Molecular fractionation with conjugated caps density matrix with pairwise interaction correction for protein energy calculation

Pairwise interaction correction (PIC) is introduced to account for electron density polarization due to short-range interactions such as hydrogen bonding and close contact between molecular fragments in the molecular fractionation with conjugated caps density matrix (MFCC-DM) approach for energy calculation of protein and other polymers [Chen et al., J. Chem. Phys. 122, 184105 (2005)]. With this PIC, the accuracy of the calculated protein energy and other electronic properties are improved, and the MFCC approach can be applied to study real proteins with short-range structural complexity. In the present MFCC-DM-PIC approach, the short-range interresidual interactions are represented by a pair of small molecules (interacting units) which are made from the two residues that fall within a certain distance criterion. The density matrices of fragments, concaps, interacting units and pairs are calculated by conventional Hartree-Fock or density functional theory methods and are combined to construct the full density matrix which is finally employed to calculate the total energy, electron density, electrostatic potential, dipole moment, etc., of the protein. Numerical tests on seven conformationally varied peptides are presented to demonstrate the accuracy of the MFCC-DM-PIC method.     

The generalized molecular fractionation with conjugate caps/molecular mechanics method for direct calculation of protein energy

A generalized molecular fractionation with conjugate caps/molecular mechanics (GMFCC/MM) scheme is developed for efficient linear-scaling quantum mechanical calculation of protein energy. In this GMFCC/MM scheme, the interaction energy between neighboring residues as well as between non-neighboring residues that are spatially in close contact are computed by quantum mechanics while the rest of the interaction energy is computed by molecular mechanics. Numerical studies are carried out to calculate torsional energies of six polypeptides using the GMFCC/MM approach and the energies are shown to be in general good agreement with the full system quantum calculation. Among those we tested is a polypeptide containing 396 atoms whose energies are computed at the MP26-31G* level. Our study shows that using GMFCC/MM, it is possible to perform high level ab initio calculation such as MP2 for applications such as structural optimization of protein complex and molecular dynamics simulation.     

Electrostatic embedding in large-scale first principles quantum mechanical calculations on biomolecules

Biomolecular simulations with atomistic detail are often required to describe interactions with chemical accuracy for applications such as the calculation of free energies of binding or chemical reactions in enzymes. Force fields are typically used for this task but these rely on extensive parameterisation which in cases can lead to limited accuracy and transferability, for example for ligands with unusual functional groups. These limitations can be overcome with first principles calculations with methods such as density functional theory (DFT) but at a much higher computational cost. The use of electrostatic embedding can significantly reduce this cost by representing a portion of the simulated system in terms of highly localised charge distributions. These classical charge distributions are electrostatically coupled with the quantum system and represent the effect of the environment in which the quantum system is embedded. In this paper we describe and evaluate such an embedding scheme in which the polarisation of the electronic density by the embedding charges occurs self-consistently during the calculation of the density. We have implemented this scheme in a linear-scaling DFT program as our aim is to treat with DFT entire biomolecules (such as proteins) and large portions of the solvent. We test this approach in the calculation of interaction energies of ligands with biomolecules and solvent and investigate under what conditions these can be obtained with the same level of accuracy as when the entire system is described by DFT, for a variety of neutral and charged species.     

Electrostatic energy calculations for potassium chloroplatinite and related compounds

A new method of estimating the effective charges on palladium and platinum in the square planar complexes PdCl₄²⁻, PtCl₄²⁻, PdBr₄²⁻ and PtBr₄²⁻ is described in detail. It turns out that Cl is more electronegative than Br and Pt is more electronegative than Pd in these complexes. It is shown further that the strikingly long PtPt distance between the complexes in K₂PtCl₄ is caused by the K⁺ ions, probably together with dispersion forces, but is not the result of electrostatic repulsion between the PtCl₄²⁻ units. The analysis accounts for the “blue shift” of the ¹A_1g−³E_g transition in the absorption spectrum of the PtCl₄²⁻ complex observed in high pressure spectroscopic experiments by Interrante et al.     

Electrostatic isopotential maps for large biomolecules

A new method for the calculation of electrostatic isopotential maps, based on completely transferable bond orbitals, is proposed. According to the simple form of the potential term the amount of computational work is proportional to the first power of the number of valence electrons. Slater-type atomic orbitals are used to construct the bond orbitals, thus all integrals can be calculated explicitly. The subtilisin charge-relay system is studied as an example. It is found, in agreement with the results of Beppu and Yomosa, that the two protons transfer in a stepwise manner.     

Ab initio and molecular mechanics (MM2 and MM3) calculations of positively charged conjugated nitrogen-containing compounds

Ab initio calculations have been carried out on s-trans-N-vinylmethyleneammonium, pyridinium, and related compounds to obtain rotational barriers, structures, and vibrational frequencies. The restricted Hartree-Fock (RHF) level of theory with 6-31G** basis set was used for these calculations. In addition, the MM2(91) and MM3(94) force fields have been parameterized to calculate these positively charged nitrogen-containing compounds. A bond order term was incorporated in the force field to reproduce accurately the rotational barriers of s-trans-N-vinylmethyleneammonium and related compounds. Molecular mechanics geometries and vibrational frequencies compare well with those calculated by ab initio methods. © 1996 by John Wiley & Sons, Inc.     

Adaptively biased molecular dynamics for free energy calculations

We present an adaptively biased molecular dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using nonequilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential and is inspired by the metadynamics method. The ABMD method has several useful features, including a small number of control parameters and an O(t) numerical cost with molecular dynamics time t. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures and/or collective variables. This is beneficial not only in terms of the speed and accuracy of a calculation, but also in terms of the amount of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Nme peptide in a gaseous and solvated environment.     

Local density calculations of correlation energy for molecular systems

The correlation energy for various diatomic molecules is calculated employing the expressions recently proposed (R.L. Boada and V.V. Karasiov, Chem. Phys. Lett., 175 (1990) 641; R.L. Boada, J. Maldonado and V.V. Karasiov, J. Mol. Struct. (Theochem), 287 (1994) 11). The calculated correlation energies are compared with results obtained from other methods.     

Electrostatic free energy of weakly charged macromolecules in solution and intermacromolecular complexes consisting of oppositely charged polymers

When oppositely charged polyelectrolytes are mixed in water, attraction between oppositely charged groups may lead to the formation of polyelectrolyte complexes (associative phase separation, complex coacervation, interpolymer complexes). Theory is presented to describe the electrostatic free energy change when ionizable (annealed) (macro-)molecules form a macroscopic polyelectrolyte complex. The electrostatic free energy includes an electric term as well as a chemical term that is related to the dissociation of the ionic groups in the polymer. An example calculation for complexation of polyacid with polybase uses a cylindrical diffuse double layer model for free polymer in solution and electroneutrality within the complex and calculates the free energy of the system when the polymer is in solution or in a polyelectrolyte complex. Combined with a term for the nonelectrostatic free energy change upon complexation, a theoretical stability diagram is constructed that relates pH, salt concentration, and mixing ratio, which is in qualitative agreement with an experimental diagram obtained by Bungenberg de Jong (1949) for complex coacervation of arabic gum and gelatin. The theory furthermore explains the increased tendency toward phase separation when the polymer becomes more strongly charged and suggests that complexation of polyacid or polybase with zwitterionic polymer (e.g., protein) of the same charge sign (at the "wrong side" of the iso-electric point) may be due (in part) to an induced charge reversal of the protein.     

Electron and energy transfer in donor-acceptor systems with conjugated molecular bridges

Electron and energy transfer reactions in covalently connected donor-bridge-acceptor assemblies are strongly dependent, not only on the donor-acceptor distance, but also on the electronic structure of the bridge. In this article we describe some well characterised systems where the bridges are pi-conjugated chromophores, and where, specifically, the interplay between bridge length and energy plays an important role for the donor-acceptor electronic coupling. For any application that relies on the transport of electrons, for example molecule based solar cells or molecular scale electronics, it will be imperative to predict the electron transfer capabilities of different molecular structures. The potential difficulties with making such predictions and the lack of suitable models are also discussed.     

B-spline method for energy minimization in grid-based molecular mechanics calculations

A method is described for molecular mechanics calculations based on a cubic B-spline approximation of the potential energy. This method is useful when parts of the system are allowed to remain fixed in position so that a potential energy grid can be precalculated and used to approximate the interaction energy between parts of a molecule or between molecules. We adapted and modified the conventional B-spline method to provide an approximation of the Empirical Conformational Energy Program for Peptides (ECEPP) potential energy function. The advantage of the B-spline method over simpler approximations is that the resulting B-spline function is C2 continuous, which allows minimization of the potential energy by any local minimization algorithm. The standard B-spline method provides a good approximation of the electrostatic energy; but in order to reproduce the Lennard–Jones and hydrogen-bonding functional forms accurately, it was necessary to modify the standard B-spline method. This modification of the B-spline method can also be used to improve the accuracy of trilinear interpolation for simulations that do not require continuous derivatives. As an example, we apply the B-spline method to rigid-body docking energy calculations using the ECEPP potential energy function. Energies are calculated for the complex of Phe-Pro-Arg with thrombin. For this system, we compare the performance of the B-spline method to that of the standard pairwise summation in terms of speed, accuracy, and overhead costs for a variety of grid spacings. In our rigid-body docking calculations, the B-spline method provided an accurate approximation of the total energy of the system, and it resulted in an 180-fold reduction in the time required for a single energy and gradient calculation for this system. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 71–85, 1998     

Electrostatic stretching of a charged vesicle

We present a closed-form solution of electrostatic potential self-induced by a uniformly charged micro/nanovesicle and the corresponding elastic deformation of the vesicle membrane due to Maxwell stress. At equilibrium, the electrostatic force induced on both sides of the membrane is balanced by the elastic force of the stretched membrane. We develop differential and integral solutions of the coupled Poisson-Boltzmann system for a spherical vesicle and demonstrate that the integral solution is relatively flexible in formulating asymmetric configurations. Analytical results are formulated in terms of vesicle size, Debye length, and the surface charge density. The membrane stretching is characterized by the dimensionless group that defines the relative strength of the net electric force with respect to the membrane stiffness. We found that the self-induced electrostatic interaction will lead to a pre-stressed membrane although the small displacement is often negligible compared with the vesicle size. Quantitative analysis also reveals that the electric force can assist the vesicle in recovering its opening pore.     

Approaches to charge calculations in molecular mechanics. 2 Resonance effects in conjugated systems

A previously published scheme of estimating atomic charges in haloalkanes is extended to include olefines, alcohols amines, acids, ethers, and amides. In the conjugated systems the effects of mesomeric transfer of charge are explicitly included. Generally good agreement with the observed dipole moments of these compounds and their substituted derivatives is found. The atomic charges so obtained are compared with those of other semiempirical and quantum-mechanical calculations for the amide group. The charges so obtained fall within the range of values obtained by these other schemes, supporting the general validity of this approach.     

Reference program for molecular calculations with Slater-type orbitals

A program for computing all the integrals appearing in molecular calculation with Slater-type orbitals is reported. The program is mainly intended as a reference for testing and comparing other algorithms and techniques. An analysis of the performance of the program is presented, paying special attention to the computational cost and the accuracy of the results. Results are also compared with others obtained with Gaussian basis sets of similar quality. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1284–1293, 1998     

Recent applications of density functional theory calculations to biomolecules

The purpose of this overview is to highlight the broad scope and utility of current applications of density functional theory (DFT) methods for the study of the properties and reactions of biomolecules. This is illustrated using examples selected from research carried out within our research group and in collaboration with others. The examples include the hyperfine coupling constants of amino acid radicals, the use of an amino acid as a chiral catalyst for the formation of carbon–carbon bonds in the aldol reaction, hydrogen-bond mediated catalysis of an aminolysis reaction, radiation-induced protein–DNA cross-links, and the mechanism by which an antitumor drug cleaves DNA. We demonstrate that DFT-based methods can be applied successfully to a broad range of problems that remain beyond the scope of conventional electron-correlation methods. Furthermore, we show that contemporary computational quantum chemistry complements experiment in the study of biological systems. Received: 19 December 2001 / Accepted: 8 April 2002 / Published online: 4 July 2002     

Some practical aspects of free energy calculations from molecular dynamics simulation

In this work, we address two critical aspects of calculation of the free energy differences in molecular systems from molecular simulations. The first aspect involves checking whether the calculated free energy difference depends significantly on the extent of perturbation used for accomplishment of a given transformation. The second aspect of interest is to verify if the sampling errors in calculating the free energy differences between the wild-type molecule and a mutated one in its free state and in a complex are similar, or not, for a finite-length dynamic simulation. The reliability of the free energy estimates obtained from molecular simulations using thermodynamic cycles depends in part on this fact. For investigating these aspects, we use a self-transformation scheme in which a transformation of a part of a molecular system into itself is considered. We perform MD simulations of DNA fragments in which a part of a specific base is subjected to such a self-transformation. Results indicate that the estimated free energy differences do not depend significantly on the extent of perturbation used to achieve the transformation. Interestingly, the variation in the cumulative free energy difference, ΔA, with the coupling parameter, λ, depends significantly on the extent of perturbation. We examine the physical basis of the observed nature of the variation of the accumulated free energy difference, ΔA, against the λ value in the case of a self-transformation. In a thermodynamic cycle, the sampling errors due to the finite-length simulation for the molecular system are found to be similar to each other for the two perturbations (free and in a complex) justifying the use of such approach in calculating ΔΔA in molecular complexes. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 877–885, 1999     

Identifying low variance pathways for free energy calculations of molecular transformations in solution phase

Improving the efficiency of free energy calculations is important for many biological and materials design applications, such as protein-ligand binding affinities in drug design, partitioning between immiscible liquids, and determining molecular association in soft materials. We show that for any pair potential, moderately accurate estimation of the radial distribution function for a solute molecule is sufficient to accurately estimate the statistical variance of a sampling along a free energy pathway. This allows inexpensive analytical identification of low statistical error free energy pathways. We employ a variety of methods to estimate the radial distribution function (RDF) and find that the computationally cheap two-body "dilute gas" limit performs as well or better than 3D-RISM theory and other approximations for identifying low variance free energy pathways. With a RDF estimate in hand, we can search for pairwise interaction potentials that produce low variance. We give an example of a search minimizing statistical variance of solvation free energy over the entire parameter space of a generalized "soft core" potential. The free energy pathway arising from this optimization procedure has lower curvature in the variance and reduces the total variance by at least 50% compared to the traditional soft core solvation pathway. We also demonstrate that this optimized pathway allows free energies to be estimated with fewer intermediate states due to its low curvature. This free energy variance optimization technique is generalizable to solvation in any homogeneous fluid and for any type of pairwise potential and can be performed in minutes to hours, depending on the method used to estimate g(r).     

Aggregation mechanism of polyglutamine diseases revealed using quantum chemical calculations, fragment molecular orbital calculations, molecular dynamics simulations, and binding free energy calculations

Polyglutamine (polyQ) diseases, including Huntington’s disease (HD), are caused by expansion of polyQ-encoding repeats within otherwise unrelated gene products. The aggregation mechanism of polyQ diseases, the inhibition mechanism of Congo red, and the alleviation mechanism of trehalose were proposed here based on quantum chemical calculations and molecular dynamics simulations. The calculations and simulations revealed the following. The effective molecular bonding is between glutamine (Gln) and Gln (Gln + Gln), between Gln and Congo red (Gln + Congo red), and between Gln and trehalose (Gln + trehalose). The bonding strength is −13.1 kcal/mol for Gln + Gln, −24.4 kcal/mol for Gln + Congo red, and −12.0 kcal/mol for Gln + trehalose. In the polyQ region, both the number of intermolecular Gln + Gln formations and the total calories generated by the Gln + Gln formation are proportional to the number of repetitions of Gln. We propose an aggregation mechanism whose heat generated by the intermolecular Gln + Gln formation causes the pathogeny of polyQ disease. In our aggregation mechanism, this generated heat collapses the host protein and promotes fibrillogenesis. Without contradiction, our mechanism can explain all the experimental results reported to date. Our mechanism can also explain the inhibition mechanism by Congo red as an inhibitor of polyglutamine-induced protein aggregation and the alleviation mechanism by trehalose as an alleviator of that aggregation. The inhibition mechanism by Congo red is explained by the strong interaction with Gln and by the characteristic structure of Congo red.     

Internal energy deposition in doubly charged tungsten hexacarbonyl on charge exchange at high energy with atomic and molecular targets

Charge exchange of W(CO) in the course of 7 keV collisions with various targets results in singly charged ions which are highly internally excited. The distribution of internal energies estimated by the degree to which consecutive fragmentations by CO loss occur is broad and includes very high energies (up to 15 eV). This result is inconsistent with exclusive operation of long-range electron transfer e.g. by either a curve crossing or Demkov mechanism; rather it suggests that direct excitation to a high-energy repulsive state of the products also occurs perhaps by an electronic excitation mechanism. The nature of the internal energy distribution suggests mechanistic analogies with simple collisional activation. Different target gases give rise to different average internal energy depositions monatomics and diatomics yielding higher energy depositions than do polyatomic targets. There is an approximate correlation between energy deposition and target ionization energy which is consistent with the proposed excitation mechanism considering the shapes of the potential energy surfaces. When the detailed internal energy distributions are compared characteristic differences are seen for individual targets. The efficiencies of the various targets at producing charge exchange were also compared. Large differences were found with the polyatomic targets having the greatest efficiencies. In addition a rough correlation was observed between the extent of charge exchange and target ionization energy and this is interpreted in terms of greater contributions from the long-range electron transfer process for targets of lower ionization energy. All the results are also consistent with contributons from more violent collisions which involve electron transfer at relatively small internuclear distances where the shapes of the potential surfaces are strongly dependent on distance.