Predicting protein-ligand binding affinities: a low scoring game?

We have investigated the performance of five well known scoring functions in predicting the binding affinities of a diverse set of 205 protein-ligand complexes with known experimental binding constants, and also on subsets of mutually similar complexes. We have found that the overall performance of the scoring functions on the diverse set is disappointing, with none of the functions achieving r(2) values above 0.32 on the whole dataset. Performance on the subsets was mixed, with four of the five functions predicting fairly well the binding affinities of 35 proteinases, but none of the functions producing any useful correlation on a set of 38 aspartic proteinases. We consider two algorithms for producing consensus scoring functions, one based on a linear combination of scores from the five individual functions and the other on averaging the rankings produced by the five functions. We find that both algorithms produce consensus functions that generally perform slightly better than the best individual scoring function on a given dataset.     

Scoring protein–protein docked structures based on the balance and tightness of binding

One main issue in protein-protein docking is to filter or score the putative docked structures. Unlike many popular scoring functions that are based on geometric and energetic complementarity, we present a set of scoring functions that are based on the consideration of local balance and tightness of binding of the docked structures. These scoring functions include the force and moment acting on one component (ligand) imposed by the other (receptor) and the second order spatial derivatives of protein-protein interaction potential. The scoring functions were applied to the docked structures of 19 test targets including enzyme/inhibitor, antibody/antigen and other classes of protein complexes. The results indicate that these scoring functions are also discriminative for the near-native conformation. For some cases, such as antibody/antigen, they show more discriminative efficiency than some other scoring functions, such as desolvation free energy (deltaG(des)) based on pairwise atom-atom contact energy (ACE). The correlation analyses between present scoring functions and the energetic functions also show that there is no clear correlation between them; therefore, the present scoring functions are not essentially the same as energy functions.     

How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: energy levels of coupled systems with as many as 16 coordinates

In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.     

Configuration interaction matrix elements. III. Spin functions relating the unitary and symmetric group approaches

Spin functions that are compatible with orbital ordering and geminal antisymmetry conditions are investigated. It is shown that two widely used classes of spin functions, namely, the spin-bonded functions and Yamanouchi–Kotani (or, equivalently, Gelfand–Tsetlin) functions possess these properties. The relationship of the latter with Young–Yamanouchi spin functions is also outlined using graphical techniques of spin algebras. These techniques are also used to rederive the Hamiltonian matrix elements between spin-bonded functions and to show the relationship among the various schemes used in this case.     

A multidimensional discrete variable representation basis obtained by simultaneous diagonalization

Direct product basis functions are frequently used in quantum dynamics calculations, but they are poor in the sense that many such functions are required to converge a spectrum, compute a rate constant, etc. Much better, contracted, basis functions, that account for coupling between coordinates, can be obtained by diagonalizing reduced dimension Hamiltonians. If a direct product basis is used, it is advantageous to use discrete variable representation (DVR) basis functions because matrix representations of functions of coordinates are diagonal in the DVR. By diagonalizing matrices representing coordinates it is straightforward to obtain the DVR that corresponds to any direct product basis. Because contracted basis functions are eigenfunctions of reduced dimension Hamiltonians that include coupling terms they are not direct product functions. The advantages of contracted basis functions and the advantages of the DVR therefore appear to be mutually exclusive. A DVR that corresponds to contracted functions is unknown. In this paper we propose such a DVR. It spans the same space as a contracted basis, but in it matrix representations of coordinates are diagonal. The DVR basis functions are chosen to achieve maximal diagonality of coordinate matrices. We assess the accuracy of this DVR by applying it to model four-dimensional problems.     

Study of pair correlation in three-electron atoms

Pair correlation in the ground state of the Li isoelectronic sequence is studied through four approximate wave functions which incorporate inter- and intrashell pair correlation. Of these functions, two possess symmetry appropriate to a three-electron system, while two do not. The functions are not variational functions in the usual sense. They are instead fixed linear combinations of products of orbitals and pair functions for the appropriate states of two-electron atoms. They are considered here as zero-order approximations to the exact wave functions, and the corresponding zero-order Hamiltonians are obtained. The simplest of these functions is improved by the introduction of a screening parameter for the “outer” electron. This latter function is found to be a satisfactory compromise between accuracy and simplicity and is proposed for study via higher-order perturbation theory.     

Protein function prediction using functional inter-relationship

With the growth of high throughput sequencing techniques, the generation of protein sequences has become fast and cheap, leading to a huge increase in the number of known proteins. However, it is challenging to identify the functions being performed by these newly discovered proteins. Machine learning techniques have improved traditional methods’ efficiency by suggesting relevant functions but fails to perform well when the number of functions to be predicted becomes large. In this work, we propose a machine learning-based approach to predict huge set of protein functions that use the inter-relationships between functions to improve the model’s predictability. These inter-relationships of functions is used to reduce the redundancy caused by highly correlated functions. The proposed model is trained on the reduced set of non-redundant functions hindering the ambiguity caused due to inter-related functions. Here, we use two statistical approaches 1) Pearson’s correlation coefficient 2) Jaccard similarity coefficient, as a measure of correlation to remove redundant functions. To have a fair evaluation of the proposed model, we recreate our original function set by inverse transforming the reduced set using the two proposed approaches: Direct mapping and Ensemble approach. The model is tested using different feature sets and function sets of biological processes and molecular functions to get promising results on DeepGO and CAFA3 dataset. The proposed model is able to predict specific functions for the test data which were unpredictable by other compared methods. The experimental models, code and other relevant data are available at https://github.com/richadhanuka/PFP-using-Functional-interrelationship.     

Concerning the theory of chirality functions

Within the scope of the theory of chirality functions, qualitatively complete chirality functions are subject to restrictions concerning both generality and applicability. In contrast thereto, the concept of qualitative supercompleteness results in less restrictive requirements for chirality functions. Consequently, the applicability of qualitatively supercomplete chirality functions is unlimited with respect to the number of ligand kinds. Given this concept, a group theoretical treatment is performed supplying the formal conditions of qualitative supercompleteness. Subsequently a construction rule for qualitiatively supercomplete chirality functions is presented, which is elaborated in detail in the appendix. On combining physical considerations with the requirement of qualitative supercompleteness the resulting chirality functions appear to include all the possible interactions within and/or between ligands and skeleton. From both a mathematical and a physical point of view these chirality functions should be adequate for describing the chiroptical properties of molecules belonging to a given skeletal class. Nevertheless, all the other critical objections to the theory of chirality functions remain.     

A Mathematical Discussion on Density and Shape Functions, Vector Semispaces and Related Questions

Density and shape functions are studied from the point of view of vector semispace structure and properties. Useful characteristics based on the shell structure of vector semispaces are used to analyze some properties of both functions. Fukui functions and quantum similarity indices are also studied when basic applications of the theory are discussed. Construction of approximate density functions and pseudo wave functions is also outlined. Finally, the original DFT variational theorem is reformulated within the frame of the shape function.     

Properties of atoms in molecules

Atomic charges calculated by the population analysis method for three types of semi-empirical wave functions have been compared with charges obtained by integrating the corresponding electronic density functions over individual atomic regions. It was found that the two sets of charges compare quite well for CNDO wave functions and for extended-Hückel functions which are in terms of orthogonalized basis orbitals. However only the CNDO charges are reasonably close to those obtained by integrating near-Hartree-Fock electronic density functions.     

Energies of the first row atoms from quantum Monte Carlo

All-electron variational and diffusion quantum Monte Carlo calculations of the ground state energies of the first row atoms (from Li to Ne) are reported. The authors use trial wave functions of four types: single-determinant Slater-Jastrow wave functions, multideterminant Slater-Jastrow wave functions, single-determinant Slater-Jastrow wave functions with backflow transformations, and multideterminant Slater-Jastrow wave functions with backflow transformations. At the diffusion quantum Monte Carlo level and using their multideterminant Slater-Jastrow wave functions with backflow transformations, they recover 99% or more of the correlation energies for Li, Be, B, C, N, and Ne, 97% for O, and 98% for F.     

Solvation of nonpolar groups of biomacromolecules

The method of correlation functions used in the theory of liquids is suggested for investigation of the solvation shell structure in the vicinity of nonpolar surfaces. An approach making it possible to obtain the molecule correlation functions in the solvation shell is worked out. In this approach the correlation functions are the product of the interatomic correlation functions.     

Fourier transforms of atomic orbitals. II. Convolution theorems

Analytic expressions have been derived for tensor convolutions of basis functions of exponential class (the reduced Bessel functions, the Slater, and the hydrogen-like functions). For reduced Bessel functions, in particular, the method suggested in the paper is a more clear and compact alternative to the laborious proof of the important Filter and Steinborn convolution theorem.     

Application of Maclaurin Series in Relating Interatomic Potential Functions: A Review

This paper provides a short review on the application of Maclaurin series in relating potential functions within the same category of interatomic interaction. The potential functions covered are those commonly adopted in computational chemistry softwares. While various mathematical approaches have been employed in generating relationships amongst potential functions, the use of Maclaurin series has been prevalent recently due to the increasing application of polynomial series-type potential functions. In the case of covalent bond-stretching, the Maclaurin series for the exponential function is used to transform the Morse potential into the polynomial series potential, and vice versa. For covalent bond-bending, Maclaurin series for the sine and cosine functions were employed to extract polynomial angle series potential from the Fourier series and harmonic cosine potential functions, and vice versa. Finally, both the exponential and the 1/(1 – x) expressions in Maclaurin series were used in obtaining the exact relationship for the repulsive terms between two potential functions.     

Nuclear attraction and electron interaction integrals of exponentially decaying functions and the Poisson equation

Summary The technique proposed by O-Ohata and Ruedenberg (J Math Phys 7:547 (1966)) and by Silver and Ruedenberg (J Chem Phys 49:4306 (1968)) of computing nuclear attraction and electron interaction integrals by solving an inhomogeneous Laplace equation can also be applied ifB functions (Filter E, Steinborn EO (1978) Phys Rev A 18:1) are used as basis functions in atomic and molecular calculations. It is shown that because of the remarkable mathematical properties ofB functions the derivation of compact explicit expressions for the multicenter integrals mentioned above is particularly simple. These results are also of interest in the context of other exponentially decaying functions, since all the other commonly occurring exponentially decaying functions as, for instance, Slater functions or bound state hydrogen eigenfunctions can be expressed as simple linear combinations ofB functions. Consequently, their multicenter integrals can also be expressed in terms of multicenter integrals ofB functions.Dedicated to Prof. Klaus Ruedenberg on the occasion of his 70th birthday     

Test of the consistency of various linearized semiclassical initial value time correlation functions in application to inelastic neutron scattering from liquid para-hydrogen

The linearized approximation to the semiclassical initial value representation (LSC-IVR) is used to calculate time correlation functions relevant to the incoherent dynamic structure factor for inelastic neutron scattering from liquid para-hydrogen at 14 K. Various time correlations functions were used which, if evaluated exactly, would give identical results, but they do not because the LSC-IVR is approximate. Some of the correlation functions involve only linear operators, and others involve nonlinear operators. The consistency of the results obtained with the various time correlation functions thus provides a useful test of the accuracy of the LSC-IVR approximation and its ability to treat correlation functions involving both linear and nonlinear operators in realistic anharmonic systems. The good agreement of the results obtained from different correlation functions, their excellent behavior in the spectral moment tests based on the exact moment constraints, and their semiquantitative agreement with the inelastic neutron scattering experimental data all suggest that the LSC-IVR is indeed a good short-time approximation for quantum mechanical correlation functions.     

One-Parameter double-zeta atomic functions for the hartree–fock energies of helium isoelectronic sequence

The double-zeta atomic functions are characterized by the nuclear charge z of the two-electron atomic system. The Hartree–Fock total energies and the corresponding orbital energies are calculated using various atomic wave functions for the helium isoelectronic sequence. The expectation values rⁿ of various wave functions are also examined. It is found that the accuracy of our one-parameter double-zeta functions corresponds to the accuracy of the usual five-parameter double-zeta functions.     

The spin degeneracy problem in the AMO method

The use of different spin functions in the AMO method was investigated for benzene, the ring of six H atoms, and fulvene. The additional improvement in the energy obtained by the use of a linear combination of different spin functions is quite small for the singlet ground state. It is pointed out that there are great differences in energy between functions having the same spatial function but different spin functions.     

Calculation of molecular quadrupole moments and a demonstration of the importance of overlap densities in the theory of polyatomic molecules

A computational method for calculating quadrupole moments from molecular wave functions in a Slater orbital basis set is described. Using both IEHT and CNDO wave functions quadrupole moments for a series of polyatomic molecules are calculated. They are compared with experimental results and the IEHT wave functions are found to give agreement with experiment while CNDO wave functions do not. The importance of bicentric densities (overlap densities) in the calculation of multipole moments is shown. This is followed by a discussion of the usefulness of these wave functions for a quantitative characterization of the electronic structure of large molecules.     

A detailed view of the Gaussian–Lorentzian sum and product functions and their comparison with the Voigt function

The Gaussian–Lorentzian sum (GLS) and product (GLP) functions remain important in X-ray photoelectron spectroscopy (XPS) peak fitting. Here, we present a detailed view of these functions, comparing them with each other and with the Voigt function (the “LA(m)” function). First, we show the GLS, GLP, and LA(m) functions as a function of their mixing parameters, m, which reveals differences between them. We then illustrate the use of these functions to fit a series of spectra acquired at different pass energies (resolutions). Next, we show the underlying Gaussian and Lorentzian components of a series of GLS and GLP functions as a function of m, which confirms that the GLS is a simple linear combination of Gaussian and Lorentzian functions. However, one of the two functions used to make the GLP can be very wide, that is, at its extremes, one of these functions has infinite width. We then discuss a plot of the areas of the GLS, GLP, and LA(m) functions as a function of m, which reveals the expected, linear increase in area of the GLS, but nonlinear changes in the areas of the other two functions. Finally, to better understand them, we fit these functions to each other. These results indicate that the GLS and GLP better match the LA(m) function at lower and higher values of the mixing parameter, respectively.