Chapter 5

Writing Rate Laws for Binding Processes

Binding is ubiquitous in biological systems. We define binding to be any non-covalent linking of one chemical species to another. Examples of binding are 1) hormone - receptor binding, 2) neurotransmitter - receptor binding, 3) cytokine - receptor binding, 4) protein (hetero- or homo-) dimerization, 5) enzyme - substrate interactions, 6) ion - protein binding, 7) allosteric binding of a cellular metabolite to a regulated enzyme, 8) transcription factor binding to regulatory elements of DNA, and many others. Often, we are concerned with binding of a small molecule or ion to a protein. In such cases, the small molecule can be considered the ligand and the protein can be considered the receptor. If we let L be the concentration of ligand, and R be the concentration of receptor, then the simplest form of the fundamental rate law for the binding process is

where k₁ is a second order rate constant. Both sides would be multiplied by V, the volume in which the binding takes place, if the flux was to be written in mol/sec. If we denote the ligand-receptor complex as C, then the flux for the unbinding process is first order, and has the form

We examined the equilibrium behavior of a binding reaction a few pages ago, so now we can extend this paradigm to see why the phenomenon of binding results in saturation behavior in many biological processes. And this is exactly what we will do in the next chapter when we develop the rate law for an enzymatic process. There are, however, several important features of binding processes that are frequently encountered in modeling projects. The first of these is cooperativity.

Cooperativity is the word used to describe a response that is out of proportion to its stimulus. To understand this, recall the binding equation from the previous chapter, or if you prefer, the Michaelis-Menten equation for the catalytic flux of a simple enzyme.

If h is a hormone, and r is its receptor, then the cellular response to this hormone will be proportional to [hr]. This means that for a three fold change in [h], you will get a three-fold change in response, at least as long as [h] is less than or equal to K_D. At concentrations greater than the K_D, a 3-fold change will result in less than a 3-fold change in [hr], and consequently, less than a 3-fold change in the response. This is just another way of saying that the response saturates.

For many hormonal stimuli, however, the cellular response is many fold greater than the factor by which [h] changes. For example, the production of InsP₃ increases 5 to 10-fold in response to a doubling of norepinephrine concentration outside a vascular smooth muscle cell. Clearly, the simple binding equation is incapable of this behavior. It could produce no more than a two-fold change in response for a doubling of the hormone concentration.

In a classic study of the kinetics of oxygen binding to hemoglobin, A.V. Hill formulated a modification of the usual binding equation that permits it to account for cooperative behavior. He introduced an exponent that has come to be known as the Hill coefficient. We recommend that you use the following expression for these situations:

Of course, when n = 1, the equation reverts to the classical form. For n>1 the response will display what is called positive cooperativity, and for n<1, the response is negatively cooperative. This has the useful consequence that 3-fold changes in [h] are no longer limited to 3-fold changes in [hr] or 3-fold changes in response. In fact, as shown in figure 5-1, the response to a doubling of [h] can be huge.

To understand the significance of figure 5-1, notice that the quantity plotted on the vertical axis, the "Hill Fraction", is equal to

where C is the hormone concentration, and K_D is the dissociation constant. This ratio, just like the corresponding fraction without the Hill coefficient, always falls in the range from zero to one, and represents the fraction of receptors occupied by hormone. On the horizontal axis, we've plotted the ratio of hormone concentration to its dissociation constant. Consequently, this axis reveals whether the current concentration is greater or less than the concentration at half-maximal binding, and a value of 0.1 on the horizontal axis means simply that the current concentration is ten times less than the K_D.

Figure 5-1 is plotted log-log to emphasize the ability of cooperative mechanisms to produce more than 2-fold changes in response for only a 2-fold change in the stimulus. To see this, recognize that the stimulus is the hormone concentration on the horizontal axis, and the response will be proportional to the fraction of receptors bound to hormone, as plotted on the vertical axis. As a specific example, notice that for a Hill coefficient of 2, a doubling of stimulus (from 0.2 to 0.4) yields a 10-fold increase in response (from 0.01 to 0.1).

Cooperativity of this magnitude is commonplace in biological systems.