Molecular Dynamics Secondary article Jaros˚aw Meller, Cornell University, Ithaca, New York, USA Nicholas Copernicus University, Article Contents Torun´, Poland . Introduction . Computer Simulation Is a Powerful Research Tool Molecular dynamics is a technique for computer simulation of complex systems, modelled . Atomic Force Field Model of Molecular Systems at the atomic level. The equations of motion are solved numerically to follow the time . Molecular Dynamics Algorithm evolution of the system, allowing the derivation of kinetic and thermodynamic properties . Numerical Integration of the Equations of Motion of interest by means of ‘computer experiments’. Biologically important macromolecules . Force Calculation and Long-range Interactions and their environments are routinely studied using molecular dynamics simulations. Molecular Dynamics Is a Statistical Mechanics Method . Limitations of Molecular Dynamics . Molecular Modeller Kit Introduction . Studies on Conformational Changes in Proteins . Studies on Substrate/Inhibitor Binding to Proteins There is a complex network of chemical entities that evolve dynamically creating life at the molecular level. For problems. However, they require more computational example, proteins and nucleic acids fold (adopting specific resources. At present only the classical MD is practical for structure consistent with their function), ions are trans- simulations of biomolecular systems comprising many ported through membranes, enzymes trigger cascades of thousands of atoms over time scales of nanoseconds. In the chemical reactions, etc. Because of the complexity of remainder of this article the classical MD will simply be biological systems, computer methods have become referred to as MD. increasingly important in the life sciences. With faster and more powerful computers larger and more complex systems may be explored using computer modelling or computer simulations. Computer Simulation Is a Powerful Molecular dynamics (MD) emerged as one of the first Research Tool simulation methods from the pioneering applications to the dynamics of liquids by Alder and Wainwright and by Experiment plays a central role in science. It is the wealth of Rahman in the late 1950s and early 1960s. Due to the experimental results that provides a basis for the under- revolutionary advances in computer technology and standing of the chemical machinery of life. Experimental algorithmic improvements, MD has subsequently become techniques, such as X-ray diffraction or nuclear magnetic a valuable tool in many areas of physics and chemistry. resonance (NMR), allow determination of the structure Since the 1970s MD has been used widely to study the and elucidation of the function of large molecules of structure and dynamics of macromolecules, such as biological interest. Yet, experiment is possible only in proteins or nucleic acids. conjunction with models and theories. There are two main families of MD methods, which can Computer simulations have altered the interplay be- be distinguished according to the model (and the resulting tween experiment and theory. The essence of the simula- mathematical formalism) chosen to represent a physical tion is the use of the computer to model a physical system. system. In the ‘classical’ mechanics approach to MD Calculations implied by a mathematical model are carried simulations molecules are treated as classical objects, out by the machine and the results are interpreted in terms resembling very much the ‘ball and stick’ model. Atoms of physical properties. Since computer simulation deals correspond to soft balls and elastic sticks correspond to with models it may be classified as a theoretical method. On bonds. The laws of classical mechanics define the dynamics the other hand, physical quantities can (in a sense) be of the system. The ‘quantum’ or ‘first-principles’ MD measured on a computer, justifying the term ‘computer simulations, which started in the 1980s with the seminal experiment’. work of Car and Parinello, take explicitly into account the The crucial advantage of simulations is the ability to quantum nature of the chemical bond. The electron density expand the horizon of the complexity that separates function for the valence electrons that determine bonding ‘solvable’ from ‘unsolvable’. Basic physical theories in the system is computed using quantum equations, applicable to biologically important phenomena, such as whereas the dynamics of ions (nuclei with their inner quantum, classical and statistical mechanics, lead to electrons) is followed classically. equations that cannot be solved analytically (exactly), Quantum MD simulations represent an important except for a few special cases. The quantum Schro¨ dinger improvement over the classical approach and they are equation for any atom but hydrogen (or any molecule) or used in providing information on a number of biological the classical Newton’s equations of motion for a system of ENCYCLOPEDIA OF LIFE SCIENCES / & 2001 Nature Publishing Group / www.els.net 1 Molecular Dynamics more than two point masses can be solved only approxi- motions) configuration of the heavy nuclei. The nuclei, in mately. This is what physicists call the many-body turn, move in the field of the averaged electron densities. As problem. a consequence, one may introduce a notion of the potential It is intuitively clear that less accurate approximations energy surface, which determines the dynamics of the become inevitable with growing complexity. We can nuclei without taking explicit account of the electrons. compute a more accurate wave function for the hydrogen Given the potential energy surface, we may use classical molecule than for large molecules such as porphyrins, mechanics to follow the dynamics of the nuclei. which occur at the active centres of many important Identifying the nuclei with the centres of the atoms and biomolecules. It is also much harder to include explicitly the adiabatic potential energy surface with the implicit the electrons in the model of a protein, rather than interaction law, we obtain a rigorous justification of the representing the atoms as balls and the bonds as springs. intuitive representation of a molecule in terms of interact- The use of the computer makes less drastic approximations ing atoms. The separation of the electronic and nuclear feasible. Thus, bridging experiment and theory by means of variables implies also that, rather than solving the computer simulations makes possible testing and improv- quantum electronic problem (which may be in practice ing our models using a more realistic representation of infeasible), we may apply an alternative strategy, in which nature. It may also bring new insights into mechanisms and the effect of the electrons on the nuclei is expressed by an processes that are not directly accessible through experi- empirical potential. ment. The problem of finding a realistic potential that would On the more practical side, computer experiments can be adequately mimic the true energy surfaces is nontrivial but used to discover and design new molecules. Testing it leads to tremendous computational simplifications. properties of a molecule using computer modelling is Atomic force field models and the classical MD are based faster and less expensive than synthesizing and character- on empirical potentials with a specific functional form, izing it in a real experiment. Drug design by computer is representing the physics and chemistry of the systems of commonly used in the pharmaceutical industry. interest. The adjustable parameters are chosen such that the empirical potential represents a good fit to the relevant regions of the ab initio Born–Oppenheimer surface, or they may be based on experimental data. A typical force field, Atomic Force Field Model of Molecular used in the simulations of biosystems, takes the form shown in eqn [2]. Systems X a X b Uðr ; ÁÁÁ; r Þ¼ i ðl À l Þ2 þ i ð À Þ2 The atomic force field model describes physical systems as 1 N 2 i i0 2 i i0 bonds angles collections of atoms kept together by interatomic forces. In X c particular, chemical bonds result from the specific shape of þ i ½1 þ cos ðn! À Þ 2 i i torsions the interactions between atoms that form a molecule. The "#8 9 8 9 X 12 6 interaction law is specified by the potential U(r1, _, rN), > > > > :> ij;> :> ij;> which represents the potential energy of N interacting þ 4"ij À rij rij atoms as a function of their positions r 5 (x , y , z ). Given atom pairs i i i i X q q the potential, the force acting upon ith atom is determined þ k i j ½2 r by the gradient (vector of first derivatives) with respect to atom pairs ij atomic displacements, as shown in eqn [1]. In the first three terms summation indices run over all the bonds, angles and torsion angles defined by the covalent @U @U @U structure of the system, whereas in the last two terms F i ¼Àrri Uðr1; ÁÁÁ; rN Þ¼À ; ; ½1 @xi @yi @zi summation indices run over all the pairs of atoms (or sites The notion of ‘atoms in molecules’ is only an approxima- occupied by point charges qi), separated by distances tion of the quantum-mechanical picture, in which mole- rij 5 |ri 2 rj| and not bonded chemically. cules are composed of interacting electrons and nuclei. Physically, the first two terms describe energies of Electrons are to a certain extent delocalized and ‘shared’ by deformations of the bond lengths li and bond angles yi many nuclei and the resulting electronic cloud determines from their respective equilibrium values li0 and yi0. The chemical bonding. It turns out, however, that to a very harmonic form of these terms (with force constants ai and good approximation, known as the adiabatic (or Born– bi) ensures the correct chemical structure, but prevents Oppenheimer) approximation and based on the difference modelling chemical changes such as bond breaking. The in mass between nuclei and electrons, the electronic and third term describes rotations around the chemical bond, nuclear problems can be separated. which are characterized by periodic energy terms (with The electron cloud ‘equilibrates’ quickly for each periodicity determined by n and heights of rotational instantaneous (but quasistatic on the time scale of electron barriers defined by ci).