Hello,
I'll do what I can to try and explain what's going on here.
The free energy of solvation is the difference in the energy that is
required to immerse a substance in a solvent of a given dielectric constant.
Because the free energy is a state function, we can construct any
thermodynamic pathway we want to get from our substance in air (or vacuum,
with dielectric constant = 1) to solution (in solvent, with dielectric
constant = X, which is ~80 for water). The way that's typically chosen is
to create a given charge distribution (in the case of MM, that's the
arrangement of point charges in a given conformation of your system) in the
electric potential that's present in both environments (air and water, for
instance). The difference between these 2 potentials is often referred to
as the reaction field potential. Once you have the reaction field
potential, you can find the free energy of solvation via the equation:
Delta_G = 1/2 Integral( rho(X) * phi(X) dX)
[[ This is the 'magical equation' you were looking for. It comes from basic
electrostatics, which is probably why it isn't mentioned often -- Energy =
potential * charge summed over a continuous distribution, at its most basic.
It's certainly easy to get lost here when all you're doing is looking
through literature that assumes this is common knowledge and the buck stops
with calculating the reaction field ;) ]]
Here, rho(X) is your charge distribution (just point charges here) and
phi(X) is this reaction field potential I described above, dependent on
atomic positions and integrated over all space. (The 1/2 is to account for
double-counting in the double sum/integral). Since we know the charge
distribution, the problem now becomes that we need to solve for the reaction
field potential, where we solve for the electrostatic potential in both
environments. The PBE is the equation that we use to solve for this
potential. As you've pointed out, the electrostatic potential itself
depends on the charge distribution (which doesn't change in our MM framework
since everything's a fixed point charge; ignoring polarizable force fields,
but will affect the electron density in a QM framework if taken into account
properly). Therein lies the difficulty (solving the PBE). The generalized
Born approximation is an approximation based on a "trivial" solution to the
PBE that attempts to provide an efficient, analytic way of calculating the
solvation free energy. Good references I think are Chris Cramer's book and
Andrew Leach's book, as this is only a very, very broad overview.
The problem with just adding up pairwise potentials, oversimplified, is that
it neglects any screening effects caused by a dielectric solvent. Because
solvent partial charges can react to solute partial charges, electrostatic
effects are screened compared to what they'd be in vacuum (this is the
effect of a dielectric).
Someone else may be able to explain this a little more succinctly, but I
would suggest trying to work through the texts that I mentioned (the author
itself should be enough to find them).
HTH,
Jason
On Wed, Aug 10, 2011 at 7:58 AM, Jan-Philip Gehrcke <jgehrcke.googlemail.com
> wrote:
> Hello,
>
> I am new to the field and after reading quite some literature about
> MMPBSA, I am still wondering about the specific step of calculating the
> coulomb energy of the solvent within the electrostatic potential PHI(r)
> caused by the charge distribution of the solute rho(r).
>
> I understand that by solving the PBE with rho(r) as input, we get
> PHI(r). Now, we want to know (at least in my picture of the things) the
> sum of the electrostatic energies of all solvent molecules placed into
> this potential. Is that correct so far? This would be some
> method/function taking PHI(r) and solvent properties as input and
> providing the energy as output.
>
> As implicit solvent models are used, I doubt that we simply sum up the
> coulomb energy of the partial charges of all solvent molecules within
> PHI(r) (what would be wrong with this approach, by the way?).
>
> This step was left unexplained in each paper I've read. If you have a
> good reference for this, I would be very happy to read it. And if it is
> so obvious that it is not worth explaining it in one of the basic
> papers, then I would be happy to know what the obvious is :)
>
> I think "the secret" is behind this sentence of the AmberTools manual:
>
> "The charging free energy is a function of the electrostatic potential
> PHI".
>
> It suggests that the problem is solved from the perspective of
> calculating the system's free energy change while "charging" the solute,
> but it also does not explain the details and does not give a reference.
>
> Hope that someone can help me out here.. thanks a lot!
>
> Jan-Philip
>
>
> --
> Jan-Philip Gehrcke
> PhD student
> Structural Bioinformatics Group
>
> Technische Universität Dresden
> Biotechnology Center
> Tatzberg 47/49
> 01307 Dresden, Germany
>
>
>
>
> _______________________________________________
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> AMBER.ambermd.org
> http://lists.ambermd.org/mailman/listinfo/amber
>
--
Jason M. Swails
Quantum Theory Project,
University of Florida
Ph.D. Candidate
352-392-4032
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Received on Wed Aug 10 2011 - 16:00:02 PDT
