Mathematical optimization or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. For large numbers of local optima, SA can find the global optima. It is often used when the search space is discrete. For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms such as gradient descent or branch and bound.
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 .
In numerical analysis, stochastic tunneling (STUN) is an approach to global optimization based on the Monte Carlo method-sampling of the function to be objective minimized in which the function is nonlinearly transformed to allow for easier tunneling among regions containing function minima. Easier tunneling allows for faster exploration of sample space and faster convergence to a good solution.
Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials science to study molecular systems ranging from small chemical systems to large biological molecules and material assemblies. The simplest calculations can be performed by hand, but inevitably computers are required to perform molecular modelling of any reasonably sized system. The common feature of molecular modelling methods is the atomistic level description of the molecular systems. This may include treating atoms as the smallest individual unit, or explicitly modelling protons and neutrons with its quarks, anti-quarks and gluons and electrons with its photons.
Protein design is the rational design of new protein molecules to design novel activity, behavior, or purpose, and to advance basic understanding of protein function. Proteins can be designed from scratch or by making calculated variants of a known protein structure and its sequence. Rational protein design approaches make protein-sequence predictions that will fold to specific structures. These predicted sequences can then be validated experimentally through methods such as peptide synthesis, site-directed mutagenesis, or artificial gene synthesis.
Internal Coordinate Mechanics (ICM) is a software program and algorithm to predict low-energy conformations of molecules by sampling the space of internal coordinates defining molecular geometry. In ICM each molecule is constructed as a tree from an entry atom where each next atom is built iteratively from the preceding three atoms via three internal variables. The rings kept rigid or imposed via additional restraints. ICM is used for modelling peptides and interactions with substrates and coenzymes.
Tinker, previously stylized as TINKER, is a suite of computer software applications for molecular dynamics simulation. The codes provide a complete and general set of tools for molecular mechanics and molecular dynamics, with some special features for biomolecules. The core of the software is a modular set of callable routines which allow manipulating coordinates and evaluating potential energy and derivatives via straightforward means.
The Reverse Monte Carlo (RMC) modelling method is a variation of the standard Metropolis–Hastings algorithm to solve an inverse problem whereby a model is adjusted until its parameters have the greatest consistency with experimental data. Inverse problems are found in many branches of science and mathematics, but this approach is probably best known for its applications in condensed matter physics and solid state chemistry.
In computational physics, variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system.
In the field of computational chemistry, energy minimization is the process of finding an arrangement in space of a collection of atoms where, according to some computational model of chemical bonding, the net inter-atomic force on each atom is acceptably close to zero and the position on the potential energy surface (PES) is a stationary point. The collection of atoms might be a single molecule, an ion, a condensed phase, a transition state or even a collection of any of these. The computational model of chemical bonding might, for example, be quantum mechanics.
CP2K is a freely available (GPL) quantum chemistry and solid state physics program package, written in Fortran 2008, to perform atomistic simulations of solid state, liquid, molecular, periodic, material, crystal, and biological systems. It provides a general framework for different methods: density functional theory (DFT) using a mixed Gaussian and plane waves approach (GPW) via LDA, GGA, MP2, or RPA levels of theory, classical pair and many-body potentials, semi-empirical and tight-binding Hamiltonians, as well as Quantum Mechanics/Molecular Mechanics (QM/MM) hybrid schemes relying on the Gaussian Expansion of the Electrostatic Potential (GEEP). The Gaussian and Augmented Plane Waves method (GAPW) as an extension of the GPW method allows for all-electron calculations. CP2K can do simulations of molecular dynamics, metadynamics, Monte Carlo, Ehrenfest dynamics, vibrational analysis, core level spectroscopy, energy minimization, and transition state optimization using NEB or dimer method.
This is a list of notable computer programs that are used for nucleic acids simulations.
MacroModel is a computer program for molecular modelling of organic compounds and biopolymers. It features various chemistry force fields, plus energy minimizing algorithms, to predict geometry and relative conformational energies of molecules. MacroModel is maintained by Schrödinger, LLC.
Crystal structure prediction (CSP) is the calculation of the crystal structures of solids from first principles. Reliable methods of predicting the crystal structure of a compound, based only on its composition, has been a goal of the physical sciences since the 1950s. Computational methods employed include simulated annealing, evolutionary algorithms, distributed multipole analysis, random sampling, basin-hopping, data mining, density functional theory and molecular mechanics.
David John Wales is a professor of chemical physics in the Department of Chemistry at the University of Cambridge.
PyMC is a Python package for Bayesian statistical modeling and probabilistic machine learning which focuses on advanced Markov chain Monte Carlo and variational fitting algorithms. It is a rewrite from scratch of the previous version of the PyMC software. Unlike PyMC2, which had used Fortran extensions for performing computations, PyMC relies on PyTensor, a Python library that allows defining, optimizing, and efficiently evaluating mathematical expressions involving multi-dimensional arrays. From version 3.8 PyMC relies on ArviZ to handle plotting, diagnostics, and statistical checks. PyMC and Stan are the two most popular probabilistic programming tools. PyMC is an open source project, developed by the community and fiscally sponsored by NumFOCUS.
