[gmx-users] Questions on Binding Free Energy Calculations

[Rob Carlson]

An Appeal to Gromacs Users,

I have been learning computer modeling for a while, on my own, and without 
any formal training. Having said that, I am a little confused about some 
concepts concerning free energies of binding. Therefore, I would like to 
propose a scenario and hopefully you can tell me if I am right, completely 
wrong, or on the proper track. Any response is sincerely appreciated. The 
scenario is as follows:

Suppose we have a ligand and a receptor. The ligand has 4 conformations in 
solution. The receptor has 3 conformations in solution. The complex 
(receptor+ligand) has 2 conformations in solution.

I will now define some terms. The binding free energy is G(bind). The 
complex free energy is G(complex). The ligand free energy is G(ligand). The 
receptor free energy is G(receptor).

Thus, G(bind) = G(complex) - G(receptor) - G(ligand)

To calculate G(ligand), I will use the following equation: G(ligand) = -RT 
ln Z(ligand), where Z(ligand) is the configuration integral of the ligand.

Now here is where I probably start misunderstanding everything.

In theory Z = integral over "r" of [ integral over "r" of ( EXP(-E(r)/RT) ] 
........ where E(r) is the internal energy (often called U) of the 
considered species (ligand, receptor, or complex) plus the solvation energy 
(unless explicit water molecules are used). "r" is the atomic coordinates of 
the species under consideration.

So, in my case, since I have 4, 3, and 2 conformations for the ligand, 
receptor, and complex respectively, I am thinking that the above equation 
can be simplified.

For the case of the ligand, Z(ligand) = EXP( -E(r1) / RT )  +  EXP( -E(r2) / 
RT )  +  EXP( -E(r2) / RT )  +  EXP( -E(r2) / RT ), where r1 to r4 are the 
coordinates of the 4 ligand conformations. In practice, this means that for 
each of the 4 ligand conformations I will run a 0 step dynamics simulation 
at a desired temperature to obtain the energy E of each conformation. The E 
is plugged back into that equation to get Z(ligand). The reason I run a 0 
step dynamics is that I don't want the conformation to change and I need to 
get the kinetic and potential energies. A minimization would only give me 
potential energies. Solvation energies I can get from GB or PB. If I use 
explicit water molecules, however, I don't need to do this because the 
energy E obtained from the 0 step dynamics will contain the contributions 
from the explicit waters.

I will then perform similar calculations to get Z(complex) and Z(receptor). 
Once I get all the Z values I can get the individual G values and then the 
G(bind) from the equations above.

So, hopefully my understanding is sound. But as I mentioned above, any 
comments or criticisms is appreciated, as is any recommendations for books, 
publications, or webpages that can help me better understand these concepts. 
Thank you.

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