Computational Chemistry - MD Simulations
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1 Computational Chemistry - MD Simulations P. Ojeda-May pedro.ojeda-may@umu.se Department of Chemistry/HPC2N, Umeå University, , Sweden. May 2, 2017
2 Table of contents 1 Basics on MD simulations Accelerated MD Umbrella sampling String method Coarse graining Alchemical method 2 Setting up the system, minimization, solvation, equilibration, production and analysis.
3 Simulations time scale Figure : Accuracy w.r.t. time scale for different modeling approaches.
4 Early MD simulations Figure : Nature, 253 (1975).
5 Current MD simulations Figure : Source:
6 Current MD simulations
7 Application of MD Proteins Clays Figure : Clay [JPC C, 118, 1001 (2014)]. Figure : AdK enzyme in water.
8 Application of MD Figure : Ice cream research.
9 Application of MD Figure : Asphalt research. Figure : Asphalt [Const. Build. Mat., 121, 246 (2016)].
10 Newton s equation F = U Newton s Law(1687) (1) solution of this equation requires the knowledge of an array of particles positions and velocities X = (x 1 1, x 1 2, x 1 3, x 2 1, x 2 2, x x N 1, x N 2, x N 3 ) (2) V = (v 1 1, v 1 2, v 1 3, v 2 1, v 2 2, v v N 1, v N 2, v N 3 ) (3)
11 s Figure : Source:
12 s U = bonds k bonds(r r 0 ) 2 + torsions i<j Coulomb angles V j (1 + cosjφ) j q i q j r ij + i<j VdW 1 2 k angle(θ θ 0 ) 2 { [ (σij ) 12 4ɛ ij r ij ( σij r ij ) 6 ]} (4)
13 s Figure : Energy terms. Source: (chemistry)
14 s Figure : FF for proteins comparison, PLoS ONE, 7, e32131, (2012).
15 s: Energy surface Figure : Energy surface described by U = sin(x) cos(x)
16 s Proteins and Hydrocarbons: GROMOS, OPLS-AA, AMBER, CHARMM. Clays: CLAYFF Coarse-graining: MARTINI If the parameters of your compound are not part of the force field you need to use QM approaches.
17 Water models Figure : 3-5 sites water models. Source: models.html
18 Water models Figure : See for details: JPC A, 105, 9954 (2001).
19 Protein systems Figure : 20 natural amino acids. Source: goo.gl/yryvwv
20 Protein systems Figure : PDB information of AdK. Figure : Structure of yeast AdK.
21 Periodic boundary conditions (PBC) The systems we can study with MD simulations are tiny compared to real experimental setups (10 23 particles). Figure : PBCs and minimum image convention [Allen & Tildesley, Comp. Sim. of Liquids]
22 Electrostatic interactions: Ewald method The elecrostatic energy for a periodic system can be written as 1, E = 1 2 N m Z 3 i,j=1 q i q j r ij + ml where r ij = r i r j, m refers to the periodic images. Primed summation means i = j interaction is excluded for m = 0. q x is the partial charge on atom x. (5) 1 Adv. Polym. Sci., 185, 59 (2005)
23 Electrostatic interactions: Ewald method The elecrostatic energy for a periodic system can be written as 1, E = 1 2 N m Z 3 i,j=1 q i q j r ij + ml where r ij = r i r j, m refers to the periodic images. Primed summation means i = j interaction is excluded for m = 0. q x is the partial charge on atom x. The potential is splitted such that, 1 r = f (r) + 1 f (r) r r (5) (6) 1 Adv. Polym. Sci., 185, 59 (2005)
24 Electrostatic interactions: Ewald method The elecrostatic energy for a periodic system can be written as 1, E = 1 2 N m Z 3 i,j=1 q i q j r ij + ml where r ij = r i r j, m refers to the periodic images. Primed summation means i = j interaction is excluded for m = 0. q x is the partial charge on atom x. givig rise to the total energy: (5) E = E (r) + E (k) + E (s) + E (d) (6) 1 Adv. Polym. Sci., 185, 59 (2005)
25 Electrostatic interactions E (r) = 1 2 N m Z 3 i,j=1 E (k) = 1 2V k 0 q i q j erfc(α r ij + ml ) r ij + ml (7) 4π k 2 ek2 /4α 2 ρ(k) 2 (8) E (s) = α q 2 π i (9) E (d) 2π = (1 + 2ɛ )V ( i i q i r i ) 2 (10)
26 Integration of Newton s equation We now now the force field and we know the law of motion: F = ma = U Newton s Law (11) we need to integrate this equation, here we use the leap-frog scheme [Hockney, 1970], r(t + δt) = r(t) + δtv(t + 1 δt) (12) 2 v(t δt) = v(t 1 δt) + δta (13) 2 velocities are updated according to, v(t) = 1 (v(t δt) + v(t 12 ) δt) (14)
27 Constraints Collision of two diatomic molecules Figure : Free collision. Figure : SHAKE constraint. See JCP, 112, 7919 (2000)
28 Constraints Modern approaches to deal with constraints Figure : ILVES method. See JCC, 32, 3039 (2011)
29 Techniques to speedup simulations MPI parallelization MPI+OpenMP parallelization Domain decomposition scheme Multiple communicators Figure : Nodes (MPI). do i=1,num_particles x(i) = x(i) + f(i)*dt enddo Figure : NUMA machine (OpenMP).
30 Ergodicity A obs = < A > time =< A(Γ(t)) > time = lim t obs tobs 0 A(Γ(t))dt (15) Figure : Coffee cup.
31 Statistical ensembles Microcanonical ensemble (NVE) partition function is [Allen & Tildesley, Comp. Sim. of Liquids], Q NVE = 1 1 drdpδ(h(r, p) E) (16) N! h 3N The thermodynamic potential is the negative of the entropy S/k B = ln Q NVE In the case of the Canonical ensemble (NVT) the partition function is, Q NVT = 1 1 drdp exp( H(r, p)/k B T ) (17) N! h 3N with thermodynamic potential A/k B T = ln Q NVT.
32 Statistical ensembles Isothermal-isobaric ensemble (NPT) partition function is, Q NPT = N! h 3N dv drdp exp( (H(r, p)+pv )/k B T ) V 0 (18) the corresponding thermodynamic potential is G/k B = ln Q NPT Grand-canonical ensemble (µvt) partition function is, Q µvt = 1 1 N! h 3N exp(µn/k BT ) drdp exp( H(r, p)/k B T ) N (19) the corresponding thermodynamic potential is PV /k B = ln Q µvt
33 Thermostats NVE is obtained by solving NE. NVT can be achieved with the following thermostats: Berendsen, Velocity-rescaling, Nose-Hoover. H = N i=1 p i 2m i + U(r 1, r 2,..., r N ) + p2 ξ 2Q + N f kt ξ (20) A better approach is Nose-Hoover chain. Using general and local thermostats. NPT can be simulated with Berendsen and Parrinello-Rahman methods.
34 Accelerated MD simulations The original potential energy surface V (r) is modified according to, V (r) = { V (r), V (r) E, V (r) + V (r) V (r) < E. (21) Figure : Modified potential energy surface [JCP, 120, (2004)].
35 Accelerated MD simulations the biasing term is, V (r) = (E V (r))2 α + (E V (r)) (22) Figure : Free energy landscape of Alanine dipeptide [JCP, 120, (2004)].
36 Umbrella sampling (US) simulations The potential energy is modified as follows JCP, 23, 187 (1977): E b (r) = E u (r) + w i (ξ) with w i (ξ) = K/2(ξ ξ ref i ) 2 Figure : Potential energy surface. For each window the free energy is given by, A i (ξ) = (1/β) ln P b i (ξ) w i (ξ)+f i Figure : Probability histograms.
37 String method (SM) simulations Define a set of collective variables z j and effective forces as follows k T T 0 (z j θ j (t))dt F (z) z j The free energy along the string is computed by PRB, 66, (2002), Figure : Free energy surface. F (z(α)) F (z(0)) = α 0 N i=1 dz i (α ) dα F (z(α )) z i dα
38 String method (SM) simulations Figure : Free energy surface of Alanine dipeptide.
39 Coarse-grain simulations Figure : Reduction of the degrees of freedom [Annu. Rev. Biophys., 42, 73 (2013)].
40 Coarse-grain simulations Figure : Reduction of the degrees of freedom [Annu. Rev. Biophys., 42, 73 (2013)].
41 Alchemical simulations Figure : Thermodynamic cycle for binding of two protein ligands L 1 and L 2, [JCC, 30, 1692 (2009)]. G bind L i L j = G bind L j G bind L i The Hamiltonian is modified according to, = G prot RL i RL j G solv L i L j (23) H = T x + (1 λ)v 0 + λv 1 (24)
42 Alchemical simulations Figure : Thermodynamic cycle for binding of two protein ligands L 1 and L 2, [JCC, 30, 1692 (2009)]. The free energy difference going from λ = 0 to λ = 1 is, 1 G λ=0 λ=1 = 1 β ln exp ( β(h (λ+δλ) H (λ) ) ) (25) λ=0
43 Setting up the system, minimization, solvation, equilibration, produ Installing CHARMM on Kebnekaise Load the modules module load GCC/ OpenMPI/ go to the charmm folder and type:./install.com gnu M at the end of the installation one gets the message: install.com> CHARMM Installation is completed. The CHARMM executable is /pfs/nobackup/home/
44 mpirun -np 28 /home/u/user/pfs/charmm_tutorial/charmm/exec/ -i step4_equilibration.inp > out_equilibration.dat mpirun -np 28 /home/u/user/pfs/charmm_tutorial/charmm/exec/ P. cnt=1 Ojeda-May -i step5_production.inp Computational > out_production.dat Chemistry - MD Simulations Basics on MD simulations Setting up the system, minimization, solvation, equilibration, produ Running CHARMM on Kebnekaise #!/bin/bash #SBATCH -A project-id #SBATCH -N 1 #SBATCH --time=01:00:00 #SBATCH --output=job_str.out #SBATCH --error=job_str.err #SBATCH --exclusive #SBATCH --mail-type=end module load GCC/ OpenMPI/1.10.3
45 Setting up the system, minimization, solvation, equilibration, produ CHARMM Setting up the system minimization solvation neutralization equilibration production analysis
46 Setting up the system, minimization, solvation, equilibration, produ CHARMM files *.pdb (coordinates) CHARMM parameter files *.inp (input file for simulation)
47 Setting up the system, minimization, solvation, equilibration, produ Performance of CHARMM Figure : Old CHARMM version. Figure : Comparison of CHARMM with other novel software. System of 19,609 atoms. JCC, 35, (2014)
48 Setting up the system, minimization, solvation, equilibration, produ CHARMM support contact HPC2N support
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