Coupling of SCC-DFTB, Generalized Born and Hydrophobic Models in
Description of Hydration Free Energies
Bogdan Lesyng
Interdyscyplinary Centre for Mathematicaland Computational Modelling
and Faculty of Physics, University of Warsaw (http://www.icm.edu.pl/~lesyng)
andEuropean Centre of Excellence forMultiscale Biomolecular Modelling,
Bioinformatics and Applications(http://www.icm.edu.pl/mamba)
AMM-IV
Leicester, 18-21/08/2004
Dynamics, classical and/or quantum one in
the real molecular environment
Sequences at the protein & nucleic acids levels
3D & electronicstructure
Function
Metabolic pathways & signalling
Sub-cellular
structures & processes
Cell(s), structure(s) & functions
1 RPDFCLEPPY 10 11 TGPCKARIIR 20 21 YFYNAKAGLC 30 31 QTFVYGGCRA 40 41 KRNNFKSAED 50
51 CMRTCGGA 58
In our organismswe have ~ 103
protein kinasesand phosphatases
which phosphorylate/
dephosphorylateother proteinsactivating ordisactivating
them.These arecontrollersof most ofmethabolicpathways.
A Protein Kinase Molecule with ATP (catalytic domain)
Designing inhibitors 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
Classes of Models
Microscopic models
Mesoscopic models
O
O
O
R’
OH
OR''
OH
OHOCH3
ABCD
O
X
CH3
YH
H
H
R” : H, R’ : H, OHX : H, OH, NH2
Y : H, OH, NH2
W.R.Rudnicki et al., Acta Biochim. Polon., 47, 1-9(2000)
G = 0.191 x E - 0.41
r2=0.947
5
6
7
8
9
10
30 35 40 45 50 55 60
E [kcal/mol]
G [k
cal/m
ol]
.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, 11-17, 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.
11
Protonation equilibria in proteins
M. Wojciechowski, T. Grycuk, J. Antosiewicz, B.lesyngPrediction of Secondary Ionization of the Phosphate Group in Phosphotyrosine Peptides, Biophys.J, 84, 750-756 (2003)
Active site(quantum subsystem)
Classical molecular scaffold (real molecular environment)
Solvent (real thermal bath)
Interacting quantum and classical subsytsems.
Enzymes, special case of much more general problem.
Microscopic generators of the potential energy function
• AVB – (quantum)• AVB/GROMOS - (quantum-classical)
• SCC-DFTB - (quantum)• SCC-DFTB/GROMOS - (quantum-classical)• SCC-DFTB/CHARMM - (quantum -classical)• ....
Dynamics
• MD (classical)• QD (quantum)• QCMD (quantum-classical)• ....
Mesoscopic potential energy functions
•Poisson-Boltzmann (PB)•Generalized Born (GB)•....
atomic charges
many-electron wave function representingi-th valence structure
Approximate Valence Bond (AVB) MethodSee: Trylska et al., IJQC 82, 86, 2001) and references cited
positions of the nuclei
Hamiltonian matrix in basis of valence structures
electronic ground state energy
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:
1-electron orbitals
total electrondensity
1-electronHamiltonian
(Kohn-Sham equation)
Total energy for arbitrary electronic density
has minimumat 0 (0 ) and 0 , resulting from Kohn-Sham eq.(ground state)
el. kinetic. en., el.-nuclei interaction, el.-el. Exchange and twice el.-el. electrostatic interaction
n-n inter., XCnon-local corr.and minus el.-el.electrostatic int.
(R)
(R)
TB approach:expansion of the energy functional around the ground state
density of the ground state
second and higher orderexpansion terms (SCC version)
TBDFT approximations
densities at free atoms
atom pair potentials
current atomicnet charges net charges
of free atoms
+ LCAO approximation
atomic orbitals
Mulliken charges
combination coefficients (c)
Condition for the ground state
Hamiltonian matrix
overlap matrix:
TBDFT equations:
J.Li, T.Zhu, C.Cramer, D.Truhlar,J. Phys. Chem. A, 102, 1821(1998)
New generation of charges capable reproducing electrostatic properties, in particular molecular dipole moments.
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:
and the correction function which is taken to be a second order polynomial with coefficients depending on the atom types:
which involves Meyers bond order:
0 1 2 3 4 5 6
0
1
2
3
4
5
6
Dipole moments in Debyes
MullikenCM3
Experimental
Cal
cula
ted
Mesoscopic models of the molecular electrostatic
energy
)( rnqri
iiext
kT
rqnrn
)(exp)( 0
kTIe
rrrrrqrrk
kk
22
2
2
P o i s s o n - B o l t z m a n n ( P B ) m e t h o d
i n t h e r m o d y n a m i c e q u i l i b r i u m
s o l v i n g o n a g r i d , o rw i t h fi n a l e l e m e n t s
e x t e r n a l i o n i c d e n s i t y
r i q i
D e b y e - H u c k e l s c r e e n i n g p a r a m e t e r , I - i o n i c s t r e n g t h
..int repvdisppolel VVVVV
termcrossnpfieldmean
el GGGG int
PBel
fieldmeanel GG
• Microscopic (quantum) description of intermolecular interactions:
• Mesoscopic description of intermolecular interactions (free energies)
Electrostatic Poisson-Boltzmann energy
Interaction potentials
meanVdWcavnp GGG
0 termcrossG
See eg. E.Gallicchio and R.M.Levy, J.Comput.Chem.,25,479-499(2004)
PBelG GB
elG ”GB” – Generalized Born
rdconstVGex
kk
repmeanVdW
rr3
6
1
k
kcav AkG Ak - van der Waals surface area of atom k
k - surface tension parameter assigned to
atom 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)
G e n e r a l i z e d B o r n ( G B )
G e lG B = G e l
0 + G e ls o l
ji ijin
jioel r
qqG
21
ji ij
ji
ex
f
in
solel f
qqeG
i j
,
121
G e l – T o t a l e l e c t r o s t a t i c e n e r g y
C o u l o m b i c i n t e r a c t i o n e n e r g yb e t w e e n a t o m s
E l e c t r o s t a t i c i n t e r a c t i o n e n e r g y ( s o l v a t i o n e n e r g y ) o f t h e m o l e c u l a r s y s t e mw i t h d i e l e c t r i c e n v i r o n m e n t ( e g . w a t e r ) .
r i j – d i s t a n c e b e t w e e n a t o m s
– D e b y e - H u c k e l p a r a m e t e r
R i – B o r n r a d i u s i – V a n d e r W a a l s r a d i u s
ji
ijjiijij RR
rRRrf
4exp
22
w h e r e :
The same atoms are characterized by diff erent Bornradii. Their values depend on geometry of themolecular system, and on localization of the atoms inthe system (geometrical property). The Born radii are large inside, and are close to VdW radii on the surface.
Born radiiand
Van der Waals radii
Molecular area
E x p r e s s i o n s f o r B o r n r a d i i
rdrR solventi
34
1
4
11
3
1
36
1
4
31
rd
rR solventi
3
1
314
31
n
solventn
i
rdr
nR
3.0
033
32.4
solv
solute
n
A . O n u f r i e v , D . B a s h f o r d , D . C a s e , J . P h y s . C h e m . B , 1 0 4 , 3 7 1 2 - 3 7 2 0 ( 2 0 0 0 )
T . G r y c u k , J . C h e m . P h y s , 1 1 9 , 4 8 1 7 - 4 8 2 6 ( 2 0 0 3 )
M . W o j c i e c h o w s k i , B . L e s y n g , J . C h e m . P h y s , s u b m i t t e d
1
233
1
714
ex
inex
exo
i
ED
ACAC
R
4
1
3747
141
41
rd
rRA
inVdW
M.Feig, W.Im, C.L.Brooks, J.Chem.Phys.,120,903-911(2004)
(I)
(II)
(III)
(IV)
Coulomb Field appr.
Kirkwood Model
Ratio of the GB solvation enery to the Kirkwood solvation energy
Ratio of the GB solvation enery to the Kirkwood solvation energy(zooming)
case IV
in/ex
The optimal value of the exponent
3.0
033
32.4
ex
in
n
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.
Corrections to the ionic strength
77.053.0 ijij ff ee
Coupling of GB and SCC-DFTB
• computing the CM3/ SCC-DFTB charges• computing precise Born radii• computing Gel
sol
• computint the diff erence Eexp – Gelsol
• fi tting the nonpolar term to this diff erence
Minnesota solvation data base.Reproducing PB.
======================================
SASA A2
CHARMM
SASA A2 CHARMM
SASA A2 Fit 1
SASA A2
Fit 2
kkk t
k
t
k
tk
sR
sR
sSASA ,0,1
2
2,2
11
F i t t i n g t h e n o n p o l a r c o n t r i b u t i o n
kkk t
k
t
k
tk
np gR
gR
gG ,0,1
2
2,2
11
w h e r e : g – fi t t e d c o e ffi c i e n t s ,k – a t o m n u m b e r s ,t – a t o m t y p e s .
k waterkk
kk
VdWcavfit
np
rR
constk
GGG
A 3
Following Gallicchio & LevyJ.Comput.Chem.,25,479-499(2004)
Fitting the nonpolar solvation energy with the cavity and VdW components(preliminary)
expnpG
7.0
*/12.0 2
k
kAmolkcal
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 refi ned version of the GB model seems to be at the moment the best one.I t reproduces very well the PB resultsf or smaller systems and quitewell f or proteins (f or large systemsthere are some technical problems toquickly compute the GB radii).
• The experimental nonpolar contribution to the hydration energy is fi ttedeither with short polynomials depending onreciprocal values of the GB radii, or on the sum of the cavity and mean VdW contributions.
• Eff ective, mesoscopic interaction potentials should noticeably increase our research capabilities of structuresand f unctions of complex biomolecularsystems (hopefully).
Acknowledgements
PhD students:
Jarek KalinowskiMichał WojciechowskiPiot KmiećMagda Gruziel
Collaboration:
Dr. T. Frauenheim SCC-DFTBDr. M. Elstner
Dr. D. Truhlar CM3-chargesDr. J. Thompson Minnesota Solvation Data BaseDr. C. Cramer
Studies supported by ”European CoE for Multiscale Biomolecular Modelling, Bioinformatics and Applications”.