Help with Hill Equn. needed

Giovanni Maga maga at
Tue Nov 7 06:59:27 EST 1995

In article <tday-0611950901490001 at>, tday at
(Tony Day) wrote:

> Unfortunately, this protein unfolds irreversibly, and has several
> unfolding transitions visible in DSC.  So it is also possible that it
> binds to a folded intermediate (or the fully folded form) and destabilises
> that, rather than stabilising the unfolded form.  This still begs the
> question, if the binding were non-specific, wouldn't the data be a poor
> fit to a n=1 binding equn.?
> Tony

No, if even multiple (i.e non-specific) binding events are independent and
equivalent to each other. The Hill's equation apply to the case when, in
the presence of multiple binding sites, the affinity of an empty site for
the incoming ligand is modified (either increased or reduced) when another
site is already occupied. If you fit the equation to a nH=1 it means that
either you have a single site or all the binding events are equivalent (no
cooperativity). Indeed, with nH=1 the Hill equation reduces to a simple
Michelis-Menten equation. Anyway, even if you do not have cooperativity,
but simply nonequivalent binding sites, the Hill equation should give you
a negative cooperativity (nH<1). Usually, the classical approach is to
test before if you have apparent cooperativity and then use the Hill
equation to determine the nH. One way to do that is to determine your
inactivation rates at different ligand concentration, then plot it as
Eadie-Hofstee plot. If you have linear fitting, cooperativity is unlikely.
If you have non-linear fitting, you can test cooperativity. However,
non-linearity can arise from different reasons, but linearity means that
your ligand binds a single site or that there are multiple sites, but they
are equivalent and independent (inependent but non-equivalent binding
sites should give negative cooperativity). If I am correct, you incubate
the protein at fixed T, fixed time with increasing amount of ligand, then
use an activity assay as a measure of inactivation. Thus (excluding
interferences of your ligand in the activity assay), if you have linear
rate change dependence from the ligand concentration, that does not allow
you to conclude that there is a single binding event for each protein
molecule, but simply that there is no cooperativity in binding.
There is anyway a further complication. Since you are usign an indirect
assay, the inactivation (decrease in rate of the reaction) that you
observe is due to the decrease of the active fraction of the enzyme that
partecipates in the reaction. Thus, the linearity of your inactivation
rate would refer to the rate of decrease of your active enzyme
concentration, but can you relate it directly to the rate of binding of
your ligand? What I mean is that, if your enzyme is inactive just as a
first unfolded intermediate, it can still bind other ligand molecules that
will affect the further unfolding (accelerating it probably), without
anyway affecting the enzymatic activity (that is already impaired). Thus,
all you can say from these experiments is that the rate of ligand binding
that leads to the enzyme inactivation is linear, but this does not mean
that your inactive protein does not bind still other ligand molecules.
Since you are interested in measuring the rate of unfolded intermediates
formation in the presence of your ligand, you should be able to compare
the relative activities of the different unfolded intermediates (maybe
performing reactions at different temperatures) and look if all the
intermediates forms at the same rate in the presence or in the absence of
your ligand molecule. Since you can detect the different unfolded
intermediates directly by DSC, you can try to see if at any given
temperature the kinetic of accumulation of these intermediates is the same
with or without the ligand.
In order to verify that you have a 1:1 ratio of ligand-protein, you have
to determine the actual concentration of both in your assay and then to
correct it for the fraction of both that truly partecipate in the
You can try to isolate the protein and the protein/ligand complex by gel
filtration to see if it forms aggregates and from the MW estimate the
ratio ligand/protein. Indeed, it can be that your ligand acts as an
anti-chaperon, binding unfolded intermediates and stabilizing them. Did
you check the effect of temperature on the binding affinity? (i.e.
partially unfolding the protein and then adding the ligand to see if any
of the intermediates you see has higher affinity?).
I apologize for this very long and maybe not clear letter. Good Luck.
G.Maga, PhD
Institute of Veterinary Biochemistry
University of Zuerich (CH)

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