Virial

Coefficients

of Charged Protein Solutions




Lab Meeting
NIH, NIDDK, LBG
Travis Hoppe, Allen Minton, Di Wu

Main Idea

Predict from structural information


against


pH Dependence

Concentration

Charge anisotropy

Relevance?


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

An equation of state expanded in powers of density



is the pairwise interaction of two molecules
is the pairwise interaction of three molecules
...


From the equation of state you can calculate most thermodynamic properties!

Virial Coefficients

Why work with this expansion?

We can measure them!

usually with light scattering

Virial Coefficients


For rotationally invariant molecules


but in general...

How do you model a

protein?



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


Important terms:

  1. Volume exclusion
  2. Electrostatics
  3. Non-specific interactions (London/dispersion forces)

How do you model a

protein?


What are we ignoring?


  1. Non spherical geometries
  2. Polarization
  3. Internal conformational energies
  4. Solvent effects


Must decide if this approximation is valid for the system.

modeling the

Excluded Volume

(it's easy!)

Hard spheres

overlap energy is either or .




modeling the

Electrostatics

(not so easy)

Electrostatic field

Coulomb's Law (point charge)


Correction for dielectrics?


What to do with 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


Crystallized PDB Structure

Adaptive Poisson-Boltzmann Solver



Typically (in absence of ions)

The Process


Electrostatic field

The Process


Excluded volume

The Process


Spherical Harmonic decomposition

The Process



Best fit charges

The Process


Monte-Carlo Simulation

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.

What's next?

(work in progress)

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,


If the sampling works, then we should be able to calculate the non-ideality of a protein molecule after subtracting both the excluded volume and the electrostatics portions.


This would allow us to predict phase behaviors in at different pH values, protein concentrations, binary mixtures, and salt concentrations.

That would be really neat.

Thanks, you.

How were these slides made?


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## What does Markdown look like?

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