Advanced Molecular Molecular Dynamics

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Sheryl Nelson
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1 Advanced Molecular Molecular Dynamics Technical details May 12, 2014

2 Integration of harmonic oscillator r m period = 2 k k and the temperature T determine the sampling of x (here T is related with v 0 ) The time scales as p m Some people don t realize this I have seen papers where people scale masses and accelerate MD!

3 Energy comparisons Euler not stable VV & LF very stable Euler not reversible VV+LF symplectic Largest timestep: 2 energy potential kinetic half step kinetic full step time step This is 3.14 steps per period!!! Advantage: great stability! Disadvantage: too (?) great stability

4 The kinetic energy is LF and VV With the same initial conditions LF and VV produce the same trajectory sequence x i v i = (v i-1/2 + v i+1/2 )/2 K = 1/2 m v 2 unless equal: v i2 < [(v i-1/2 + v i+1/2 )/2] 2 The LF velocity are the real steps in the trajectory LF <K> should be close to <U> VV <K> might underestimate <U>

5 Integrator conclusions Not considering v, LF and VV are identical LF gives a good K VV has x and v at the same time but with VV you can also do half steps for v and get the better K Use whatever you want, but be aware of the K issue with VV

6 Today Discuss technical aspects required for MD simulations

7 How to calculate pressure? The pressure is the derivative of the free-energy wrt the volume Pressure definition: Virial definition: P = 2 V (E kin W ) W (r) = 3 2 V du dv potential energy For isotropic scaling: The viral sum: dr i dv = r i 3V W (r) = 1 2 X (virial definitions can contain different factors) r i F i For pair interactions: W (r) = 1 2 i X r ij F ij i<j

8 How to calculate interactions U(r) = X bonds U bond (r)+ X U angle (r)+x angles dihs X X q i q j + + A ij 4 i j>i 0 r ij rij 12 U dih (r) B ij r 6 ij We often only need: F i = du dr i For bonded interactions: simply loop over items in the sum and calculate F (and U) Non-bonded for small molecules: do the double loop Cost O(N 2 ): prohibitive for large systems

9 Non-bonded cut-off 3 oxygen atom pair in water 2 V Coulomb /100 Cut-off interactions beyond a radius V (kj/mol) 1 0 V LJ Fine for LJ Not fine for Coulomb r (nm) Potential should be the integral of the force V co (r) = V (r) V (rc ) r<r c 0 r r c

10 Cut-off effect on Lennard-Jones Lennard-Jones potential decays a r -6 But one atom sees many others For constant density beyond the cut-off, the missing LJ energy is: U LJ = Z 1 4 r 2 1 N C 6 ij r r 6 dr =3 1 N C 6 ij c This missing attraction can be added: long-range or dispersion correction Virial correction is identical, but adds a factor of 6 The pressure correction can be significant r 3 c

11 Inhomogeneous dispersion Uniform correction does not work for inhomogeneous systems e.g. phase boundaries and lipid membranes

12 Cut-off & force fields The basis, including LJ parameters, of most biomolecular force fields is decades old Simulation were done with cut-off s of 0.8/0.9 nm Force-fields were parametrized to give the correct density and H vap with the cut-off used Using a larger cut-off, or dispersion correction will thus result in a too high density To correct this: re-parametrize all LJ interactions Advice: use the right cut-off for the force field!

13 Non-zero force at cut-off With a plain cut-off: F(r c )!=0 This could give integration errors A huge problem for Coulomb No real issue for LJ larger issues for small r Normal Force Shifted Force Shift Function Solution: f(r) 0.5 switch F to 0 shift V to r

14 Coulomb cut-off Coulomb interactions decay as 1/r Cut-off can t be used What to do? >1 r One option: High dielectric: r c weak electrostatics =1 r use a reaction-field

15 Reaction-field Developed for dipoles Linear dipole reaction F V rf = q iq j 4 0 apple 1 + kr r ij 2 c ij k = 1 r 3 c rf 1 2 rf +1 For charges: 3 rf Additional constant c = 1 r c 2 rf required for V(r c )=0 Implicit assumption: uniform background energy (kj/mol) Coulomb RF, c=0 RF with c charge r (nm)

16 Reaction-field cntd Issue: F(r c )!=0 Solution: use rf = 1 F rf (r c )= q iq j r 2 c apple 1 2 rf 2 2 rf +1 conducting or tin-foil boundary condition But shouldn t you match the dielectric of the solvent? Integration errors often worse than deviation in dielectric Also, mismatch goes as: 1/ r

17 Boundary conditions One option: end the system Spherical boundary But what happens at the boundary? apolar liquids OK water problematic What effects on the pressure?

18 Periodic boundary conditions Minimum image convention Much simpler: use r c <L/2

19 Periodic unit or primitive cell triclinic rectangular compact Different shapes can represent the same periodic boundary conditions What matters is not the shape but the periodic shift vectors Different shapes useful for different purposes

20 3D triclinic unit-cells

21 Calculating periodic interactions j j j j i j i i j j Calculating all interactions of one i with many j: you can find which j-image you need easier: move i to different periodic shifts More on this later...

22 Electrostatics in periodic systems X X X NX NX q i q j V = f 2 n x n y n z i j r ij,n We can write down the sum over all charge pairs in all periodic images But as Coulomb goes as 1/r, this sum is only conditionally convergent Direct sums have bad convergence

23 Ewald summation 8 6 1/r direct space reciprocal space 4 V = V dir + V rec + V V dir = f 2 NX i,j X n x X n y X n z q i q j erfc( r ij,n ) r ij,n V rec = f 2 V NX i,j q i q j X m x X m y X m z exp ( m/ ) 2 +2 im (r i r j ) m 2 V 0 = f X N p q 2 i i

24 Particle mesh methods Ewald summation is slow: O(N 2 ) Solution: do the reciprocal part on a mesh Particle-Particle Particle-Mesh (PPPM/P3M) Particle-Mesh Ewald (PME) Most popular SPME (smooth) by Darden et al. spread charges on grid 3D FFT solve in fourier space 3D FFT q gather forces from grid

25 PME PME Parameters cut-off smoothing parameter B spreading order S grid size: M x,m y,m z Accuracy determined by: real space error beyond cut-off: erfc(b r c )/r c spreading accuracy spreading order smoothing parameter grid spacing Computational cost: direct: O(r c3 ) spread: O(#charges*S 3 ) 3D FFT: O(N log(n)) Complex, for SPME no simple analytical formula But, important for performance and accuracy of simulations!

PME settings in practice Complex, but also costly! Use what others use with your system and/or software Typical settings: order 4: spread 4 3 =64 points cut-off 0.

26 PME settings in practice Complex, but also costly! Use what others use with your system and/or software Typical settings: order 4: spread 4 3 =64 points cut-off 0.9 nm grid spacing 0.12 nm #grid point similar to #particles Soon: tools to set parameters based on force accuracy But what does a force accuracy of 0.1 kj/mol/nm 2 mean?

27 Long-range electrostatics methods All methods calculate the same potential and forces computational cost pre-factor communication cost Ewald summation PPPM / Particle Mesh Ewald Fast Multipole Method Multigrid Electrostatics O(N 3/2 ) small (FFT) high O(N log N) small (FFT) high O(N) larger low O(N) larger low

28 Fast multipole method

29 Multigrid electrostatics

30 PME and charged systems With net system charge: implicit assumption: uniform background charge no effect on F, effect on U Problems with nonuniform dielectric Always safer to neutralize with ions!

31 Adding (counter-)ions Simply add ions Na + or Cl - to neutralize Adding just a few ions can lead to sampling problems Better: add Na + and Cl - at physiological concentration, if possible What is physiological concentration locally? Some systems might need different ions DNA/RNA: Na + or Mg 2+?

32 Electrostatics at surfaces One surface is impossible, at least 2 To use PME with PBC: Add 2/3 of vacuum Use dipole correction Yeh&Berkowitz (JCP 111,3155) U z = 2 V M 2 z, F z,i = 4 q i V M z

33 Algorithms to do efficient MD

34 Recap. What do we need to calculate? bonded interactions: cheap non-bonded interactions: expensive maybe PME: expensive integration: cheap

interactions for n steps within cut-off r c When to update the list?

35 Calculating non-bonded interactions Using a Verlet list: Make a Verlet pair list using radius: r list = r c + r buf Calculate interactions for n steps within cut-off r c When to update the list? Option: when a particle moved more than r buf /2 Becomes expensive for large systems

36 Charge groups In the early years of MD cut-offs were used: bad with cut-off electrostatics! partial remedy: use neutral charge groups e.g. group 3 atoms in water: only dipole Remnants of this still in the Gromacs package often used without a Verlet list bad for energy conservation with thermostat fine for most purposes

37 Order of particles & interactions Without particle ordering interactions are randomly distributed in memory: bad performance Sorting the particles on a grid groups interactions good for performance & parallelization

38 Parallel Molecular Dynamics Particle or force decomposition bad memory access characteristics Spatial or domain decomposition good memory access & communication half shell eighth shell midpoint r c r c 1 2 r c (a) (b) (c)

39 Domains & load balancing 7 3 r b r c d r c For inhomogeneous systems load balancing is required Gromacs has full 3D dynamic load balancing

40 Reading Read Frenkel&Smit part V.F on saving CPU time I will put an exercise on the site this afternoon Next lecture: November 10, 10:00 at FB55 No lecture on November 15

Computer simulation methods (2) Dr. Vania Calandrini

Computer simulation methods (2) Dr. Vania Calandrini in the previous lecture: time average versus ensemble average MC versus MD simulations equipartition theorem (=> computing T) virial theorem (=> computing