
Chapter 5
Writing Rate Laws for Binding Processes
Binding is ubiquitous in biological systems. We define binding
to be any noncovalent 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 k_{1} 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 ligandreceptor 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
MichaelisMenten 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 threefold
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 3fold change will result in less than a 3fold change in [hr],
and consequently, less than a 3fold 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 InsP_{3} increases 5 to
10fold 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 twofold 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
3fold changes in [h] are no longer limited to 3fold changes in
[hr] or 3fold changes in response. In fact, as shown in figure
51, the response to a doubling of [h] can be huge.
To understand the significance of figure 51, 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 halfmaximal
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 51 is plotted loglog to emphasize the ability of
cooperative mechanisms to produce more than 2fold changes in
response for only a 2fold 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 10fold increase in response (from 0.01 to 0.1).
Cooperativity of this magnitude is commonplace in biological
systems.
 