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Energetics of protein structure. Energetics of protein structures. Molecular Mechanics force fields Implicit solvent Statistical potentials. Energetics of protein structures. Molecular Mechanics force fields Implicit solvent Statistical potentials. What is an atom?.

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energetics of protein structures
Energetics of protein structures
  • Molecular Mechanics force fields
  • Implicit solvent
  • Statistical potentials
energetics of protein structures1
Energetics of protein structures
  • Molecular Mechanics force fields
  • Implicit solvent
  • Statistical potentials
what is an atom
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)
slide5

Example of atom definitions: CHARMM

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

slide6

Example of residue definition: 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

atomic interactions
Atomic interactions

Non-bonded

pair

Torsion angles

Are 4-body

Angles

Are 3-body

Bonds

Are 2-body

forces between atoms
Forces between atoms

Strong bonded interactions

b

All chemical bonds

q

Angle between chemical bonds

Preferred conformations for

Torsion angles:

- w angle of the main chain

- c angles of the sidechains

(aromatic, …)

f

forces between atoms vdw interactions
Forces between atoms: vdW interactions

r

1/r12

Lennard-Jones potential

Rij

1/r6

slide12

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)

energetics of protein structures2
Energetics of protein structures
  • Molecular Mechanics force fields
  • Implicit solvent
  • Statistical potentials
potential of mean force
Potential of mean force

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

The potential energy U can be decomposed as:

UP(X): protein-protein interactions

US(X): solvent-solvent interactions

UPS(X,Y): protein-solvent

interactions

potential of mean force1
Potential of mean force

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

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

WT(X) is called thepotential of mean force.

potential of mean force2
Potential of mean force

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

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).

the sa model
The SA model

Surface area potential

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

surface area potentials which surface
Surface area potentialsWhich surface?

Accessible

surface

Molecular

Surface

hydrophobic potential surface area or volume
Hydrophobic potential:Surface Area, or Volume?

Surface effect

(Adapted from Lum, Chandler, Weeks,

J. Phys. Chem. B, 1999, 103, 4570.)

Volume effect

“Radius of the molecule”

For proteins and other large bio-molecules, use surface

slide22

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
Elementary Electrostatics in vacuo

Some basic notations:

Divergence

Gradient

Laplacian

elementary electrostatics in vacuo1
Elementary Electrostatics in vacuo

Coulomb’s law:

The electric force acting on a point charge q2 as the result of the presence of

another charge q1 is given by Coulomb’s law:

r

u

q1

q2

Electric field due to a charge:

By definition:

q1

E “radiates”

elementary electrostatics in vacuo2
Elementary Electrostatics in vacuo

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, r(X)=qd(X-X0)

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

elementary electrostatics in vacuo3
Elementary Electrostatics in vacuo

Energy and potential:

- The force derives from a potential energy U:

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

Potential produced by q1 at

at a distance r:

For two point charges in vacuo:

elementary electrostatics in vacuo4
Elementary Electrostatics in vacuo

The cases of multiple charges: the superposition principle:

Potentials, fields and energy are additive

For n charges:

qN

qi

X

q1

q2

elementary electrostatics in vacuo5
Elementary Electrostatics in vacuo

Poisson equation:

Laplace equation:

(charge density = 0)

slide29

Uniform Dielectric Medium

Physical basis of dielectric screening

An atom or molecule in an externally imposed electric field develops a non

zero net dipole moment:

-

+

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

The field generated by these induced dipoles runs against the inducing

field the overall field is weakened (Screening effect)

The negative

charge is

screened by

a shell of positive

charges.

slide30

Uniform Dielectric Medium

Electronic polarization:

Under external

field

-

-

-

-

-

-

-

-

-

-

-

-

-

-

+

+

-

-

-

-

-

-

-

-

-

-

-

-

Resulting dipole moment

Orientation polarization:

Under external

field

Resulting dipole moment

slide31

Uniform Dielectric Medium

Polarization:

The dipole moment per unit volume is a vector field known as

the polarization vector P(X).

In many materials:

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

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

slide32

Uniform Dielectric Medium

Some typical dielectric constants:

slide33

Uniform Dielectric Medium

Modified Poisson equation:

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

slide34

Uniform Dielectric Medium

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 a

reaction potentialf that is proportional to q for a linear response.

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

For N charges:

Free energy

slide35

System with dielectric boundaries

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

If we can solve this equation, we have the potential, from which we can derive

most electrostatics properties of the system (Electric field, energy, free energy…)

BUT

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

slide36

The Poisson Boltzmann Equation

r(X) is the density of charges. For a biological system, it includes the charges

of 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):

The potential f is itself influenced by the redistribution of ion charges, so the

potential and concentrations must be solved for self consistency!

slide37

The Poisson Boltzmann Equation

Linearized form:

I: ionic strength

slide38

Solving the Poisson Boltzmann Equation

  • 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
slide39

Linear Poisson Boltzmann equation:

Numerical solution

  • Space discretized into a
  • cubic lattice.
  • Charges and potentials are
  • defined on grid points.
  • Dielectric defined on grid lines
  • Condition at each grid point:

ew

eP

j : indices of the six direct neighbors of i

Solve as a large system of linear

equations

slide40

Electrostatic solvation energy

The electrostatic solvation energy can be computed as an energy change

when solvent is added to the system:

The sum is over all nodes of the lattice

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

slide41

Approximate electrostatic solvation energy:

The Generalized Born Model

Remember:

For a single ion of charge q and radius R:

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:

slide42

Approximate electrostatic solvation energy:

The Generalized Born Model

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:

DGGB is correct when rij 0 and rij∞

ai: Born radius of charge i:

Assuming that the charge i produces a Coulomb potential:

slide43

Approximate electrostatic solvation energy:

The Generalized Born Model

DGGB

further reading
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)
energetics of protein structures3
Energetics of protein structures
  • Molecular Mechanics force fields
  • Implicit solvent
  • Statistical potentials
slide47

Counts

Energy

Ile-Asp

Ile-Asp

r(Ǻ)

r(Ǻ)

Ile-Leu

Ile-Leu

r(Ǻ)

r(Ǻ)

slide48

The Decoy Game

Finding near native conformations

1CTF

Score

cRMS (Ǻ)