Introduction

ã 1999 GraphPad Software Inc.


Introducing dose-response curves
From curvefit.com.  Copyright 1999 by GraphPad Software, Inc.  All Rights Reserved.
What is a dose-response curve?
Dose-response curves can be used to plot the results of many kinds of experiments. The X-axis plots concentration of a drug or hormone. The Y-axis plots response, which could be almost anything. For example, the response might be enzyme activity, accumulation of an intracellular second messenger, membrane potential, secretion of a hormone, heart rate or contraction of a muscle.
The term "dose" is often used loosely. The term "dose" strictly only applies to experiments performed with animals or people, where you administer various doses of drug. You don't know the actual concentration of drug -- you know the dose you administered. However, the term "dose-response curve" is also used more loosely to describe in vitro experiments where you apply known concentrations of drugs. The term "concentration-response curve" is a more precise label for the results of these experiments. The term "dose-response curve" is occasionally used even more loosely to refer to experiments where you vary levels of some other variable, such as temperature or voltage.
An agonist is a drug that causes a response. If you administer various concentrations of an agonist, the dose-response curve will go uphill as you go from left (low concentration) to right (high concentration). A full agonist is a drug that appears able to produce the full tissue response.  A partial agonist is a drug that provokes a response, but the maximum response is less than the maximum response to a full agonist. An antagonist is a drug that does not provoke a response itself, but blocks agonist-mediated responses. If you vary the concentration of antagonist (in the presence of a fixed concentration of agonist), the dose-response curve will run downhill.
The shape of dose-response curves
Many steps can occur between the binding of the agonist to a receptor and the production of the response. So depending on which drug you use and which response you measure, dose-response curves can have almost any shape. However, in very many systems dose-response curves follow a standard shape, shown below.
Dose-response experiments typically use 10-20 doses of agonist, approximately equally spaced on a logarithmic scale. For example doses might be 1, 3, 10, 30, 100, 300, 1000, 3000, and 10000 nM. When converted to logarithms, these values are equally spaced: 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0.
Note: The logarithm of 3 is actually 0.4771, not 0.50. The antilog of 0.5 is 3.1623. So to make the doses truly equally spaced on a log scale, the concentrations ought to be 1.0, 3.1623, 10.0, 31.623 etc.
Since the linkage between agonist binding and response can be very complex, any shape is possible. It seems surprising, therefore, that so many dose-response curves have shapes identical to receptor binding curves. The simplest explanation is that the link between receptor binding and response is direct, so response is proportional to receptor binding. However, in most systems one or more second-messenger systems link receptor binding to response. For example, agonist binding activates adenylyl cyclase, which creates the second-messenger cAMP. The second messenger can then bind to an effector (such as a protein kinase) and initiate a response.
What do you expect a dose-response curve to look like if a second messenger mediates the response? If you assume that the production of second messenger is proportional to receptor occupancy, the graph of agonist concentration vs. second messenger concentration will have the same shape as receptor occupancy (a hyperbola if plotted on a linear scale, a sigmoid curve with a slope factor of 1.0 if plotted as a semilog graph). If the second messenger works by binding to an effector, and that binding step follows the law of mass action, then the graph of second messenger concentration vs. response will also have that same standard shape. It isn't obvious, but Black and Leff (see The operational model of agonist action) have shown that the graph of agonist concentration vs. response will also have that standard shape (provided that both binding steps follow the law of mass action). In fact, it doesn't matter how many steps intervene between agonist binding and response. So long as each messenger binds to a single binding site according to the law of mass action, the dose-response curve will follow the same hyperbolic/sigmoid shape as a receptor binding curve.
The EC50
A standard dose-response curve is defined by four parameters: the baseline response (Bottom), the maximum response (Top), the slope, and the drug concentration that provokes a response halfway between baseline and maximum (EC50).
It is easy to misunderstand the definition of EC50. It is defined quite simply as the concentration of agonist that provokes a response half way between the baseline (Bottom) and maximum response (Top).  It is impossible to define the EC50 until you first define the baseline and maximum response.
Depending on how you have normalized your data, this may not be the same as the concentration that provokes a response of Y=50. For example, in the example below, the data are normalized to percent of maximum response, without subtracting a baseline. The baseline is about 20%, and the maximum is 100%, so the EC50 is the concentration of agonist that evokes a response of about 60% (half way between 20% and 100%).  
Don't over interpret the EC50. It is simply the concentration of agonist required to provoke a response halfway between the baseline and maximum responses. It is usually not the same as the Kd for the binding of agonist to its receptor.
The steepness of a dose-response curve
Many dose-response curves follow exactly the shape of a receptor binding curve. As shown below, 81 times more agonist is needed to achieve 90% response than a 10% response.
Some dose-response curves however, are steeper or shallower than the standard curve. The steepness is quantified by the Hill slope, also called a slope factor. A dose-response curve with a standard slope has a Hill slope of 1.0. A steeper curve has a higher slope factor, and a shallower curve has a lower slope factor. If you use a single concentration of agonist and varying concentrations of antagonist, the curve goes downhill and the slope factor is negative.