Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems

We rigorously study a novel type of trigonometric Fourier collocation methods for solving multi-frequency oscillatory second-order ordinary differential equations (ODEs) \(q^{\prime \prime }(t)+Mq(t)=f(q(t))\) with a principal frequency matrix \(M\in \mathbb {R}^{d\times d}\). If \(M\) is symmetric and positive semi-definite and \(f(q) = -\nabla U(q)\) for a smooth function \(U(q)\), then this is a multi-frequency oscillatory Hamiltonian system with the Hamiltonian \(H(q,p)=p^{T}p/2+q^{T}Mq/2+U(q),\) where \(p = q'\). The solution of this system is a nonlinear multi-frequency oscillator. The new trigonometric Fourier collocation method takes advantage of the special structure brought by the linear term \(Mq\), and its construction incorporates the idea of collocation methods, the variation-of-constants formula and the local Fourier expansion of the system. The properties of the new methods are analysed. The analysis in the paper demonstrates an important feature, namely that the trigonometric Fourier collocation methods can be of an arbitrary order and when \(M\rightarrow 0\), each trigonometric Fourier collocation method creates a particular Runge–Kutta–Nyström-type Fourier collocation method, which is symplectic under some conditions. This allows us to obtain arbitrary high-order symplectic methods to deal with a special and important class of systems of second-order ODEs in an efficient way. The results of numerical experiments are quite promising and show that the trigonometric Fourier collocation methods are significantly more efficient in comparison with alternative approaches that have previously appeared in the literature.     

Apparent NAC effect in chorismate mutase reflects electrostatic transition state stabilization

The catalytic reaction of chorismate mutase (CM) has been the subject of major current attention. Nevertheless, the origin of the catalytic power of CM remains an open question. In particular, it has not been clear whether the enzyme works by providing electrostatic transition state stabilization (TSS), by applying steric strain, or by populating near attack conformation (NAC). The present work explores this issue by a systematic quantitative analysis. The overall catalytic effect is reproduced by the empirical valence bond (EVB) method. In addition, the binding free energy of the ground state and the transition state is evaluated, demonstrating that the enzyme works by TSS. Furthermore, the evaluation of the electrostatic contribution to the reduction of the activation energy establishes that the TSS results from electrostatic effects. It is also found that the apparent NAC effect is not the reason for the catalytic effect but the result of the TSS. It is concluded that in CM as in other enzymes the key catalytic effect is electrostatic TSS. However, since the charge distribution of the transition state and the reactant state is similar, the stabilization of the transition state leads to reduction in the distance between the reacting atoms in the reactant state.     

The Kirkwood-Buff integrals for one-component liquids

The Kirkwood-Buff integrals (KBIs) for one-component systems are calculated from either the pair correlation functions or from experimental macroscopic quantities. As in the case of mixtures, the KBIs provide important information on the local densities around a molecule. In the low density limit (rho-->0) one can extract from the KBI some information on the strength of the intermolecular forces. No such information may be extracted from the KBIs at higher densities. We used experimental data on densities and isothermal compressibilities to calculate the KBIs for various liquids ranging from inert molecules, to hydrocarbons, alcohols, and liquid water.     

One-dimensional model for water and aqueous solutions. I. Pure liquid water

Two simplified one-dimensional models for waterlike particles are studied. One is referred to as the primitive model which is a simplified version of a model introduced by Ben-Naim in 1992 [Statistical Thermodynamics for Chemists and Biochemists (Plenum, New York, 1992)]. The second, referred to as the primitive cluster model, is a simplified version of the model used by Lovett and Ben-Naim in 1969 [J. Chem. Phys. 51, 3108 (1969)]. The two models are shown to be nearly equivalent and both exhibit some of the most characteristic behavior of liquid water. It is argued that a key feature of the molecular interactions--the correlation between the strong binding energy and low local density--is essential for the manifestation of the anomalous behavior of liquid water. It is also essential for the understanding of the outstanding behavior of liquid water.     

Through the channel and around the channel: Validating and comparing microscopic approaches for the evaluation of free energy profiles for ion penetration through ion channels

Microscopic calculations of free energy profiles for ion transport through biological ion channels present a very serious challenge to modern simulation approaches. The main problem is due to the major convergence problems associated with the heterogeneous landscape of the electrostatic environment in ion channels and with the need to evaluate the profile associated with the transfer of the ion from bulk water to the channel environment. This problem is compounded by the lack of reliable and relevant benchmarks that can discriminate between alternative approaches. The present study is aimed at reducing the above problems by defining benchmarks that are directly relevant to ion channels and can also give converging results. This is done by constructing a series of models of a truncated gramicidin channel with different numbers of water molecules and by comparing the profiles for going around the channel and through the channel. These discriminating models are then used to validate and compare the adiabatic charging free energy perturbation (FEP) approach combined with an umbrella sampling approach (Warshel, A. J. Phys. Chem. 1982, 86, 2218) and the potential of mean force (PMF) approach used frequently in studies of ion channels. It is found that both approaches work quite well until one moves to the case of the fully solvated channel. In this limit, the PMF approach may give different results for the overall work of going through the channel and around the channel, while the FEP approach gives physically consistent results. The present benchmark also indicates that the weighted histogram analysis method (WHAM) approach does not offer a significant advantage over earlier approaches at least as much as studies of ion channels are concerned. Finally, it is concluded that the FEP approach may be more useful in evaluating the overall barrier for moving ions from water to ion channels and that in some cases it might be beneficial to use the FEP approach for selective points along the channel and then to connect these points by PMF calculations.     

Asymptotic solvers for ordinary differential equations with multiple frequencies

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms,focusing on the case of multiple,non-commensurate frequencies.We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question.Numerical examples illustrate the effectiveness of the method.     

Gravitational migration of fuel in porous media

Fuel migration in a water flowing through a porous medium generally occurs parallel to porous strata, which may not be horizontal. In this case, gravity tends to cause vertical segregation of fluids, depending on their densities. This phenomenon can exert a strong effect on fuel migration. The gravitational force creates the buoyancy force which acts upon the fuel, and may be either parallel or anti-parallel to the water flow direction.In this study, the above effects are investigated using the one-dimensional model of Pistineret al. We go beyond the latter investigation in describing the influence of the gravitational forces upon the movement of fuel saturation fronts in a vertical porous layer against and along the water flow direction.It is found that when the directions of the buoyancy force and of the water flow are anti-parallel, fuel migrates in the direction of the buoyancy force, provided the latter is strong enough. However, in the case of a weak buoyancy force, the direction of migration of the fuel depends on its mass. Small fuel masses move mainly in the direction of the water flow. However, big fuel slugs possessing large masses will move mainly in the direction of the buoyancy force. Slugs, characterized by intermediate masses, have no preferable moving direction and are almost stagnant.     

Numerical Stability in the Presence of Variable Coefficients

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric differentiation matrices for univariate finite-difference methods. We prove that, in order to sustain a skew-symmetric differentiation matrix of order \(p\ge 2\), a grid must satisfy \(2p-3\) polynomial conditions. Moreover, once it satisfies these conditions, it supports a banded skew-symmetric differentiation matrix of this order and of the bandwidth \(2p-1\), which can be derived in a constructive manner. Some applications require not just skew-symmetry, but also that the growth in the elements of the differentiation matrix is at most linear in the number of unknowns. This is always true for our tridiagonal matrices of order 2 but need not be true otherwise, a subject which we explore further. Another subject which we examine is the existence and practical construction of grids that support skew-symmetric differentiation matrices of a given order. We resolve this issue completely for order-two methods. We conclude the paper with a list of open problems and their discussion.