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					  Biophysical Chemistry G4170:
Introduction to Molecular Dynamics



              Ruhong Zhou



     IBM Thomas Watson Research Center
     Yorktown Heights, NY 10598
Polarizable Force Fields
                  Solvent-Induced Dipole
    m = 1.85 D                                            m ~ 2.5 D



H                H   DU = -9.9 kcal/mol
                                                     H                 H
       O                                                      O

    Gas phase                                          Aqueous phase

       It takes energy to change from gas-phase to liquid-phase charges:
                         for H2O, DUpol = 2-5 kcal/mol
               Conformational Effects
                                                           C terminus
 -0.42     O

 +0.42
           C

                       N      -0.20


                +
                       H     +0.20
                                                           N terminus

  Amide bond: m = 3.5 D                             a-helix: m = 5.0 D

Wada, Adv. Biophys., 9, 1 (1976); Duijnen and Thole, Biopolymers, 21, 1749 (1982)
         Polarizable Force Fields
   Atomic charges adjust to different chemical
    environments
   Electrostatic interactions are long-ranged interactions,
    accurate models needed
   Needed to calculate many-body interactions
   Hopefully a better transferability

                           A

                                          C

                           B

    EAB (C): Energy between A & B depends on C’s position
         Polarizable Force Fields
   Friesner & Berne: Polarizable OPLSAA
        Fluctuating charges and fluctuating dipoles

   Kollman & Case: AMBER2002
        Dipole polarizability

   Ponder: TINKER Force Field
       Dipole polarizability, and higher multipoles such as
       quadruples
       Fluctuating Charge Model
                                        1
        U polarization  Vi Dqi   Dqi J ij Dq j
                         i         i, j 2
   Dqi: change in partial charge on atom i
   Vi: applied electrostatic potential at atom i
   Jij: coefficient representing interaction between partial
    charges at sites i,j – depends on nuclear configuration

   Minimize Upolarization to find Dqi, yielding a set of linear
    equations.
   Alternatively, treat Dqi as dynamical variables and
    propagate them along with the coordinates {xi,yi,zi,qi}

S. Rick, S. Stuart, and B. Berne, J. Chem. Phys. 1994
    Dipolar Polarizability Models
                       1       
    U polarizati  -  Ei  m i
                                                         ai: polarizability of
               on
                       2 i                                atom i
                            
    m i  a i   Ei -  J ij  m j                     mi: induced dipole
                                                        on atom i
                       j i        
                                                         Ei: applied electric
  After defining ai = 1/Jii, we can rewrite it into
                                                          field at atom i
                                                         Jij: dipole interaction
                                                          tensor representing
                           1              
   U polarization    - Ei  m i   m i J ij m j        interaction between
                       i           2 i, j                 dipoles at sites i,j

Thole 1981; Rullmann & van Duijnen 1988; Cieplak & Kollman 1990;
Bernardo, Ding, Krogh-Jerpersen, & Levy 1994
  Combined Fluctuating Charge & Dipole Model
                                                                                                                                
  Each atom can have both a partial charge and a                                               U polarization     VA Dq A - E B  m B 
  dipole, so it might have up to four variables:                                                                       A          i

                                                                                                                1
  one charge and three dipole moments                                                                             ' Dq A J AA'DqB 
                                                                                                                2 AA
                                                                                                                              
    Charges on atoms A and dipoles on atoms B.
                                                                                                                  Dq A J AB  m B 
                                                                                                                AB '

                                                                                                                1       
                                                                                                                   m B J BB' m B '
                                                                                                                2 BB'
  All models may be written succinctly in matrix form:
                                                                                                                          1
    where vectors f={V, Ex, Ey, Ez} and                                                           U polarization  φ  q  q  Jq
    q={q, mx, my, mz}                                                                                                     2

    Minimize to determine charges and/or dipole
    Moments on each atom
                                                                                                            Jq  -φ
J. Bank, G. Kaminski, R. Zhou, D. Mainz, B. Berne, R. Friesner, J. Chem. Phys. 110, 741, 1999
H. Stern, G. Kaminski, J. Banks, R. Zhou, B. Berne, R. Friesner, J. Phys. Chem. B103, 4730, 1999
G Kaminski, H. Stern, B. Berne, R. Friesner, Y. Cao, R. Murphy, R. Zhou, J. Comput. Chem. 23, 1515, 2002
G. Kaminski, R. Friesner, R. Zhou, J. Comput. Chem. 24, 267, 2003
Polarizable FF Fitting Philosophy
   Polarization:
       Treat long-range interactions by Coulomb’s law. Scale short-
        range interactions by adjustable parameters
       Apply a series of electrostatic perturbations to a molecule
       For each perturbation, compute the change in the
        electrostatic potential at a series of grid points from ab initio
        calculations on the unperturbed and perturbed molecules
       Fit the parameters of the model so as to best reproduce these
        changes when the same perturbation are applied
   Gas-phase electrostatics: choose fixed charges so that
    the total electrostatic potential of the model best
    reproduces high-level ab initio gas-phase calculations
   Intramolecular, Lennard-Jones, and torsional terms:
    take from OPLSAA. Refit key torsions to ab initio
    relative conformational energies
Three-body Energies for molecules
        with two probes




                       E(3) = E123 – E12 –E23 –
                        E13 + E1 + E2 +E3
                        3-body energies are all
                        zero in standard force
                        fields
                        RMS errors are from
                        comparisons to high level
                        QM calculations
        Cases where fluctuating charge
                 model fails




   Two cases that point-charge-only model fails for three-
    body energies
       Bifurcated hydrogen bond
       Probes above or below aromatic rings, out-of-plane
        polarization
Relative Conformational Energy
    Summary on Polarizable OPLSAA
   Force fields incorporating explicit polarization have
    been developed that accurately predict many-body
    effects
   Polarizable FF dramatically improves the prediction of
    relative conformational energies for small peptides
   Dipolar model can correct errors in fluctuating charge
    model alone for cases with out-of-plane polarization
    (aromatic rings) or bifurcated hydrogen bonds (O, S
    atoms)
   Parameterization was systematic and transferable
II. Solvation Models
                  Solvation Models
   Explicit solvent models
       Fixed charge models: SPC, SPC/E, TIP3P, TIP4P,
        TIP5P, ST2,…
       Polarizable water models: TIP4P/FQ, POL5,
        MCDHO,…
   Implicit Solvent models
       Poisson-Boltzman solver (Delphi, Honig)
       Generalized Born Model (Still)
       Karplus’ EEF1 model
       Benoit Roux’s Spherical Solvent Boundary Potential
        (SSBP)
  Explicit Water models
SPC, SPC/E, TIPnP, POL5
Water Model Geometries
   Water Model Parameters
• SPC, SPC/E (Berendsen)
• TIP3P, TIP4P, TIP5P (Jorgensen)
• TIP4P/FQ, POL5 (Berne)
Properties of Water Models
Water density maximum
     Water structure comparison




M. Mahoney and W. L. Jorgensen, J. Chem. Phys. 112, 8910, 2000
POL5 Model
        Gas-phase electrostatic properties

              POL5     TIP4P/FQ   TIP5P   Experiment
mz(D)         1.854     1.860     2.292     1.855
Qxx (D Å)     -2.335    -1.785    -1.48      -2.5
Qyy (D Å)     2.337     1.882     1.65       2.63
Qzz (D Å)     -0.002    -0.098    -0.17     -0.13
axx (Å)       1.060      0.0       0.0      1.415
ayy (Å)       1.494      2.55      0.0      1.528
azz (Å)       1.320      0.82      0.0      1.468
                  Water dimer properties

                                 r                  f

                       q
              POL5         TIP4P/FQ   TIP5P    Ab initio   Experiment
U(kcal/mol)    -4.96         -4.50     -6.78     -4.96      -5.4±0.7
r (Å)          2.896         2.924     2.676     2.896        2.98
q (degrees)    4.694         0.173    -1.610     4.754         0±6
f (degrees)   62.638        27.170    50.222    57.281        58±6
m (D)          2.435         3.430     2.920     2.683        2.643
<m> (D)        2.063         2.055     2.292      2.1           ?
                  Trimer

                                        16
                                        14
                                        12
                                               POL5
                               binding 10
                                               TIP4P/FQ
                                energy   8
                              (kcal/mol) 6     TIP5P
                                         4     ab initio
                                         2
                                         0




           2.95                          1.4
            2.9                          1.2
average H- 2.85   POL5                     1   POL5
                              net dipole 0.8
   bond           TIP4P/FQ                     TIP4P/FQ
            2.8                moment
 distance         TIP5P                  0.6   TIP5P
           2.75                  (D)
    (A)           ab initio              0.4   ab initio
            2.7
                                         0.2
           2.65                            0
                   Tetramer

                                        30
                                        25
                                               POL5
                               binding 20
                                               TIP4P/FQ
                                energy 15
                              (kcal/mol) 10    TIP5P
                                               ab initio
                                         5
                                         0




           2.85                          2.5
                                        2.45
            2.8                average 2.4
average H-        POL5                         POL5
           2.75               molecular
   bond           TIP4P/FQ              2.35   TIP4P/FQ
                                dipole
 distance 2.7     TIP5P        moment 2.3      TIP5P
    (A)           ab initio             2.25   ab initio
           2.65                  (D)
                                         2.2
            2.6                         2.15
                 Pentamer

                                        39
                                        38
                                        37
                                              POL5
                              binding 36
                                        35    TIP4P/FQ
                               energy
                                        34    TIP5P
                             (kcal/mol) 33
                                              ab initio
                                        32
                                        31
                                        30




           2.9                          1.4
          2.85                          1.2
average H- 2.8   POL5                     1   POL5
                             net dipole 0.8
   bond   2.75   TIP4P/FQ                     TIP4P/FQ
                              moment
 distance 2.7    TIP5P                  0.6   TIP5P
                                (D)
    (A)   2.65   ab initio              0.4   ab initio
           2.6                          0.2
          2.55                            0
Hexamers
                  Book hexamer

                                          47
                                          46
                                          45
                                          44    POL5
                                binding 43
                                                TIP4P/FQ
                                 energy 4241
                               (kcal/mol) 40    TIP5P
                                          39    ab initio
                                          38
                                          37
                                          36




                                         2.5
           2.82
            2.8
           2.78                            2
           2.76    POL5                         POL5
average H- 2.74                net dipole 1.5
                   TIP4P/FQ     moment          TIP4P/FQ
   bond    2.72
 distance 2.7      TIP5P          (D)       1   TIP5P
           2.68    ab initio
           2.66                          0.5
           2.64
           2.62                            0
                 Prism hexamer

                                          46
                                          45
                                          44
                                          43   POL5
                                binding 42
                                               TIP4P/FQ
                                 energy 41
                               (kcal/mol) 40   TIP5P
                                          39   ab initio
                                          38
                                          37
                                          36




          2.84                           3.5
          2.82                             3
average H- 2.8     POL5                  2.5   POL5
                               net dipole 2
   bond            TIP4P/FQ                    TIP4P/FQ
          2.78                  moment
 distance          TIP5P                 1.5   TIP5P
          2.76                    (D)
    (A)            ab initio               1   ab initio
          2.74
                                         0.5
          2.72                             0
                 Liquid-state properties

                    POL5       TIP4P/FQ         TIP5P     Experiment
U (kcal/mol)     -9.92±0.01    -9.89±0.02    -9.87±0.01       -9.92
 (g/cm3)       0.997±0.001   0.998±0.001   0.999±0.001       0.997
m (D)           2.712±0.002        2.6           2.29     ? (2.5—3.2)
0                  98±8          79±8          82±2          78.3
•              1.689±0.001   1.592±0.003         1           1.79
D (10-9 m2/s)    1.81±0.06       1.9±0.1     2.62±0.04         2.3
NMR (ps)          2.6±0.1       2.1±0.1       1.4±0.1         2.1
Water density revisited
Implicit Solvent Models
       PBF, GB
Continuum Solvent Model


         continuum solvent
               =80


            =1-4
            protein
Molecular Surfaces

              Dotted line: Solvent
               Accessible Surface (SAS)
              Solid line: molecular
               surface (MS)
              Shaded grey area: van der
               Waals surface
R. Levy, JCC 2002
Molecular Surface Colored by Potential




  The molecular surface of acetyl choline esterase molecule color coded by electrostatic potential. the
  view is directly into the active site and acetyl choline is present in a bond representation. note the
  depth of the pocket, its negative nature corresponding to the positive charge on the acetyl choline.
       Trp-cage Folding: Kinetics




• OPLS united atom Force Field
• Continuum Solvent GBSA                        B: MD simulation
• Langevin dynamics                             C: NMR structure
• Water viscosity g=91/ps                       2.1 A Ca RMSD
                                                Folding time 1.5ms (3.0 A cutoff)
                                                 to 8.7 ms (2.5 A cutoff)
M. Snow, B. Zagrovic, V. Pande, JACS 124, 14548, 2002
      Trp-cage Folding: Structure




                                                             Blue: MD simulation
• AMBER99 Force Field                                        Grey: NMR structure
• Continuum Solvent GBSA                                     0.97 A Ca RMSD
• NVT ensemble                                               1.4 A RMSD heavy atoms


C. Simmerling, B. Strockbine, A. Roitberg, JACS 124, 11258, 2002
Protein (un)Folding Example: a b-hairpin




         Protein G (2gb1)




            Res. 41-56
     GEWTYDDATKTFTVTE

V. Munoz, P. Thompson, J. Hofrichter, W. Eaton, Nature, 390, 196, 1997
R. Zhou, B. Berne and R. Germain, PNAS, 98, 14931, 2001
b-hairpin Folding in Various Models




                            a.     OPLSAA/SPC (explicit)
                            b.     OPLSAA/SGB
                            c.     OPLSAA/PB



                       R. Zhou, B. J. Berne, PNAS 99, 2002
                       R. al, PNAS G. Krilov, B. J. Berne, JPC, 2004
                 R. Zhou, et Zhou, 98, 2001
                 R. Zhou, and B. Berne, PNAS 99, 2002
    Lowest free energy structures




       Explicit              SGB                     PB

   Erroneous salt-bridges exist in all continuum solvent models
   Overly strongly salt-bridge effects expelled F50 out of the
    hydrophobic core in SGB
   PB models behaves significantly better than the GB model
   Both PB and GB models need improvements
          Computational expense
Simulations using standard Ewald summation and 256 molecules




   Model                      Relative Expense
   TIP4P                      0.76
   TIP4P/FQ                   0.82
   TIP5P                      1.00
   POL5                       2.02

				
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posted:10/11/2011
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