The Molecular Dynamics Method Thermal motion of a lipid bilayer Water permeation through channels Selective sugar transport Potential Energy (hyper)surface What is Force? Energy U(x) F = " d dx U(x) Conformation (x) 1
Classical Molecular Dynamics r ( t +! = r( + v(! t v ( t +! = v( + a(! t a ( t ) = F( / F =! m d U (r) dr Classical Molecular Dynamics 1 qiq j U ( r) = 4!" 0 rij Coulomb interaction 12 &, R ), min, ij R U ( r) =. $ * ' - 2* ij $ % + rij ( + r van der Waals interaction min, ij ij 6 ) # '! (! " 2
Classical Molecular Dynamics U(r) = 1 + q i q $ j + # ij - R min,ij 4"# 0 r ij - & % r, ij ' ) ( 12 % F(r) = '" 1 q i q j 2 ' 4#$ 0 r & ij * 2 R 6 $ '. min,ij 0 & % r ) ij ( 0 / "12 $ ij r ij + % - - ',& R min,ij r ij ( * ) 12 " R 6 % (.( min,ij 0 * ' & r * ij ) 0 * ˆ /) r ij Classical Molecular Dynamics 3
Classical Molecular Dynamics Bond definitions, atom types, atom names, parameters,. What is a Force Field? In molecular dynamics a molecule is described as a series of charged points (atoms) linked by springs (bonds). To describe the time evolution of bond lengths, bond angles and torsions, also the non-bonding van der Waals and elecrostatic interactions between atoms, one uses a force field. The force field is a collection of equations and associated constants designed to reproduce molecular geometry and selected properties of tested structures. 4
Energy Functions U bond = oscillations about the equilibrium bond length U angle = oscillations of 3 atoms about an equilibrium bond angle U dihedral = torsional rotation of 4 atoms about a central bond U nonbond = non-bonded energy terms (electrostatics and Lenard-Jones) Energy Terms Described in the CHARMm Force Field Bond Angle Dihedral Improper 5
Interactions between bonded atoms V angle = K " ( ) 2 " #" o V bond = K b ( ) 2 b " b o V dihedral = K ( 1+ cos( n"! )) " # Energy function: Classical Dynamics F=ma at 300K used to determine the force on each atom: yields a set of 3N coupled 2 nd -order differential equations that can be propagated forward (or backward) in time. Initial coordinates obtained from crystal structure, velocities taken at random from Boltzmann distribution. Maintain appropriate temperature by adjusting velocities. 6
Langevin Dynamics Langevin dynamics deals with each atom separately, balancing a small friction term with Gaussian noise to control temperature: The most serious bottleneck steps Rotation of buried sidechains Local denaturations Allosteric transitions s ms 10 15 10 12 µs 10 9 (year) Hinge bending Rotation of surface sidechains Elastic vibrations ns ps 10 6 (day) 10 3 SPEED LIMIT Bond stretching Molecular dynamics timestep fs 10 0 δt t = 1 fs 7
Potential Energy (hyper)surface r ( t +! = r( + v(! t Energy U(x) Conformation (x) ( b b ) 2 V = K! bond b o Chemical type K bond b o C-C 100 kcal/mole/å 2 1.5 Å C=C 200 kcal/mole/å 2 1.3 Å C=C 400 kcal/mole/å 2 1.2 Å Bond Energy versus Bond length 400 Potential Energy, kcal/mol 300 200 100 Single Bond Double Bond Triple Bond 0 0.5 1 1.5 2 2.5 Bond length, Å Bond angles and improper terms have similar quadratic forms, but with softer spring constants. The force constants can be obtained from vibrational analysis of the molecule (experimentally or theoretically). 8
Dihedral Potential V dihedral = K ( 1+ cos( n"! )) " # Dihedral energy versus dihedral angle 20 Potential Energy, kcal/mol 15 10 5 K=10, n=1 K=5, n=2 K=2.5, N=3 0 0 60 120 180 240 300 360 Dihedral Angle, degrees δ = 0 van der Waals interaction *# R & " min,ij ij,, % $ r ( + ij ' 12 ) 2 R 6 # & - min,ij / % $ r ( ij ' /. Short range 9
CHARMM Potential Function PDB file geometry Topology PSF file parameters Parameter file File Format/Structure The structure of a pdb file The structure of a psf file The topology file The parameter file Connection to potential energy terms 10
PDB Files a little information Simulations start with a crystal structure from the Protein Data Bank, in the standard PDB file format. PDB files contain standard records for species, tissue, authorship, citations, sequence, secondary structure, etc. Some of the available fields atom name (N, C, CA) residue name (ALA, HIS) residue id (integer) coordinates (x, y, z) occupancy (0.0 to 1.0) temp. factor (a.k.a. beta) segment id (6PTI) No hydrogen atoms! (We must add them ourselves.) PDB File (available from www.rcsb.org if structure of biopolymer solved) REMARK FILENAME="bpti19.pdb REMARK PROTEINASE INHIBITOR (TRYPSIN) 13-MAY-87 6PTI REMARK BOVINE PANCREATIC TRYPSIN INHIBITOR REMARK BOVINE (BOS TAURUS) PANCREAS REMARK A.WLODAWER REMARK DATE:26-Jun-00 21:34:42 created by user: ATOM 1 HT1 ARG 1 13.150-7.331 10.849 1.00 0.00 BPTI ATOM 2 HT2 ARG 1 11.747-7.115 11.780 1.00 0.00 BPTI etc etc etc ATOM 554 CA GLY 56 15.319 0.828 11.790 1.00 17.33 BPTI ATOM 555 C GLY 56 16.029-0.385 12.375 1.00 18.91 BPTI ATOM 556 OT1 GLY 56 15.443-1.332 12.929 1.00 21.00 BPTI ATOM 557 OT2 GLY 56 17.308-0.138 12.617 1.00 21.95 BPTI END 11
PSF Files atomic properties (mass, charge, type) Every atom in the simulation is listed. Provides all static atom-specific values: atom name (N, C, CA) atom type (NH1, CT1) Ala residue name (ALA, HIS) residue id (integer) HA segment id (6PTI) atomic mass (in atomic mass units) partial charge (in electronic charge units) What is not in the PSF file? CA coordinates (dynamic data, initially read from PDB file) velocities (dynamic data, initially from Boltzmann distribution) force field parameters (non-specific, used for many molecules) N O C HN CB HB1 HB3 HB2 MASS HS 1.0080! thiol hydrogen MASS C 12.0110! carbonyl C, peptide backbone MASS CA 12.0110! aromatic C... (missing data here)!----------------------------------------------------------- AUTOGENERATE ANGLES=TRUE DIHEDRALS=TRUE END!----------------------------------------------------------- RESIDUE ALA Example of PSF File GROUP ATOM N TYPE=NH1 CHARGE= -.4700 END! ATOM HN TYPE=H CHARGE=.3100 END! N--HN ATOM CA TYPE=CT1 CHARGE=.0700 END! HB1 ATOM HA TYPE=HB CHARGE=.0900 END! / GROUP! HA-CA--CB-HB2 ATOM CB TYPE=CT3 CHARGE= -.2700 END! \ ATOM HB1 TYPE=HA CHARGE=.0900 END! HB3 ATOM HB2 TYPE=HA CHARGE=.0900 END! O=C ATOM HB3 TYPE=HA CHARGE=.0900 END! GROUP! ATOM C TYPE=C CHARGE=.5100 END ATOM O TYPE=O CHARGE= -.5100 END!END GROUP BOND CB CA BOND N HN BOND N CA BOND O C BOND C CA BOND CA HA BOND CB HB1 BOND CB HB2 BOND CB HB3 DONOR HN N ACCEPTOR O C END {ALA } N CA HA C O HN HB1 CB HB2 HB3 12
Steps in a Typical MD Simulation 1. Prepare molecule Read in pdb and psf file 2. Minimization Reconcile observed structure with force field used (T = 0) 3. Heating Raise temperature of the system 4. Equilibration Ensure system is stable 5. Dynamics Simulate under desired conditions (NVE, NpT, etc) Collect your data 6. Analysis Evaluate observables (macroscopic level properties) Or relate to single molecule experiments Preparing Your System for MD Solvation Biological activity is the result of interactions between molecules and occurs at the interfaces between molecules (protein-protein, protein- DNA, protein-solvent, DNA-solvent, etc). Why model solvation? many biological processes occur in aqueous solution solvation effects play a crucial role in determining molecular conformation, electronic properties, binding energies, etc How to model solvation? explicit treatment: solvent molecules are added to the molecular system implicit treatment: solvent is modeled as a continuum dielectric 13
Classical Molecular Dynamics r ( t +! = r( + v(! t v ( t +! = v( + a(! t a ( t ) = F( / F =! m d U (r) dr Maxwell Distribution of Atomic Velocities 14
Analysis of E kin, T (free dynamics) The atomic velocities of a protein establish a thermometer, but is it accurate? ubiquitin in water bath! 2 = T 3N 2 2 15