Biochemistry - The Chemical Reactions of Living Cells, Volume 1 - D. Metzler 1980

How molecules bind to each other
Quantitative evaluation of binding strength
Data analysis

When conducting biochemical research, the need frequently arises to evaluate the binding affinity between various molecules. For instance, to elucidate the physiological significance of processes such as hormone-Cell membrane interactions or feedback Enzyme Inhibition, data on the binding strength between a hormone and a membrane or an enzyme and an inhibitor can be invaluable. Therefore, It is important to understand how binding affinity can be determined and what challenges may arise in the process.

The ability of molecule X to bind to molecule P [Eq. (4-1)] can be assessed by varying the concentrations of X and P and determining the resulting change in the concentration of the PX complex. First, one must select a measurable property that quantitatively distinguishes the complex from its uncomplexed components. For example, the complex might be colored while the starting components are colorless. Most commonly, the complex and its components exhibit different Light absorption at a specific wavelength; occasionally, they display distinct circular dichroism spectra or different chemical shifts in NMR signals. If P is an enzyme, only the dissociation of the PX complex leads to the formation reaction products. Sometimes, The rate of PX (enzyme-substrate complex) dissociation is slow compared to the rate at which equilibrium is established among X, P, and PX. In this case, the observed rate of product formation is proportional to the complex concentration.

Regardless of which property's change is measured experimentally, the magnitude of this change must increase as the concentration of X increases while the concentration of P remains constant. Typically, the molar concentration of P in experiments is low, whereas the concentration of X is varied over a very wide range. Under these conditions, at sufficiently high values of [X], the majority of P molecules are converted into PX, and The change in the property (let us denote it as ∆A) ceases to increase. This phenomenon, known as saturation, has been observed in the majority of binding studies as well as in many physiological processes.

The property or property change being quantified (∆A) reaches its maximum value at saturation—i.e., when all of substance P is converted into PX—denoted as ∆A_max. The ratio of [PX] to the total concentration of all forms of P present in the solution, [P]_total, is called the degree of saturation and is designated by . If molecule P has more than one binding site for X, then the quantity characterizes the fraction of occupied sites out of the total number of binding sites. If we let n denote the number of binding sites per molecule, the total number of binding sites will be equal to n ∙ [P]. The quantity is frequently taken to be equal to ∆A/∆A_max. For macromolecules with Multiple binding sites, this equality holds true only if The addition of each successive molecule of X leads to an identical change in A. This condition is not always met, but when it is, the following relationship must also hold:

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The index i denotes the number of ligands X bound to P; it can range from 0 to n. At n = 1, the degree of saturation and the value ∆A can be expressed in terms of the unbound X concentration and the formation constant. The corresponding relationships are of the form

Fig. 4-1 shows the curve illustrating the dependence of (or ∆A) on [X] for a hypothetical reaction (this curve is sometimes called an adsorption isotherm, since reliable results require experiments to be conducted at a constant Temperature). From Fig. 4-1 and equation (4-9), it can be seen that when

the value of [X] is precisely equal to 1/K_f (or K_d), then . It is also evident that saturation is approached slowly, and even at the point corresponding to the highest concentration of X (8/K), the degree of saturation does not exceed 90%. Because experiments typically measure ∆A , it is difficult to determine the limiting value ∆A_max from a curve of this type (except when K_f is very large). However, determining K_f requires precisely knowing ∆A_max. Consequently, plots similar to the one presented in Fig. 4-1 are rarely used in practice, and it is included here primarily to illustrate the Structure/97.html">Definitions introduced. The curve depicted in Fig. 4-1 is an equilateral hyperbola in coordinates ; saturation curves of this type are therefore frequently referred to as hyperbolic. This designation highlights the difference between the adsorption isotherm and slightly Different types of binding curves (Section B.7) that display a sigmoidal (S-shaped) character in the same coordinates.

FIG. 4-1. Adsorption isotherm: the dependence of the degree of saturation (or the change in a property ∆A) on the concentration of a substance [X] that reversibly binds to a macromolecule. The curve is hyperbolic in nature;

FIG. 4-2. Saturation curve on a semilogarithmic scale, plotted using the same data points as the curve in Fig. 4-1.

It is often more convenient to plot graphs in alternative coordinates, namely (Fig. 4-2). Let us list the reasons why these coordinates are more advantageous. 1. The curve becomes symmetrical about the midpoint, where lg[X] = —lg K_f. 2. Regardless of how wide the range of used X concentrations is, a scale can always be chosen such that all points fit on a single sheet of paper. 3. The distances between points on a curve plotted in these coordinates are approximately uniform, unlike a graph plotted in coordinates (compare the plots shown in Fig. 4-1 and Fig. 4-2, where the experimental points correspond to the exact same data, with each successive [X] value being twice the preceding one). 4. A logarithmic scale of this type can be used for all compounds regardless of their binding affinity, and the curve shapes for all 1:1 complexes are identical. The slope of the curve at its midpoint is 0.576; the change in lg[X] upon going from 10% to 90% saturation is 1.81. Similar curves are well known to chemists because they resemble titration curves in shape, with pH substituted for —lg[X]. When transitioning from a weaker complex to a stronger one, the curve shifts to the left, and vice versa. Curves of this type are very conveniently described mathematically using hyperbolic Functions [4a].

To graphically represent saturation data, researchers frequently employ another coordinate system known as Scatchard plots¹ (Fig. 4-3). In this approach, is plotted along one axis, causing the data points to fall on a straight line. As follows directly from equation (4-10)—which is derived straight from equation (4-9)—the intercept on the abscissa and the slope of the line directly yield The values of ∆A_max and K_t.

Before attempting to generate saturation curves, everyone should read Deranleau's excellent paper [5], and to better comprehend the material presented hereafter, one should review the work by Dowd and Riggs [6].

FIG. 4-3. Scatchard plot generated from the same data points as the curves in Fig. 4-1 and 4-2. This is the most convenient linear plot for analyzing binding data.

¹ The Scatchard plot is the best of all linear transformations of the saturation equation, specifically outperforming the double-reciprocal plot (Fig. 6-3).