Medicinal Chemistry Applet

dose-response relationships

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Introduction

Like graphs generated from the Michaelis-Menten equation (eq. 1) (see Michaelis-Menten exercise), a typical dose-response plot also resembles a rectangular hyperpolic curve.  Their similarities are not coincidental.  Both involve two molecules reversibly binding and possibly leading to a change (product formation for enzymes and response for a receptor).  The derivation of the dose-response relationship as described by Clark's Occupancy Theory (eq. 2) closely resembles the related derivation of the Michaelis-Menten equation.  Incidentally, the Michaelis-Menten equation predated Clark's work by approximately a decade.

     (1)

     (2)

Clark's Occupany Theory contains a number of assumptions.  First, the drug (D) and receptor (R) bind reversibly to form a drug-receptor complex (D-R).  The D-R complex generates a response (E) (eq. 3).  Second, the magnitude or intensity of the response is directly proportional to the concentration of the D-R complex (eq. 4).  Third, the maximum possible response (E_max) is achieved when all the receptors (R_T or R_total) are bound in a D-R complex (eqs. 5).

     (3)

     (4)

     (5)

Expressing the reverse of Equation 3 as an equilibrium of dissociation generates Equation 6.  The total receptor concentration (R_T) is equivalant to the concentration of unbound receptor (R or R_free) and bound receptor (D-R or R_bound) (eq. 7).  Substitution and rearrangement affords Equations 8 through 11, and ultimately Equation 2.

     (6)

     (7)

     (8)

     (9)

     (10)

     (11)

     (2)

Note that according to Equation 6, the units of K_D are concentration.  If Equation 11 is solved for the value of K_D when E equals 1/2 E_max, K_D equals [D].  Therefore, at 50% maximal response, the drug concentration will be equal to the dissociation equilibrium constant.  K_D is equal to the concentration at which the drug causes 50% maximal response (EC₅₀) (eq. 12).  This relates to the Michaelis constant, K_m, being equal to the substrate concentration at 1/2 V_max.

     (12)

Clark's treatment of receptor theory was fairly general, but it was not perfect.  The model works best for compounds that are able to affect a full response with sufficiently high concentration.  These compounds are called full agonists.  Clark's theory has trouble with certain drugs that elicit a response, but regardless of concentration, the response never reaches E_max.  These compounds are called partial agonists.  Another class of compounds, called antagonists bind to the receptor but do not elicit any response.  When mixed with an agonist, antagonists decrease the level of response caused by the agonist.  Ariens and Stephenson independently modified Clark's Occupancy Theory in the 1950s by adding another term, &alpha or intrinsic activity, and forming a new equation (eq. 13).  In Clark's original equation, &alpha = 1, and this is true for full agonists.  For partial agonists, &alpha is less than 1, but greater than 0.  Antagonists elicit no response, so they have an &alpha value of 0.  Intrinsic activity and &alpha are from Ariens modifications; Stephenson used efficacy and e.

     (13)

A problem with dose-response plots is that the interesting part of the graph is always compressed to the left.  Most of the graph shows a nearly level line that approaches the y-value of 100% response.  A method of graphing that better demonstrates differences in agonists and partial agonists is obtained by plotting log(dose) vs response.

Applet

This applet plots % response vs [D] data for up to three compounds.  For each compound, the ED₅₀ and e (efficacy or intrinsic activity) must be specified.  The units of EC₅₀ and [D] are concentration, often µM or nM.  Efficacy/intrinsic acitivity are unitless.  ED₅₀ values are commonly encountered in scientific notation, e.g. 1.5x10^-3.  Currently, this applet only accepts numbers in decimal format, e.g. 0.0015.

Problem information

The data for three agonists and partial agonists are shown in the table below.  Use this data to answer the following questions.

Problems

1. Using the applet, generate a dose-response plot for all three drugs.  Between drug A and B, which is the better agonist?  Explain your answer.
2. Between drug A and C, which is the better agonist?  Why?
3. The two questions above approach "better" and "worse" from only a pharmacodynamic consideration.  There are many other perspectives.  Based on ideas surrounding pharmacokinetics, come up with at least two reasons why your answer in question 2 might change.