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”.