![]() | This article may be too technical for most readers to understand.(February 2011) |
Local elevation is a technique used in computational chemistry or physics, mainly in the field of molecular simulation (including molecular dynamics (MD) and Monte Carlo (MC) simulations). It was developed in 1994 by Huber, Torda and van Gunsteren [1] to enhance the searching of conformational space in molecular dynamics simulations and is available in the GROMOS software for molecular dynamics simulation (since GROMOS96). The method was, together with the conformational flooding method, [2] the first to introduce memory dependence into molecular simulations. Many recent methods build on the principles of the local elevation technique, including the Engkvist-Karlström, [3] adaptive biasing force, [4] Wang–Landau, metadynamics, adaptively biased molecular dynamics, [5] adaptive reaction coordinate forces, [6] and local elevation umbrella sampling [7] methods. The basic principle of the method is to add a memory-dependent potential energy term in the simulation so as to prevent the simulation to revisit already sampled configurations, which leads to the increased probability of discovering new configurations. The method can be seen as a continuous variant of the Tabu search method.
The basic step of the algorithm is to add a small, repulsive potential energy function to the current configuration of the molecule such as to penalize this configuration and increase the likelihood of discovering other configurations. This requires the selection of a subset of the degrees of freedom, which define the relevant conformational variables. These are typically a set of conformationally relevant dihedral angles, but can in principle be any differentiable function of the cartesian coordinates .
The algorithm deforms the physical potential energy surface by introducing a bias energy, such that the total potential energy is defined as
The local elevation bias depends on the simulation time and is set to zero at the start of the simulation () and is gradually built as a sum of small, repulsive functions, giving
where is a scaling constant and is a multidimensional, repulsive function with .
The resulting bias potential will be a sum of all the added functions
To reduce the number of added repulsive functions, a common approach is to add the functions to grid points. The original choice of is to use a multidimensional Gaussian function. However, due to the infinite range of the Gaussian as well as the artifacts that can occur with a sum of gridded Gaussians, a better choice is to apply multidimensional truncated polynomial functions [8] . [9]
The local elevation method can be applied to free energy calculations as well as to conformational searching problems. In free energy calculations the local elevation technique is applied to level out the free energy surface along the selected set of variables. It has been shown by Engkvist and Karlström [3] that the bias potential built by the local elevation method will approximate the negative of the free energy surface. The free energy surface can therefore be approximated directly from the bias potential (as done in the metadynamics method) or the bias potential can be used for umbrella sampling (as done in metadynamics with umbrella sampling corrections [10] and local elevation umbrella sampling [7] methods) to obtain more accurate free energies.
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanics force fields. The method is applied mostly in chemical physics, materials science, and biophysics.
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.
Chemistry at Harvard Macromolecular Mechanics (CHARMM) is the name of a widely used set of force fields for molecular dynamics, and the name for the molecular dynamics simulation and analysis computer software package associated with them. The CHARMM Development Project involves a worldwide network of developers working with Martin Karplus and his group at Harvard to develop and maintain the CHARMM program. Licenses for this software are available, for a fee, to people and groups working in academia.
Molecular mechanics uses classical mechanics to model molecular systems. The Born–Oppenheimer approximation is assumed valid and the potential energy of all systems is calculated as a function of the nuclear coordinates using force fields. Molecular mechanics can be used to study molecule systems ranging in size and complexity from small to large biological systems or material assemblies with many thousands to millions of atoms.
Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the maximization of the real-valued function is equivalent to the minimization of the function .
Parallel tempering in physics and statistics, is a computer simulation method typically used to find the lowest free energy state of a system of many interacting particles at low temperature. That is, the one expected to be observed in reality. It addresses the problem that at high temperature one may have a stable state different from low temperature, whereas simulations at low temperature may become "stuck" in a metastable state. It does this by using the fact that the high temperature simulation may visit states typical of both stable and metastable low temperature states.
Mulliken charges arise from the Mulliken population analysis and provide a means of estimating partial atomic charges from calculations carried out by the methods of computational chemistry, particularly those based on the linear combination of atomic orbitals molecular orbital method, and are routinely used as variables in linear regression (QSAR) procedures. The method was developed by Robert S. Mulliken, after whom the method is named. If the coefficients of the basis functions in the molecular orbital are Cμi for the μ'th basis function in the i'th molecular orbital, the density matrix terms are:
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system in order to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.
In nuclear and materials physics, stopping power is the retarding force acting on charged particles, typically alpha and beta particles, due to interaction with matter, resulting in loss of particle kinetic energy. Its application is important in areas such as radiation protection, ion implantation and nuclear medicine.
The Gross–Pitaevskii equation describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.
In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates, (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.
Free energy perturbation (FEP) is a method based on statistical mechanics that is used in computational chemistry for computing free energy differences from molecular dynamics or Metropolis Monte Carlo simulations.
Umbrella sampling is a technique in computational physics and chemistry, used to improve sampling of a system where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Torrie and Valleau in 1977. It is a particular physical application of the more general importance sampling in statistics.
In computational fluid dynamics, shock-capturing methods are a class of techniques for computing inviscid flows with shock waves. The computation of flow containing shock waves is an extremely difficult task because such flows result in sharp, discontinuous changes in flow variables such as pressure, temperature, density, and velocity across the shock.
Thermodynamic integration is a method used to compare the difference in free energy between two given states whose potential energies and have different dependences on the spatial coordinates. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead a function of the Boltzmann-weighted integral over phase space, the free energy difference between two states cannot be calculated directly from the potential energy of just two coordinate sets. In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the Kirkwood's coupling parameter method.
The Wang and Landau algorithm, proposed by Fugao Wang and David P. Landau, is a Monte Carlo method designed to estimate the density of states of a system. The method performs a non-Markovian random walk to build the density of states by quickly visiting all the available energy spectrum. The Wang and Landau algorithm is an important method to obtain the density of states required to perform a multicanonical simulation.
A polymer field theory is a statistical field theory describing the statistical behavior of a neutral or charged polymer system. It can be derived by transforming the partition function from its standard many-dimensional integral representation over the particle degrees of freedom in a functional integral representation over an auxiliary field function, using either the Hubbard–Stratonovich transformation or the delta-functional transformation. Computer simulations based on polymer field theories have been shown to deliver useful results, for example to calculate the structures and properties of polymer solutions, polymer melts and thermoplastics.
The Widom insertion method is a statistical thermodynamic approach to the calculation of material and mixture properties. It is named for Benjamin Widom, who derived it in 1963. In general, there are two theoretical approaches to determining the statistical mechanical properties of materials. The first is the direct calculation of the overall partition function of the system, which directly yields the system free energy. The second approach, known as the Widom insertion method, instead derives from calculations centering on one molecule. The Widom insertion method directly yields the chemical potential of one component rather than the system free energy. This approach is most widely applied in molecular computer simulations but has also been applied in the development of analytical statistical mechanical models. The Widom insertion method can be understood as an application of the Jarzynski equality since it measures the excess free energy difference via the average work needed to perform, when changing the system from a state with N molecules to a state with N+1 molecules. Therefore it measures the excess chemical potential since , where .
Metadynamics is a computer simulation method in computational physics, chemistry and biology. It is used to estimate the free energy and other state functions of a system, where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Alessandro Laio and Michele Parrinello in 2002 and is usually applied within molecular dynamics simulations. MTD closely resembles a number of recent methods such as adaptively biased molecular dynamics, adaptive reaction coordinate forces and local elevation umbrella sampling. More recently, both the original and well-tempered metadynamics were derived in the context of importance sampling and shown to be a special case of the adaptive biasing potential setting. MTD is related to the Wang–Landau sampling.
Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies, adsorption, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions.