agenda: motivation basic knowledge on electrostatics...

Sfb765, 29. January 2009 electrostatic energy computations of complex molecular systems 1

Institute of Chemistry and Biochemistry Freie Universität Berlin Takustrasse 36A14195 Berlin, Germany

email: [email protected]

Computation of electrostatic energies for complex molecular systems

Agenda:

motivation

basic knowledge on electrostatics

Application:

Compute electrostatic energies in proteins to evaluate pKa values


motivation

Electrostatic interactions are ubiquitous and most relevant not only in all kinds of molecular systems but more general everywhere in the universe where there is not too densely pact matter of low enough energy.In the sun, of course, the most important interactions are gravitation and nuclear interactions. But, already in the earth atmosphere ionic interactions are most important being for instance responsible for aurora borealis.

Electrostatic interactions in chemistry are responsible for van der Waals interactions about 80% of the H-bond interaction energies and consequently they determine melting and boiling points of substances.

They govern the specificity in molecular recognition in: base pairing in DNA and RNA, protein substrate interactions, protein complex formation, immune response and cell-cell interactions. This list is by far not complete.


motivation

hydrophobicity versus hydrophilicity:

Charged and polar molecular groups are generally considered to be

hydrophilic (i.e. they like water).

Water with its large permanent dipole

interacts strongly with such molecular

groups by reorienting its dipoles around a

charge. Hence, the hydrophilicity is here

governed by electrostatic interactions.

For the same reason bulk water interacts

strongly among each other. On the average it forms 3 of 4 possible

H-bonds per molecule. There are many different H-bond pattern in

bulk water, which vary dynamically on picosecond timescale. That is

why water is a low viscosity liquid. The large number of H-bond

pattern involve a large amount of entropy.

++

+

- +

+

-

++

-

+

+

- +

+- +-

+


motivation

hydrophobicity versus hydrophilicity:

What about hydrophobic groups like CH4? They do not strongly

interact with water, but disturb the H-bond structure of bulk water.

Specifically they force water in the neighborhood to be locked in a

rigid H-bond pattern reducing the entropy of bulk water.

Bulk water fights back by pushing these molecules into separate

clusters (the “oil drop in water” phenomenon).

This is the classical hydrophobic effect related to bulk water entropy.

But what about ions?

Are they not disturbing the H-bond structure of bulk water?

Yes, they are, but, the electrostatic effect is stronger and leads to

efficient solvation in spite of the hydrophobic effect.



Interaction of point charges:

Coulomb interaction: q1 q2

1 2

1 2

1 2

q qW(| r - r |) =

| r - r |

� �

� �

1r�

2r�

0in a homogeneous dielectric:

1 2

1 2

1 2

q qW(| r - r |) =

| r - r |ε

� �

� �

Interaction of a unit charge

q =1 with a charge cloud ρρρρ’( ): r′� q ρρρρ’( )

r�

r′�

0

ρ (r )(r) = dr

| r - r |

′ ′′ ′φ

′∫

�

� �

� �

φφφφ’ is the electrostatic potential generated by the charge cloud ρρρρ’


The electrostatic potential is the fundamental quantity of electrostatics to

calculate interaction energies between different charge clouds ρ and ρ’:

Disadvantage: more difficult to evaluate φ

Advantage: allows to consider inhomogeneous dielectrics

as they occur in protein membrane water systems.

An alternative way to calculate the electrostatic potential Φ

induced by a charge cloud ρ is the Poisson equation:

( ) ( ) ( )r r 4 r∇ε ∇φ = − πρ� ��

i

where

tf (r) f (r) f (r)

f(r) =x y z, ,

∂ ∂ ∂∇

∂ ∂ ∂

� � ��


(r) (r )W = dr (r) (r) = dr dr

| r r |

''′ρρ

′ρ ρ′ρ φ

′∫ ∫ ∫

-

� �

� � � � �

� �

( )rε�


What is a dielectric medium?

A dielectric considers a material consisting of microscopic dipoles, which

can reorient and are continuously distributed (cloud of dipoles).

These implicit dipoles react on charges: they reorient or in other words

they are polarized.

For molecular systems there are two such contributions:

Nuclear polarization due to permanent dipoles and

Electronic polarization due to distortion of the electronic wave function.

Typical values of the dielectric constant ε are

ε = 2 for electronic polarization (the only contribution for fast processes)

ε > 2 for nuclear polarization (for water ε = 80).

Note that the dielectric constant will be ε = 1, if all components of a

molecular system are considered explicitly, i.e. the Schrödinger

equation is solved in all detail. Whatever is not considered explicitly

can be considered approximately by a dielectric medium.



streptococcal protein G

pKa predictions in proteins with pH adapted global conformers

application


AH A + H+

=

+

A

A HK

AH

( )

= −

∆ → =

A

OA

ApK pH log

AH

G AH A ln10 RT pK− −

− −

= +−

=+

A

A

A

ln10(pH pK )

ln10(pH pK )

xpK pH log

1 x

ex

1 e

( )∆ = − AG(pH,T) ln10 RT pH pK

application pKa, what is it?


pKA of titratable groups in proteins

ASH AS + H+S

APH AP + H+S

H+

H+

∆GS,P(AH) ∆GS,P(A)

∆GP(AH�A)

∆GS(AH�A)

∆GP(AH�A) = ∆GS(AH�A) + ∆GS,P(A) – ∆GS,P(AH)

∆∆GS,P

∆∆GS,P = ∆∆Gborn + ∆∆Gback + ∆∆Gint

εw=80 εw=80

εp=4 εp=4

application computation of pKa in proteins


pKA of titratable groups in proteins

∆GP(AH�A) = ∆GS(AH�A) + ∆∆Gborn + ∆∆Gback + ∆∆Gint

( ) ( )

( ) ( )

Φ − Φ

− Φ − Φ

∑

∑

i

i

N

i P i S ii

N

i P i S ii

1Q r,A r,A

2

1Q r,AH r,AH

2

µ µ µ ν µνµ µ ν µ ν= = = ≠

∆ = − + ∆∆∑ ∑ ∑N N N

n n int r n nA,

1 1 1,

g x RT ln10 (pH pK ) x x W

interaction of titratable groupsin protein (enthalpy)

µ

µ

−∆

=

−∆

=

=∑

∑

Nn

Nn

2n g /RT

n 12

g /RT

n 1

x ex

e

application computation of pKa in proteins


data set of experimental pKA: 15 different proteins in 22 crystals, 199 experimental pKA

B. amyloliquefaciens barnase

- 1A2P, 1B20 (14)

bovine pancreatic trypsin inhibitor

- 4PTI (14)

intestinal bovine Ca2+-binding protein

- 3ICB (19)

T-lymphocyte adhesion glycoprotein

- 1HNG (14)

hen egg-white lysozyme

- 2LZT, 1B0D (19)

turkey ovomucoid inhibitor

- 1PPF, 1CHO, 3SGB (15)

streptococcal B1 IG-binding protein G

- 1PGA (15)

ribonuclease A

- 3RN3, 7RSA (16)

ribonuclease H1RNH, 2RN2, 1RDD (25)

ribonuclease T3RNT (4)

HIV protease- 1HPX (8)

xylanase- 1XNB (10)

experimental pKa in proteins

thioredoxin- 1ERT (17)

cryptogein- 1BEO (3)

ribosomal protein L9- 1DIV (6)


computed vs. experimental pKA shifts

computed vs. experimental pKa

weak ARG salt bridge (#contacts <=2)


pKa of residues in salt bridges

salt bridges


energetics of arginine salt bridges

OD2

OD1

+2

-2

protonation of Asp

deprotonation of Arg

salt bridges


Including conformational flexibility

µ µ µνµ µ ν= ≠

∆ = − + ∆∆ + ∆∑ ∑N N

n,l n,l int r,l l lA, conf

1

g x RT ln10 (pH pK ) W G

∆ = − = ∆ + ∆∆ + ∆∆l l r l l lconf conf conf hom.elec. S NESG G G G G G

∆ lconfG

∆ lhom.elec.G

∆ rSG ∆ r

NESG ∆ lSG ∆ l

NESG

conformational flexibility


generation of global conformers – perturbing ion pairs

1. Identify salt bridge on protein surface

- distance cutoff (4.0 Å)

- SASA cutoff (>30 Å2)

2. Generate 30 global conformations by random perturbation of dihedrals & minimization with constraints toward crystal structure

3. Identify lowest energy conformations

- CHARMM22 force field energy including bonded and non-bonded (electrostatic and vdW) energy terms

- vacuum or GBSW

4. Save coordinates & repeat N times with different random seed

- 5-8 conformers used in sampling

streptococcal protein G, 1PGA



generation of global conformers – perturbing ion pairs

1. Identify salt bridge on protein surface

- distance cutoff (4.0 Å)

- SASA cutoff (>30 Å2)

2. Generate 30 global conformations by random perturbation of dihedrals & minimization with constraints toward crystal structure

3. Identify lowest energy conformations

- CHARMM22 force field energy including bonded and non-bonded (electrostatic and vdW) energy terms

- vacuum or GBSW

4. Save coordinates & repeat N times with different random seed

- 5-8 conformers used in sampling




generation of global conformers – self-consistency

PDB

self-consistentPQR

protonation pattern

KarlsbergMonte Carlo

pKint∆∆W

TAPBSelectrostatics

PQR-Hcoordinates, charges, radii

CHARMM

repeat untilprot. patterndoes not change

Rabenstein et al (1998) Eur Biophys J 27: 626-637



generation of global conformers – adapting to pH

PDB

self-consisten t PQR

protonation pattern

Karlsberg

pK int

∆ ∆W

TAPBS

PQR-H

CHARMM

repeat until

prot. patterndoes not change

PDBPDB



protonation pattern

Karlsberg

protonation pattern

KarlsbergKarlsbergKarlsberg

pK int

∆ ∆W

TAPBS

pK int

∆ ∆W

pK int

∆ ∆W

pK int

∆ ∆W

TAPBSTAPBS

PQR-H

CHARMM

PQR-HPQR-H

CHARMMCHARMMCHARMM

repeat until


repeat until


PDB


protonation pattern

Karlsberg

pK int

∆ ∆W

TAPBS

PQR-H

CHARMM

repeat until


PDBPDB



protonation pattern

Karlsberg

protonation pattern


pK int

∆ ∆W

TAPBS

pK int

∆ ∆W

pK int

∆ ∆W

pK int

∆ ∆W

TAPBSTAPBS

PQR-H

CHARMM

PQR-HPQR-H

CHARMMCHARMMCHARMM

repeat until


repeat until


PDB


protonation pattern

Karlsberg

pK int

∆ ∆W

TAPBS

PQR-H

CHARMM

repeat until


PDBPDB



protonation pattern

Karlsberg

protonation pattern


pK int

∆ ∆W

TAPBS

pK int

∆ ∆W

pK int

∆ ∆W

pK int

∆ ∆W

TAPBSTAPBS

PQR-H

CHARMM

PQR-HPQR-H

CHARMMCHARMMCHARMM

repeat until


repeat until


self-consistent with respect to pH -8

PDB

5x 5x

1x

self-consistent with respect to pH 7

self-consistent with respect to pH 20

5 global

conformers

5 global

conformers

hydrogenconformer



results – occupancies of 4 different protein conformations


results conformational flexibility


results – occupancies of 4 different protein conformations




results – conformer occupancies

barnase, 1A2P



comparison of results – scatter plots

Tyr53

Asp66



comparison of results – RMSD

5.903.322.901.959.26

(3.45)

11.51

(4.68)

MAXERR

|pKa|>=1.8

2.491.311.241.052.73

(1.64)

3.85

(2.69)

RMSD

|pKa|>=1.8

5.904.862.903.219.26

(5.25)

11.51

(9.47)

MAXERR

|pKa|>=1.0

1.781.230.991.052.15

(1.81)

3.12

(2.76)

RMSD

|pKa|>=1.0

5.904.864.403.269.26

(8.51)

11.51

(10.86)MAXERR

1.030.911.021.101.84

(1.74)

2.62

(2.51)RMSD

NULLPROPKAbMCCEamc(pH,sb)mc(pH)scpH7

a Georgescu et al., Biophysical Journal 2002, 83, 1731-48. b Li, Robertson, Jensen, Proteins, 2005, 61, 704-721.

comparison with other methods


comparison of results – PROPKA

results other methods - PROPKA

Li, Robertson, Jensen, Proteins, 2005, 61, 704-721.


Team: Francesco Bettella, Gernot Kieseritzky, Ane L Gamiz, Artur Galstjan,

Hiroshi Ishikita, Alok Junea, Björn Kleier, Björn Kolbeck, Jorge Numata,

Dr. Björn Rabenstein, Dawid Rasinski, Henning Riedesel,

Tobias Schmidt-Gönner, Marcel Schmidt am Busch

Emmy Noether young researcher group of Dr. Thomas Renger:

Julia Adolphs, Dr. Mohamed El Amine Madjet, Dr. Frank Müh,

Grzegorz Raszewski

Funding: Berlin, Sfb 498, 2 Graduiertenkollegs, DAAD, Studienstiftung

agenda: motivation basic knowledge on electrostatics...

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