page 1 2011JH Steinbach - 2011A brief introduction to the Hill equation and kinetic models.I. The concentration-effect relationshipWe often analyze a "dose-response" or "concentration-effect" relationship. This relationship describes how much effect (receptor activation, for example) is produced by various concentrations of a drug. Steady-stateSteady-state conditions mean that the system is not changing with time. Of course, in biological studiesa true (infinite) steady-state will not be present. Indeed, often it is the case that the system is far from steady-state, perhaps over a limited time it is "pseudo-steady-state." However, we will be considering steady-state responses.The basic parameter for a dose-response curve is the fraction of the maximal activation which is produced. The fraction can be expressed in two basic forms. Experimentally it is usually the response relative to the largest response which can be measured. Theoretically, however, it is possible to calculate the fraction of the maximal response which would be produced if all the receptors were fully activated. The difference between these two fractions will be clear later on.II. Kinetic modelsA kinetic model is a short-hand description of a particular idea we might have for how a receptor works. It shows "states" of the receptor (for example, a resting receptor with no ligands bound, or a receptor with 2 ligands bound and an open channel). The states are connected by arrows that show what can happen (a ligand can bind or unbind, a channel can open or close). The states show what we are allowing the receptor to do, and the arrows show how the states are connected. II.A. A single site occupancy modelThis is the simplest type of model: binding directly results in gating. The simplest form is to say that a receptor (R) has a single site to bind an activator (A), and when the site is occupied the receptor is active (R*). κ1[A][R]A+R⇄AR*λ1[AR*]The association rate constant is κ1 and the dissociation rate constant λ1. R receptor with no drug boundAR* receptor with drug bound (defined as active for this model)Rtot total receptor, Rtot = R + AR*A free drugAR* drug bound to receptorAtot total drug, Atot = A + AR*κ1[A][R] association rate; units s-1λ1[AR*] dissociation rate; units s-1κ1association rate constant; units M-2s-1λ1 dissociation rate; units M-1s-1K1dissociation constant, K1 = λ1/κ1 (units M)page 2 2011To calculate the steady-state concentration-response relationship for this model, we need to determine the fraction of the total receptors (R + AR*) that is active (AR*). At steady state, there is no change in AR* with time - the rate for entering AR* is the same as for leaving:λ1[AR*] = κ1[A][R] = κ1([Atot] - [AR*])([Rtot] - [AR*])In most experimental conditions there is a vast excess of free activator (that is [A] >>> [AR*]) available for reaction, and so we assume that [A] is constant at Atot. (On some occasions this might not be true. However, we will use this assumption for the rest of this handout.)λ1[AR*] = κ1[A] ([Rtot] - [AR*])[AR*]/[Rtot] = κ1[A]]/λ1 / {1+ κ1[A]/λ1}This is the Michaelis-Menten or Langmuir equation. The dissociation constant (K1 = λ1/κ1) is usually used so it looks like[AR*]/[Rtot] = [A]/K1 / {1+ [A]/K1}This is not a very useful equation for the receptors we study, but essential as the first step. II.B. A two site occupancy modelIn words, this model is:- there are 2 binding sites for a drug on each receptor- when a receptor has both sites occupied by drug the receptor is activated- when the receptor is activated we see a response.This is the simplest model for "cooperative" activation (a "two site occupancy model") in which an occupied receptor is active.The model looks like:κ1[A][R] κ2[A][AR]A+R⇄AR + A⇄AR*Aλ1[AR] λ2[AR*A]The new rate constants are κ2 and λ2, giving dissociation constant K2..The occupancies of the various states are:[AR] = [R][A]/K1[AR*A] = [AR][A]/K2 = [R][A]/K1 x [A]/K2soAR*A/Rtot = {[A]/K1 x [A]/K2 } / {1 + [A]/K1 + [A]/K2 + [A]/K1 x [A]/K2} = {[A]2/(K1 K2 )}/ {1 + [A]/(K1 + K2) + [A]2/(K1 K2 )}Numbers of binding site - A brief digressionThe number of bound agonists necessary to produce an activated receptor determines the steepness of the concentration-response curve. This is illustrated for a multi-site occupancy model in Fig. II-1. Only receptors with all sites occupied are active.page 3 2011II.C. Binding/gating modelsA wealth of experiments has shown that occupancy models only are accurate for competitive antagonists (drugs which act by binding to the binding site but having no other effect). An occupancy model is not able to explain the actions of agonists (drugs which bind to the receptor and then activate it). It is necessary to separate the binding step from the activation step. del Castillo and Katz proposed the first version of this type of model (del Castillo and Katz. Proc R Soc Lond B Biol Sci. 1957;146:369-81.).In words, a binding/gating model might be:- there are 2 binding sites for a drug on each receptor- when a receptor has both sites occupied by drug then it can become activated- a doubly-occupied receptor activates and inactivates- when the receptor is activated we see a response.κ1[A][R] κ2[A][AR] βA+R⇄AR + A⇄ARA⇄AR*Aλ1[AR] λ2[AR*A] αThe new rate constants are the "activation" rate constant, β and "inactivation" rate constant, α. The equilibrium opening constant is W2 = β/α (W2 because two molecules of A are bound).[AR] = [R] [A]/K1 + [R] [A]/K2[ARA] = [R] [A]/K1 x [A]/K2 [AR*A] = [R] [A]/K1 x [A]/K2 x W2soAR*A/Rtot = {[A]/K1x[A]/K2xW2} / {1 + [A]/K1 + [A]/K2 + [A]/K1x[A]/K2 + [A]/K1x[A]/K2xW2}N site occupancy modelnA+R <-> ... <-> AnR*[drug]101102103AnR/Rtot0.00.51.0n=1n=2n=3n=5n=50N site occupancy modelnA+R <-> ... <-> AnR*n[drug]101102103AnR/Rtot0.00010.0010.010.11n=1n=2n=3n=5n=50Fig. II-1. Increasing numbers of sites. In the left panel the fraction of receptors that are active is plotted against the drug concentration for a number of values for the number of binding site. Remember that only receptors with ALL sites occupied are active! The foot of the curve gets progressively more curved as n increases. On the log-log plot (right panel), it can be seen that the steepness of the curve approaches the number of sites, for small responses (the inset shows straight lines of increasing steepness). The