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

Xavier Periole x.periole at rug.nl
Fri Mar 24 13:32:08 CET 2006

```Rob Carlson wrote:

> 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.
>
You are about right but here Z would be the Partition function. The
following simplification that would be really too
easy if it was thrue. We all would be doing very different things. The
idea is correct expect that you have to have the
total partition function to be able to do so and the complete partition
function implies that you sample the full
conformational space !!
In principle if you know the proportion of the different states you
could weight them ubt in practice this will not
give a realistic result, or may be you are super lucky.

I am not a specialist of this so you could llok up some books or paper
that would talk about binding affinity
and molecular dynamics. Alan Mark, Wvan Gunsteren have done a lot of
work on this subject.

> 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.

Best continuation
XAvier

--
----------------------------------
Xavier Periole - Ph.D.

Dept. of Biophysical Chemistry / MD Group
Univ. of Groningen
Nijenborgh 4
9747 AG Groningen
The Netherlands

Tel: +31-503634329
Fax: +31-503634398
email: x.periole at rug.nl
web-page: http://md.chem.rug.nl/~periole
----------------------------------

```