Molecular dynamics simulation of Aquaporin-1. 4 nm
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1 Molecular dynamics simulation of Aquaporin-1 4 nm
2 Molecular Dynamics Simulations Schrödinger equation t (r, R) =H (r, R) Born-Oppenheimer approximation H e e(r; R) =E e (R) e(r; R) Nucleic motion described classically Empirical Force field 1
3 Molecular Dynamics Simulations Interatomic interactions
4 Force- Field
5 Molecular Dynamics Simulation Molecule: (classical) N-particle system Newtonian equations of motion: with Integrate numerically via the leapfrog scheme: with Δt 1fs! (equivalent to the Verlet algorithm)
6 Aquaporin water channel
7 Human hemoglobin
8 Lipid membranes
9 Today s lecture Protein structures Notes on force calculations Setup of a simulation Organize force field parameters Algorithms used during simulation Energy minimization and equilibration of initial structure Analysis of a simulation
10 Protein structures: primary structure 20 different amino acids encoded in the DNA 3-letter and 1-letter codes Primary structure = amino acid sequence From N- to C-terminus Lysozyme www2.chemistry.msu.edu KVFGRCELAAAMKRHGLDNYRGYSLGNWVC AAKFESNFNTQATNRNTDGSTDYGILQINSRW WCNDGRTPGSRNLCNIPCSALLSSDITASVNC AKKIVSDGNGMNAWVAWRNRCKGTDVQAWI RGCRL
11 Protein structures: secondary structure Secondary structure = 3D fold of local AA segments alpha helix Lysozyme: alpha-helices, beta sheets, connected by loops beta sheet Turns, 310-helix,
12 Protein structures: tertiary structure Tertiary structure = 3D fold of one polypeptide chain Mainly alpha-helical
13 Protein structures: tertiary structure Tertiary structure = 3D fold of one polypeptide chain Mainly beta sheets
14 Protein structures: tertiary structure Tertiary structure = 3D fold of one polypeptide chain OmpX (pdb 2M06)
15 Protein structures: ter-ary structure Alpha helices and beta sheets
16 Protein structures: ter-ary structure Alpha helices and beta sheets
17 Protein structures: quaternary structure Arrangement of multiple folded polypeptides Example: Haemoglobin four subunits Interesting: Cooperative oxygen binding through quaternary transitions
18 Multiple Time Stepping H. Grubmüller, H. Heller, A. Windemuth, K. Schulten; Mol. Sim. 6 (1991) 121
19 i j Multipole Methods O(N 2 ) 1. Taylor expansion i j Exact for infinite multipole series
20 Fast Multipole Method (FMM) à O(N) + arbitrary accuracy - high order expansions required to achieve moderate accuracy L. Greengard and V. Rokhlin, J. Comp. Phys. 73 (1987) 325
21 Fast structure-adapted multipole methods: O(N) M. Eichinger, H. Grubmüller, H. Heller, P. Tavan, J. Comp. Chem. 18 (1997) 1729
22 Ewald summation Another very popular method to efficiently compute Coulomb forces of without simple cutoffs (applicable for periodic systems) Charge density: q q q x x x Point charges Idea: Rewrite the charge density as a sum of two terms: Quickly varying density: Potential can be computed accurately with cut-offs ( direct space calculation ) Slowly varying density: potential can be efficiently computed in reciprocal space using the Fast Fourier Transform (FFT); O(N log(n)) Fourier transform of charge density Ewald, Ann. Phys. 64: (1921)
23 Simulation system setup 1 Get PDB structure and check for missing atoms/groups inaccuracies (flipped histidine ring) missing ligands chemical plausibility mutations (e.g., to facilitate crystallization) read the paper!! Choose force field all-atom or united-atom, e.g. CH2, CH3 as one atom implicit or explicit hydrogen atoms polarizable force field required? QM methods required (chemistry?) Add hydrogen atoms to protonable ( titratable ) groups (Histidine!)
24 Simulation system setup 2 Choose periodic boundary conditions or not
25 Role of environment - solvent explicit or implicit solvation? box or droplet? Typical: box with periodic boundary conditions, avoid surface artefacts
26 Surface (tension) effects? periodic boundary conditions and the minimum image convention
27 Simulation system setup 2 Choose periodic boundary conditions or not if membrane protein: add lipid membrane atoms add water molecules add ions as counter ions (if possible, according to Debye- Hückel) ~x i (t = 0) done!
28 Simulation system setup 3 Define V(x1,...xN) via force field bond parameters angle parameters b (i) 0,K(i) b for all bonds (j) 0,K(j) for all angles dihedrals, extraplanars partial charges Van-der-Waals parameters (Lennard-Jones potential) q i apple V LJ =4 r for all atoms i, i for all atoms 12 r 6
29 Simulation system setup 4 For frequently reoccurring chemical motifs define atom types, e.g.: hydrogen HC carbon CH2 parameter file: list properties of atom types and their bonds, angles,... HC q=+0.2 m=1.0 # charge, mass CH2 q=-0.4 m=12.0 HC -CH2 K=200 b=1.1 # bonds CH2-CH2 K=500 b=1.5 HC-CH2-HC K= # angles HC-CH2-CH2...
30 Simulation system setup 5 Topology file: defines atoms bonds angles dihedrals etc. of the simulation system [ atoms ] ; nr type name 1 HC HA1 2 HC HA2 3 HC HB1 4 HC HB2 5 CH2 CA 6 CH2 CB [ bonds ] 1 5 HC-CH2 2 5 HC-CH2 3 6 HC-CH2 4 6 HC-CH2 5 6 CH2-CH2 [ angles ] HC-CH2-HC HC-CH2-CH2...
31 Simulation phase - algorithms Integration of Newton s equations of motion where Integrate numerically via the leapfrog scheme: with Δt 1fs! (equivalent to the Verlet algorithm)
32 Simulation phase - algorithms Integration of Newton s equations of motion Constrain bond lengths (LINCS, SHAKE) idea: eliminate fastest vibrations (C-H) to increase the integration time step from 1fs to 2fs side-effect: better descriptions of QM vibrations Remove overall translation (and rotation): Avoid drift of the molecule: remove translation (and rotation) of the entire simulation system: Remove overall momentum: Remove angular momentum analogously ~P = ~p i 0 NX atoms i=0 ~p i = ~p i m i M ~ P
33 Simulation phase - algorithms Remove overall translation (and rotation): Avoid drift of the molecule: remove translation (and rotation) of the entire simulation system: Numerical instability: Accumulation of kinetic energy in to one degree of freedom. (Flying ice cube problem) Coordinate (nm) Center of mass Time (ps) Potential (kj/mol) Time (ps)
34 Simulation phase - algorithms Choose thermodynamic ensemble NVE (microcanonical ensemble) NVT (canonical ensemble, isochoric): T-coupling NPT (canonical ensemble, isobaric): T-coupling and P-coupling T-coupling, e.g. Berendsen thermostat After each step Δt: τ = coupling time constant T0 = target temperature ~v i ~v i s1 T = 2 3 P-coupling: analogous, by scaling volume Write out coordinates at some frequency 1 Nk B t NX i=1 T 1 T 0 m 2 v2 i
35 Mimimization/equilibration: 1) Energy minimization Reduce the steric strain by a moving along the steepest descent in V (~x 1,...,~x N ) Notes: Protein moves in to local minimum Attention: proteins don t tend towards the local minimum in V(x), but towards the global minimum in the free energy! Entropy/ensembles are important!
36 BPTI: Minimization
37 Mimimization/equilibration: 2) Thermalization Heat the system to, e.g. 300K by assigning Maxwell-distributed velocities p(v x ) / e mv 2 x 2k B T, p(v y ) / Trick to avoid distortion of the protein: assign velocities to to the system keep protein backbone restrained equilibrate for ~100ps
38 Mimimization/equilibration: 3) Equilibration How long? Multiple checks: Convergence of energy contributions (particularly Coulomb and Lennard-Jones) and box dimensions Room-mean square deviation (RMSD) from the crystal/nmr structure RMSD(t) = 1 N X N i=1 [~x i(t) ~x i (0)] 2 1/2 0.15? Typically: RMSD (nm) picosecond jump conformational sampling Time (ns)
39 Mimimization/equilibration: 3) Equilibration Reasons for RMSD increase/drift: Fast fluctuations picosecond jump slow conformational motions nanosecond drift Conformational transitions stairs Structural drift due to - bad X-ray structure - inaccurate force field - software bug - OK OK OK NOT OK
40 Mimimization/equilibration: 3) Equilibration Judgement of RMSD: RMSD does not converge simulation is not OK. But: RMSD converges simulation is OK. Better check, e.g., PCA projections (see later lecture)
41 e.g times Flow chart of MD simulation Preparation Simulation Specify simulation parameters (time step, temperature, ) Compute forces using your force field Set initial velocities Update atom positions & velocities ( integration step ) Energy minimisation Take care of pressure and temperature Choose force field Prepare simulation system (add hydrogen atoms, water, ions) Get initial positions of atoms (e.g., from the PDB) Update time step t! t + t Repeat up to requested simulation time
42 Simulation analysis Available after simulation: Positions: ~x 1 (t i ),...,~x N (t i ), t i =0, t, 2 t,..., T e.g., T = 10ns, N = , Δt = 2fs Byte = 6 TByte! Velocities ~v 1 (t i ),...,~v N (t i ) Temperature T (t i )= 1 (3N 6)k B NX m i vi 2 (t i ) i=1 Potential energies: V bond (t i ),V angle (t i ),V dih (t i ),V Coul (t i ),V LJ (t i ), Anything you can program
43 Simulation analysis Observables that may be interesting: everything that can be measured Size of atomic fluctuations h(~x j h~x i j ) 2 i 1 MX 2 ~x j (t i ) x j M Note: ensemble average time average i=1 X M x j = M 1 ~x j (t i ) i=1 Anything that helps to understand the protein function: - Movie (!), motion of groups - interaction energies, hydrogen bonds, radial distribution functions, transition rates, change in secondary structure
44 BPTI: Molecular Dynamics (300K)
45 Opening transition of the enzyme ATCase
46 Molecular dynamics simulation of Aquaporin-1 4 nm
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