Biochemistry I Fall Term, 2001

Lecture 12: Ligand Binding Measurements

Assigned reading in Campbell: none.
Background reading: "Ligand Binding Equations and Analysis" by Dr. Gordon Rule.
This more extensive treatment is available as an eight-page PDF image.
Key Terms:

Ligand Binding

Measurements of the binding affinity and binding stoichiometry between molecules and macromolecules provide essential information on topics that range from mechanism to specificity. For example, ligands are nearly always competing with other ligands for a specific site, especially in vivo. In addition, binding events are frequently coupled to conformational changes, catalysis, and cell-cell interactions. 

A. Basic treatment of simple binding data

This section repeats the derivation of the Scatchard equation found in the Ligand Binding Equations and Analysis handout. The origin of this equation in the equilibrium equations and the definitions that allow a few algebraic manipulations will be covered in lecture as follows:

1. M + L <==> ML

2. K_d = [M][L]/[ML]       K_a = 1/K_d

3. n = [L]_bound/[M]_total = [ML]/([M] + [ML])

4. n = [L]/(K_d + [L])
From the saturation equation (eqn. 4), the following useful rerrangements can be made:

1. n/[L] = 1/K_d - n/K_d       Scatchard equation.
2. 1/n = K_d/[L] + 1       Double reciprocal equation.


3. n/[L] = n/K_d - n/K_d       Multiple binding sites.
4. logq = -logK_d + n_Hlog[L]       Hill Equation.

B. Experimental Measurements of Ligand Binding
 

Model reaction:     ML <=> M + L Schematically:  <=>  +  Dissociation constant: Kd = [M][L]/[ML] The following figures show how the equilibrium dialysis experiment can be used to determine the concentrations of M, L, and ML at binding equilibrium. At the start of the measurement, the protein (M) is present only in the left cell of the dialysis chamber. The small molecule (L) is present only in the right cell. The left and right cells are separated by a semipermeable membrane, through which only the ligand can pass. [For this illustration we will use concentration units of "balls/Box" (b/B) instead of M (mol/liter).] Starting concentrations (b/B):  Left cell: [ML] = 0;   [L] = 0;   [M] = 4.  Right cell: [ML] = 0;   [L] = 12;   [M] = 0.   When equilibrium is reached, the concentration of free ligand will be the same in both cells. However, because the protein can bind the ligand, the concentration of total ligand will be higher in the left cell. Equilibrium concentrations (b/B):  Left cell: [ML] = 2;   [L] = 5;   [M] = 2.  Right cell: [ML] = 0;   [L] = 5;   [M] = 0. The above illustrations are static "snapshots" of the molecules during the experiment. Open a window that simulates the Molecular Motion at Binding Equilibrium.  (The animation requires the Shockwave plug-in.) With the above results we can calculate the Kd for this cartoon binding reaction: Kd = [M][L]/[ML] = (2 b/B)¥(5 b/B)/(2 b/B) = 5 b/B. Note the following features of this binding equilibrium:   [L] = 5 b/B in both cells.   [L]total = 7 b/B in the left cell.   [L]total - [L] = 2 b/B = [ML] in the left cell. Thus, the difference in [L]total between the two cells is the measurement of [ML] that we need to add one datum to a Scatchard plot. A Scatchard plot consists of equilibrium dialysis measurements done at several starting concentrations of ligand. (See the Ligand Binding page for an example of the calculations and graphing procedures.) For example, with the Kd calculated above, what do you predict for the equilibrium concentrations of all species if the starting concentration of ligand had been 33 b/B? or 5 b/B?

C. Examples of Data Analysis

Experimental data, e.g. from an equilibrium dialysis experiment, can be analyzed to yield values for K_d and n in several ways. Computer programs are available that provide automated and objective estimates of these parameters. The processed data are then published in one or more of the following graphical formats:

1. Scatchard Plots: single site (n = 1) and multiple sites (n >1).
2. Fraction Saturation Plots: Y vs. log[L].
3. Double Reciprocal Plots: single site (n = 1) and multiple sites (n >1).
4. Hill Plots: Cooperative binding to multiple sites (n_H <n).

The class handout, Ligand Binding Curves, is shown on a separate page.

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