Big thanks for your response, Jason! Comments below.
On 08/11/2011 12:55 AM, Jason Swails wrote:
> 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.
Thanks, let me summarize:
1) Calculate phi_vac(x) by solving PBE with rho(x) and epsilon_vac
2) Calculate phi_sol(x) by solving PBE with rho(x) and epsilon_sol
3) Take the difference of phi_vac(x) and phi_sol(x) to get phi_reaction(x).
4) Calculate the energy E of rho(x) in the reaction field potential.
E is then considered to be the electrostatic contribution to solvation
free energy.
Now, I still have to think/read about what this reaction field potential
means and why E is the energy we are looking for. It's still a bit
paradox for me that rho(x) is placed into a potential that was
calculated by using rho(x).
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).
While adding up pairwise coulomb energies, screening effects could be
included by using a (maybe distance dependent) dielectric constant,
right? So that does not convince me :-) Any other important reason, why
the result of that sum is not what we are looking for?
>
> 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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>
>
>
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Received on Thu Aug 11 2011 - 02:00:02 PDT
