A Simplified Representation of

Anisotropic Charge Distributions



Drexel Physics Colloquium

Travis Hoppe , Allen Minton

NIH, NIDDK, LBG

Main Idea

Capture structural features of protein solutions.


Phase Changes &

Virial Coefficients

via

pH Dependence

Concentration Dependence

Prior Work


Coarse-grained models:

Hard spheres, point charges/dipoles, LJ liquids


Fine-grained models:

All-atom molecular dynamics, quantum ab-initio


Liquid theory

Ornstein-Zernike equation, Percus-Yevick closures...


Anisotropy


Directional (angular) dependence


eg. polarizations, nematic phases, chirality, ...

The Process

Crystallized PDB Structure

The Process

Electrostatic field

The Process

Spherical Harmonic decomposition

The Process

Best fit charges

The Process

Monte-Carlo Simulation

Electrostatic field

Coulomb's Law (point charge)

\phi = \frac{1}{4 \pi \epsilon_0}\frac{q}{r}


Correction for dielectrics?

\phi = \frac{1}{4 \pi \epsilon_0 \epsilon_r }\frac{q}{r}


What to do with the solvent?

Yukawa Potential


\phi = \frac{1}{4 \pi \epsilon_0 \epsilon_r }\frac{q}{r} e^{-\kappa r}


First order approximation to screening effects.


Charge strength decays exponentially due to ions.

Poisson Boltzmann


\vec{\nabla}\cdot\left[\epsilon(\vec{r})\vec{\nabla}\phi(\vec{r})\right] = -\rho^{f}(\vec{r}) - \sum_{i}c_{i}^{\infty}z_{i} q \lambda(\vec{r}) \exp \left[{\frac{-z_{i}q\phi(\vec{r})}{k_B T}}\right]

Describes the electrostatic interaction between a charge distribution and an ionic solution.


Assumes ions are Boltzmann-distributed in the solution.


Can be linearized and solved on a computer efficiently.


Splits space into regions of discrete \epsilon_r .

Adaptive Poisson-Boltzmann Solver

Typically (in absence of ions)

\epsilon_{\text{water}} = 80, \epsilon_{\text{protein}} = 4

Simplify!


Build a basis set


Represent the data with a set of operators with the same boundary conditions. In effect, reduce millions of data points to dozens .

1D: Harmonics

1D: Bessel Functions

Can be made into 2D and 3D

3D: Spherical Harmonics

3D: Spherical Harmonics

3D: Spherical Harmonics

Approximate \phi as a function of the spherical harmonics...


+
+
+
+
+ ...

Accuracy Test


Define an error between the target potential \Psi

and the generated potential \phi


\chi = \frac{\int |\psi - \phi|^2}{\int{|\psi|}^2}

Does the Spherical Harmonics work?

not really...

Why do the Spherical Harmonics fail?


They are expansions of a field \phi \approx 1/r


\phi = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell \alpha_{\ell m} Y_{\ell,m} (\theta, \phi)


When we consider the exponential decay \phi \approx e^{-\kappa r}/r


\phi = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell \alpha_{\ell m} k_\ell(\kappa r) Y_{\ell,m} (\theta, \phi)


We need to use Bessel functions

Ovalbumin, I=0.15M

Lysozyme, I=0.15M

Human Serum Albumin, I=0.15M

Ovalbumin, I=0.30M

Results degrade with increasing ionic strength

Ovalbumin, I=0.45M

Results degrade with increasing ionic strength

Linearization e \phi \ll kT , breaks down

Protein Caricatures


Good approximation of the near field, poor up close


Captures the anisotropic field
especially near the isoelectric point


Macrocharge approximations make for reasonable
models of large protein solutions

Charge fitting


Fit discrete macrocharges to match harmonics


Potential is linear in charge magnitude -
only coordinates need to be fit

Why not use multipoles directly?


Extremely complicated for anything > quadrupole


Macrocharges mapped to protein coordinates,

useful in it's own right?

Fitting Problems


Charges must be constrained

Outside the expansion, the fits are not valid


Creation of artificial dipoles

Unstable w.r.t. fits, constant q d results in extreme magnitude fluctuations


Effective charge error, Lysozyme

\alpha, \beta are unit normalized coefficients for APBS and effective charges

Effective charge error

Clockwise from top: Ovalbumin, Lysozyme, Human Serum Albumin
Clockwise from top: Ovalbumin, Lysozyme, Human Serum Albumin

How many charges needed?

Logarithmic dependence observed
Logarithmic dependence observed

Human Serum Albumin

Human Serum Albumin

Lysozyme

Lysozyme

Ovalbumin

Ovalbumin


So Far...


Built a coarse-grained representation of a protein in an ionic solution for a given pH.


What's next?

Interaction Energy

Second Virial Coeff, B_{22}

B_{22} \approx -2\pi \int \left ( \left < e^{-\Phi_{12}/kT} \right > - 1 \right ) r^2 dr

Predictions

Phase separations

Leibler (2004)

Phase separations lead to sudden fundamental changes in liquid structure and local density


This is usually really important


Predictions

Radial distributions & B2

Basic Science


Match up experiential data with results of computational models


Classify common protein solution behavior from macrocharges or multipoles


Extrapolate to mutations and other unknown proteins


Thank you

How were these slides made?


Math Rendering: \text{\LaTeX}


JavaScript: reveal.js


Custom Markdown: Travis Hoppe

What does Markdown look like?

A text-based human-readable markup.

Equation rendering is simple e^{i \pi} = -1 .

The code for this particular slide looks like this:


## What does Markdown look like?
A **text-based** human-readable markup. 
Equation rendering is simple $e^{i \pi} = -1$.
The code for this *particular slide* looks like this: