Dear Allstat,
I would like to thank everyone who replied to my query about the Hill
coefficient.
Here is a summary of the references and an explanation given by John
Wood.
Regards
Lynne Holton
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You will find it in the new Encyclopedia of Biostatistics (Ed Armitage &
Colton) - John Wiley & Sons. Volume 4 page 3363
Graham Dunn
You will find it mentioned in my book
Statistical Issues in Drug Development, Wiley, 1997, pages 296-297.
Stephen Senn
There is a discussion of the Hill equation, the Hill plot and the Hill
coefficient in the context of ligand-receptor interaction on pp 354, 355
in Models in Biology by D Brown &
P Rothery , John Wiley & Sons 1993. ISBN 0 471 93322 8.
R A Brown.
Take a look at J. Wong, Kinetics of Enzyme Mechanisms, Academic Press,
1975.
Ron Kenett
Dose-response curves of this sort are usually
represented by logistic curves:
y=a + c/(1 + exp[-n(x - m)] )
where x is ln(conc.)
Here, n is the Hill slope.
There are a number of circumstances when this
would be expected to be 1, namely when the first
step of activity is binding of the drug to a receptor (say),
the response is simply (eg linearly will do) related to
the concentration of bound receptor, and the binding step
follows the law of mass action.
This latter is the simplest model of reversible binding,
and just says that the rate of formation of bound complex
is proportional to the product of the concentrations of the free
reactants:
= (kf).[A].[R] where kf is the forward rate constant
and the rate of dissociation is first order:
= (kb).[AR]
Thus we have: d[AR]/dt = (kf).[A].[R] - (kb).[AR]
and at equilibrium: [AR] = [A].[R] / K
wher K is the ratio of the 2 rate constants and is
known as the dissociation constant.
Noting that [R] and [AR] together must add up to
the total concentration of receptor leads to a rectangular
hyperbola of [AR] as a function of [A], which is essentially
all the logistic curve with n=1 is, if we move from using
ln(conc.) to conc. on the natural scale.
John Wood
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