Bioinformatics 2 -- lecture 17

Molecular surfaces

Electrostatic maps

The Hydrophobic Effect

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 surfaceThe 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 modelthe free energy of solvation.

Surfaces have:•size/area•electrostatic properties•shape properties

protein

water

M + nH2O M.nH2OM in a vacuum M in water

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 inthe triangle has a different brightness,depending on where it is.

Surfaces maybe described as a set of connected triangles

Phong-shaded cow. Shading give theillusion of higher resolution.

A cow-shaped manifold made of triangles.

The Connolly surfaceConceptually, 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

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.

rolling probe sphere

Coloring by atom, by shape

or by shape. Yellow parts are ‘reentrant’.

Surfaces maybe shaded by partial charge.

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

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.

no, not this kind...

If a space is not big enough for solvent, it is a void.

If we compare two surface-surface 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

higher energy

Distance profile of the void volumeWhat happens to the surface and volume as two hydrophobic atoms come together??

Distance profile of the void volumeWhite area is excluded volume. Water can't go closer thanone radius (1.4Å).

SES

Distance profile of the void volumeFirst excluded surface (green) is approx compensated by new"toroidal" surface (brown).

Distance profile of the void volumeToroidal surface grows faster than atom surface is lost. SES goes up.

Distance profile of the void volumeAt short D, toroidal surface stops growing, atom surface shrinks faster.

Is there a barrier to hydrophobic collapse?

Distance

SES

When hydration shells first touch, surface area goes up, so solvation free energy goes up, but volume goes down.

black: G calculations using explicit watersblue: SESgreen: excluded volume

Vol

G

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.

Amino Acid Hydrophobicity Scales

Low entropy waters fill reentrant holes on the surface.

Electrostatic potential maps/surfaces

Coulomb's Law

E=qiqjεrij

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 byrotating 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=qiqjεrij

(must be ≥ 1.000)

Electric field for one chargeAn 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).

++ +

membrane

water

(a) (b)(c)

isopotential lines

Coulomb's law applies Coulomb's law doesn't apply

Electrostatic map for more than one charge.

-

+

-

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.

E(x) =qiεrix

∑

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 equationIf every point has a charge density (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.

∇2φ r( ) =−ρ r( )e0ε

the charge density map

the constant dielectric

the unitsthe 3D 2nd deriv. (curvature) of...

the electrostatic potential at r

..continuedThis equation allows us to express the electrostatic potential as a function of any distribution of charges.

∇2φ r( ) =−ρ r( )e0ε

"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 , we can get .

The problem is....

(1) Mobile charges will redistribute themselves in the electric field.

(2) We can't assume that the dielectric is constant.

When an ion moves to a position of opposite electrostatic potential, it changes the charge distribution (r), which in turn changes the electrostatic potential.

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 distributionBoltzmann states that the number density of ions in an electric field or map is related to the electrostatic potential as follows:

ρ r( ) =Ie−φ r( )kbT

number density of ions (charges)

bulk number density ≈ ionic strength

energy to bring a charge to point r from infinity

a constant, depending on the units

So.... if we know , we can get .

NOTE: The Boltzmann distribution is a generalization of the equation for equilibrium using Gibbs Free Energy: G = -RTlog(Keq)

Variable dielectricGenerally, 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

∇ ⋅ε r( )∇φ r( ) −κφ r( )=−4πρ r( )What's this? •It has position specific dielectric ((r)).•It depends on the position specific charge density (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 (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

0

3

5

2

6

1

4

1

2

3

5

4

6

(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!

q0

Projecting the electrostatic map onto the surface

The results of Poisson-Boltzmann 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 theGrid Analyzer from a saved file. You can start theGrid Analyzer by typingrun ‘gridshow.svl’in the command line. Or, set a function key to typethis command, using Window-->Options-->function keys