Neuropharmacology: the Hill-Langmuir Equation (Dr. Moss) Flashcards
Consider the simple model A + R = AR.
What is k+1 ?
What is k-1 ?
Both are rate constants (k+1 in M-1.s-1 and k-1 in in s-1)
k+1 [A][R] = rate at which the AR complex is formed = association rate constant
k-1 [AR] = rate at which the AR complex is broken down into A and R = dissociation rate constant
What does the law of mass action state ?
That the rate of a reaction is proportional to the concentration of its reactants.
What is KA ?
KA = k-1/k+1 = dissociation equilibrium constant
What is the Hill-Langmuir (H-L) equation ?
pAR = [A] / KA + [A]
Who was A.V. Hill ?
Archibald Vivian Hill was an English physiologist, one of the founders of the diverse disciplines of biophysics and operations research.
What was A.V. Hill’s contribution to the Hill-Langmuir equation innovatory ?
A.V. Hill (26 Sept, 1886 - 3 June, 1977) was the first to apply the law of mass action to the relationship between drug concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached.
Who was I. Langmuir ?
Irving Langmuir (Janu 31, 1881 – Aug 16, 1957) was an American chemist and physicist. His most noted publication was the famous 1919 article “The Arrangement of Electrons in Atoms and Molecules” in which, building on Gilbert N. Lewis’s cubical atom theory and Walther Kossel’s chemical bonding theory, he outlined his “concentric theory of atomic structure”.
What was I. Langmuir’s contribution to the Hill-Langmuir equation ?
The physical chemist I. Langmuir showed a few years later than Hill that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption (the adhesion of atoms, ions, or molecules from a gas, liquid, or dissolved solid to a surface) of gases to the surface of a metal.
Why is it unnecessary, when deriving the H-L equation, to make a distinction between the free concentration of agonist [A] and the total concentration of agonist [A]T ?
Because we only apply this equation in cases where [A] does not changes considerable when [AR] is formed. In effect, the drug is considered to be present in such excess that it is scarcely depleted by combination of a little of it with the receptors. Thus [A]T ≈ [A].
What is the relationship between receptor occupancy and drug concentration [A] on a linear scale ?
On a log scale ?
Linear scale = rectangular hyperbola
Log scale = sigmoidal (elongated “S” shape)
What happens if KA = [A] ?
pAR = 0.5
Thus, if KA = [A], half the receptors are occupied
Consider the equation:
log(pAR/1-pAR) = log[A] - logKA
What do you get if you plot log[pAR/(1 - pAR)] against log[A] ?
Whe should get a straight line of unity slope
(y-intercept = -logKA), described as a Hill plot, again after A.V. Hill who was the first to employ it, and is often used when pAR is measured directly with a radiolabelled ligand. In practice, the slope of the line is not always unity and is referred to as the Hill coefficient (nH) or Hill slope.
What oversimplified assumption did A.J. Clark make when considering the relationship between receptor occupancy and tissue response ?
How can this be modelled mathematically ?
Clark made the tentative assumption that the relationship might be one of direct proportionality (though he was well aware that this was almost certainly an oversimplification, as we now know it often is):
γ/γ(max) = pAR
Consider the equation:
log(γ/γ(max)-γ) = log[A] - logKA where we set γ(max) = 100
What do you get if you plot log(γ/100-γ) against log[A] ?
A.J. Clark was the first to test this using the responses of isolated tissues, and this equation provides a good fit to the experimental values, w/ the slopes of the Hill plots close to unity. However, later studies with a wide range of tissues have shown that some concentration-response relationships cannot be fitted by equation this equation. In particular, the Hill coefficient is generally, if not always, greater than unity for responses mediated by ligand-gated ion channels
Give another reason why there is not a relation of direct proportionality between receptor occupancy ans tissue response.
It is now known that in many tissues the maximal response (e.g. contraction of intestinal smooth muscle) may occur when an agonist such as ACh occupies less then 1/10th of the available receptors. By the same token, when an agonist is applied at the concentration needed to give a half maximal response (termed the [A]50 or EC50), receptor occupancy may be as little as 1% in some tissues, rather than the 50% to be expected were the response to be directly proportional to occupancy.
