Virial

Coefficients

of Charged Protein Solutions

NIH, NIDDK, LBG
Travis Hoppe, Allen Minton, Di Wu
(deck source)

Main Idea

against

pH Dependence

Concentration

Charge anisotropy

intra-protein interactions

Human serum albumin, Ovalbumin, Lysozyme, Bovine Serum Albumin respectively.

How do we model the interaction between proteins on a larger scale? Can we predict aggregates?

Mixtures

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

This is usually really important.

Basic Science

Match experimental data with results
of computational models.

Classify common protein solution behavior
from coarse-grained models.

Predict behavior of mutations and unknown solutions.

Virial Coefficients

Virial Coefficients

For rotationally invariant molecules

but in general...

Virial Coefficients

Experimental measurements (light scattering) possible.

How do you model a

protein?

Need an expression for the interaction energy
This is not the free energy, but the enthalpy

Important terms:

• Volume exclusion
• Electrostatics
• Non-specific interactions (London/dispersion forces)

How do you model a

protein?

Second-order effects?

• Non spherical geometries
• Polarization
• Internal conformational energies
• Solvent effects

Must decide if this approximation is valid for the system.

modeling the

Excluded Volume

Hard spheres

modeling the

Electrostatics

Electrostatic field

Coulomb's Law (point charge)

Correction for dielectrics?

What about the solvent?

Yukawa Potential

First order approximation to screening effects.
Charge strength decays exponentially due to ions.

Poisson Boltzmann

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

Can be linearized and solved on a computer efficiently.

Splits space into regions of discrete .

The Process

Start with the crystallized PDB Structure (HSA)

Adaptive Poisson-Boltzmann Solver

Typically (in the absence of ions)
,

The Process

Electrostatic field

The Process

Determine a region of excluded volume.

The Process

Spherical Harmonic decomposition for large distances.

The Process

Best fit macrocharges to replicate the field.

This works

J. Chem. Phys. 2013

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.

So Far...

Excluded volume for modeled as a hard sphere.

Representation of the protein electrostatics
in an ionic solution for a given pH.

Remember this?

Our charge distributions are not isotropic anymore,
we must to compute this:

Sampling woes

There are many pairwise orientations.
Blind sampling may miss specific interactions.
Need to know at different if we want to scale the model.

Density of states

counts the number of ways
we can get a particular energy,

Calculate the non-ideality of a protein molecule after including both the excluded volume and electrostatics.

Predict the second-virial coefficient as a function of pH values, protein concentrations, binary mixtures, and salt concentrations.

If experimental results agree, use the model in higher-order simulations to predict phase behavior via Gibb's ensembles.

Results for Ovalbumin

Future work

results for square-well potentials shown

Thanks, you.

How does it work?

A text-based human-readable markup.
SVG equations , and links!
The code for this particular slide looks like this:

## How does it work?

A *text-based* human-readable markup. 
Equation rendering is simple $e^{i \pi} = -1$.
and [links](http://thoppe.github.io/)!
The code for this _particular slide_ looks like this: