Re: [AMBER] Hill coef in constant ph calculation fit

From: Jason Swails <>
Date: Tue, 19 Nov 2013 13:16:52 -0500

On Tue, 2013-11-19 at 15:23 -0200, Fabrício Bracht wrote:
> Hello. I think I need some help to better understand what the Hill
> coefficient means when I calculate the pKa using the constant pH module of
> amber.
> f(x) = 1 / (1 + 10**(n*(pka -x)))
> I guess that "traditionally" the Hill coef "n" describes the cooperativity
> of the binding. But I don't see where this would lead me when describing
> protonation states. Actually, I am not sure if it would mean anything
> physically significant. For instance, I have, for 2 of the three ASP
> residues I used for the calculation, "n" < 1, and for a third I have "n" >
> 1 . I apologize if this is the kind of question that the answer lies right
> in front of me, but I am to slow to get it.

The Hill equation was actually originally used to describe cooperativity
in ligand binding. Think of the 'ligand' in this case as the titrating

When the Hill coefficient is 1, that means each residue obeys the normal
Henderson-Hasselbalch titration behavior, so it is clearly not impacted
by the titration of other titrating residues (the protein environment
exerts a 'mean-field' effect on the titrating residue that does not
change as the pH---and therefore the protonation states of the other

When the Hill coefficient is greater than 1, this indicates cooperative
binding of protons. That is, as other protons bind (elsewhere, on other
titrating residues), the affinity for the proton you're looking at goes
up. Therefore, as you decrease the pH, the protonation fraction
increases rapidly since other protons are appearing elsewhere which
helps the proton you're looking at bind more easily. (Note, an
increasing protonation fraction implies a decreasing deprotonation
fraction, which is the f(x) in your equation above). Likewise, as the
pH increases, the rate at which your proton dissociates increases with
respect to non-cooperative binding since other protons are coming off

When the Hill coefficient is less than 1, this indicates
anti-cooperativity. That is, the proton you're looking at is _more_
likely to bind when another proton dissociates. You can use the same
line of reasoning I described above for cooperativity to explain why the
curve in this case is shallower than a non-cooperative binding event.

As a disclaimer here, this description applies to a _real_ Hill
coefficient that is not 1. The Hill coefficient being significantly
deviated from 1 could indicate that your CpHMD simulation is not
converged at all (or any) of the pHs you simulated, or it could indicate

I discussed this to some extent in one of my papers
[] where I showed instances
where a Hill coefficient != 1 indicates poor sampling and another where
the Hill coefficient indicates binding cooperativity (or rather,
anticooperativity as would make sense in a direct Asp -- Glu

Hope this helps,

Jason M. Swails
Rutgers University
Postdoctoral Researcher
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Received on Tue Nov 19 2013 - 10:30:02 PST
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