From Crystallography of Biomoleculesto

More Detailed Understanding of their Structure and Function

Bogdan LesyngICM and Faculty of Physics, University of Warsaw

(http://www.icm.edu.pl/~lesyng)and

European Centre of Excellence forMultiscale Biomolecular Modelling,

Bioinformatics and Applications(http://www.icm.edu.pl/mamba)

Łódź, 4/10/2004

Chapter 4.5,page 72

W.Saenger & K.H.Sheit, J.Mol.Biol., 50, 153-169(1970)

B.Lesyng & W.Saenger, Z.Naturforsch. C,36, 956-960(1981)

Crystallized from water

Crystallized from butyric acid !

Structures in the crystalline state can be interpreted in terms of packing forces, properties of hydrogen bonds, a kind of consensus between the intramolecular energy and the intermolecular interaction energy, etc.

B.Lesyng, G.A.Jeffrey, H.A.Maluszynska, A Model for the Hydrogen-bond-length Probability Distributions in the Crystal Structures of Small-molecule Components of the Nucleic Acids, Acta Crystallog., B44, 193-8(1988)

However, this problem can also be seen in a different, more abstract way, namely as minimization of the

free energy of a selected molecular system in its real molecular environment – in this particular case this is

the environment formed by surrounding molecules with imposed constraints resulting from the

symmetry.

Towards generalization of experimentally observedstructural changes

Fields are equally important as structures !

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

Determination of biomolecular structures

X-ray and neutrondiffraction data

NMR

Molecular quantum mechanics.Minimization of the B.-O.

energy

Homology analysisand structure prediction

”Ab intio” methods

Minimization of the MM-energy or free energy

Experimental and ”data-mining” approaches

0

RE

EHBO

BOel

0

RF MM

eFF

eFzyx

lkhi

lkhlkh

lkh

lzkyhxi

lkhV

,,

,,,,

,,

)(2

,,

1,,

R ,

Towards global minimum of the free energy(Gibbs & Boltzmann – equilibrium properties,

Kramers & Eyring - kinetics)

Homology analysisand

structure prediction.

Making use of molecular evolution conceptsand Darwinian-type

approach.

Optimal sequence alignment, followed by a 3D structure alignment, results in predictionof a correct, 3D-hierarchical biomolecular structure.

”Optimal” – consistent withcurrent evolutionary concepts.

Wrong sequence alignmenttypically results in a wrong structure.

.Multiscale modelling methods, the approach to refine structures

and to understend functioning of complex biomolecular systems and processes

• Virtual titration- J. Antosiewicz, E. Błachut-Okrasińska, T. Grycuk, J. Briggs, S. Włodek, B.

Lesyng, J.A. McCammon, Prediction of pKas of Titratable Residues in Proteins Using a Poisson-Boltzman Model of the Solute-Solvent System, in “Computational Molecular Dynamics: Challenges, Methods, Ideas”, Lecture Notes in Computational Science and Engineering, vol. 4, Eds. P.Deuflhard et al, Springer-Verlag, Berlin, Heidelberg, pp. 176-196,1999

– 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.

- 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)

• Quantum forces and dynamics in complex biomolecular systems.– P. Bala, P. Grochowski, B. Lesyng, J.A. 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)

14

Protonation equilibria in proteins

Protonation equilibria - microstates

,..0,1,1,0,1,1,0aix

Tk

aiaiB

aixGe

Z xx1

)( 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

The model group – a reference state

GGGG HinoHiiiipHpH

,,mod,

This differenceassumed to bepurely electrostatic

Ensamble -role of a reference state (”model group”)

Z

ZGH

noHRT ln GGGG HnoHpHpH

mod

The microstate energy

GGxGxG ijjiji

ii

iiioaixxpH

,2

1

Phosphotyrosine in phospholipase C-

SH2 domain of phospholipase C-1

(pdb: 2PLE)

S.M.Pascal,A.U.Singer,G.Gish,T.YamazkiS.E.Shoelson, T.Pawson, L.E.Kay, J.D.Forman-Kay, Nuclear Magnetic Resonance Structure Of An Sh2 2ple Domain Of Phospholipase C-Gamma1 Complexed With A High Affinity Binding Peptide, Cell, 77,461-472(1994)

phosphotyrosine

phosphoglucomutase(pdb: 3PMG)

W.J.Ray, Junior, Y.Liu, S.Baranidharan,Structure of Rabbit Muscle Phosphoglucomutase at 2.4 Angstroms Resolution. Use of Freezing Point Depressant and Reduced Temperature to Enhance Diffractivity, to be published

phosphoserine

Phosphoserine in phosphoglucomutase

Open and close forms of PKA

Typical results for phosphorylated proteins

molecule prediction experimental

phopsphotyrosine

tetrapeptide 1 5.36 5.9

dodecapeptide 5.66 6.1

phospholipase C-1 3.71 4.0

phosphoserine

tetrapeptide 2 5.7 6.1

phosphoglucomutase   <4

phosphothreonine

tetrapeptide 3 6.1 6.1

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.

Quantum-classical dynamics in simulations of enzymatic processes (phospholipase A2 – a case study)

Conclusions• I n general, experimental structures

should be refi ned, which in particular, requires application of virtual titrationprocedures.

• Knowledge of intra- and intermolecular fi elds (electrostatic, hydrophobic, etc.) are required f or better understanding ofmolecular and f unctional properties. I nfl uence of the real molecular environment can be modelled by such fi elds.

• Development and applications of multiscalemolecular modelling methods (f romquantum to mezoscopic ones) allow to much better describe biomolecular mechanisms and logic of their f unctioning(a f ew ongoing projects in my group).

• Development and applications of a variety of mathematical models and theories makes such studies to be not boaring !

Acknowledgements

PhD students:Marta HallayJarek KalinowskiPiotr KmiećMagda GruzielMichał WojciechowskiŁukasz WalewskiFranek RakowskiJanek Iwaszkiewicz

Coworkers:Prof. J.AntosiewiczProf. P.BałaDr. P.Grochowski

Collaboration:Prof. J.A.McCammonProf. W.SaengerProf. D.Truhlar

Studies are supported by”European CoE for Multiscale Biomolecular

Modelling, Bioinformatics and Applications” and

Polish State Committee for Scientific Research.

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)•....

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 . P h y s C h e m , i n p r e s s

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