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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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
Example of residue definition: CHARMM
RESI ALA 0.00GROUP ATOM N NH1 0.47 ! ATOM HN H 0.31 ! HNNATOM CA CT1 0.07 !  HB1ATOM HA HB 0.09 !  /GROUP ! HACACBHB2ATOM 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
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
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, 392393 (2001)
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): proteinprotein interactions
US(X): solventsolvent interactions
UPS(X,Y): proteinsolvent
interactions
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.
The potential of mean force can be rewritten 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).
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 biomolecules, use surface
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”
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(XX0)
 Coulomb’s law for a charge can be retrieved from Gauss’s law
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:
The cases of multiple charges: the superposition principle:
Potentials, fields and energy are additive
For n charges:
qN
qi
X
q1
q2
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.
Electronic polarization:
Under external
field














+
+












Resulting dipole moment
Orientation polarization:
Under external
field
Resulting dipole moment
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
Some typical dielectric constants:
Modified Poisson equation:
Energies are scaled by the same factor. For two charges:
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
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!!
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 (DebyeHuckel theory):
The potential f is itself influenced by the redistribution of ion charges, so the
potential and concentrations must be solved for self consistency!
Solving the Poisson Boltzmann Equation
Linear Poisson Boltzmann equation:
Numerical solution
ew
eP
j : indices of the six direct neighbors of i
Solve as a large system of linear
equations
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.
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:
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: