Conformational energy of the 5-membered ring. Implication of geometric and energetic properties in the conformational characteristics

In a previous article (8) a geometrical study of the five-membered ring showed that: a) for the case of the 20 symmetrical C₂ and C_s conformations, the pseudorotation formulae for the torsion angles are a geometrical property of the ring; b) geometrical considerations alone are unable to define the puckering amplitude, the bond angle values, and the pathway between two symmetrical conformations. Here we examine how the energy equations enable us to define the deformation amplitude _m, establish the bond angles expressions and check the energy invariability along the pseudorotation circuit. The problem is next developed fully in the case where the bond and torsional energy only are considered: the literal expression¹ of _m is then given as a function of the bond angle which cancels out the bond angle energy. A numerical application is carried out on cyclopentane and the values of the parameters K^t, K¹ and used in the Conformational energy calculations are considered.Notations used 1 _i bond lengths 1 in the case of the regular ring - _i torsional angles - _i bond angles - 3/5 = 108 - 4/5 = 144 - , _{i i} – = complement to the 108 bond angle _i - _T - E Conformational energy of the 5-membered ring - E Conformational energy difference between planar and deformed ring - A _n Coefficients of the energy development in terms of - E _i ^l Bond energy relative to atom i (associated with angle _i) - K _i ^l Bond constant relative to atom i (associated with angle _i) - E _i ^l Torsional energy relative to the i ^th bond (associated with angle _i) - k _i ^l Torsional constant relative to the i ^th bond (associated with angle _i) - _i Angle _i value corresponding to zero bond energy E _i ^l (when the 5 atoms of the ring are identical, _i ) - r _ij Distance between atoms i and j - q _i Charge carried by atom i - e Constant of proportionality including the effective dielectric constant - A _ij, B_ij, d_ij Coefficients dependent on the nature of the atoms i and j and accounted for in the Van der Waals energy and hydrogen bond expressions - S (r _ij) Electrostatic contribution to the hydrogen bond energy - P Pseudorotation phase angle - _m Maximum torsional angle value characterising the deformation amplitudeM     

Prediction of Side Chain Orientations in Proteins by Statistical Machine Learning Methods

Abstract

We develop ways to predict the side chain orientations of residues within a protein structure by using several different statistical machine learning methods. Here side chain orientation of a given residue i is measured by an angle Ω_i between the vector pointing from the center of the protein structure to the C^α _i atom and the vector pointing from the C^α _i atom to the center of its side chain atoms. To predict the Ω_i angles, we construct statistical models by using several different methods such as general linear regression, a regression tree and bagging, a neural network, and a support vector machine. The root mean square errors for the different models range only from 36.67 to 37.60 degrees and the correlation coefficients are all between 30% and 34%. The performances of different models in the test set are, thus, quite similar, and show the relative predictive power of these models to be significant in comparison with random side chain orientations.     

Molecular Mechanics Studies of Molybdenum Disulphide Catalysts Parameterisation of Molybdenum and Sulphur

Abstract

A method for the parameterisation of molybdenum disulphide is presented which reproduces the crystal structure accurately. The method involves calculating parameters such that there is no net force contribution from any individual term of the potential on any atom. Ideal bond lengths and bond angles are taken from the X-ray crystal structure; stretching and bending force constants are calculated from a combination of spectroscopic data and quantum mechanics calculations, whereby the energy function with bond length or bond angle is calculated and fitted with an harmonic potential. For the non-bonded Lennard-Jones parameters, the dispersion coefficient C was calculated by an interpolation of existing published parameters using a multiple regression and then the crystal energy was minimised with respect to the van der Waals radius r₀ using a fixed crystal fragment.

These parameters were tested for various models of the hexagonal and rhombohedral forms of MoS₂. RMS fits between structures minimised with molecular mechanics and experimental models ranged from 0.006 Å to 0.012 Å.     

Transformation of the ϕ-ψ plot for proteins to a new representation with local helicity and peptide torsional angles as variables

A new representation of protein structure is obtained by the angular coordinate transformations η_i = (?_i+1?ψ_i)/2 and ξ_i = ?_i+1+ψ_i with careful mathematical attention to the cyclical boundary conditions of all of the variables involved. From published ?-ψ data it is possible to obtain a new η-ξ plot. As the angle ξ_i is varied from – 180° through 0° to + 180° in this plot, the local helicity of the polypeptide chain changes continuously and contiguously without sudden reversals in handedness. The variable, η_i, gives the torsional position of the ith peptide group. Some peptide groups in proteins, such as the second peptide residue in a type II β-turn, are nonhydrogen-bonded and can undergo considerable torsional oscillation. In such cases the η angle should be represented by a line whose length reflects the allowed dynamical variations in the peptide torsional position. Certain peptide residues in proteins may be able to undergo a complete torsional rotation of 360°. Such residues would be represented on the η-ξ plot as a straight line across the plot parallel to the abscissa. Other examples of the possible usefulness of this plot are also given.     

Conformational behavior of α,α-dialkylated peptides

The preferred conformations of N-acetyl-N′-methyl amides of some dialkylglycines have been determined by empirical conformational-energy calculations; minimum-energy conformations were located by minimizing the energy with respect to all the dihedral angles of the molecules. The conformational space of these compounds is sterically restricted, and low-energy conformations are found only in the regions of fully extended and helical structures. Increasing the bulkiness of the substituents on the C^α, the fully extended conformation becomes gradually more stable than the helical structure preferred in the cases of dimethylglycine. This trend is, however, strongly dependent on the bond angles between the substituents on the C^α atom: In particular, helical structures are favored by standard values (111°) of the N-C^α-C′ angle, while fully extended conformations are favored by smaller values of the same angle, as experimentally observed, for instance, in the case of α,α-di-n-propylglycine.     

Solution Structure of a Parallel Left-handed Double-helical Gramicidin-A Determined by 2D1H NMR

The structure of a parallel left-handed double-helical form of gramicidin was detected by circular dichroism spectroscopy and determined using 500 and 600 MHz NMR in CaCl₂/methanol solution. Measurements of TOCSY, DQF-COSY and NOESY spectra were converted into 604 distance and 48 torsional angle constraints for structure calculations. Stereospecific assignments and χ₁angles were calculated using³J_αβ, d_αβ(i,i), d_Nβ(i,i) and d_Nγ(i,i). χ₂angles were determined using d_αβ(i,i), d_Nβ(i,i), d_βδ(i,i), d_Nγ(i,i) and d_αγ(i,i). The calculations of initial structures were performed using the distance geometry/simulated annealing method in XPLOR. The initial structures were further refined and energy minimized using simulated annealing/molecular dynamics methods. Back-calculations for every generated structure were also performed to check their consistency with the experimental data.187 final structures with no violations above the threshold conditions (0.05 Å, 5°, 5°, 0.5 Å and 5° for bonds, angles, improper, NOE and cdihe, respectively) were produced from the 200 initial structures. Twenty structures with the lowest NOE energies were used for further analysis. The average r.m.s. deviations for the 20 structures are 0.64 Å for backbone and 1.1 Å for all non-hydrogen atoms.Gramicidin in this form, with approximately 5.7 residues per turn, is a parallel double helical dimer. The length along the helix axis is about 30 Å and the inner pore diameter varies from 1 to 2 Å. It is different from all other gramicidin structures determined to date. The presence of Ca^{2 +}stabilises a conformation that prevents the binding of monovalent cations. It is likely that this structure is related to a non-channel, antibiotic role of gramicidin.     

Geometry of experimentally observed RNA residues in tRNA and dinucleoside monophosphates: the effect of small variations in the backbone angles.

The polymerization of various experimentally observed conformers of RNA from tRNA and some dinucleoside monophosphates have been examined with a program that computes the basic helix parameters directly from the six backbone torsion angles ω′, ?′, ψ′, ψ, ?, ω to give n (= 360/θ), the number of residues per turn; h, the rise per residue; and r, the radius of the phosphate atoms from the helix axis. The single-stranded regions of tRNA that have A-form residues have a notably lower value of n than the double-stranded regions. The G-U “wobble” base pair is shown to be an energetically strained left-handed form. The A-form dinucleoside monophosphates also have a low value of n. A model of UpAl polymerized as a fourfold left-handed helix with the bases on the outside and phosphates on the inside is investigated for its sharp 90° turn angle characteristics. UpA2 cannot be polymerized due to a low values of h (1.31 Å) and r (2.72 Å), which cause steric hindering. An eightfold model of poly(rA) is discussed as are the nonhelical residues of tRNA. Finally, the effects of small changes in dihedral angles and bond lengths and angles on the helical parameters are investigated and discussed by way of explaining this behavior.     

An Unbalanced Two-way Model With Random Effects Having Unequal Variances

Consider the model y_ijk=u ± a_i ± b_i ± c_ij ± e_ijk i=1, 2,…, t; j=1, 2,…b; k=1, 2,…,n_ij where μ is a constant and a_i, b_i, c_ij are distributed independently and normally with zero means and variances Δ₂ Δ2/bd_ij and δ₂ respectively. It is assumed that d_i's, and d_ij's are known (positive) constants (for all i and j). In this paper procedures for estimating the variance components (Δ₂, Δ²_b and Δ²_a) and for testing the hypothesis H_oc:Δ²_c/Δ² = y₃ and H_oa:Δ²_b/Δ² = y₄ (where y₂, y₃, and y₄, are specified constants) are presented. A generalization for the mixed model case is discussed in the last section.     

Distance-constraint approach to higher-order structures of globular proteins with empirically determined distances between amino acid residues

An analysis of higher-order structures of globular proteins by means of a distance-constraint approach is presented. Conformations are generated for each of 21 test proteins of small and medium sizes by optimizing an objective functionf=w _ij(d _ij–d _ij)², whered _ij is a distance between residuesi andj in a calculated conformation, d _ij is an assigned distance to the (ij) pair of residues which is determined based on the statistics of known three-dimensional structures of 14 proteins in the earlier study, andw _ij is a weighting factor. d _ij involves information about hydrophobicity and hydrophilicity of each amino acid residue and about connectivity of a polypeptide chain. In these calculations, only the amino acid sequence is used as input data specific to a calculated protein. With respect to higher-order structures regenerated in the optimized conformations, the following properties are analyzed: (a) N14 of a residue, defined as the number of residues surrounding the residue located within a sphere of radius of 14 Å; (b) root-mean-square differences of the global and local conformations from the corresponding X-ray conformations; (c) distance profiles in the short and medium ranges; and (d) distance maps. The effects of supplementary information about locations of secondary structures and disulfide bonds are also examined to discuss the potential ability of this methodology to predict the three-dimensional structures of globular proteins.     

An Unbalanced Nested Model with Random Effects Having Unequal Variances

Consider the model Y_ijk=μ + a_i + b_ij + e_ijk (i=1, 2,…, t; j=1,2,…, B_i; k=1,2…,n_ij), where μ is a constant and a¹,b_ij and e_ijk are distributed independently and normally with zero means and variances σ²_ad_ij and σ², respectively, where it is assumed that the d_i's and d_ij's are known. In this paper procedures for estimating the variance components (σ², σ²_a and σ²_b) and for testing the hypothesis σ²_b = 0 and σ²_a = 0 are presented. In the last section the mixed model y_ijk, where x_ijkkm are known constants and β_m's are unknown fixed effects (m = 1, 2,…,p), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β_m's.     

Correlated Variations of Bond Lengths in Pseudorotating Furanose Rings

Abstract

Correlated variations of bond lengths in pseudorotating furanose rings are investigated by a theoretical method. At first, matrix equations are proposed to determine the spatial coordinates of the ring atoms from the bond lengths, the bond angles, and the pseudorotation parameters. Secondly, a necessary functional form of the variations of the bond lengths of five-membered rings is derived from a consideration of symmetry. Finally, demonstrations are performed on a furanose ring whose bond angle variations have been precisely determined by experimental analyses. The resulting bond length variations are:

δR_i = β_iCos(8/5π · (i-2)+2P)

where δR_i is the variation of the bond length between atoms i and i+1, P is the pseudorotation phase, and β_iis a negative constant about ?0.01 Å. These bond length variations are balanced on the apparent strains of the bond lengths and the bond angles.     

Synthesis and enzymatic evaluation of phosphoramidon and its β anomer: Anomerization of α-l-rhamnose triacetate upon phosphitylation

A novel and efficient strategy for the synthesis of phosphoramidon and its β anomer has been developed by manipulating the anomerization of α-l-rhamnose triacetate upon phosphitylation. The experimental results suggest that proton transfer, bond rotation, and N atom are the key factors for the anomerization. The determined K_i and K_d values establish that phosphoramidon prepared by this method possesses excellent biological activity, and indicate that the contacts of rhamnose moiety with the enzyme have limited contribution to the binding.     

The structure of native cellulose

Native cellulose has been shown to consist of a crystalline array of parallel chains, based on the X-ray diffraction data for specimens from the sea alga Valonia ventricosa. The unit cell is monoclinic with dimensions a = 16.34 Å, b = 15.72 Å, c = 10.38 Å (fiber axis), and β = 97.0°. The space group is P2₁ and the cell contains disaccharide segments of eight chains. Models containing chains with the same sense (parallel) or alternating sense (antiparallel) were refined against the intensity data using rigidbody least squares procedures. The results show a preference for a parallel chain structure with specific chain polarity with respect to the c axis. The refinement places the ? CH₂OH side chains approximately 20′ from the so-called tg conformation, with a result that an 02′? H…06 intramolecular bond is formed. The structure also contains an 03? H…05′ intramolecular bond and an 06? H…03 intermolecular bond along the a axis. All these bonds lie in the 020 planes, and the structure is an array of hydrogen-bonded sheets. A major consequence of this work is that regular chain folding can be ruled out and cellulose is seen as extended chain polymer single crystals.     

Directional structural features of globular proteins

We describe a novel presentation of the conformation of the backbone atoms for proteins of known structure. Given the C^α atom cartesian co-ordinates from X-ray crystallography, a matrix is calculated, where the ijth element of the matrix is the cosine of the angle between the direction of the chain at residue i and the direction of the chain at residue j. These “direction matrices” have distinctive patterns which correspond to α-helix, extended structure, straight or bent segments, “superhelix”, and many other important structural features. We discuss the direction matrices for a number of proteins, and make some generalizations on the basic principles of protein folding.     

ω‐Turn: A novel β‐turn mimic in globular proteins stabilized by main‐chain to side‐chain CH···O interaction

Mimicry of structural motifs is a common feature in proteins. The 10‐membered hydrogen‐bonded ring involving the main‐chain C?O in a β‐turn can be formed using a side‐chain carbonyl group leading to Asx‐turn. We show that the N? H component of hydrogen bond can be replaced by a C^γ‐H group in the side chain, culminating in a nonconventional C? H···O interaction. Because of its shape this β‐turn mimic is designated as ω‐turn, which is found to occur ～three times per 100 residues. Three residues (i to i + 2) constitute the turn with the C? H···O interaction occurring between the terminal residues, constraining the torsion angles ?_{i + 1}, ψ_{i + 1}, ?_{i + 2} and χ′_{1(i + 2)} (using the interacting C^γ atom). Based on these angles there are two types of ω‐turns, each of which can be further divided into two groups. C^β‐branched side‐chains, and Met and Gln have high propensities to occur at i + 2; for the last two residues the carbonyl oxygen may participate in an additional interaction involving the S and amino group, respectively. With Cys occupying the i + 1 position, such turns are found in the metal‐binding sites. N‐linked glycosylation occurs at the consensus pattern Asn‐Xaa‐Ser/Thr; with Thr at i + 2, the sequence can adopt the secondary structure of a ω‐turn, which may be the recognition site for protein modification. Location between two β‐strands is the most common occurrence in protein tertiary structure, and being generally exposed ω‐turn may constitute the antigenic determinant site. It is a stable scaffold and may be used in protein engineering and peptide design. Proteins 2015; 83:203–214. © 2014 Wiley Periodicals, Inc.     

Modeling of the phase equilibria of polystyrene in methylcyclohexane with semi-empirical quantum mechanical methods I

A method for calculating interaction parameters traditionally used in phase-equilibrium computations in low-molecular systems has been extended for the prediction of solvent activities of aromatic polymer solutions (polystyrene+methylcyclohexane). Using ethylbenzene as a model compound for the repeating unit of the polymer, the intermolecular interaction energies between the solvent molecule and the polymer were simulated. The semiempirical quantum chemical method AM1, and a method for sampling relevant internal orientations for a pair of molecules developed previously were used. Interaction energies are determined for three molecular pairs, the solvent and the model molecule, two solvent molecules and two model molecules, and used to calculated UNIQUAC interaction parameters, a _ij and a _ji . Using these parameters, the solvent activities of the polystyrene 90,000 amu+methylcyclohexane system, and the total vapor pressures of the methylcyclohexane+ethylbenzene system were calculated. The latter system was compared to experimental data, giving qualitative agreement. Figure Solvent activities for the methylcylcohexane(1)+polystyrene(2) system at 316 K. Parameters a _ij (blue line) obtained with the AM1 method; parameters a _ij (pink line) from VLE data for the ethylbenzene+methylcyclohexane system. The abscissa is the polymer weight fraction defined as ₂(x ₁)=(1–x ₁)M ₂/[x ₁ M ₁+(1–x ₁)M ₂], where x ₁ is the solvent mole fraction and M _i are the molecular weights of the components.An erratum to this article can be found at     

Conformational energy studies on N-methylated analogs of thyrotropin releasing hormone, enkephalin, and luteinizing hormone-releasing hormone

Empirical conformational energy calculations have been carried out for N-methyl derivatives of alanine and phenylalanine dipeptide models and N-methyl-substituted active analogs of three biologically active peptides, namely thyrotropin-releasing hormone (TRH), enkephalin (ENK), and luteinizing hormone-releasing hormone (LHRH). The isoenergetic contour maps and the local dipeptide minima obtained, when the peptide bond (ω) preceding the N-methylated residue is in the trans configuration show that (1) N-methylation constricts the conformational freedom of both the ith and (i + 1)th residues; (2), the lowest energy position for both residues occurs around ? = ?135° ± 5° and ψ = 75° ± 5°, and (3) the α_L conformational state is the second lowest energy state for the (i + 1)th residue, whereas for the ith residue the C₅ (extended) conformation is second lowest in energy. When the peptide bond (ω_i) is in the cis configuration the ith residue is energetically forbidden in the range ? = 0° to 180° and ψ = ?180° to +180°. Conformations of low energy for ω_i = 0° are found to be similar to those obtained for the trans peptide bond. In all the model systems (irrespective of cis or trans), the α_R conformational state is energetically very high. Significant deviations from planarity are found for the peptide bond when the amide hydrogen is replaced by a methyl group. Two low-energy conformers are found for [(N-Me)His²]TRH. These conformers differ only in the ? and ψ values at the (N-Me)His² residue. Among the different low-energy conformers found for each of the ENK analogs [D -Ala²,(N-Me)Phe⁴, Met⁵]ENK amide and [D -Ala²,(N-Me)Met⁵]ENK amide, one low-energy conformer was found to be common for both analogs with respect to the side-chain orientations. The stability of the low-energy structures is discussed in the light of the activity of other analogs. Two low-energy conformers were found for [(N-Me)Leu⁷]LHRH. These conformations differ in the types of bend around the positions 6 and 7 of LHRH. One bend type is eliminated when the active analog [D -Ala⁶,(M-Me)Leu⁷]LHRH is considered.     

Limiting similarity and the form of the competition coefficient

The question of whether there is a limit to the similarity of competing species has previously been investigated by a number of authors. These studies have all used the Lotka-Volterra model of competition, and have assumed that the competition coefficient α_ij may be calculated using the expression, α_ij = ∝ U_i(R) U_j(R) dR/∝ (U_i(R))² dR. In this paper, the generality of this formula is questioned and two alternative expressions for α_ij are proposed. When these expressions are used in an analysis of limiting similarity, qualitatively different conclusions emerge regarding the existence and nature of this limit, using either deterministic or stochastic models. The relevance of these findings to theories of character convergence and similarity barriers is discussed. The available field evidence does not strongly support the validity of the formula for α_ij used in previous studies. Since a given method of calculating α_ij must be derived from a higher level model, it is suggested that the Lotka-Volterra model is not sufficient in an investigation of limiting similarity.     

Geometry of the five-membered ring mathematical demonstration of the pseudorotation formulae

The geometrical relations between the 15 typical parameters (bond lengths and angles, torsion angles) of a five-membered ring are derived for any ring then for a regular one. It is demonstrated that for the case of the 20 symmetrical C ₂ and C _sconformations, only geometrical considerations are needed to obtain the pseudorotation formulae for the torsion angles. However, the puckering intensity as well as the bond angle values cannot be expressed from geometrical constraints alone but would require energetical considerations.     

Interaction of molecules with nucleic acids. II. Two pairs of families of intercalation sites, unwinding angles, and the neighbor-exclusion principle

Intercalation-site geometries are generated for a tetramer duplex extracted from B-DNA. Glycosidic angles and puckers of the deoxyribose sugar groups bonded to base pairs BP₁ and BP₄, namely, those at either end of the tetramer duplex, are assumed to be those of B-DNA to insure continuity. All possible geometrical conformations for combinations of C(2′)-endo, C(3′)-endo, C(2′)-exo, and C(3′)-exo sugar puckers are determined for the tetranucleotide backbone. Those with minimum energy are selected as candidates for intercalation sites. Calculations reveal two pairs of physically meaningful families of intercalation sites which occur in two distinct regions, I and II, of helical angles which orient BP₂ relative to BP₃ and with the helical axis disjointed between these base pairs. For each site I and II within BP₂ and BP₃, there are two distinct backbone conformations, A and B, connecting BP₃ to BP₄ or BP₁ to BP₂ which do not disrupt backbone conformations connecting BP₂ to BP₃. Hence two pairs, IA and IB, and IIA and IIB, of intercalation sites exist in which the sugar puckers along the backbone of the tetramer alternate from C(2′)-endo to C(3′)-endo on the backbone (5′p3′) connecting BP₂ to BP₃. The glycosidic angles of the C(3′)-endo sugar χ₃^γ are, coincidentally, 80° ± 2° for both conformations γ = A and B connecting BP₃ to BP₄ along the phosphate backbone (5′p3′). Consistent with the theoretical results, the experimental unwinding angles can be grouped into two categories with absolute values of 18° and 26°. The theoretical unwinding angles for sites IA and IB of 16° and for sites IIA and IIB of 20° occur for a displacement of -0.8 Å in the helical axes of BP₂ and BP₃ and for a 100% G·C composition, with a decrease depending on the amount of A·T base pairs present. Ratios of theoretical unwinding angles of sites I and II, which range from 0.75 to 0.84 for the two principal sites, compare well with the experimental value of 0.71. The theoretical results, in agreement with experimental observation, provide a new interpretation of the nature and conformation of the possible binding sites. Conformations obtained from these studies of intercalation sites in a tetramer duplex are used to rationalize the well-known neighbor-exclusion principle. The possibility of violation of this principle is demonstrated by the existence of two families of physically meaningful conformations. Conformations of unconstrained dimer duplexes are also obtained, one of which corresponds to the experimental crystal structure of ethidium–dinucleoside complexes, but these cannot be joined to the B-DNA structure. Backbone conformations of the tetramer duplex can be constructed until the base-pair separation reaches 8.25 Å, which may limit the molecules that can intercalate.