Seminar in Particles Simulations. Particle Mesh Ewald. Theodora Konstantinidou

Transcription

1 Seminar in Particles Simulations Particle Mesh Ewald Theodora Konstantinidou

2 This paper is part of the course work: seminar 2 Contents 1 Introduction Role of Molecular Dynamics - MD Molecular potential 3 3 Modelling the physical system Limitations Necessity of the calculation of the electrostatic potential Periodic boundary conditions The Ewald method for the calculation of the potential experienced by an ion Reciprocal space Reciprocal vectors Fourier Analysis Application to Ewald method Particle Mesh Ewald method Application of the method Results of the parallelization of the method in CBM d Orleans 19 8 Advantages of the PME method 20

3 1 Introduction 3 Molecular dynamics (MD) according to the definition given by [1] is a computer simulation technique where the time evolution of a set of interacting atoms is simulated by integrating their equations of motion. In molecular dynamics we follow the laws of classical mechanics, and most notably Newton s law: F i = m i a i, where m i is the atom mass, a i its acceleration, and F i the force acting upon it, due to the interactions with other atoms. 1.1 Role of Molecular Dynamics - MD Molecular dynamics could be applied to a vast area of scientific research. The following are certain areas where MD simulations have been applied with good results. Scientists are optimistic for further and greater contribution of MD in the future, especially with regards to the amazing development of hardware. [1] Liquids Defects Fracture Surfaces Friction Clusters Biomolecules Electronic properties and dynamics 2 Molecular potential In MD simulations one is interested in simulating the trajectories of interacting atoms, thus obtaining a good overview of the phenomena taking place. Most thermodynamic processes or chemical reactions are associated with specific motions of molecules. In most cases the discretization of Newton s laws gives a good approximation of the real motion. For the calculation of trajectories it is important to compute the potential (energy in a sense) of each atom as it influences considerably each atom s motion. A typical empirical force field formulation

4 of the potential energy for N atoms at positions r 1,...,r N, can be written as: 4 a i E = bonds 2 (l i l i0 ) n n angles 4ε i j [ ( σ i j b i 2 (θ i θ i0 ) 2 + ) 12 ( σ i j ) 6 + i=1 j 1 r i j r i j }{{} Lennard Jonespotential torsions V n (1 + cos(nω γ)) (1) 2 ] + 1 n n q i q j 2 i=1 j 1 r i j }{{} Coulombpotential Figure 1 illustrates the different potentials between bonds in atoms. Figure 1: Potentials appearing in a molecule [5] We have to mention here, that in MD simulations the input is not taken from real experiments, but it is the computer which generates it. In the equation 1 for example the values of the parameters a i,b i,v n should be derived from experiments whenever a different molecule is to be simulated. The characteristics of bonds among atoms however, are in most cases the same no matter to which molecule they belong. Introducing empirical values consequently, does not entail the danger of simulating a different system. Molecular dynamics is not the only approach to examine such systems. Quantum mechanics is a more rigorous one. The drawback in quantum mechanics is that it is the most computationally expensive method. In practice scientists simulate small molecules and they make the assumption that the same behavior holds for larger ones. It is a fact however, that such an assumption holds for sure for well known types of molecules, whereas there is no guarantee

5 for the rest. The molecule in figure 2 illustrates the possible movements of bonds. In most biomolecular simulations we are interested in the behavior of very big molecules as 5 Figure 2: representation of a molecule and the acting potentials [3] shown in figure 3, comprised of thousands of atoms.

6 6 Figure 3: The bovine pancreatic trypsin inhibitor protein[4] 3 Modelling the physical system As mentioned above, the physical system is completely determined by its potential, constructed from the relative positions of the atoms with respect to each other, rather than from the absolute positions. Forces are obtained as the gradients of the potential with respect to atomic displacements: F i = ri E(r i,...,r N ) Energies can be calculated using either molecular mechanics or quantum mechanics methods. It is clear that Newton s laws cannot describe situations including drastic changes in electronic structure such as bond making/breaking (see section Limitations ). For such systems a quantum mechanics non-equilibrium simulation is used. Knowledge of the atomic forces and masses can then be used to solve for the positions of each atom along a series of extremely small time steps (on the order of femtoseconds = seconds). The resulting series of snapshots of structural changes over time is called a trajectory. The use of this method to compute trajectories can be more easily seen when Newton s equation is expressed in the form, m i d 2 r i dt 2 = F i = de dr i. (2) Equation 2 gives the accelerations for each atom. In practice (for the systems we simulate) we do not know the analytical solution and hence we can only we compute numerically the velocities and trajectories from equations 3 and 4 respectively

7 7 dv i dt = a i, (3) dr i dt = v i. (4) The leapfrog method is a common numerical approach to calculating trajectories based on Newton s equation. Characteristic feature in this method is the use of two different grids, one for the velocity and one for the position. The steps can be summarized as follows: de dr i = F i = m i a i (t), v i (t + t/2) = v i (t t/2) + a i (t) t, r i (t + t) = r i (t) + v i (t + t/2) t. 3.1 Limitations The de Broglie thermal wavelength is used to define the limits between MD and Quantum Mechanics simulations [1]. 2πh Λ = 2 Mk B T Λ: Broglie thermal wavelength Λ << α. where M is the atomic mass and T the temperature h Plank s constant, k B Boltzman s constant and a the mean nearest neighbor separation. When the fraction Λ α is at maximum 0.1 one can apply Newtons laws. However, when the distance between atoms is very small (a) this fraction becomes larger which means that the system is in a non equilibrium state. As a result an MD simulation is not recommended. The above test refers to the energy of the system. One should in addition control the stiffness of the system. When simulating a stiff system we confront the problem of simulating the rapidly vanishing modes as well as the normal phenomena which finally prevail. One should then focus on the prevailing frequencies and not end the simulation before the relaxation of the quantities we are interested in. The size of the system can cause inaccuracies, in terms of space correlation functions. It can happen that the size of the simulation cell will be such, that the correlations will be totally ignored or misinterpreted. This problem can be partially reduced by a method

8 known as finite size scaling. This consist of computing a physical property A using several boxes with different sizes L, and then fitting the results on a relation 8 A(L) = A 0 + c L n L:length of each side of simulation cell. 4 Necessity of the calculation of the electrostatic potential The accurate calculation of macromolecular structures and dynamics remains a challenge [13]. Basically the inaccuracies are due to the simplifications we do on the conformation of the simulated system. In addition to that problem molecular dynamics simulations using explicit solvent molecules (that means that the solvent, most often water, is treated as discrete molecules whereas implicit solvents are treated as continuum) often exhibit unrealistic behavior on the time scale they take place. Recent work has shown that simulations using current force fields without truncation of Coulombic interactions do not exhibit this unrealistic behavior. In the article of [16] it is mentioned that molecular dynamics simulations involving explicit solvent molecules have usually been performed under one of the following boundary conditions on the Coulombic interactions: Periodic boundary conditions using a finite cutoff, similar to the approach of Verlet for Lennard Jones system (At this approach the only interactions taken into consideration are within the area of a circle whose radius Rcutoff is defined empiricaly) Periodic boundary conditions using Ewald summation Periodic boundary conditions together with a reaction field Nonperiodic boundary conditions together with some treatment of the system environment interface.

9 4.1 Periodic boundary conditions 9 The boundary conditions of the simulated system cause intrinsic difficulties. If for example the system simply terminates at the boundary then the surface effects occur, which causes problems when simulating a region from the interior of a system. An solution to this problem is to use periodic boundary conditions (PBC). When using PBC, the particles are enclosed in a simulation cell which is replicated to all three dimensions until infinity. That means that the whole space is comprised of copies of our simulation cell. This excludes any interaction with the surroundings. In other words, if a particle is located at position r in the box, we assume that this particle really represents an infinite set of particles located at: r + la + mb + nc, l,m,n (, ) integers, a,b,c : vectors along cell axis. Advantages: Avoid surface effects all physical properties remain invariant of simulation cell translation It is important to understand that this periodicity is in space not in time. Fig. visualization of this periodicity. 4 gives a Figure 4: Periodic boundary conditions. The central box is outlined by a thicker line. [6]

10 10 5 The Ewald method for the calculation of the potential experienced by an ion The electrostatic potential caused by charged bodies (i.e. atoms, ions) is given according to Coulobs law by formula 5 (we make the assumption that interactions are always between two charges). The problem with this formula is that it is only conditionally convergent. That means that if one would change the order of summation the series might not converge. E(r 1,r 2,...,r N ) = 1 2 n i j q i q j r i r j + n (5) For solving this problem, Ewald proposed a conversion (based on Adel s theorem) of the non converged series using Gaussian distributions as convergence factors. Practically, he added and then subtracted charge in the form of Gaussian distribution and thus obtained a convergent series. Then he split the resulting integral (summation) into two; One entailing only Gaussian distributions and the second one entailing both Gaussian distributions and delta functions of the charge. The first summation is taking place in the reciprocal space (see 5.1) where the Poisson s equation 2 φ a = 4πρ can very easily be solved. In the following we give the fundamental concepts of Fourier transformation. It is obvious that calculating the second derivative of exponentials provides a significant gain of computation time. The second summation is solved in normal space since the the Fourier expansion of a periodic array of delta functions do not decay for large reciprocal vectors. 5.1 Reciprocal space For a better understanding of Ewald s method we give here an overview of reciprocal space [7]. Consider the non-orthogonal lattice illustrated in figure 5. If a function f is periodic on the lattice, it means that f (r) and f at all of the points Q share the same value. The function f (r) depends only on r s location relative to the cell. The function f thus needs to throw away all the parts of r that have to do with translation, and just focus on the remaining bit of r. Thus we seek to write r in terms of our basis vectors a and b. It should be clear that r can always be written as a linear combination of a and b (i.e.,thata and b span the space), but how do we find the particular linear combination with r = αa + βb? The job of finding this linear combination is made easier by constructing the vectors a and b. a is constructed so that it is perpendicular to b and is scaled so that when dotted with

11 11 Figure 5: one of the benefits of reciprocal space is that lattice does not need to be orthogonal as shown in the picture[7] a it yields 1. The fact that, in dimensioned variables, a has the units of reciprocal length, contributes to the name reciprocal space.) Now if we dot both side of the above equation with a, the term drops out (since it is perpendicular to b) and since a a =1 we have: α = ra β = rb Now f needs to ignore the whole number parts of α andβ, as the fraction parts state where r is in the base rhombus. One way of ignoring the whole number parts is via periodic functions like the trigonometric functions sine and cosine. For example, cos[2π(nα + mβ)] = cos[(n2πa + m2πb )r] = cos[(na + mb )r] = cos[gr] 5.2 Reciprocal vectors The basic reciprocal vectors can easily be derived from the normal ones by the following formulas: a = b c ab c b = c a ab c c = a b ab c ba = 2πδ ab ca = 2πδ ca bc = 2πδ bc G = υ 1 a + υ 2 b + υ 3 c G reciprocal vector

12 12 n(r + T) = n G exp(igr)exp(igt) G since exp(igt) = 1, exp(igt) = exp[i(υ 1 a + υ 2 b + υ 3 c )(u 1 a + u 2 b + u 3 c)] n(r + T) = n(r) = exp[i2π(υ 1 u 1 + υ 2 u 2 + υ 3 u 3 )] (u i,υ j integers) Periodic functions on the real lattice can be represented on the reciprocal space by their frequencies. E.g a wave of type ψ(r) = e ikr shown in figure 6 in real space and in reciprocal or k-space Figure 6: illustration of the simplification the k-space offers [10] 5.3 Fourier Analysis In Fourier analysis the fundamental principle is that every periodic function can be represented as a linear combination of the cosine and sine functions [9]. When a function is periodic with period T the following is true: In one dimension the Fourier expansion is n(r + T) = n(r)fort R d. n(x + α) = n 0 + [C p cos(2πpx/α) + S p sin(2πpx/α)] = n 0 + [C p cos(2πpx/α + 2πp) + S p sin(2πpx/α + 2πp)] p>0 = n 0 + [C p cos(2πpx/α) + S p sin(2πpx/α)] = n(x). p>0

13 We can rewrite the above with the use of exponential function as n(x) = n p exp(i2πp/α). p 13 In three dimension we use the concept of reciprocal vectors G ana the Fourier expansion becomes n(r) = n G exp(igr) G: reciprocal vector of the 3D lattice. G The inverse Fourier Transformation is n G = Vc 1 dv n(r) exp( igr). cell 5.4 Application to Ewald method As already mentioned, Ewald just split the electrostatic potential [9] caused by the charge of atoms (or ions), into two potentials the real and the reciprocal in order to preserve convergence. The following computations reffer to the potential of a single atom at a certain position. φ = φ 1 + φ 2 φ 1 is the reciprocal space potential and is computed as the difference of the sum of the potential caused by the charge of all atoms φ a minus the potential of the charge of the atom itself φ b (see fig. 7) φ 1 = φ a φ b Figure 7 displays the splitting. In the figure, lattice 1 is the real space potential, and lattice 2 the reciprocal. Figure 7: Ewald summation [8]

14 The figure illustrates the splitting that Ewald used to achieve convergence. The parameter of the chosen distribution will not affect the result since their sum is zero. A good choice of parameters can enhance the speed of convergence. We express first the potential φ a and the charge density which causes it as a sum of Fourier series: φ a = c G exp(igr) G ρ = ρ G exp(igr). G Poisson s equation, 6, relates the potential to the charge distribution = G 14 2 φ a = 4πρ (6) G 2 c G exp(igr) = 4π ρ G exp(igr). G Details about the derivation of φ 1 (i) can be found in [9], we obtain φ 1 (i) = 4π S(G)G 2 exp( G 2 /4η) 2q i (η/π) 1/2 G The index i signifies a position in the lattice. The potential of the real space is the sum of three potentials: the point potential, the potential of the Gaussian distribution of the charge density that is in the sphere of radius r l over the l-th atom position and the potential of the Gaussian distribution of the charge density that is out the sphere of radius r l. The last term is a correction term known as background charge density. q l [ 1 r l 1 r l rl 0 ρ(r)dr r l Summing this over all atoms of the unit cell yields φ 2 = l q l r l F(η 1/2 r l ), f ρ(r) ] r dr where F(x) = (2/π 1/2 ) exp( s 2 )ds. x Finally we obtain for the potential of the atom at position i φ(i) = 4π S(G)G 2 exp( G 2 /4η) 2q i (η/π) 1/2 G }{{} φ 1 + q l F(η 1/2 r l ). l r l }{{} φ 2

15 6 Particle Mesh Ewald method 15 The Particle Mesh Ewald method is based on the Partical Mesh algorithm. The basic idea is, to solve Poisson s equation in PBCs taking advantage of fast Fourier transform (FFT) for calculating discrete Fourier transforms. The potential caused by Coulombic interactions is treated as one of a short range describing the influence of the neighboring atoms and the long range potential (in Fourier space ) as a situation in the whole domain. The four principal steps for the particle mesh calculation are: 1. Assign charge to the mesh 2. Solve the field equation on the mesh 3. Calculate the mesh defined force field 4. Interpolate to find forces on the particles In Particle Mesh Ewald method the steps are similar, the difference is on how the charge density is approximated. Here, the complex exponentials (because of PBC) of the charge are just locally interpollated on the grid points whereas in Particle Mesh method the Gaussian distributions are approximated. 1. Assign charge to the mesh using a weighting function 2. Grid transformation - Fourier space representation of the discrete charge distribution 3. Calculate the Potential in Fourier space 4. Transformation to real space 5. Interpolate the potential to the atom and differentiate for calculation of force (Smooth PME method) Figure 8 illustrates the method applied for the computation of the real space term of the potential. As one can observe only the atoms in the area defined by the radius Rcutoff are assumed to interact with the atom whose potential we wish to compute. Figure 9 illustrates graphically the above mentioned steps.

16 16 Figure 8: The real space term of the potential is calculated with R cuto f f methods [15] Figure 9: The steps a,b,c,d illustrate the time sequence of the method.[15] 6.1 Application of the method Suppose there are N point charges q 1,q 2,...,q N at positions r 1,r 2,...r N within the unit cell U satisfying q 1 + q q N = 0. The vectors a α,α = 1,2,3 which need not be orthogonal define the edges of the unit cell. The conjugate reciprocal vectors a α are defined by the relation a α a β = δ αβ. The point charge q i at position r i has fractional coordinates s αi,α = 1,2,3 defined by s αi = a αr i. The charges interact according to Coulob s law with periodic boundary conditions. Thus a point charge q i at position r i interacts with other charges q j at positions r j as well as with all of their periodic images at positions r j +n 1 α 1 +n 2 α 2 +n 3 α 3 for all integers n 1,n 2,n 3. It also interacts with its own periodic images at r i + n 1 α 1 + n 2 α 2 + n 3 α 3 non zero. The

17 electrostatic energy of the unit cell can be written : E(r 1,r 2,...,r N ) = 1 2 n i j q i q j r i r j + n We define the reciprocal lattice vectors m by m = m 1 a 1 + m 2 a 2 + m 3 a 3 and the structure factor S(m) of the unit cell given as 17 N N S(m) = q j exp(2πm r j ) = q j exp[2πi(m 1 s 1 j + m 2 s 2 j + m 3 s 3 j ]. (7) j=1 j=1 Equation 9 gives the part of the energy computed in normal (or direct) space, equation 10 is the part of the energy computed in reciprocal space (the result will be transformed back with the inverse Fourier transformation) and the energy computed by equation 11 represents a correction term. The correction term is subtracted from the total because it represents the interactions between atoms of a molecule that are very close to each other. Since this energy was added to the first two parts it should then be subtracted. E dir = 1 2 n E rec = 1 2 m N i, j er f c(β r j r i + n ) r j r i + n exp( π 2 m 2 /β 2 ) m 2 S(m)S( m) (9) 0E cor = 1 er f c(β r j r i + n ) 2 β N r (i, j) M j r i + n π i=1q 2 i (10) In order to compute the force acting on a atom at position i one has to add the above given energies and then differetiate with respect to r i. Since we have periodic boundary conditions (similar to crystal stucture), the electrostatic structure factors are approximated by interpolation of complex exponentials. As shown in figure 9, we start with a random distribution of charges, which we then interpolate at the grid points. On this gridded charge distribution we calculate the reciprocal energy. This energy is differentiated in order to get the forces. In order to compute the force acting on an atom at a position r i one has interpolate back to the atoms position. Given K 1,K 2,K 3 are positive integers and a point r in the unit cell we define the scaled fractional coordinates by u α = K α a α r for α = 1,2,3 and 0 u α < K α ( ) ( 2πim1 u 1 2πim2 u 2 exp(2πimr) = exp exp K 1 K 2 ) exp ( ) 2πim3 u 3 Let [u] denote integer part of u then we use the following equation for the interpolation of the exponentials ( ) ( ) ( ) 2πimα u α 2πimα 2πimα exp (1 (u α [u α ]))exp [u α ] +(u α [u α ])exp ([u α ]+1) K α K α K α K 3 (8) (11)

18 W 2 (u) = 1 u f or 1 u < 1 W 2 (u) = 0 f or u > 1 ( ) ( ) 2πimα u α 2πimα exp K α W 2 (u a k) exp k k= K α 18 For higher order interpolation the terms W i, i = 1,2,3 are read as W 2p (u) = j=p 1 j= p, j p (u + j k) j=p 1 j= p, j p ( j k) k u k + 1 k = p, p + 1,..., p 1 The index p defines the order of the interpolation. The linear case is for p = 1. S(m) S(m) = = K 1 1 k 1 =0 K 2 1 k 2 =0 K 3 1 k 3 =0 ( ) ( ) 2πimα u α 2πimα exp K α W 2p (u a k) exp k k= K α N q i i=1 k 1 = k 2 = k 3 = ( ) ( ) 2πim1 2πim2 exp k 1 exp k 2 K 1 K 2 [ ( m1 k 1 Q(k 1,k 2,k 3 )exp 2πi where F(Q)Fourier Transformation of Q Q(k 1,k 2,k 3 ) = N i=1 W 2p (u 1i k 1 ) W 2p (u 2i k 2 ) W 2p (u 3i k 3 ) ( ) 2πim3 exp k 3 K 3 K 1 + m 2k 2 K 2 + m 3k 1 K 3 )] = F(Q)(m 1,m 2,m 3 ) n 1,n 2,n 3 q i W 2p (u 1i k 1 n 1 K 1 ) W 2p (u 2i k 2 n 2 K 2 ) W 2p (u 3i k 3 n 3 K 3 ) E rec Ẽ rec = 1 2πV exp( π 2 m 2 /β 2 ) m 0 m 2 F(Q)(m 1,m 2,m 3 )F(Q)( m 1, m 2, m 3 ) (12) The symbol in equation 12 is justified by the interpolation. The resulting reciprocal energy will be transformed back to cartesian space.

19 19 7 Results of the parallelization of the method in CBM d Orleans The following results come from the university of Orlean (Centre Biophysique moleculaire) [15]. The simulation is of water molecules whose behavior is considered like the one of atoms. One can observe that the more the processors the less accurate the method. The ideal case for this method is for small simulations where the number of processors does not go over 8. #5000 molecules of water (15000 atoms) #Length of simulation box =10 #Number of grid points for each grid direction=25 #Interpolation order =4 #Rcutoff = 1.25 P=4 P:number of processors Energy of direct space = Energy of reciprocal space= Energy of correction term= P=8 P:number of processors Energy of direct space = Energy of reciprocal space= Energy of correction term= P=16 P:number of processors Energy of direct space = Energy of reciprocal space= Energy of correction term= P=32 P:number of processors Energy of direct space = Energy of reciprocal space= Energy of correction term= P=64 P:number of processors Energy of direct space = Energy of reciprocal space= Energy of correction term= time= time= time=time= time= time= time= time= time= time= time= time= time= time= time= time=

20 8 Advantages of the PME method 20 High accuracy: The method can provide high accuracy results ( relative force) with a relatively small augmentation to computtation time [12]. Ease of implementation: The PME method can be efficiently implemented into conventional MD Continuity: Both potentials computed with the PME method are continuous functions of positions which alleviates the problem of integrating discontinuous functions. Efficiency: The PME method is fast. For large macromolecular systems, the PME requires only about a 40% overhead over conventional truncated list-based methods to obtain relative force accuracies of References [1] Furio Ercolessi A molecular dynamics primer [2] Celeste Sagui and Thomas A.Darden Molecular Dynamics simulations of biomolecules: Long Range Electrostatic Effects, Annu.Rev. Biomol.Struct. 28: (1999) [3] molecular_mechanics_document.html [4] [5] [6] [7] T.Kirkman [8] [9] Charles Kittel, Einfuerung in Festkoerperphysic (1991) [10] official/m.mehring/condmat/reciprocal.htm [11] R. Hockney, and J. Eastwood Computer simulation using particles McGraw-Hill, New York. (1981) [12] T.Darden, D.York and L Pederson Particle mesh Ewald: An Nlog(N) for Ewald sums in large systems J.Chem.Phys.98 (1993) page 10089

21 [13] Ulrich Essman, Lalith Perera, and Max L.Berkowitz- Department of Chemistry, University of North Carolina A smoothe particle mesh Ewald method J.Chem.Phys. (1995) page 8577 [14] Brock A.Luty, Ilario G.Tironi and Wilfred F. van Gunsteren Lattice-sum methods for calculating electrostatic interactions in molecular simulations J.Chem.Phys.103(1995) page 3014 [15] /ihperf98/ihperf.htm. [16] M. Allen and D. Tildesley Computer Simulation of Liquids Oxford Science, London, (1990) 21