Transcript PowerPoint Presentation - Bioinformatics 2 -

Bioinformatics 2 -- lecture 17
Molecular surfaces
Electrostatic maps
The Hydrophobic Effect
What is a molecular surface?
A molecular surface closed 3D
"manifold".
What's a "manifold"?
Here is a 2D manifold.
A cell, for example, is a 3D
manifold. It is continuous, closed,
non-intersecting. It has an inside and
an outside.
Solvent accessible surface
The solvent accessible surface is the
interface betweena molecule and its
solvent. Solvent molecules on the surface
may behave differently that bulk solvent.
Therefore surfaces can be used to model
the free energy of solvation.
Surfaces have:
•size/area
•electrostatic properties
•shape properties
M + nH2O
M in a vacuum
M.nH2O
M in water
water
protein
Rendering a surface
•A surface of any shape may be composed of 3D triangles,
that is, 3 sets of xyz coordinates, one for each vertex.
•To display the surface on the screen, each triangle is rotated and
translated according to the current frame of reference.
•Continuous triangles make a continuous surface.
•Then, each pixel is assigned a brightness according to the angle
between the triangle and the light source.
•Phong shading may be applied to
simulate curvature. In this case, each pixel in
the triangle has a different brightness,
depending on where it is.
Surfaces maybe described as a set of
connected triangles
A cow-shaped manifold made
of triangles.
Phong-shaded cow. Shading give the
illusion of higher resolution.
The
Connolly
surface
Conceptually, roll a
probe sphere over the
molecule...
•Everywhere the center
of the sphere goes is
the Solvent Accessible
Surface (SAS).
•Everywhere the
sphere touches
(including empty
space) is the Solvent
Excluded (or
"Connolly") Surface
(SES).
atomic coordinates
atom radius
rolling probe sphere
Surface shapes
Green: convex-convex,
contact surface of probe with
atom.
Brown: convex-concave,
toroidal surface when touches
two atoms.
Yellow: concave-concave,
reentrant surface when
touches three atoms.
Coloring by atom, by shape
Surfaces maybe shaded by
partial charge.
or by shape. Yellow parts are
‘reentrant’.
How do surfaces interact?
Molecular surfaces dictate what the function of a molecule is,
whether it is a binding (signaling) protein, an enzyme, a
channel, or an integral membrane protein.
•Shape interactions
•Charge interactions
Surfaces interact through shape
and charge complementarity
Complementary surfaces
leave relatively less unfilled
(void) space.
+
-
-
Void space is unfavorable
+
Proteins may be complimentary to ligands, other
proteins, or the transition states of reactions.
Binding determines function.
Nature abhors a vacuum
no, not this kind...
There is only one way to make space
empty, but many ways to fill it.
The entropy is the number of different
states available to the system.
Lower entropy implies higher free energy.
If a space is not big enough for
solvent, it is a void.
higher
energy
If we compare two surfacesurface interactions, all else
being equal, the one with
more void spaces has a
higher free energy, and is
therefore less favorable.
The one that fills space better
has a lower energy.
lower
energy
Distance profile of the void volume
What happens to the surface and volume as two hydrophobic
atoms come together??
Distance profile of the void volume
White area is excluded volume. Water can't go closer than
one radius (1.4Å).
SES
Distance profile of the void volume
First excluded surface (green) is approx compensated by new
"toroidal" surface (brown).
Distance profile of the void volume
Toroidal surface grows faster than atom surface is lost. SES
goes up.
Distance profile of the void volume
At short D, toroidal surface stops growing, atom surface
shrinks faster.
Is there a barrier to hydrophobic collapse?
black: G calculations using explicit waters
blue: SES
green: excluded volume
G
SES
Vol
Distance
When hydration shells first touch, surface area goes up,
so solvation free energy goes up, but volume goes down.
Amino Acid
Hydrophobicity
Scales
1. J. Janin, Surface and Inside Volumes in Globular Proteins, Nature, 277(1979)491-492.
2. R. Wolfenden, L. Andersson, P. Cullis and C. Southgate, Affinities of Amino Acid Side Chains for
Solvent Water, Biochemistry 20(1981)849-855.
3. J. Kyte and R. Doolite, A Simple Method for Displaying the Hydropathic Character of a Protein, J.
Mol Biol. 157(1982)105-132.
4. G. Rose, A. Geselowitz, G. Lesser, R. Lee and M. Zehfus, Hydrophobicity of Amino Acid Residues
in Globular Proteins, Science 229(1985)834-838.
Low entropy waters fill reentrant
holes on the surface.
Electrostatic potential maps/surfaces
Coulomb's Law
E
qi q j
 rij
Where qi is the charge on particle i, qj is the
charge on particle j, rij is the distance between
them, and  is the dielectric.
This is the energy required to bring two charges
from infinite distance to the distance rij in a
medium with dielectric .
Die Electric!
"Dielectric" is the degree to which the medium
disperses charge. Charge dispersion can happen by
rotating dipoles (like water) or by induced polarization.
In a higher dielectric, the charges feel less force. The
energy to pull two like charges together is less, and the
energy to push two opposite charges apart is less.
+
E
+
qi q j
 rij
(must be ≥ 1.000)
Electric field for one charge
An electric field or electrostatic map is the energy
landscape for a point charge.
Examples: (a) the electrostatic map of a system with a
single charge, with low dielectric. (b) High dielectric.
(c) Variable dielectric (on the surface of a membrane).
Coulomb's law
Coulomb's law applies
doesn't apply
isopotential lines
water
+
+
(a)
(b)
+
membrane
(c)
Electrostatic map for more than one charge.
+
qi
E(x)  
rix
Assuming a constant , the energy for putting a charge (+) at
any position r is summed using Coulomb's Law. We can
draw contours for surfaces where E is constant (isopotential).
This is like a topological map.
Partial charges
Quantum mechanical calculations can tell us the electron
distributions around atom nuclei. If there are more or less
electrons that belong to a given atom nucleus, it has a non-zero
charge.
•Formal charge is the net charge of an ion or ionized group.
•Partial charge is the charge introduced by polarization of
electron clouds.
Insight's Forcefield-->set potentials function defines the
partial charges of each atom. It requires all atoms to be present
with the right valances.
Poisson's equation
If every point has a charge density r(r), and the sum over all
points is zero (no net charge), and the energy is related to the
charge inverse-linearly, then we can state the following
relationship between the charge density and the curvature of
the electrostatic potential, Poisson's equation.
the charge
density map
2
rr
 r  
e0 
the 3D 2nd deriv.
(curvature) of...
the units
the electrostatic potential at r
the constant
dielectric
..continued
This equation allows us to express the electrostatic potential
as a function of any distribution of charges.
rr
 r  
e0 
2
"del" means (d/dx)+(d/dy)+(d/dz)
"phi" is the electrostatic potential.
DelPhi is a program that calculates the electrostatic potential map.
So.... if we know r, we can get .
The problem is....
(1) Mobile charges will redistribute themselves in the electric
field.
When an ion moves to a position of opposite electrostatic
potential, it changes the charge distribution r(r), which in turn
changes the electrostatic potential.
(2) We can't assume that the dielectric is constant.
The dielectric is the ability of the medium to disperse the
charge. Solid or semisolid objects have a much lower
dielectric than liquids, especially polar liquids like water.
Boltzmann distribution
Boltzmann states that the number density of ions in an
electric field or map is related to the electrostatic
potential as follows:
energy to bring a
charge to point r



r

number density of ions (charges)
from infinity
rr  Ie
k bT
a constant,
depending on the
units
bulk number density ≈ ionic strength
So.... if we know , we can get r.
NOTE: The Boltzmann distribution is a generalization of the equation for
equilibrium using Gibbs Free Energy: G = -RTlog(Keq)
Variable dielectric
Generally, we assume there
is a  for inside the protein
(green, =2 or 4), and a a
for outside in the solvent
(=80).
Poisson-Boltzmann equation
  r  r   r   4rr
What's this?
•It has position specific dielectric ((r)).
•It depends on the position specific charge density r(r).
•It has a -dependent correction for Boltzmann distribution,  (r)
•It expresses the relationship between the electrostatics and the
charge distribution.
•This equality must hold for any two functions r
(the charge distribution) and  (the electrostatic potential) at all
points r.
•The P-B equation has to be solved numerically.
Solving the P-B Equation
5
5
6
6
4
q0
1 0
4
3
3
1 2
2
(1) First we calculate the charge density qi based on the electrostatic potential .
For all grid points i, using neighboring boxes only.
(2) Then we calculate the electrostatic potential  based on the charge density q,
using neighboring boxes only. Repeat until converged!
Projecting the electrostatic map
onto the surface
The results of PoissonBoltzmann is a grid of the
electrostatic potential  at
each point in a box.
Points on the surface
though this grid is assigned
the nearest  value
(interpolated).
This is mapped to a color
from display.
Projecting the electrostatic map
onto the surface
Active sites and binding
surfaces can be found by
coloring the surface by the
electrostatic potential. Here
we igore all grid points
except the ones closest to
the surface.
A ligand should have a
complementary
electrostatic surface AND a
complementary shape!
Run the MOE Electrostatics
tutorial.
Help-->Tutorials-->Implicit solvent electrostatics
Upload the final MOE file as Exercise 7.
http://www.bioinfo.rpi.edu/~bystrc/courses/biol4550/homework.html
NOTE: there is no menu item to start up the
Grid Analyzer from a saved file. You can start the
Grid Analyzer by typing
run ‘gridshow.svl’
in the command line. Or, set a function key to type
this command, using Window-->Options-->function keys