2 References: 1)A.R. Leach, “Molecular Modeling”, second edition, Prentice Hall pp ) D. Frenkel and B. Smit, “Understanding Molecular Simulations from Algorithms to Applications”, second edition, Academic Press. Chapters 4 and 6. 3) M.P. Allen and D. J. Tildesley, “Computer Simulation of Liquids”, 1991, Oxford. Chapter 3.
3 Introduction Biological processes are commonly studied by experimental techniques (X-ray, NMR, etc.). However, to gain deeper insights in terms of atomic interactions we try to model biological macromolecules (proteins, DNA, carbohydrates, etc.) and simulate their behavior by Monte Carlo (MC) methods or molecular dynamics (MD) techniques that obey the rules of physics. Before discussing MD let us refresh some basic notions from high school physics.
4 Forces Examples of F (F vector; F=|F|) : 1) Stretching a spring by a distance x: F= -fx, Hook’s Law f- spring constant. F negative because it is opposite to x. 2) Gravitation force: F= kMm/r 2 - m and M masses with distance r; k - constant. On earth (R,M large), g=kM/R 2 F=mg 3) Coulomb law: F=kq 1 q 2 /r 2, q 1 and q 2 charges. Newton’s second law (F - resultant force): a, v, and x vectors; t time
5 Mechanical work W: If a constant force is applied along distance d, W=Fd (F=|F|). More general, W= ! F. dx. Potential energy: If mass m is raised to height h negative work is done, W = –mgh and the mass gains potential energy,E p = -W = +mgh - the ability to do mechanical work: when m falls dawn, E p is converted into kinetic energy, E k = mv 2 /2, where v 2 /2=gh (at floor). A spring stretched by d: E p = -W = f ! xdx = fd 2 /2 Two charges: E p = kq 1 q 2 /r In a closed system the total energy, E tot = E p + E k is constant but E p /E k can change; e.g., oscillation of a mass hung on a spring.
6 Linear momentum p=mv The total momentum of a system of particles is equal to the momentum of a single particle with the total mass of the system moving with the velocity of the center of mass of the system, v CM v CM = Σm i v i / Σm i In a system with no external forces the total momentum is conserved - the center of mass moves in a straight line with constant speed. Notice, the force exerted on a particle is the negative derivative of the potential energy with respect to the coordinates. F = dp/dt = - dE p /dx
7 MD simulations We treat N argon atoms of mass m enclosed in an isolated container. Each pair interacts via Lennard-Jones potential energy (E tot = const.) r =σ ε r The force in x 1 direction between (x 1, y 1, z 1 ) and (x 2, y 2, z 2 )
8 Assume that at t=0 atom i is positioned at coordinates x i (0) and has initial velocity v i (0) (i=1,N); one can solve numerically Newton’s equations obtaining the positions x i (t) and velocities v i (t) at time t. This is the essence of MD. Integration of the equations of motion by a finite differences method with the popular Verlet algorithm: Taylor expansions to 3 rd order for i Adding these equations gives [up to order (δt) 4 ] Independent of velocities. a(t) is calculated from F/m; m – mass.
9 The velocities (required for the kinetic energy and temperature) are: i.e., correct up to (δt) 2. They can also be estimated at half step, t+(1/2)δt, The process is iterative. Starts with initial coordinates and velocities; then t+δt t and t t- δt etc. Irrespective of initial conditions the particles get mixed according the laws of statistical mechanics. Most of computer time is spent on calculation of the forces, f = a/m. The Verlet algorithm satisfies time reversal r(t + δt) = r(t - δt).
10 The procedure is approximate if δt is not small enough numerical instabilities and drift in the total energy can occur. For proteins, typically δt=1/2 - 4 femtoseconds (1 fs = seconds) large protein systems (~10 4 atoms) are limited to ~1 ms simulations. Very small changes in the initial conditions will lead to different trajectories. However, we are not interested in the trajectory per se. We seek to have a reliable trajectory from the thermodynamics point of view, i.e., one which leads to the correct statistical mechanics averages of properties such as potential energy, end-to-end distance of a polymer, etc. Verlet’s algorithm maintains constant energy for relatively long times. Calculation of the velocities (i.e. kinetic energies) is not precise enough.
11 Many other algorithms have been developed. Some are equivalent to Verlet’s method - leap-frog (Hockney, 1970) and velocity Verlet (Swope, Andersen, Berens& Wilson, 1982), where v is more accurate Here, first r(t+ δt) is calculated from v(t) and a(t). v(t+ δt ) is calculated in two stages, first at mid-step, i.e., v(t+ δt/2) Finally, a(t+ δt) is calculated and the corresponding force, which lead to v(t+ δt)
12 MD and statistical mechanics The argon atoms in the isolated box (i.e., of constant energy, E tot ) are described in statistical mechanics by a microcanonical ensemble; each system configuration (x N,v N ) = (x 1,..,x N,v 1,…,v N ) has the same probability and total energy, E tot. One can calculate ensemble averages of potential & kinetic energy + =E tot in phase space, Ω E(tot) One can calculate in this ensemble the distribution of the velocity component, v x of atom i (with mass m); it is the Maxwell-Boltzmann (Gaussian) probability (T – temperature; k B - Boltzmann const.)
13 The microcanonical ensemble provides a static probabilistic picture, while MD defines a deterministic dynamical picture. In other words, with MD (like in real experiments) measurements are carried out in time and it would be beneficial if the two pictures could be reconciled, i.e., if the MD time averages would lead to the ensemble averages. Under the ergodic hypothesis a long MD run does not depend on the initial conditions and it leads to the ensemble averages. For E p, Where the bar denotes time average and the brackets ensemble average. In practice, an MD trajectory after equilibration visits only a very limited part of phase space, consisting, however, of typical coordinates and velocities. The velocities are distributed Maxwell Boltzmann, which constitutes a criterion to check that equilibration has occurred.
A starting non-typical a typical random configuration configuration at high T While the probability of (x 1,..,x N,v 1,…,v N ) of picture 1 is equal to that of picture 2, the number of randomly distributed configurations (as in 2) is much larger than the cluster-like conf. of fig. 1; therefore, after equilibration the system will “always” be found in a typical (random) configuration (2) and the ensemble averages will be obtained. Calculation of temperature In a microcanonical system (N,V,E constant - V volume) T can be defined after MD equilibration, using the relation, T = /k B > for one degree of freedom or from the total kinetic energy.
15 The relation between the total average kinetic energy E k and T is: T/2=E k /k B (3N-N c ) Where N c is the number of constraints. For example, if the system is defined with periodic boundary conditions, it might be beneficial to choose initial velocities for which the velocity of the center of mass is zero. In this case N c =3. Periodic boundary conditions: the atom that should have left the box to position (x o +Δx,y), appears instead on the other side at position (Δx,y) with the same velocity. Δx xoxo The total kinetic energy is estimated from the values E k (t) obtained from a sample of n snapshots taken at constant time intervals t:
16 Efficiency MD is a robust method, which is used extensively in all kinds of systems, fluids, polymers, biological macromolecules etc. In an MD simulation two successive snapshots at times t and t+Δt are correlated if Δt is not large enough. If the correlations are strong the system will not span the required regions in phase space and the averages calculated will be incorrect, i.e., the ergodic assumption unsatisfied. Indeed, in many cases, such as in glasses, systems under a phase transition, and proteins, the correlations are strong even for very large Δt and sophisticated techniques are required to handle such systems. Processes in nature (e.g., protein folding) and in experiments might occur on relatively large time scales (seconds and above) that are beyond the reach of MD with the present computers (nano - microseconds).
17 Movie 216 argon atoms run by velocity Verlet algorithm at constant energy (NVE). Temperature 200 K (T critical 150 K) – dense gas. Time step δt= 5 femtoseconds Each frame is taken after 20 time steps (0.1 ps) Total run time for 2500 frames is 250 ps. To demonstrate the independence of the equilibrium behavior on the initial conditions (ergodic theorem) the simulation starts from a non- typical configuration where the atoms are arranged at the corner with a density of liquid argon (i.e., 8 times higher than the system density after equilibration).
18 Potential energy calculated from the movie frames
22 MD simulations in the canonical ensemble As we have seen, in the microcanonical ensmble (N,V,E), an MD run is completely mechanistic. However, in the canonical ensemble, where the temperature, T replaces the energy, E (i.e.,the variables are N,V,T) one has to provide a “thermostat”, i.e., a procedure to keep T constant. The most common thermostats are those of Berendsen and Andersen. In the canonical ensemble the system is in contact with a large (infinite) heat bath with constant T bath, which exchanges energy with the system. At equilibrium the average temperature of the system is also T bath, but this temperature slightly fluctuates around its average value (T bath ), the larger the system the smaller the fluctuations. According to the Berendsen thermostat at each time step the velocities are rescaled by the factor δt - time step t T - parameter
23 For one degree of freedom assuming δt = t T one obtains For δt < t T the change in velocities is more moderate and t T = 0.4 ps has been found to be appropriate. Evaluation: easy to implement, works well, but doest not obey the canonical distribution (the velocities are not distributed according to Maxwell-Boltzmann). This procedure can lead to local hot and cold regions in the system while T system is ok, i.e. T system ~T bath.
24 With the Andersen thermostat every predefined time interval an atom is selected at random and its velocity is redefined by drawing a new velocity from a Maxwell-Boltzmann distribution. Thus, during each interval the system moves at constant energy, which is changed from interval to interval. The interval is proportional to, is the thermal conductivity Evaluation: The velocities are distributed as needed - according to Maxwell-Boltzmann. The trajectory is not smooth - a collection of short microcanonical stretches. If the interval is large the distribution is not canonical (close to microcanonical). If the interval is small the velocities are changed too frequently and the fluctuation of the kinetic energy is incorrect. Methods exist where several or all the particles are treated at once.
25 Dynamics MD not only provides statistical mechanics averages, but also allows calculating dynamical properties. One can calculate from an MD trajectory time correlation functions that lead to dynamic parameters such as diffusion coefficients. For example, the autocorrelation function of the velocity can be estimated from an MD trajectory, where the velocities are measured n times in time intervals t In practice, the measurements start at time (m+1)t and go back m time intervals. Contribution from all particles are considered and averaged.
26 In the same way one can estimate 2 by averaging over all particles. It can be shown (Einstein, Green - Kubo) that Where v and r denote particle velocity and position vectors. D is the self diffusion coefficient. Notice that t should be large and in practice the length of the trajectory might be insufficient. Other correlation functions lead to corresponding transport properties.
27 Biological macromolecules - proteins The typical potential energy function (force field) is: E FF = bonds K r (r - r eq ) 2 + angles K ( - eq ) 2 + dihedrals V n /2 [1 + cos(n - )] + i
28 Typically the protein is immersed in a ‘box’ of water molecules where the system consists of n ~10,000 atoms (or more). Most of computer time is spent on calculating the forces related to water-water protein- protein and water-protein interactions. Notice that the “spring constants” K r and K are strong (leading to fast motions) and would require small time step, δt, relatively short trajectories. Therefore, in most studies part or all of the bond lengths are constrained by procedures such as SHAKE and RATTLE. For proteins MD is significantly more efficient than Monte Carlo (MC) techniques, because with MD the atoms move according the calculated forces while with MC the atomic positions are changed at random which leads to high energy structures that are rejected in the process.
29 In an MD simulation of a protein at room temperature the system can get trapped for a long simulation time in the potential energy well of the starting structure and the move to more stable parts of conformational space is extremely slow, unreachable within a feasible simulation time. This problem can somewhat be resolved by methods such as parallel tempering, which is a hybrid of MD and MC. EpEp conformations initial well most stable well barriers
30 Homework: Show that the leap-frog and velocity-Verlet algorithms are equivalent to the Verlet algorithm. (Check the reference books, especially 2). Send to or leave in mailbox in BST 1041w