# [gmx-users] Questions on Binding Free Energy Calculations

Rob Carlson icanthearanything at hotmail.com
Fri Mar 24 02:00:10 CET 2006

```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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