Energetics of protein structure

Energetics of protein structure

Energetics of protein structures

• Molecular Mechanics force fields

• Implicit solvent

• Statistical potentials





What is an atom?

• Classical mechanics: a solid object

• Defined by its position (x,y,z), its shape (usually a ball) and its mass

• May carry an electric charge (positive or negative), usually partial (less than an electron)

MASS 20 C 12.01100 C ! carbonyl C, peptide backboneMASS 21 CA 12.01100 C ! aromatic CMASS 22 CT1 12.01100 C ! aliphatic sp3 C for CHMASS 23 CT2 12.01100 C ! aliphatic sp3 C for CH2MASS 24 CT3 12.01100 C ! aliphatic sp3 C for CH3MASS 25 CPH1 12.01100 C ! his CG and CD2 carbonsMASS 26 CPH2 12.01100 C ! his CE1 carbonMASS 27 CPT 12.01100 C ! trp C between ringsMASS 28 CY 12.01100 C ! TRP C in pyrrole ring

Example of atom definitions: CHARMM

RESI ALA 0.00GROUP ATOM N NH1 -0.47 ! |ATOM HN H 0.31 ! HN-NATOM CA CT1 0.07 ! | HB1ATOM HA HB 0.09 ! | /GROUP ! HA-CA--CB-HB2ATOM CB CT3 -0.27 ! | \ATOM HB1 HA 0.09 ! | HB3ATOM HB2 HA 0.09 ! O=CATOM HB3 HA 0.09 ! |GROUP !ATOM C C 0.51ATOM O O -0.51BOND CB CA N HN N CA BOND C CA C +N CA HA CB HB1 CB HB2 CB HB3 DOUBLE O C

Example of residue definition: CHARMM

Atomic interactions

Torsion anglesAre 4-body

AnglesAre 3-body

BondsAre 2-body

Non-bondedpair

Forces between atoms

Strong bonded interactions

20 )( bbKU

20)( KU

))cos(1( nKU

b

All chemical bonds

Angle between chemical bonds

Preferred conformations forTorsion angles: - angle of the main chain - angles of the sidechains

(aromatic, …)

Forces between atoms: vdW interactions

612

2)(r

R

r

RrE ijij

ijLJ

1/r12

1/r6

Rij

r

Lennard-Jones potential

jiijji

ij

RRR

;

2

Example: LJ parameters in CHARMM

Forces between atoms: Electrostatics interactions

r

Coulomb potential

qi qj

r

qqrE ji

04

1)(

Some Common force fields in Computational Biology

ENCAD (Michael Levitt, Stanford)

AMBER (Peter Kollman, UCSF; David Case, Scripps)

CHARMM (Martin Karplus, Harvard)

OPLS (Bill Jorgensen, Yale)

MM2/MM3/MM4 (Norman Allinger, U. Georgia)

ECEPP (Harold Scheraga, Cornell)

GROMOS (Van Gunsteren, ETH, Zurich)

Michael Levitt. The birth of computational structural biology. Nature Structural Biology, 8, 392-393 (2001)





SolventExplicit or Implicit ?

Potential of mean force

dXdYe

eYXP

kT

YXU

kT

YXU

,

,

),(

A protein in solution occupies a conformation X with probability:

X: coordinates of the atoms of the protein

Y: coordinates of the atoms of the solvent

),()()(),( YXUYUXUYXU PSSP

The potential energy U can be decomposed as: UP(X): protein-protein interactions

US(X): solvent-solvent interactions

UPS(X,Y): protein-solvent interactions


dYYXPXPP ),()(

We study the protein’s behavior, not the solvent:

PP(X) is expressed as a function of X only through the definition:

dXe

eXP

kT

XW

kT

XW

P T

T

)(

)(

)(

WT(X) is called the potential of mean force.


The potential of mean force can be re-written as:

)()()( XWXUXW solPT

Wsol(X) accounts implicitly and exactly for the effect of the solvent on the protein.

Implicit solvent models are designed to provide an accurate and fast

estimate of W(X).

++

Solvation Free Energy

Wnp

Wsol

VacchW

SolchW

cavvdWvac

chsol

chnpelecsol WWWWWWW

The SA model

Surface area potential

N

kkkvdWcav SAWW

1

Eisenberg and McLachlan, (1986) Nature, 319, 199-203

Surface area potentialsWhich surface?

MolecularSurface

Accessiblesurface

Hydrophobic potential:Surface Area, or Volume?

(Adapted from Lum, Chandler, Weeks, J. Phys. Chem. B, 1999, 103, 4570.)

“Radius of the molecule”

Volume effect

Surface effect

For proteins and other large bio-molecules, use surface

Protein Electrostatics

• Elementary electrostatics• Electrostatics in vacuo• Uniform dielectric medium• Systems with boundaries

• The Poisson Boltzmann equation• Numerical solutions• Electrostatic free energies

• The Generalized Born model

Elementary Electrostatics in vacuo

Some basic notations:

2

2

2

2

2

22

z

f

y

f

x

fffgraddivf

z

f

y

f

x

ffgradf

z

F

y

F

x

FFdivF zyx

Divergence

Gradient

Laplacian


ur

qqF

20

21

4

Coulomb’s law:

The electric force acting on a point charge q2 as the result of the presence ofanother charge q1 is given by Coulomb’s law:

q1

r

u

Electric field due to a charge:

By definition:

ur

q

q

FE

20

1

2 4

q2

q1

E “radiates”


0

qdAE

0

)())((

X

X Ediv

Gauss’s law:

The electric flux out of any closed surface is proportional to the total charge enclosed within the surface.

Integral form: Differential form:

Notes:- for a point charge q at position X0, (X)=q(X-X0)

- Coulomb’s law for a charge can be retrieved from Gauss’s law


UgradF

gradE

Energy and potential:

- The force derives from a potential energy U:

- By analogy, the electric field derives from an electrostatic potential :

For two point charges in vacuo:

r

qqU

0

21

4

r

q

0

1

4

Potential produced by q1 atat a distance r:


The cases of multiple charges: the superposition principle:

Potentials, fields and energy are additive

For n charges:

ji i

ji

N

ii

i

i

N

i i

i

XX

qqU

uXX

qXE

XX

qX

0

12

0

1 0

4

4)(

4

q1

q2

qi

qN

X


0

2

0

graddiv

Ediv

Poisson equation:

Laplace equation:

02 (charge density = 0)

+-

Uniform Dielectric MediumPhysical basis of dielectric screening

An atom or molecule in an externally imposed electric field develops a nonzero net dipole moment:

(The magnitude of a dipole is a measure of charge separation)

The field generated by these induced dipoles runs against the inducingfield the overall field is weakened (Screening effect)

The negativecharge is screened bya shell of positivecharges.

+

Uniform Dielectric MediumElectronic polarization:

- ---

----

----

-

+

- ----

----

---

-Under external

field

Resulting dipole moment

Orientation polarization:

Under externalfield

Resulting dipole moment

Uniform Dielectric MediumPolarization:

The dipole moment per unit volume is a vector field known asthe polarization vector P(X).

In many materials: )(4

1)()( XEXEXP

is the electric susceptibility, and is the electric permittivity, or dielectric constant

The field from a uniform dipole density is -4P, therefore the total field is

applied

applied

EE

PEE

4

Uniform Dielectric Medium

Some typical dielectric constants:

Molecule Dipole moment (Debyes) of a

single molecule

Dielectric constant of the

liquid at 20°C

Water 1.9 80

Ethanol 1.7 24

Acetic acid 1.7 4

Chloroform 0.86 4.8


Modified Poisson equation:

0

2 graddiv

Energies are scaled by the same factor. For two charges:

r

qqU

0

21

4


The work of polarization:

It takes work to shift electrons or orient dipoles. A single particle with charge q polarizes the dielectric medium; there is areaction potential that is proportional to q for a linear response.

The work needed to charge the particle from qi=0 to qi=q:

q

Rii

q

iiR qCqdqqCdqqW0

2

0 2

1

2

1

CqR

For N charges:

N

iiRiqW

12

1 Free energy

System with dielectric boundaries

The dielectric is no more uniform: varies, the Poisson equation becomes:

0

)()(

XXXXgradXdiv

If we can solve this equation, we have the potential, from which we can derivemost electrostatics properties of the system (Electric field, energy, free energy…)

BUT

This equation is difficult to solve for a system like a macromolecule!!

The Poisson Boltzmann Equation

(X) is the density of charges. For a biological system, it includes the chargesof the “solute” (biomolecules), and the charges of free ions in the solvent:

The ions distribute themselves in the solvent according to the electrostatic potential (Debye-Huckel theory):

N

iiiions XnqX

1

)()(

ion i typeon charge :

eunit volumper i typeof ions ofnumber :

i

i

q

nkT

Xq

i

ii

en

Xn )(

0

)(

)()()( XXX ionssolute

The potential f is itself influenced by the redistribution of ion charges, so thepotential and concentrations must be solved for self consistency!

N

i

kT

Xq

ii

i

enqX

XX1

)(0

00

1)(

The Poisson Boltzmann Equation

Linearized form:

IkT

qnkT

XXXX

XX

N

iii

01

20

0

2

2

0

21

)()()()(

I: ionic strength

• Analytical solution

• Only available for a few special simplification of the molecular shape and charge distribution

• Numerical Solution• Mesh generation -- Decompose the physical

domain to small elements;• Approximate the solution with the potential value

at the sampled mesh vertices -- Solve a linear system formed by numerical methods like finite difference and finite element method

• Mesh size and quality determine the speed and accuracy of the approximation

Solving the Poisson Boltzmann Equation

Linear Poisson Boltzmann equation:Numerical solution

P

w

• Space discretized into a cubic lattice.

• Charges and potentials are defined on grid points.

• Dielectric defined on grid lines

• Condition at each grid point:

6

1

22

0

6

1

jijijij

i

jjij

i

h

hq

j : indices of the six direct neighbors of i

Solve as a large system of linearequations

Electrostatic solvation energy

The electrostatic solvation energy can be computed as an energy change when solvent is added to the system:

i

NSSii

RFielec iiqiqW )()(

2

1)(

2

1

The sum is over all nodes of the lattice

S and NS imply potentials computed in the presence and absence of solvent.

Approximate electrostatic solvation energy:The Generalized Born Model

Remember:

N

i

RFiielec qG

12

1

For a single ion of charge q and radius R:

1

1

8 0

2

R

qGBorn Born energy

For a “molecule” containing N charges, q1,…qN, embedded into spheres or radii R1, …, RN such that the separation between the charges is large compared to the radii, the solvation energy can be approximated by the sum of the Born energy and Coulomb energy:

N

i

N

ij ij

jiN

i i

ielec r

qq

R

qG

1 01 0

2

11

42

11

1

8


The GB theory is an effort to find an equation similar to the equation above, that is a good approximation to the solution to the Poisson equation.The most common model is:

N

i

N

jaa

r

jiij

jiGB

ji

ij

eaar

qqG

1 142

02

11

8

1

ai: Born radius of charge i:

GGB is correct when rij 0 and rij ∞

Assuming that the charge i produces a Coulomb potential:

iRr

i r

dV

a 44

11


ijr

11

1

GGB

Further reading

• Michael Gilson. Introduction to continuum electrostatics. http://gilsonlab.umbi.umd.edu

• M Schaefer, H van Vlijmen, M Karplus (1998) Adv. Prot. Chem., 51:1-57 (electrostatics free energy)

• B. Roux, T. Simonson (1999) Biophys. Chem., 1-278 (implicit solvent models)

• D. Bashford, D Case (2000) Ann. Rev. Phys. Chem., 51:129-152 (Generalized Born models)

• K. Sharp, B. Honig (1990) Ann. Rev. Biophys. Biophys. Chem., 19:301-352 (Electrostatics in molecule; PBE)

• N. Baker (2004) Methods in Enzymology 383:94-118 (PBE)

http://gilsonlab.umbi.umd.edu/





Statistical Potentials

r

a

b

r(Ǻ)

f(r)

)(

)(ln),,( ),(

rP

rPrbaE ba

Ile-Asp Ile-Asp

Ile-Leu Ile-Leu

Counts Energy

r(Ǻ) r(Ǻ)

r(Ǻ) r(Ǻ)

cRMS (Ǻ)

Sco

re1CTF

The Decoy Game Finding near native conformations

ji ij

ijji

ji rP

raaPjiEE

)(

),,(ln),(

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Energetics of protein structure