Download Statistical Potentials

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?
• 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)
Example of atom definitions: CHARMM
MASS
MASS
MASS
MASS
MASS
MASS
MASS
MASS
MASS
20
21
22
23
24
25
26
27
28
C
CA
CT1
CT2
CT3
CPH1
CPH2
CPT
CY
12.01100
12.01100
12.01100
12.01100
12.01100
12.01100
12.01100
12.01100
12.01100
C
C
C
C
C
C
C
C
C
!
!
!
!
!
!
!
!
!
carbonyl C, peptide backbone
aromatic C
aliphatic sp3 C for CH
aliphatic sp3 C for CH2
aliphatic sp3 C for CH3
his CG and CD2 carbons
his CE1 carbon
trp C between rings
TRP C in pyrrole ring
Example of residue definition: CHARMM
RESI ALA
GROUP
ATOM N
NH1
ATOM HN
H
ATOM CA
CT1
ATOM HA
HB
GROUP
ATOM CB
CT3
ATOM HB1 HA
ATOM HB2 HA
ATOM HB3 HA
GROUP
ATOM C
C
ATOM O
O
BOND CB CA N
BOND C CA C
DOUBLE O C
0.00
-0.47
0.31
0.07
0.09
-0.27
0.09
0.09
0.09
0.51
-0.51
HN N CA
+N CA HA
!
!
!
!
!
!
!
!
!
!
|
HN-N
|
HB1
|
/
HA-CA--CB-HB2
|
\
|
HB3
O=C
|
CB HB1
CB HB2
CB HB3
Atomic interactions
Torsion angles
Are 4-body
Non-bonded
pair
Angles
Are 3-body
Bonds
Are 2-body
Forces between atoms
Strong bonded interactions
b

U  K (b  b0 )2
All chemical bonds
U  K (   0 ) 2
Angle between chemical bonds

U  K (1  cos( n ))
Preferred conformations for
Torsion angles:
- w angle of the main chain
- c angles of the sidechains
(aromatic, …)
Forces between atoms: vdW interactions
r
1/r12
Lennard-Jones potential
  Rij 12  Rij 6 
ELJ ( r )   ij     2  
 r 
 r  

Rij 
Ri  R j
2
;  ij   i j
Rij
1/r6
Example: LJ parameters in CHARMM
Forces between atoms: Electrostatics
interactions
r
Coulomb potential
qi
qj
1 qi q j
E (r) 
40 r
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 structures
• Molecular Mechanics force fields
• Implicit solvent
• Statistical potentials
Solvent
Explicit or Implicit ?
Potential of mean force
A protein in solution occupies a conformation X with probability:
e
P( X , Y ) 
e


U  X ,Y 
kT
U  X ,Y 
kT
dXdY
The potential energy U can be decomposed as:
U ( X , Y )  U P ( X )  U S (Y )  U PS ( X , Y )
X: coordinates of the atoms
of the protein
Y: coordinates of the atoms
of the solvent
UP(X): protein-protein interactions
US(X): solvent-solvent interactions
UPS(X,Y): protein-solvent
interactions
Potential of mean force
We study the protein’s behavior, not the solvent:
PP ( X )   P( X , Y )dY
PP(X) is expressed as a function of X only through the definition:
PP ( X ) 
e
e


WT ( X )
kT
WT ( X )
kT
dX
WT(X) is called the potential of mean force.
Potential of mean force
The potential of mean force can be re-written as:
WT ( X )  U P ( X )  Wsol ( X )
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
Wsol
+
+
Sol
ch
W
W
Vac
ch
Wnp
Wsol  Welec  Wnp  W
sol
ch
W
vac
ch
  W
vdW
 Wcav 
The SA model
Surface area potential
N
Wcav  WvdW   k SAk
k 1
Eisenberg and McLachlan, (1986) Nature, 319, 199-203
Surface area potentials
Which surface?
Accessible
surface
Molecular
Surface
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
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:

 Fx  Fy  Fz
  F  div F 


x
y
z
 f
f  grad  f   
 x

f
y

f 
z 
2
2
2

f

f

f
2
  f  div grad  f    f  2  2  2
x
y
z
Divergence
Gradient
Laplacian
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
q1
q2
u
q1q2
F
u
2
40 r
Electric field due to a charge:
By definition:
F
q1
E

u
2
q2 40 r
q1
E “radiates”
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:
q
 E  dA  
Notes:
0
(X )
div ( E ( X )) 
0
- for a point charge q at position X0, (X)=qd(X-X0)
- Coulomb’s law for a charge can be retrieved from Gauss’s law
Elementary Electrostatics in vacuo
Energy and potential:
- The force derives from a potential energy U:
F   grad U 
- By analogy, the electric field derives from an electrostatic potential :
E   grad  
For two point charges in vacuo:
q1q2
U
40 r
Potential produced by q1 at
at a distance r:

q1
40 r
Elementary Electrostatics in vacuo
The cases of multiple charges: the superposition principle:
Potentials, fields and energy are additive
For n charges:
qN
N
qi
 X   
i 1 40 X  X i
X
N
E( X )  
i 1
U 
i j
qi
qi
40 X  X i
qi q j
40 X  X i
u
2 i
q1
q2
Elementary Electrostatics in vacuo
Poisson equation:


div E 
0



div grad             
0
2
Laplace equation:
 0
2
(charge density = 0)
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.
Uniform Dielectric Medium
Electronic polarization:
-
-
-
-
-
- -
-
-
-
-
- +
- -
-
+
-
Under external
field
-
-
-
-
Resulting dipole moment
Orientation polarization:
Under external
field
Resulting dipole moment
Uniform Dielectric Medium
Polarization:
The dipole moment per unit volume is a vector field known as
the polarization vector P(X).
In many materials:
P( X )  c E ( X ) 
 1
E( X )
4
c is the electric susceptibility, and  is the electric permittivity, or dielectric constant
The field from a uniform dipole density is -4P, therefore the total field is
E  E applied  4 P
E
E applied

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
Uniform Dielectric Medium
Modified Poisson equation:



div grad        
 0
2
Energies are scaled by the same factor. For two charges:
U
q1q2
40r
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 potential  that is proportional to q for a linear response.
R  Cq
The work needed to charge the particle from qi=0 to qi=q:
W   R qi dqi  C 
q
q
0
0
1 2 1
qi dqi  Cq  qR
2
2
For N charges:
1 N
W   qiiR
2 i 1
Free energy
System with dielectric boundaries
The dielectric is no more uniform:  varies, the Poisson equation becomes:
 X 
div   X grad  ( X )       X  ( X )  
0


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
(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:
 ( X )   solute( X )  ions ( X )
The ions distribute themselves in the solvent according to the electrostatic
potential (Debye-Huckel theory):
ni ( X )
e
0
ni
 qi ( X )
kT
ni : number of ions of type i per unit volum e
qi : charge on type i ion
N
ions ( X )   qi ni ( X )
i 1
The potential f is itself influenced by the redistribution of ion charges, so the
potential and concentrations must be solved for self consistency!
The Poisson Boltzmann Equation
 X  1
0
    X  ( X )  
  qi ni e
0
 0 i 1
N

qi ( X )
kT
Linearized form:
 X 
    X  ( X )  
  ( X ) 2 ( X ) ( X )
0
N
1
2
2
0 2
 
ni qi 
I

 0kT i 1
 0kT
I: ionic strength
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
Linear Poisson Boltzmann equation:
Numerical solution
• Space discretized into a
cubic lattice.
• Charges and potentials are
defined on grid points.
w
• Dielectric defined on grid lines
• Condition at each grid point:
P
j : indices of the six direct neighbors of i
6
i 
 ij j 
j 1
6

j 1
ij
qi
 0h
  h
2 2
ij ij
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:
Welec
1
1
RF
  qi (i )   qi S (i )  NS (i ) 
2 i
2 i
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:
Gelec
1 N
  qiiRF
2 i 1
For a single ion of charge q and radius R:
1 
GBorn 
  1
80 R  

q2
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:
Gelec
 1  1 N N qi q j  1 

  1  
  1
 2 i 1 j i 40 rij  

i 1 80 Ri  
N
qi2
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:
GGB
1 1  N N

  1
80  
 i 1 j 1
GGB is correct when rij
0 and rij
qi q j
rij2  ai a j e

rij2
4 ai a j
∞
ai: Born radius of charge i:
Assuming that the charge i produces a Coulomb potential:
1
1

ai 4
dV
rRi r 4
Approximate electrostatic solvation energy:
The Generalized Born Model
1  1
  1

 rij
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)
Energetics of protein structures
• Molecular Mechanics force fields
• Implicit solvent
• Statistical potentials
Statistical Potentials
a
f(r)
r
b
r(Ǻ)
Counts
 P( a ,b ) ( r ) 
E (a, b, r )   ln 

 P( r ) 
Energy
Ile-Asp
Ile-Asp
r(Ǻ)
r(Ǻ)
Ile-Leu
Ile-Leu
r(Ǻ)
r(Ǻ)
The Decoy Game
Finding near native conformations
Score
1CTF
cRMS (Ǻ)
 P( ai , a j , rij ) 

E   E (i, j )    ln 
 P( r ) 
i j
i j
ij

