinates as variables, free from the functional form of potential energy. We can analyze complex systems such as a multiple molecular system including solvents or a transition state of chemical reactions. In addition, mixed use of the potentials of molecular mechanics and quantum chemistry is possible.     

Density functional studies on dinuclear {Ni(II)Gd(III)} and trinuclear {Ni(II)Gd(III)Ni(II)} complexes: magnetic exchange and magneto-structural maps

Theoretical calculations using density functional methods have been performed on two dinuclear {Ni(II)-Gd(III)} and two trinuclear {Ni(II)-Gd(III)-Ni(II)} complexes having two and three μ-OR (R = alkyl or aromatic groups) bridging groups. The different magnetic behaviour, having moderately strong ferromagnetic coupling for complexes having two μ-OR groups and weak ferromagnetic coupling for complexes having three μ-OR groups, observed experimentally is very well reproduced by the calculations. Additionally, computation of overlap integrals MO and NBO analysis reveals a clear increase in antiferromagnetic contribution to the net exchange for three μ-OR bridged {Ni-Gd} dimers and also provides several important clues regarding the mechanism of magnetic coupling. Besides, MO and NBO analysis discloses the role of the empty 5d orbitals of the Gd(III) ion on the mechanism of magnetic coupling. Magneto-structural correlations for Ni-O-Gd bond angles, Ni-O and Gd-O bond distances, and the Ni-O-Gd-O dihedral angle have been developed and compared with the published experimental {Ni-Gd} structures and their J values indicate that the Ni-O-Gd bond angles play a prominent role in these types of complexes. The computation has then been extended to two trinuclear {Ni(II)-Gd(III)-Ni(II)} complexes and here both the {Ni-Gd} and the {Ni-Ni} interactions have been computed. Our calculations reveal that, for both structures studied, the two {NiGd} interactions are ferromagnetic and are similar in strength. The {Ni-Ni} interaction is antiferromagnetic in nature and our study reveals that its inclusion in fitting the magnetic data is necessary to obtain a reliable set of spin Hamiltonian parameters. Extensive magneto-structural correlations have been developed for the trinuclear complexes and the observed J trend for the trinuclear complex is similar to that of the dinuclear {Ni-Gd} complex. In addition to the structural parameters discussed above, for trinuclear complexes the twist angle between the two Ni-O-Gd planes is also an important parameter which influences the J values.     

Metadynamics in essential coordinates: free energy simulation of conformational changes

We propose an approach that combines an extraction of collective motions of a molecular system with a sampling of its free energy surface. A recently introduced method of metadynamics allows exploration of the free energy surface of a molecular system by means of coarse-grained dynamics with flooding of free energy minima. This free energy surface is defined as a function of a set of collective variables (e.g., interatomic distances, angles, torsions, and others). In this study, essential coordinates determined by essential dynamics (principle component analysis) were used as collective variables in metadynamics. First, dynamics of the model system (explicitly solvated alanine dipeptide, Ace-Ala-Nme) was simulated by a classical molecular dynamics simulation. The trajectory (1 ns) was then analyzed by essential dynamics to obtain essential coordinates. The free energy surface as a function of the first and second essential coordinates was then explored by metadynamics. The resulting free energy surface is in agreement with other studies of this system. We propose that a combination of these two methods (metadynamics and essential dynamics) has great potential in studies of conformational changes in peptides and proteins.     

The preparation and characterisation of hetero- and homobimetallic complexes containing bridging naphthalene-1,8-dithiolato ligands

Homo- and heterobimetallic complexes of the form [(PPh(3))(2)(mu(2)-1,8-S(2)-nap){ML(n)}] (in which (1,8-S(2)-nap)=naphtho-1,8-dithiolate and {ML(n)}={PtCl(2)} (1), {PtClMe} (2), {PtClPh} (3), {PtMe(2)} (4), {PtIMe(3)} (5) and {Mo(CO)(4)} (6)) were obtained by the addition of [PtCl(2)(NCPh)(2)], [PtClMe(cod)] (cod=1,5-cyclooctadiene), [PtClPh(cod)], [PtMe(2)(cod)], [{PtIMe(3)}(4)] and [Mo(CO)(4)(nbd)] (nbd=norbornadiene), respectively, to [Pt(PPh(3))(2)(1,8-S(2)-nap)]. Synthesis of cationic complexes was achieved by the addition of one or two equivalents of a halide abstractor, Ag[BF(4)] or Ag[ClO(4)], to [{Pt(mu-Cl)(mu-eta(2):eta(1)-C(3)H(5))}(4)], [{Pd(mu-Cl)(eta(3)-C(3)H(5))}(2)], [{IrCl(mu-Cl)(eta(5)-C(5)Me(5))}(2)] (in which C(5)Me(5)=Cp*=1,2,3,4,5-pentamethylcyclopentadienyl), [{RhCl(mu-Cl)(eta(5)-C(5)Me(5))}(2)], [PtCl(2)(PMe(2)Ph)(2)] and [{Rh(mu-Cl)(cod)}(2)] to give the appropriate coordinatively unsaturated species that, upon treatment with [(PPh(3))(2)Pt(1,8-S(2)-nap)], gave complexes of the form [(PPh(3))(2)(mu(2)-1,8-S(2)-nap){ML(n)}][X] (in which {ML(n)}[X]={Pt(eta(3)-C(3)H(5))}[ClO(4)] (7), {Pd(eta(3)-C(3)H(5))}[ClO(4)] (8), {IrCl(eta(5)-C(5)Me(5))}[ClO(4)] (9), {RhCl(eta(5)-C(5)Me(5))}[BF(4)] (10), {Pt(PMe(2)Ph)(2)}[ClO(4)](2) (11), {Rh(cod)}[ClO(4)] (12); the carbonyl complex {Rh(CO)(2)}[ClO(4)] (13) was formed by bubbling gaseous CO through a solution of 12. In all cases the naphtho-1,8-dithiolate ligand acts as a bridge between two metal centres to give a four-membered PtMS(2) ring (M=transition metal). All compounds were characterised spectroscopically. The X-ray structures of 5, 6, 7, 8, 10 and 12 reveal a binuclear PtMS(2) core with PtM distances ranging from 2.9630(8)-3.438(1) A for 8 and 5, respectively. The napS(2) mean plane is tilted with respect to the PtP(2)S(2) coordination plane, with dihedral angles in the range 49.7-76.1 degrees and the degree of tilting being related to the PtM distance and the coordination number of M. The sum of the Pt(1)coordination plane/napS(2) angle, a, and the Pt(1)coordination plane/M(2)coordination plane angle, b, a+b, is close to 120 degrees in nearly all cases. This suggests that electronic effects play a significant role in these binuclear systems.     

Evaluation of configurational entropy methods from peptide folding-unfolding simulation

A 4-micros molecular dynamics simulation of the second beta-hairpin of the B1 domain of streptococcal protein G is used to characterize the free energy surface and to evaluate different configurational entropy estimators. From the equilibrium folding-unfolding trajectory, 200 000 conformers are clustered according to their root-mean-square deviation (RMSD). The height of the free energy barrier between pairs of clusters is found to be significantly correlated with their pairwise RMSD. Relative free energies and relative configurational entropies of the clusters are determined by explicit evaluation of the partition functions of the different clusters. These entropies are used to evaluate different entropy estimators for the largest 20 clusters as well as a subensemble comprising exclusively extended conformers. It is found that the quasi-harmonic entropy estimator operating in dihedral angle space performs better than the one using Cartesian coordinates. A recent generalization of the quasi-harmonic approach that computes Shannon entropies of probability distributions obtained by projecting the conformers along the eigenvectors of the covariance matrix performs similarly well. For the best entropy estimators, a linear correlation coefficient between 0.92 and 0.97 is found. Unexpectedly, when correlations between dihedral angles are neglected, the agreement with the reference entropies improved.     

Microwave spectroscopy measurements of rotational spectra and DFT calculations for two distinct structural isomers of 1,1'-dimethylferrocene

Microwave spectra were obtained for two distinct structural isomers of 1,1'-dimethylferrocene, an eclipsed synperiplanar isomer (phi = 0 degrees, the E0 isomer), with A = 1176.9003(2) MHz, B = 898.3343(2) MHz, C = 668.7469(2) MHz, and an eclipsed synclinal isomer (phi = 72 degrees, the E72 isomer) with A = 1208.7117(14) MHz, B = 806.4101(12) MHz, and C = 718.7179(8) MHz. The b-dipole, asymmetric-top spectra of both structural isomers were measured in the frequency range of 5-12 GHz using a Flygare-Balle type of spectrometer. A very good fit to observed transitions, with small distortion constants, was obtained for the E0 conformer, indicating that this conformer is nearly rigid. The deviations obtained in a similar least-squares fit for the E72 confomer are significantly larger, indicating possible fluxional behavior for this conformer. In addition, 7 out of the 26 transitions observed for the E72 isomer conformer clearly exhibit very small splittings, giving further evidence for internal motion. DFT calculations for the different possible conformations of 1,1'- dimethylferrocene arising from rotation of one methyl cyclopentadienyl ligand relative to the other about the nominal C5 axis by an angle phi (dihedral angle) were performed using the B3PW91 functional. The calculations converged and were optimized for five structures on this torsional potential energy surface corresponding to different dihedral angles phi; three yielded energy minima, and two gave energy maxima, corresponding to transition states. The experimental results are in very good agreement with the results of the DFT calculations.     

Application of geometric algebra for the description of polymer conformations

In this paper a Clifford algebra-based method is applied to calculate polymer chain conformations. The approach enables the calculation of the position of an atom in space with the knowledge of the bond length (l), valence angle (theta), and rotation angle (phi) of each of the preceding bonds in the chain. Hence, the set of geometrical parameters {l(i),theta(i),phi(i)} yields all the position coordinates p(i) of the main chain atoms. Moreover, the method allows the calculation of side chain conformations and the computation of rotations of chain segments. With these features it is, in principle, possible to generate conformations of any type of chemical structure. This method is proposed as an alternative for the classical approach by matrix algebra. It is more straightforward and its final symbolic representation considerably simpler than that of matrix algebra. Approaches for realistic modeling by means of incorporation of energetic considerations can be combined with it. This article, however, is entirely focused at showing the suitable mathematical framework on which further developments and applications can be built.     

Oriented ensembles in ultrafast electron diffraction.

Electron scattering expressions are presented which are applicable to very general conditions of implementation of anisotropic ultrafast electron diffraction (UED) experiments on the femto- and picosecond time scale. "Magic angle" methods for extracting from the experimental diffraction patterns both the isotropic scalar contribution (population dynamics) and the angular (orientation-dependent) contribution are described. To achieve this result, the molecular scattering intensity is given as an expansion in terms of the moments of the transition-dipole distribution created by the linearly polarized excitation laser pulse. The isotropic component (n=0 moment) depends only on population and scalar internuclear separations, and the higher moments reflect bond angles and evolve in time due to rotational motion of the molecules. This clear analytical separation facilitates assessment of the role of experimental variables in determining the influence of anisotropic orientational distributions of the molecular ensembles on the measured diffraction patterns. Practical procedures to separate the isotropic and anisotropic components of experimental data are evaluated and demonstrated with application to reactions. The influence of vectorial properties (bond angles and rotational dynamics) on the anisotropic component adds a new dimension to UED, arising through the imposition of spatial order on otherwise randomly oriented ensembles.     

Knowledge-based potential for the polypeptide backbone

A knowledge-based potential for the polypeptide backbone as a function of the dihedral angles is developed and tested. The potential includes correlations due to the conformations and compositions of adjacent residues. Its purpose is to serve as a major component of a coarse-grained protein potential by including the most relevant local interactions while averaging out nonbonded ones. A probability density estimation algorithm and a multi-resolution probability combination procedure are developed to produce smooth probability distributions and dihedral angle potentials. The potential is described by a set of two-dimensional dihedral angle surfaces involving the various combinations of amino acid triplets and duplets. Several tests are carried out to evaluate the quality of the potential. Monte Carlo simulations, using only the dihedral angle potential and a coarse-grained excluded volume potential, show that the resulting dihedral angle distributions and correlations are consistent with those extracted from protein structures. Additional simulations of unfolded proteins are carried out to measure the NMR residual dipolar coupling (RDC). Significant correlations are obtained between the simulations and the corresponding experiments consistent with other simulations in the literature. Finally, the dihedral angle entropies are calculated for the 20 amino acids. In particular, the entropy difference between alanine and glycine agrees with the ones computed from molecular dynamics simulations ( approximately 0.4 kcal/mol).     

Relations between edge lengths,dihedral and solid angles in tetrahedra

The tetrahedron, fundamental in organic chemistry, is examined in view of two important kinds of angles: to each tetrahedron edge belongs a dihedral angle (internal intersection angle between two faces having the edge as common side) and to each tetrahedron vertex a solid angle (area of the surface inside the tetrahedron on the unit sphere with the vertex as center). Based on preliminary lemmas, these angles are expressed in terms of edge lengths by an essential use of determinants. The resulting formulae enable to specify angle properties by edge lengths, especially with regard to equality and inequality of single solid angles or certain sums of dihedral angles. A special kind of equal solid angles leads to symmetry aspects. Finally, it is shown that by a particular rearrangement of edges in tetrahedra of a specific class some derived angle properties will be preserved.     

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.     

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.     

Calculation of intramolecular force fields from second-derivative tensors

A practical procedure (FUERZA) to obtain internal force constants from Cartesian second derivatives (Hessians) is presented and discussed. It allows a systematic analysis of pair atomic interactions in a molecular system, and it is fully invariant to the choice of internal coordinates of the molecule. Force constants for bonds or for any pair of atoms in general are defined by means of the eigenanalysis of their pair interaction matrix. Force constants for the angles are obtained from their corresponding two-pair interaction matrices of the two bonds or distances forming the angle, and the dihedral force constants are similarly obtained using their corresponding three-pair interaction matrices. © 1996 John Wiley & Sons, Inc.     

Using redundant internal coordinates to optimize equilibrium geometries and transition states

A redundant internal coordinate system for optimizing molecular geometries is constructed from all bonds, all valence angles between bonded atoms, and all dihedral angles between bonded atoms. Redundancies are removed by using the generalized inverse of the G matrix; constraints can be added by using an appropriate projector. For minimizations, redundant internal coordinates provide substantial improvements in optimization efficiency over Cartesian and nonredundant internal coordinates, especially for flexible and polycyclic systems. Transition structure searches are also improved when redundant coordinates are used and when the initial steps are guided by the quadratic synchronous transit approach. © 1996 by John Wiley & Sons, Inc.     

Free energy surfaces for the alpha(1 --> 4)-glycosidic linkage: implications for polysaccharide solution structure and dynamics

We present a potential of mean force surface for rotation about phi and psi dihedral angles of the alpha(1 --> 4)-glycosidic linkage in the maltose disaccharide (4-O-alpha-d-glucopyranosyl-d-glucopyranose) in aqueous solution. Comparison of the vacuum and solution free energy surfaces for maltose shows the principal effects of water to be an increase in the rotational freedom of the alpha(1 --> 4) linkage brought about by lowering the energy barrier for syn to anti conformational changes as well as expansion of the range of low-energy phi,psi conformations. This free energy analysis thus provides a thermodynamic and conformational rationale for the effects of water on alpha(1 --> 4)-linked polysaccharides and carbohydrate glasses.