I am not the only one! Getting past the original GBSA

The Generalized Born Surface Area implicit solvation model is one of the most important continuum solvation models used in deciphering protein-ligand interactions in drug discovery from an energetic standpoint. For a pretty long time, I had considerable trouble, to put it mildly, understanding the original JACS paper by Clark Still. A couple of days, I decided to give up all pretext at understanding the math, and decided to think purely from a physical and logical viewpoint, which should be natural for any chemist. To my surprise, I found that it then became rather easy to understand the crux of the model.

Then yesterday, I came across notes prepared by Matt Jacobson for his class at UCSF. Here is his opening statement about the GB model:

"Introduction to GB"

• Originally developed by Clark Still.
Original paper is virtually impossible to understand;
GB approximation remains a bit mysterious.

Ha! I stand vindicated. The original paper also contains a term which Still seems to have plucked out of nowhere, and Jacobson also mentions this in his notes. Needless to say, the notes are pretty good...

Anyway, so I thought I would put down my two cents worth and qualitative understanding of the model, without mentioning any equation (although one of the equations is "just" Coulomb's law)

The basic goal is simple; to ask what is the solvation energy of an ion or solute in solution (most commonly water). This in turn related to what must be the energy needed to transfer that ion from vacuum to the solvent.
First, let's neglect the charge of the solute. Now for transferring anything into a solvent, we obviously need to create a cavity in the solvent. Thus there needs to be a term that represents the free energy of cavity formation.

Call this ∆G (cav)

Then there is also the Van der Waals interaction that the atoms of the solute will have with the solvent atoms in this cavity. This will lead to a free energy of Van der Waals interaction between the two.

Call this ∆G (vdw)

Now, intuitively, both these quantities will depend on the surface area of the solute, or more accurately, the solvent-exposed surface area. Greater the surface area, bigger the size of the cavity and more the ∆G (cav). Also, greater the surface area, more the vdw interactions with the solvent atoms, so more the ∆G (vdw).

Thus, ∆G (cav) + ∆G (vdw) are proportional to the solvent exposed area (SA).

So ∆G (cav) + ∆G (vdw) = k * SA

where k is a constant of proportionality. Of course, the term to the right will be summed up over all solvent-solute atoms (as are all the other terms). The constant k can be found out from doing a controlled experiment, that is, by transferring an uncharged solute into solvent, for example a hydrocarbon. Thus, one can experimentally determine this constant by transferring various hydrocarbons into solvents and measuring the free energy changes. So there, we have k.

Now let's turn to the other component, the interaction due to the charge. Naturally, the term that comes to mind is a Coulombic energy term, that goes as 1/r where r is the solvent atom-solute atom distance.

Call this ∆G (coul.) which is proportional to the charges, and distance between the solute and solvent particles.

But that cannot be all. If the Coulombic energy were the only term for charge interaction, then since it depends only on the distance between the two particles, it would be independent of size and provide no provision for accounting for the size of the particles. But we know that how well an ion is solvated depends crucially on its ionic radius (Li+ vs Cs+ for example), with small ions getting better solvated. So there needs to be a term that needs to take into account this ionic radius and the charge on the ion (because what really matters is the charge/size ratio). This is the Born energy term. But it also includes another term; since what we really care about is the change in the free energy, it would depend upon the dielectric constant of the solvent, with a high-dielectric like water solvating ions better.

Call this ∆G (Born), which is proportional to the ionic charge, the ionic radius, and the dielectric constant of the solvent.

Therefore, the total energy of solvation,

∆G (solv) = ∆G (cav) + ∆G (vdw) + ∆G (coul) + ∆G (Born)

with the proportionalities and parameters as enunciated above.

There it is, the GBSA model of implicit solvation, widely used in molecular mechanics and MD protocols. Similar continuum solvation models are also used in quantum mechanical models. The model has shortcomings obviously. It neglects the nuances provided by local changes in water molecule behaviour. The Born energy terms is also an approximation, as ideally it considers higher order terms and long-range solvation effects.

But given its relative crudeness, GBSA is surprisingly accurate quite often, such as in the MM-GBSA protocol of Schrodinger. Looking at the terms, it is not surprising to see why this is so, since the model does a good job of capturing the essential physics of solvation.