Transcript Centre of Excellence for Multi

Coupling of SCC-DFTB, Generalized Born
and Hydrophobic Models in Description of
Hydration Free Energies
Bogdan Lesyng
Interdyscyplinary Centre for Mathematical
and Computational Modelling
and Faculty of Physics, University of Warsaw
(http://www.icm.edu.pl/~lesyng)
and
European Centre of Excellence for
Multiscale Biomolecular Modelling,
Bioinformatics and Applications
(http://www.icm.edu.pl/mamba)
AMM-IV
Leicester, 18-21/08/2004
Sequences
at the protein &
nucleic acids levels
1
11
21
31
41
51
RPDFCLEPPY
TGPCKARIIR
YFYNAKAGLC
QTFVYGGCRA
KRNNFKSAED
CMRTCGGA
Metabolic
pathways &
signalling
3D & electronic
structure
Function
Dynamics,
classical and/or
quantum one in
the real
molecular
environment
10
20
30
40
50
58
Sub-cellular
structures & processes
Cell(s),
structure(s)
&
functions
In our organisms
we have ~ 103
protein kinases
and phosphatases
which
phosphorylate/
dephosphorylate
other proteins
activating or
disactivating
them.
These are
controllers
of most of
methabolic
pathways.
A Protein Kinase Molecule with ATP (catalytic domain)
Every two years we
organize international
conferences on
”Inhibitors of Protein
Kinases”, and
workshops on
„Mechanisms on
Phosphorylation
Processes”
The next one:
June 26-30, 2005
Warsaw
http://www.icm.edu.pl/
ipk2005/
Ref. To Piotr Setny’s poster
Designing inhibitors
Classes of Models
Microscopic models
Mesoscopic models
O
OH
O
OH
C
D
B
A
R’
OCH3 O
R” : H,
OH
CH3
H
O
Y
X
OR''
H
H
R’ : H, OH
X : H, OH, NH2
Y : H, OH, NH2
W.R.Rudnicki et al., Acta Biochim. Polon., 47, 1-9(2000)
Motivation for multiscale modelling
.
•
Structure formation mechanisms -> molecular recognition processes,
– M.H.V. van Regenmortel, Molecular Recognition in the Post-reductionist Era,
J.Mol.Recogn., 12, 1-2(1999)
– J.Antosiewicz, E. Błachut-Okrasińska, T. Grycuk and B. Lesyng,
A Correlation Between Protonation Equilibria in Biomolecular Systems and their
Shapes: Studies using a Poisson-Boltzmann model., in GAKUTO International
Series, Mathematical Science and Applications. Kenmochi, N., editor, vol. 14, 1117, Tokyo, GAKKOTOSHO CO, pp.11-17, 2000.
•
Quantum forces in complex biomolecular systems.
– P. Bala, P. Grochowski, B. Lesyng, J. McCammon, Quantum Mechanical Simulation
Methods for Studying Biological System, in: Quantum-Classical Molecular
Dynamics. Models and Applications, Springer-Verlag, 119-156 (1995)
– Grochowski, B. Lesyng, Extended Hellmann-Feynman Forces, Canonical
Representations, and Exponential Propagators in the Mixed Quantum-Classical
Molecular Dynamics, J.Chem.Phys, 119, 11541-11555(2003)
To understand structure & function of complex biomolecular systems.
Protonation equilibria in proteins
M. Wojciechowski, T. Grycuk, J. Antosiewicz, B.lesyng
Prediction of Secondary Ionization of the Phosphate
Group in Phosphotyrosine Peptides, Biophys.J, 84,
750-756 (2003)
11
Interacting quantum and classical subsytsems.
Enzymes, special case of much more general problem.
Active site
(quantum subsystem)
Classical molecular scaffold
(real molecular environment)
Solvent (real thermal bath)
Microscopic generators of the potential
energy function
• AVB
• AVB/GROMOS
•
•
•
•
SCC-DFTB
- (quantum)
SCC-DFTB/GROMOS - (quantum-classical)
SCC-DFTB/CHARMM - (quantum -classical)
....
Dynamics
•
•
•
•
– (quantum)
- (quantum-classical)
MD (classical)
QD (quantum)
QCMD (quantum-classical)
....
Mesoscopic potential energy functions
•Poisson-Boltzmann (PB)
•Generalized Born (GB)
•....
Approximate Valence Bond (AVB) Method
See: Trylska et al., IJQC 82, 86, 2001) and references cited
many-electron wave
function representing
i-th valence structure
Hamiltonian matrix in basis of valence structures
positions of the nuclei
electronic ground state energy
atomic charges
SCC-DFTB Method
(Self Consistent Charge Density Functional Based Tight Binding Method,
SCC DFTB, Frauenheim et al. Phys Stat. Sol. 217, 41, 2000)
basic DFT concepts:
total electron
density
1-electron orbitals
1-electron
Hamiltonian
(Kohn-Sham equation)
Total energy for arbitrary electronic density
(R)
has minimum
at 0 (0 ) and 0 , resulting
from Kohn-Sham eq.
(ground state)
(R)
n-n inter., XC
non-local corr.
and minus el.-el.
electrostatic int.
el. kinetic. en., el.-nuclei interaction,
el.-el. Exchange and twice el.-el.
electrostatic interaction
TB approach:
expansion of the energy functional around the ground state
density of the ground state
second and higher order
expansion terms (SCC version)
TBDFT approximations
densities at
free atoms
atom pair
potentials
current atomic
net charges
net charges
of free atoms
+ LCAO approximation
combination coefficients (c)
atomic orbitals
Mulliken
charges
Condition for the ground state
 TBDFT equations:
overlap matrix:
Hamiltonian matrix
New generation of charges capable
reproducing electrostatic properties, in
particular molecular dipole moments.
J.Li, T.Zhu, C.Cramer, D.Truhlar,
J. Phys. Chem. A, 102, 1821(1998)
CM3/SCC-DFTB charges
J.A. Kalinowski, B.Lesyng, J.D. Thompson, Ch.J. Cramer, D.G. Truhlar,
Class IV Charge Model for the Self-Consistent Charge Density-Functional
Tight-Binding Method, J. Phys. Chem. A 2004, 108, 2545-2549
CM3 charges are defined with the following mapping:
which involves Meyers bond order:
and the correction function which is taken to be a second order
polynomial with coefficients depending on the atom types:
Dipole moments in Debyes
6
Mulliken
5
CM3
Calculated
4
3
2
1
0
0
1
2
3
Experimental
4
5
6
Mesoscopic models of the
molecular electrostatic
energy
Poisson-Boltzmann (PB) method
 
 
  2 
rrqkr rk rr r
k
2
2
e
 I
2
kT
Debye-Huckel screening
parameter, I-ionic strength
r i qi
solving on a grid, or
with final elements
extr    qi ni (r)
i

  q ( r ) 

n ( r )  n0 exp  

kT 

external ionic density
in thermodynamic
equilibrium
Interaction potentials
• Microscopic (quantum) description of intermolecular interactions:
Vint  Vel  Vpol  Vdisp  Vv.rep.
• Mesoscopic description of intermolecular interactions (free energies)
Gint  Gelmean  field  Gnp  Gcross term
Gelmean  field  GelPB
Electrostatic Poisson-Boltzmann energy
mean
Gnp  Gcav  GVdW
See eg. E.Gallicchio and R.M.Levy, J.Comput.
Chem.,25,479-499(2004)
Gcross term  0
GelPB
G
Gcav  
A
k
GB
el
k
G
mean
VdW
k
”GB” – Generalized Born
Ak - van der Waals surface area of atom k
k
- surface tension parameter assigned to atom k
 Vrep  const 
k

ex
1
3
d
r
6
r r 
k
First papers on Born models:
•M.Born, Z.Phys., 1,45(1920)
•R.Constanciel and R.Contreas, Theor.Chim.Acta, 65,111(1984)
•W.C.Still, A.Tempczyk,R.C.Hawlely,T.Hendrikson, J.Am.Chem.Soc.,112,6127(1990)
•D.Bashford, D.Case, Annu.Rev.Phys.Chem., 51,129(2000)
Generalized Born (GB)
GelGB = Gel0 + Gelsol
Gel – Total electrostatic energy
qi q j
1
G  
2 i  j  in rij
o
el
Coulombic interaction energy
between atoms
 f
sol
el
G
1  1 e ij
  
2   in  ex
 qi q j

 i, j f
ij

Electrostatic interaction energy
(solvation energy) of the molecular system
with dielectric environment (eg. water).
where:
2



r
ij
2


f ij  rij  Ri R j exp
 4R R 
 i j
rij – distance between atoms
 – Debye-Huckel parameter
Ri– Born radius
i– Van der Waals radius
Born radii
and
Van der Waals radii
Molecular area
The same atoms are characterized by different Born
radii. Their values depend on geometry of the
molecular system, and on localization of the atoms in
the system (geometrical property). The Born radii
are large inside, and are close to VdW radii on the
surface.
Expressions for Born radii
1
1

Ri 4
1 3
d r
4

r
so lven t
Coulomb Field appr. (I)
A.Onufriev, D.Bashford,D.Case, J.Phys.Chem.B, 104,3712-3720(2000)
1  3
 
Ri  4
1 3
d r 
6

r
solvent

1
3
Kirkwood Model (II)
T.Grycuk, J.Chem.Phys, 119, 4817-4826(2003)
Ri  
1
 3 ex 
 A7
Co A4  C1 
 3 ex  2 in 
D
E
 ex  1
(III)
 1
1
A7   4 
 4RVdW 4
1 3
in r 7 d r 
1
4
M.Feig, W.Im, C.L.Brooks, J.Chem.Phys.,120,903-911(2004)
1  n3
1 3


d r 
n

Ri  4 so lven t r

1
n 3
n
4.32
  so lu te
 033


so lv


M.Wojciechowski, B.Lesyng, J.Chem.Phys, submitted
0. 3
(IV)
Ratio of the GB solvation enery to the Kirkwood solvation energy
Ratio of the GB solvation enery to the Kirkwood solvation energy
(zooming)
in/ex
case IV
The optimal value of the exponent
n
4.32
  in  033
 

ex


0.3
Corrections to the ionic strength
Conventional Born,
D.Bashford & D.Case, Annu.Rev.
Phys.Chem.,51,129-152(2000)
Srinivasan et al.,Theor.Chem.Acc.,
101,426-434(1999)
M.Wojciechowski & B.Lesyng,
Submitted to J.Phys.Chem.
f ij
0.53f ij 0.77
e
e
Coupling of GB and SCC-DFTB
• computing the CM3/SCC-DFTB charges
• computing precise Born radii
• computing Gelsol
• computint the difference Eexp – Gelsol
• fitting the nonpolar term to this difference
======================================
Minnesota solvation data base.
Reproducing PB.
SASA A2
CHARMM
SASA 
k
SASA A2
Fit 1
SASA A2
CHARMM
SASA A2
Fit 2
2




1
s  1   s


s
1,t k
0 ,t k
 2 ,t k  R 2 

R
k 
k



Fitting the nonpolar contribution
Gnp  
k
where:
2




1
g  1   g


g
2 ,t k 
1
,
t
0
,
t
2
k
k


Rk 
Rk



g – fitted coefficients,
k – atom numbers,
t – atom types.
Fitting the nonpolar solvation energy with the cavity and VdW components
(preliminary)
exp
Gnp
Following Gallicchio & Levy
J.Comput.Chem.,25,479-499(2004)
Gnpfit  Gcav  GVdW
 
k
const
 
A
k
R  r 
k
3
k
k
k
water
 k  0.12kcal / mol * A2
 k  0.7
•
•
•
•
Conclusions:
CM3/SCC-DFTB charges reproduce very
well molecular dipole moments.
They depend on conformations, which
is an adventage in comparison to other
conventional parameterizations.
Our refined version of the GB model
seems to be at the moment the best one.
It reproduces very well the PB results
for smaller systems and quite
well for proteins (for large systems
there are some technical problems to
quickly compute the GB radii).
The experimental nonpolar contribution
to the hydration energy is fitted
either with short polynomials depending on
reciprocal values of the GB radii,
or on the sum of the cavity and mean
VdW contributions.
Effective, mesoscopic interaction
potentials should noticeably increase
our research capabilities of structures
and functions of complex biomolecular
systems (hopefully).
Acknowledgements
PhD students:
Jarek Kalinowski
Michał Wojciechowski
Piot Kmieć
Magda Gruziel
Collaboration:
Dr. T. Frauenheim
Dr. M. Elstner
SCC-DFTB
Dr. D. Truhlar
Dr. J. Thompson
Dr. C. Cramer
CM3-charges
Minnesota Solvation Data Base
Studies supported by ”European CoE for Multiscale Biomolecular
Modelling, Bioinformatics and Applications”.