Atomistische Simulationen. Vorlesungen zur Molekularen Dynamik. Prof. M. Meuwly Departement Chemie Universitaet Basel
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1 Atomistische Simulationen Vorlesungen zur Molekularen Dynamik Prof. M. Meuwly Departement Chemie Universitaet Basel
2 1. Einführung 2. Elektronenstruktur 3. Kraftfelder 4. Optimierungsstrategien 5. Molekulare Dynamik 6. Statistische Mechanik von Proteinen
3 Warum Simulationen? Fundamentales Verständnis atomistischer Prozesse Interpretation von Experimenten komplexer Systeme kann sehr schwierig sein. Gefährliche Experimente Aufwendige Experimente Stellen weiterführender Fragen
4 Leitmotiv The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. Chemie-Nobelpreise 1998 (Kohn und Pople) und 2013 (Karplus, Levitt und Warshel) P. A. M. Dirac (1929)
5 Water Transport in Channels
6 Water Transport in Channels Blue curve: water-water interaction; green curve: protein-water interaction (H-bonding); black curve: sum of green and blue.
7 Chaperones
8 Proteine sind liquid-like F. Schotte et al., Science 300, 1944 (2003)
9 Charge Exchange in N N 2 N 2 + N 2 + IACS, Kolkata February 2014 Tong et al., Chem. Phys. Lett. (2012)
10 Vibrationally Induced Dissociation of Sulfuric Acid Statistics of several 1000 reactive trajectories from exciting v 9 =5 and v 9 =6 Reyes and Meuwly JPCA (2011)
11 Empirical Force Fields and their optimization Ziel: Beschreibung der Wechselwirkungen in einem komplexen System. Problem: Anzahl Freiheitsgrade (3N mit N10 4 ) Ansatz: Ausgangspunkt ist Struktur des Systems Bindungen, Valenzwinkel, Diederwinkel Parametrisierung der Wechselwirkung
12 PEF/PES: Potential Energie Funktion Potential Energy Surface Topologie ( Landkarte ) der Wechselwirkung Niedrig-dimensionale Darstellung der WW Projektion! Komplizierte Landschaft
13 Potential energy function (mathematical equations) Empirical force field (equations and parameters that relate chemical structure and conformation to energy)
14 Common empirical force fields Class I CHARMM AMBER OPLS/AMBER/Schrödinger ECEPP (free energy force field) GROMOS Class II CFF95 (Biosym/Accelrys) MM3 MMFF94 (CHARMM, Macromodel, elsewhere) UFF, DREIDING
15 Class I Potential Energy Function Intramolecular (internal, bonded terms) bonds K b impropers 2 2 b b K K1 cos( n ) K o angles 2 2 o KUB r1,3 r1,3, o o UreyBradley torsions Intermolecular (external, nonbonded terms) q i q j ij nonbonded4dr ij R min,ij r ij 12 2 R 6 min,ij r ij
16 Class II force fields (e.g. MM3, MMFF, UFF, CFF) bonds angles K b, 2 dihedrals K impropers b b o 2 K b, 3 b b o K, 2 o 3 K b, 4 b b o 4 2 K, 3 o 3 K, 4 o 4 K,1 1 cos K, 2 1 cos 2 K, 3 1 cos 3 2 bondsbonds' bondsangles K bb' K b bondsdihedrals bonds' dihedrals anglesdihedrals b b o b b o b'b o ' K ' o b b o o anglesangles' ' o ' K, b1 cos K, b2 cos 2 K, b3 cos 3 b'b o ' K, b'1 cos K, b'2 cos 2 K, b'3 cos 3 o K, 1 cos K, 2 cos 2 K, 3 cos 3 anglesangles' dihedrals o ' o ' cos
17 bonds K b K impropers Intramolecular parameters b b o 2 K o 2 K 1 cos(n ) angles o 2 K UB UreyBradley torsions r 1,3 r 1,3,o 2 Equilibrium terms b o : bonds o : angles n: dihedral multiplicity o : dihedral phase o : impropers r 1,3o : Urey-Bradley Force constants K b : bonds K : angles K : dihedral K : impropers K UB : Urey-Bradley
18 V bond K b b b o V dihedral K (1 (cosn )) V angle K o Intermolecular interactions between bonded atoms 1,2 interactions: 0 1,3 interactions: 0 1,4 interactions: 1 or scaled > 1,4 interactions: 1
19 H V improper K o 2 H H C H V UreyBradley K UB r 1,3 r 2 1,3o
20 V bond K b b b 2 o Chemic al type K bond b o C-C 100 kcal/mole/å Å C=C 200 kcal/mole/å Å C=-C 400 kcal/mole/å Å Bond Energy versus Bond length Bond length, Å
21 V dihedral K (1 (cosn )) Dihedral energy versus dihedral angle Dihedral A ngle, degrees = 0
22 Intermolecular parameters q i q j ij nonbonded4dr ij R min,ij r ij 12 2 R min,ij r ij 6 q i : partial atomic charge D: dielectric constant : Lennard-Jones (LJ, vdw) well-depth R min : LJ radius (R min /2 in CHARMM) Combining rules (CHARMM, Amber) R min,ij = R min,i + R min,j i,j = SQRT( i * j )
23 Electrostatic Energy versus Distance Interaction energy, kcal/mol q1=1, q2=1 q1=-1, q2=1-100 Distance, Å
24 Lennard-Jones Energy versus Distance 0.9 Lennard-Jones Energy Interaction Energy, kcal/mol eps,i,j i,j e=0.2,rmin=2.5 Series Rmin,i,j R min,ij Distance, Å Distance, Å ij R min,ij r ij 12 2 R 6 min,ij r ij
25 Extent of Parameter Optimization Minimal optimization by analogy (i.e. direct transfer of known parameters) Maximal optimization time-consuming requires appropriate target data (expt, calculations) Choice based on goal of the calculations Minimal database screening NMR/X-ray structure determination Maximal free energy calculations (perturbations, potential of mean force) mechanistic studies subtle environmental effects lead optimization
26 Initial Geometry Intermolecular Optimization if intermolecular change > conv.crit. Partial Atomic Charges if intramolecular and intermolecular change > conv.crit. VDW Parameters Intramolecular Optimization if intermolecular microscopic and macroscopic change < convergence criteria Bonds if intramolecular change > conv.crit. Angles Torsions Impropers, Urey-Bradley if intermolecular and intramolecular changes < convergence criteria Parameter Optimization Complete
27 1) Identify previously parameterized compounds 2) Access topology information i) Assign atom types ii) Connectivity (bonds) iii) Charges CHARMM topology (parameter files) top_all22_model.inp (par_all22_prot.inp) top_all22_prot.inp (par_all22_prot.inp) top_all22_sugar.inp (par_all22_sugar.inp) top_all27_lipid.rtf (par_all27_lipid.prm) top_all27_na.rtf (par_all27_na.prm) top_all27_na_lipid.rtf (par_all27_na_lipid.prm) top_all27_prot_lipid.rtf (par_all27_prot_lipid.prm) top_all27_prot_na.rtf (par_all27_prot_na.prm) toph19.inp (param19.inp)
28 From top_all22_model.inp RESI PHEN 0.00! phenol, adm jr. GROUP ATOM CG CA ! ATOM HG HP 0.115! HD1 HE1 GROUP! ATOM CD1 CA ! CD1--CE1 ATOM HD1 HP 0.115! // \\ GROUP! HG--CG CZ--OH ATOM CD2 CA ! \ / \ ATOM HD2 HP 0.115! CD2==CE2 HH GROUP! ATOM CE1 CA ! HD2 HE2 ATOM HE1 HP GROUP ATOM CE2 CA ATOM HE2 HP GROUP ATOM CZ CA 0.11 ATOM OH OH ATOM HH H 0.43 BOND CD2 CG CE1 CD1 CZ CE2 CG HG CD1 HD1 BOND CD2 HD2 CE1 HE1 CE2 HE2 CZ OH OH HH DOUBLE CD1 CG CE2 CD2 CZ CE1 Top_all22_model.inp contains all protein model compounds. Lipid, nucleic acid and carbohydate model compounds are in the full topology files. HG will ultimately be deleted. Therefore, move HG (hydrogen) charge into CG, such that the CG charge becomes 0.00 in the final compound. Use remaining charges/atom types without any changes. Do the same with indole
29 Intermolecular Optimization Target Data Local/Small Molecule Experimental Interaction enthalpies (MassSpec) Interaction geometries (microwave, crystal) Dipole moments Quantum mechanical Mulliken Population Analysis Electrostatic potential (ESP) based CHELPG (g98: POP=(CHELPG,DIPOLE) Restricted ESP (AMBER) Dimer Interaction Energies and Geometries (OPLS, CHARMM) Global/condensed phase (all experimental) Pure solvents (heats of vaporization, density, heat capacity, compressibility) Aqueous solution (heats/free energies of solution, partial molar volumes) Crystals (heats of sublimation, lattice parameters, interaction geometries)
30 Partial Atomic Charge Determination Additive Models: account for lack of explicit inclusion of polarizability via overcharging of atoms. RESP: HF/6-31G overestimates dipole moments (AMBER) Interaction based optimization (CHARMM, OPLS) local polarization included scale target interaction energies (CHARMM) 1.16 for polar neutral compounds 1.0 for charged compounds For a particular force field do NOT change the QM level of theory. This is necessary to maintain consistency with the remainder of the force field.
31 Comparison of analogy and optimized charges Name Type Analogy Optimized C1 CT H11 HA H12 HA H13 HA C2 C O2 O N3 NH H3 H N4 NR C5 CEL H51 HEL C6 CT H61 HA H62 HA H63 HA O NH N
32 Intramolecular optimization target data Geometries (equilibrium bond, angle, dihedral, UB and improper terms) microwave, electron diffraction, ab initio small molecule x-ray crystallography (CSD) crystal surveys of geometries Vibrational spectra (force constants) infrared, raman, ab initio Conformational energies (force constants) microwave, ab initio
33 Note that the potential energy surface about a given torsion is the sum of the contributions from ALL terms in the potential energy function, not just the dihedral term
34
35 Lead Optimization Addition of simple functional groups is generally straightforward once the full compound parameters have been optimized. H C H H O C NH 2 H N H H N H H F O C O H O CH 3 O
36 Summary: Force Fields 1) Junk in, junk out: Parameter optimization effort based on application requirements. 2) Follow standard protocol for the force field of interest (higher level QM is not necessarily better). 3) Careful parameter optimization of lead molecules 4) Simple substitutions often require minimal or no optimization.
37 Differences between Force Fields
38 Differences between Force Fields
39 Differences between Force Fields
40 Molecular Dynamics Simulations
41 Energy function: MD: Verlet Method used to determine the force on each atom: Newton s equation represents a set of N second order differential equations which are solved numerically at discrete time steps to determine the trajectory of each atom. Advantage of the Verlet Method: requires only one force evaluation per timestep
42 MD: Velocity Verlet Method Advantage of Velocity Verlet: No differences Velocities available directly t t t a t t v t t v t t a t v t t v t t a t t v t r t t r
43 Molecular Dynamics Ensembles Constant number of particles, energy, volume (NVE) (microcanonical) Constant number of particles, temperature, volume (NVT) (canonical) Constant number of particles, temperature, pressure (NPT) (isothermal-isobaric) Constant temperature, volume, chem.potential (mvt) (Grand canonical) Transformation between different ensembles via Legendre Transformation
44 Steps in Molecular Dynamics Simulations 1) Build realistic atomistic model of the system 2) Simulate the behavior of your system over time using specific conditions (temperature, pressure, volume, etc) 3) Analyze the results obtained from MD and relate to macroscopic level properties
45 Example: KscA channel solvent KcsA channel protein (in blue) embedded in a (3:1) POPE/POPG lipid bilayer. Water molecules inside the channel are shown in vdw representation. solvent
46 Simulating the system: Free MD Summary of simulations: protein/membrane system contains 38,112 atoms, including 5117 water molecules, 100 POPE and 34 POPG lipids, plus K+ counterions CHARMM26 forcefield periodic boundary conditions, PME electrostatics 1 ns equilibration at 310K, NpT 2 ns dynamics, NpT Program: NAMD2 Platform: Cray T3E (Pittsburgh Supercomputer Center)
47 MD Results RMS deviations for the KcsA protein and its selectivity filer indicate that the protein is stable during the simulation with the selectivity filter the most stable part of the system. Temperature factors for individual residues in the four monomers of the KcsA channel protein indicate that the most flexible parts of the protein are the N and C terminal ends, residues and residues Residues in the selectivity filter have low temperature factors and are very stable during the simulation.
48 Simulation Procedure Overview
49 Simulation Procedures Setup 1. PDB file Protein Databank ( 2. PSF file Generated specifically for the molecule Contains the detailed composition and connectivity of the molecule(s) of interest
50 Simulation Procedures Setup 1. Topology file information for putting molecules together, such as what atoms are to be used, which of these atoms are bonded to each other, and the sets of atoms that form bond angles 2. Parameter file physical parameters (force constants, van der Waals forces, bonds, angles, etc.)
51 Simulation Procedures Solvation Create water box or shell to enclose the molecule Minimization Minimize the energy of the system in order to reach the most favorable configuration
52 Simulation Procedures Heating Initial velocities are assigned at a low temperature. Periodically, new velocities are assigned at a slightly higher temperature and the simulation is allowed to continue. This is repeated until the desired temperature is reached.
53 Simulation Procedures Equilibration The point of the equilibration phase is to run the simulation until the structure, pressure, temperature and energy become stable with respect to time.
54 Simulation Procedures Dynamics Normal/Periodic boundary condition Single/Multiple time stepping Integrators Electrostatics
55 Converting simulations to information Computer Simulations generate information at the atomistic/microscopic level (positions, velocities, etc.). Conversion of this data to observable/macroscopic quantities (pressure, internal energy, infrared spectra, etc.) is domain of statistical mechanics. In MD the system evolves in time. Connection between computer simulation and experimental observable is provided by Gibbs theorem: A obs A lim A( t) time 0 It states that the ensemble average approaches the time average for infinite simulation time (complete, ergodic sampling) 1 dt
56 Converting simulations to information Mean energy RMS difference between two structures
57 Converting simulations to information Temperature Specific heat T C N 2K 1 p i 3Nk 3Nk m V k 1 T B B 2 E Diffusion coefficient D dt v i 0v i t B i1 NVT Infrared spectrum I dt exp it μ0μ t i 2
58 Converting simulations to information Visualization VMD MolMol
59 Shortcomings of MD Quality of the forcefield Size and Time: atomistic simulations can be performed only for systems of a few tens of angstroms (length scale) and for a few nanoseconds (time scale). Conformational freedom of the molecule: the number of possible conformations a molecule can adopt is enormous, growing exponentially with the number or rotatable bonds. Only applicable to systems that have been parameterized Connectivity of atoms: can not change during dynamics, i.e. no chemical reactions
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