Comparing approximations to spatio-temporal models for epidemics with local spread

Analytical methods for predicting and exploring the dynamics of stochastic, spatially interacting populations have proven to have useful application in epidemiology and ecology. An important development has been the increasing interest in spatially explicit models, which require more advanced analytical techniques than the usual mean-field or mass-action approaches. The general principle is the derivation of differential equations describing the evolution of the expected population size and other statistics. As a result of spatial interactions no closed set of equations is obtained. Nevertheless, approximate solutions are possible using closure relations for truncation. Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. This study is made in the context of an SIS model for plant-disease epidemics introduced in Filipe and Gibson (1998, Studying and approximating spatio-temporal models for epidemic spread and control, Phil. Trans. R. Soc. Lond. B 353, 2153–2162) of which the contact process [Harris, T. E. (1974), Contact interactions on a lattice, Ann. Prob. 2, 969] is a special case. The various methods of approximation are derived and explained and their predictions are compared and tested against simulation. The merits and limitations of the various approximations are discussed. A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.     

Density functional theory study of the potassium complexation of an unsymmetrical 1,3-alternate calix[4]-crown-5-N-azacrown-5 bearing two different crown rings

Theoretical studies of an unsymmetrical calix[4]-crown-5-N-azacrown-5 (1) in a fixed 1,3-alternate conformation and the complexes 1·K⁺(a), 1·K⁺(b), 1·K⁺(c) and 1·K⁺K⁺ were performed using density functional theory (DFT) at the B3LYP/6-31G* level. The fully optimized geometric structures of the free macroligand and its 1:1 and 1:2 complexes, as obtained from DFT calculations, were used to perform natural bond orbital (NBO) analysis. The two main types of driving force metal–ligand and cation–π interactions were investigated. NBO analysis indicated that the stabilization interaction energies (E ₂) for O…K⁺ and N…K⁺ are larger than the other intermolecular interactions in each complex. The significant increase in electron density in the RY* or LP* orbitals of K⁺ results in strong host–guest interactions. In addition, the intermolecular interaction thermal energies (ΔE, ΔH, ΔG) were calculated by frequency analysis at the B3LYP/6-31G* level. For all structures, the most pronounced changes in the geometric parameters upon interaction are observed in the calix[4]arene molecule. The results indicate that both the intermolecular electrostatic interactions and the cation–π interactions between the metal ion and π orbitals of the two pairs that face the inverted benzene rings play a significant role.     

Novel moment closure approximations in stochastic epidemics

Moment closure approximations are used to provide analytic approximations to non-linear stochastic population models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and third-order moment closure approximations which we apply to the stochastic SI and SIS epidemic models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial distribution. In addition, a closure approximation based on mixture distribution is developed in order to capture the behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass at 0 which represents the probability of extinction. Two third-order versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results shows: (1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and (2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model, in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likehood function and carry out point and interval parameter estimation.     

Hydrogen bonding interactions in noradrenaline-DMSO complexes: DFT and QTAIM studies of structure,properties and topology

The hydrogen bonding interactions between noradrenaline (NA) and DMSO were studied with density functional theory (DFT) regarding their geometries, energies, vibrational frequencies, and topological features of the electron density. The quantum theory of atoms in molecules (QTAIM) and the natural bond orbital (NBO) analyses were employed to elucidate the hydrogen bonding interaction characteristics in noradrenaline-DMSO complexes. The H-bonds involving the hydroxyls hydrogen in NA and the O atom in DMSO are dominant intermolecular H-bonds and are stronger than other H-bonds involving the methyl hydrogen of DMSO as a H-donor. The weak H-bonds also include a π H-bond which involves the benzene ring as a H-donor or H-acceptor. QTAIM identified the weak H-bonds formed between the methyl hydrogen of DMSO and the N atom in NA in some complexes (AB5, AB6 and AB7), which cannot be further confirmed by NBO and other methods, so there are probably no interactions between hydrogen and nitrogen atoms among these complexes. A good linear relationship between logarithmic electron density (lnρ _b ) at the bond critical point (BCP) and structural parameter (δR _H···Y) was found. The formations of new H-bonds in some complexes are helpful to strengthen the original intramolecular H-bond, this is attributed to the cooperativity of H-bonds in complexes and can be learned from the structure results and the NBO and QTAIM analyses. Analysis of various physically meaningful contributions arising from the energy decomposition procedures show that the orbital interactions of H-bond is predominant during the formation of the complex, moreover, both the hydrogen bonding interaction and the structural deformation are responsible for the stability of the complexes.     

Ab Initio Calculations of the Structure and Properties of Large Atomic Clusters

Abstract

A discussion is given of the problems involved in computing the total energy, using local density functional methods, of a cluster of atoms with a real space basis set of Gaussian orbitals. Particular attention is given to the methods used to evaluate the Hartree and exhange-correlation energies and their potentials. Several applications are described: molecular structures and properties, the bond lengths and dynamic properties of bulk silicon and diamond, the local vibratory mode of carbon in silicon, and the structures of H and H related complexes in diamond and gallium arsenide.     

The electronic structure of Ti(BH4)3: Photoelectron spectra and calculation of vertical ionization energies

He I and He II PE spectra of Ti(BH₄)₃ are reported and assigned by reference to density functional calculations on the molecule and cation. The performance of different functionals in predicting the first vertical ionization energy is assessed. Calculations based on hybrid functionals are found to give ionisation energies closer to the experimental value than those using pure density functionals. The accuracy of the ΔSCF method and time dependent density functional theory in calculating higher vertical ionization energies is also examined.     

Microsolvation of aminoethanol: a study using DFT combined with QTAIM

The microsolvation of aminoethanol (AE) with one, two, three or four water molecules was investigated using a density functional theory (DFT) approach. Quantum theory of atoms in molecules (QTAIM) analyses were employed to elucidate the hydrogen-bonding characteristics of AE–(H₂O) _n (n = 1–4) complexes. The results showed that AE tends to break its intramolecular OH^AE···N^AE hydrogen bond (H-bond) upon microsolvation and form intermolecular H-bonds with water molecules, while complexes that retain the intramolecular OH^AE···N^AE H-bond show reduced stabilities. The intermolecular H-bond that forms between the nitrogen atom of AE and the hydroxyl of a water molecule is the strongest one for the most stable AE–(H₂O) _n (n = 1–4) complexes, and as n increases from 1 to 4 they grow stronger. The partial covalent character of this H-bond was confirmed by QTAIM analyses. Many-body interaction analysis showed that the relaxation energies and two- and three-body energies make significant contributions to the binding energies of the complexes.     

Coordination numbers in hydrated Cu(II) ions

The potential energy surface of [Cu(H₂O)_n]²⁺ clusters with n?=?12, 16, and 18 was explored by using a modified version of the simulated annealing method. Such exploration was carried out by using the PM7 semiempirical method to obtain around 100,000 isomers, which provide candidates to be optimized with PBE0-D3, M06-2X, and BHLYP exchange-correlation functionals coupled with the 6–311++G** basis set. These methods based on the Kohn-Sham approach delivered isomers with coordination numbers of 4, 5, and 6. The analysis used to obtain coordination numbers was based on geometrical parameters and the quantum theory of atoms in molecules (QTAIM) approach. Our methodology found only one isomer with fourfold coordination and its probabilities to appear in these clusters are quite small for high temperatures. The procedure used in this article predicts important populations of fivefold and sixfold coordination clusters, in fact, the fivefold coordination dominates for PBE0-D3 and BHLYP methods, although the sixfold coordination starts to be important when the number of water molecules is increased. The nature of axial and equatorial contacts is discussed in the context of the QTAIM and the noncovalent interaction index (NCI), which gives a clear classification of such orientations. Also, these methods suggest a partial covalent interaction between the Cu²⁺ and water molecules in both positions; equatorial and axial.     

Protein surface roughness accounts for binding free energy of Plasmepsin II‐ligand complexes

The calculation of absolute binding affinities for protein‐inhibitor complexes remains as one of the main challenges in computational structure‐based ligand design. The present work explored the calculations of surface fractal dimension (as a measure of surface roughness) and the relationship with experimental binding free energies of Plasmepsin II complexes. Plasmepsin II is an attractive target for novel therapeutic compounds to treat malaria. However, the structural flexibility of this enzyme is a drawback when searching for specific inhibitors. Concerning that, we performed separate explicitly solvated molecular dynamics simulations using the available high‐resolution crystal structures of different Plasmepsin II complexes. Molecular dynamics simulations allowed a better approximation to systems dynamics and, therefore, a more reliable estimation of surface roughness. This constitutes a novel approximation in order to obtain more realistic values of fractal dimension, because previous works considered only x‐ray structures. Binding site fractal dimension was calculated considering the ensemble of structures generated at different simulation times. A linear relationship between binding site fractal dimension and experimental binding free energies of the complexes was observed within 20 ns. Previous studies of the subject did not uncover this relationship. Regression model, coined FD model, was built to estimate binding free energies from binding site fractal dimension values. Leave‐one‐out cross‐validation showed that our model reproduced accurately the absolute binding free energies for our training set (R² = 0.76; <|error|> =0.55 kcal/mol; SD_error = 0.19 kcal/mol). The fact that such a simple model may be applied raises some questions that are addressed in the article.     

The Water Dimer: Post-Hartree-Fock and Density Functional Calculations on the Potential Energy Surface

Conventional ab initio and density functional methods with extended basis sets were employed in the study of a path on the water-dimer potential energy surface. The results show that density functional methods do depend strongly on the type of exchange-correlation potential employed, as well as on the quality of the basis sets – similarly to conventional ab initio methods – and on the density of the grid. Gradient-corrected methods behave, as expected, better than uncorrected ones, the Becke–Lee–Yang–Parr (BLYP) potential being the one that gives the best results. However, too large chemical- and hydrogen-bond lengths and absolute energies, as well as too small relative total and correlation energies demonstrate that even BLYP calculations with a relative large basis set are not good as MP2 calculations of the same size. Adiabatically connected functionals (ACM), represented in this work by B3PW91, provide an improvement on the whole surface. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ab initio DFT study of structural and mechanical properties of methane and carbon dioxide hydrates

The structural and mechanical properties of methane and carbon dioxide hydrates were investigated using density functional theory simulations. Well-established equations of state of solids and exchange-correlation functionals were used for fitting the unit lattice total energy as a function of volume, and the full second-order elastic constants of these two gas hydrates were determined by energy–strain analyses. The polycrystalline elastic properties were also calculated from the unit lattice results. The final results for methane hydrate agree well with available experimental data and with other theoretical results. The two gas hydrates were found to be highly elastically isotropic, but they differed significantly in shear properties. The presented results for carbon dioxide hydrates are the first complete set reported so far. The results are a significant contribution to the ab initio material characterisation of gas hydrates required for ongoing fundamental studies and technological applications.     

Probing the geometries,relative stabilities,and electronic properties of neutral and anionic AgnSm (n + m ≤ 7) clusters

The vertical excitation energies of 3,4-dicyano-6-methoxy and 3,4-dicyano-6,7-dimethoxy carbostyril have been computed with different approximations for the time-dependent density functional theory (TD-DFT) procedure and with different implementations of the continuum solvation model COSMO. Different DFT functionals were tested in TD-DFT and Tamm-Dancoff approximations (TDA) for the excitation energies in the gas phase. TDA-B3LYP showed the best agreement with the experimental data. Then TDA-B3LYP computations were performed combined with the COSMO model of solvation comparing a linear response (LR) and a post-configuration interaction (CI) implementation of the fast solvent reorganization. The post-CI solvent model overestimates the π→π* transitions and strongly underestimates the n→π* transition. The TDA approximation in combination with the linear response implementation of the COSMO solvation model perfectly computes the experimental results. TDA-LR is the most reliable method for the computation of the vertical excitation energies in a solvent. Comparison with explicit solvent calculations shows there is only a minor effect on the energies of the electronic interaction of the solute with the solvent.     

Atomistic simulations of materials for optical chemical sensors: DFT-D calculations of molecular interactions between gas-phase analyte molecules and simple substrate models

The structures of complexes of some small molecules (formaldehyde, acetaldehyde, ammonia, methylamine, methanol, ethanol, acetone, benzene, acetonitrile, ethyl acetate, chloroform, and tetrahydrofuran, considered as possible analytes) with ethylbenzene and silanol (C₆H₅C₂H₅ and SiH₃OH, considered as models of polystyrene and silica gel substrates) and with acridine (C₁₃H₉N, considered as a model of an indicator dye molecule of the acridine series) and the corresponding interaction energies have been calculated using the DFT-D approximation. The PBE exchange-correlation potential was used in the calculations. The structures of complexes between the analyte and the substrate were determined by optimizing their ground-state geometry using the SVP split-valence double-zeta plus polarization basis set. The complex formation energies were refined by single-point calculations at the calculated equilibrium geometries using the sufficiently large triple-zeta TZVPP basis set. The calculated interaction energies are used to assess the possibility of using dyes of the acridine series adsorbed on a polystyrene or silica substrate for detecting the small molecules listed above.     

Benchmarking the DFT methodology for assessing antioxidant-related properties: quercetin and edaravone as case studies

The overall objective was to identify an accurate computational electronic method to virtually screen phenolic compounds through their antioxidant and free-radical scavenging activity. The impact of a key parameter of the density functional theory (DFT) approach was studied. Performances of the 21 most commonly used exchange-correlation functionals are thus detailed in the evaluation of the main energetic parameters related to the activities of two prototype antioxidants, namely quercetin and edaravone, is reported. These functionals have been chosen among those belonging to three different families of hybrid functionals, namely global, range separated, and double hybrids. Other computational parameters have also been considered, such as basis set and solvent effects. The selected parameters, namely bond dissociation enthalpy (BDE), ionization potential (IP), and proton dissociation enthalpy (PDE) allow a mechanistic evaluation of the antioxidant activities of free radical scavengers. Our results show that all the selected functionals provide a coherent picture of these properties, predicting the same order of BDEs and PDEs. However, with respect to the reference values, the errors found at CBS-Q3 level significantly vary with the functional. Although it is difficult to evidence a global trend from the reported data, it clearly appears that LC-ωPBE, M05-2X, and M06-2X are the most suitable approaches for the considered properties, giving the lowest cumulative mean absolute errors. These methods are therefore suggested for an accurate and fast evaluation of energetic parameters related to an antioxidant activity via free radical scavenging.     

Structure of Lennard–Jones fluid near large spherical particles: further test of the “universality” of adjustable parameter in perturbation density functional theory

Grand canonical Monte Carlo simulation is used to study the density profiles of Lennard–Jones (LJ) fluid next to a large hard sphere (mimicking a colloidal particle) of various sizes. The LJ fluid in the inhomogeneous system thus maintains equilibrium with the bulk LJ fluid. The chosen density and potential parameters for the bulk fluid correspond to the conditions situated at “dangerous” regions of the phase diagram, i.e. near the critical temperature or close to the gas–liquid coexistence curve. The aim of present extensive simulations is to provide exact data for the broad range of the bulk parameters against which the “universality” of adjustable parameter associated with a perturbation density functional approximation (DFA) can be tested. Here the term “universality” means independence of this parameter on the particular external field responsible for the generation of a non-uniform density profile of the fluid. It is shown that the “universality” of this parameter associated with a third order+second order perturbation DFA holds also in the present case of a large spherical particle as a source of external potential, similarly as established in previous studies dealing with other interaction potential and other external fields [J. Chem. Phys., 122, 064503 (2005); J. Chem. Phys., 123 124708 (2005)]. This DFA can be used as input into a recently proposed framework for the calculation of interparticle potential of mean force (PMF).     

Proton Transfer in Liquid Water II; A Semiempirical Method to Describe Chemical Reactions

Abstract

A new semiempirical method is developed to deal with the proton transfer in liquid water. In the previous work, we have shown that two- and three-body charge transfer interactions and electrostatic interactions are the most important factors to describe the potential energy surfaces (PES) of the proton transfer in liquid water [Chemical Physics 180, 239–269, 1994], In order to take account of these factors, we develop a semiempirical method imposing the principle of electronegativity equalization to the Atoms in Molecule (AIM) method. The method is free from the well-known discrepancy of the traditional AIM methods, that is, the fractional molecular charges at large molecular separation, and thus can be applied to the charge transfer reactions. Intra- and intermolecular physical quantities, such as total energies, force vectors, dipole moment vectors and intermolecular charge transfer, obtained by the present method are found to be in good agreement with those by ab initio calculation.     

How does modification of adenine by hydroxyl radical influence the stability and the nature of stacking interactions in adenine-cytosine complex?

This study reports on ab initio calculations of adenine - cytosine complexes in two different context alignments appearing in B-DNA. The influence of adenine modification by hydroxyl radical on the stability of the complexes is also discussed. The analysis was performed on over 40 crystallographic structures for each of the sequence contexts. In most cases, modification of adenine by hydroxyl radical leads to less negative intermolecular interaction energies. The issue of the influence of alteration of structural base step parameters on the stability of modified and unmodified adenine - cytosine complexes is also addressed. Analysis of the dependence of intermolecular interaction energy on base step parameters reveals that for twist and shift modification of adenine by hydroxyl radical leads to quite different interaction energy profiles in comparison with unmodified complexes. In order to elucidate the physical origins of this phenomenon, i.e. to analyze how the modification of adenine by hydroxyl radical is reflected in the change of intermolecular interaction energy components, a variational-perturbational decomposition scheme was applied at the MP2/aug-cc-pVDZ level of theory.     

Density functional study of oxygen vacancy formation and spin density distribution in octahedral ceria nanoparticles

We report plane wave basis density functional theory (DFT) calculations of the oxygen vacancies formation energy in nanocrystalline CeO _2-x in comparison with corresponding results for bulk and (111) CeO₂ surface. Effects of strong electronic correlation of Ce4f states are taken into account through the use of an effective on-site Coulomb repulsive interaction within DFT+U approach. Different combinations of exchange-correlation functionals and corresponding U values reported in the literature are tested and the obtained results compared with experimental data. We found that both absolute values and trends in oxygen vacancy formation energy depend on the value of U and associated with degree of localization of Ce4f states. Effect of oxygen vacancy and geometry optimization method on spatial spin distribution in model ceria nanoparticles is also discussed.