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Theory of Ligand Binding to Monomeric Proteins

1.1 Importance of Ligand-Binding Phenomena in Biology

Reversible interactions between or among molecules underlie nearly every aspect of biology. To understand these interactions in a chemical way means to describe them quantitatively. To do so we must be able to determine their affinity, stoichiometry, and cooperativity by carrying out ligand-binding experiments. We use the term "ligand" in a way distinct from its use to discuss coordination complexes within inorganic chemistry. In ligand-binding theory we use this term to mean any interacting partner. Although many people consider a ligand to be a small molecule that interacts with a macromolecule, in fact, either partner can be considered to be the ligand of the other. In a typical experiment the concentration of one partner is held fixed while the concentration of the other partner is incremented. In ligand-binding theory and practice we define the ligand operationally as the partner whose concentration is incremented during the experiment. Such experiments resemble pH titrations both practically and theoretically, and thus are referred to as titrations. The partner whose concentration is held fixed is referred to as the "target," and again the definition is strictly operational, that is, either a small molecule or a macromolecule can be the target, depending on how the titration is set up.

As we will show later, there are good reasons to carry out experiments with first one partner, then the other, treated as the ligand; however, depending on the chemical properties of the two partners in the reaction, there may be practical limitations to, or conceptual difficulties in, the possibility of interchanging their respective roles as the ligand and the target. In particular, if either of the reaction partners has multiple binding sites for the other, one may not obtain superimposable binding isotherms when exchanging the target and the ligand.

Affinity refers to the strength of interaction between partners. Affinity is quantitatively expressed by an equilibrium constant that we measure in our experiment or, equivalently, a free energy difference between the bound and free states of the system that we calculate from the equilibrium constant.

Stoichiometry refers to the number of molecules of each partner that participate in the binding process, and it must also be determined by our measurements. In practice, what we really mean by stoichiometry is more often molar ratio. For example, in a process involving four molecules of one kind with two of another kind, the stoichiometry is 4:2 but the molar ratio is 2:1. The determination of true stoichiometries usually requires additional information from sources other than a binding experiment (e.g., molecular weights, and state of aggregation of the target and ligand in solution).

Thermodynamic linkage is a general term that applies to ligand-binding experiments in which the same target binds two or more molecules of the same or different ligands, and each ligand modulates the affinity of the target for the other. There are at least three different types of linkage, called identical, homotropic, and heterotropic (Wyman and Gill, 1990). Identical linkage occurs when two different ligands compete for the same binding site on the target, and their binding is mutually exclusive. This type of linkage is discussed in Section 1.8 for single binding site targets, and in Section 4.8 for multiple binding site targets. Competitive enzyme inhibition is a very important case of identical linkage and is described in Section 8.6. Homotropic linkage occurs when the target can bind more than a single molecule of the same ligand, with different affinity. Homotropic linkage can occur only in targets with multiple binding sites, thus its analysis is deferred to Chapter 4. Finally, heterotropic linkage occurs when the target can bind two different ligands in a non-exclusive manner and the binding of one ligand alters the affinity of the other. It is described in Section 1.9, and for targets with multiple binding sites in Section 4.9. Important examples of heterotropic linkage are uncompetitive enzyme-inhibition (Section 8.7), and regulation of the oxygen affinity of hemoglobin by effectors, including protons (Bohr effect), diphosphoglycerate, or inositol hexaphosphate, dealt with in Chapter 7.

Homotropic and heterotropic linkage are typically regarded as an emergent property unique to proteins, but some non-protein molecules of ~500-1000 Da have been shown to exhibit cooperative binding of their ligands (Rebek, 1985). An interesting and biologically relevant example is provided by the axial ligands of iron-porphyrins (Traylor and Sharma, 1992).

Homotropic and heterotropic linkages may be either positive (if each ligand increases the affinity of the other) or negative (if each ligand decreases the affinity of the other). Cooperativity has often been used as a synonym of linkage, but unfortunately not always with the necessary precision. Often, cooperativity, or positive cooperativity, is used to indicate positive homotropic linkage, but the terms of negative cooperativity or anti-cooperativity may be used to indicate a negative homotropic or heterotropic linkage. The definition of cooperativity is sufficiently general to encompass cases in which even monomeric proteins can respond cooperatively to two different ligands, for example, the physiological ligand and an ionic component of the solution (Weber, 1992). Note that in such cases the ion must be also considered as a physiological effector. Because the general applicability of a definition is inversely related to its precision, in this book, we shall prefer the terms positive or negative homotropic or heterotropic linkage whenever precision is required. Positive cooperativity occurs in several proteins and has special relevance in physiology. For example, the binding of oxygen to hemoglobin is cooperative in that oxygen affinity becomes stronger as binding progresses, as described in detail in Chapter 7.

In this chapter we describe the theoretical bases of ligand binding under equilibrium conditions for protein:ligand complexes with 1:1 stoichiometry; in the next chapter we discuss the kinetics of the same system, and in Chapter 3 we consider some practical aspects of experimental design, and some common sources of errors.

1.2 Preliminary Requirements for Ligand-Binding Study

Whether or not a given ligand binds to a given macromolecular target may be known from prior experiment or inferred from physiological or chemical data. When no such information is available, or when the equilibrium constant for the reaction is required, then the interaction must be evaluated from titration, using a method of analysis that is suitable for the solution conditions of interest and the concentrations of partners that can be practically achieved. Although specific methods will not be detailed here, all useful methods have in common that they offer some observable that changes during, and thus reports on, the binding process. Any observable is referred to as the signal, and its change relates to a shift between the bound and free states of the system.

Whether or not a ligand binds reversibly and without any chemical transformation must also be established. Ligand-binding theory and practice generally refers only to such cases; but we shall cover in this book also some common non-conformant cases, for example, thiol reagents and enzyme-substrate reactions, that of course do begin with a ligand-binding process followed by a chemical transformation. The most common approach to determining reversibility is to simply assume it; this is not appropriate for rigorous scientific work. Reversibility is generally established by showing that the signal change is reversed when concentrations are reduced, for example, by dilution or dialysis. A more stringent criterion is to show that the separated ligand and target are recovered unchanged after their interaction, although it can be difficult to rule out a small extent or minor degree of change. One of the best ways to do so is to repeat the binding measurement itself with the recovered materials to evaluate whether the affinity is the same. However, this method can fail with labile partners.

Finally, the molar ratio and stoichiometry must be determined from the same kinds of binding experiments that are used to determine affinity and cooperativity, using strategies that will be outlined later. Without knowing the correct molar ratio the interpretation of the affinity may be plagued by high uncertainty. Another common, unfortunate, and sometimes untested, assumption is that an interaction has a 1:1 molar ratio. As we will show in Chapter 3, the experimental design required to establish molar ratio is not difficult conceptually, and usually is not practically difficult either. Thus there is no reason whatsoever to leave this very important feature to untested assumption. The analysis of binding data is simpler for the 1:1 stoichiometry case, thus we will treat this case first before expanding the treatment to cases of any molar ratio.

1.3 Chemical Equilibrium and the Law of Mass Action

Every chemical reaction, if allowed enough time, reaches an equilibrium condition in which the rate of product formation from reactants equals the rate of product degradation to reactants. When this condition is reached...