__THE BIOPHYSICAL
BASIS FOR THE GRAPHICAL REPRESENTATIONS__

Consider a molecule that interconverts between two chemical
states *B* and *A*:

Then the chemical equilibrium expression *Keq* can be described as:

If a molecule *S* binds
to both *B* and *A* forming *SB* and *SA*, then we have the
following system:

The initial binding of *S*
will perturb the initial equilibrium concentrations of *B* and *A*
by the affinity constants *K*_{1}
and *K*_{2} that *S* has for *A*
and *B* respectively. Under these
initial conditions, the amounts of *SB* and *SA* are given by the
Langmuir binding expressions:

and

If then the binding
of *S* to *A* and *B* will
depend on the initial concentrations of [*A*] and [*B*] and won't
perturb the initial ratio of [*A*]/[*B*]. If , then the initial binding of *S* to *A*
and *B* will be relatively unequal,
which will perturb [*A*]/[*B*]. This
relatively unequal binding will change the initial equilibrium as originally
expressed by Le Chatelier in his famous principle. The stress on the original
equilibrium from the binding of *S*
will deplete one side of the equilibrium (given ), which would then be compensated by a shift toward the
depleted species to restore the equilibrium. This shift would necessitate the
transfer of some amount of the species from the other side of the equilibrium
to the relatively depleted side of the equilibrium.

As an aside, if we weren't aware that *S* was binding then we would write the equilibrium
expression simply as [*A*]/[*B*]. However, if *S* was present and binding preferentially to *B*, for example, then the equilibrium concentration of
the *B* species would increase by
Le Chatelier's principle because the total *B* species would include the free *B* plus *SB*
plus the amount shifted from species *A* to relieve the stress on the original equilibrium so that the total amount of the *B* species would increase relative to what it was
initially. Also, the equilibrium concentration of species *A* would be equivalently decreased by the conservation
of matter law. The net effect would appear to shrink the equilibrium constant *Keq* in the presence of *S*. However, in fact the equilibrium constant doesn't
really change for the free species concentrations of *A* and *B*.
It only appears to change because we are now including the shift from *A* into our chemical notation for the concentration of
species *B*. The origin of this
shift is the relatively unequal binding of *S* to *A*
and *B*. Although there has been
much previous thought about how to calculate this shift, the present analysis
presents the most simple and direct method.

In order to calculate the net perturbation or shift () due to the relatively unequal binding of *S* to *B*
and *A*, we can use the
mathematically derived fundamental equation for equilibrium (link) with the
following substitutions: *b* = [*B*], *f(x)*
= *SA*, *a* = [*A*]
and *g(x)* = *SB*,

and further substituting the Langmuir binding expressions
for *SA* and *SB* gives,

and further simplifying,

we finally get,

(3)

This expression compares the two Langmuir binding functions
for *SA* and *SB* for their relative effects on [*B*] and [*A*]
by determining within the
domain *S*. This allows us to
understand how the binding of *S*
simultaneously to *A* and *B* perturbs the original chemical equilibrium between
them*.*

Apart from the fact that the fundamental equation for
equilibrium was mathematically derived, an objection may be made to the fact
that *SA* and *SB* were added to *A* and *B* rather than
subtracted. However, upon
reflection we see that the addition makes more sense. If *SA* was
subtracted from *A* instead of
added as in the following,

(4)

then the ratio would approach zero as more of *S* binds to *A.* This would reflect a decrease in the complementary numerator *A* + , which wouldn't make sense from the perspective of Le
Chatelier's principle because the increase in relative binding of *S* to *A*
should produce a greater shift in the equilibrium toward the *A* side of the equilibrium and a positive shift for not a negative
shift (note that this is true for this particular example, but isn't meant to
imply that can never be
negative). Certainly this makes
sense for the straightforward case when *S* binds only to *A* and not
to *B*.

Therefore our thinking must be correct in order to
understand this shift within the context of Le Chatelier's principle as applied
to coupled equilibria. It may be
more correct to consider that the binding of *S* to *A* increases the
potential chemical species of *A*
by including the formation of a new potential species *SA*. Therefore this increase in the potential chemical
species should be accounted for by the addition of *SA* and *SB*
to *A* and *B* rather than their subtraction (This is similar to
the ratios of probabilities in a partition function with the disjoint
probabilities being added). In this sense, the ratio on the left side of
equation (4) represents the potential chemical species of *A* compared to *B* in the presence of *S*.
Any increase in the formation of *SA*
is considered as an increase in the potential reservoir for the *A* species thereby increasing the probability that we
would find more *A* species given
the condition that .

This approach explains why there is a close correlation
between the thermodynamic coupling free energy,, for an acid-base, two-state model and the experimentally determined
efficacies for ligands binding to the 5-HT_{2A} receptor[1].
Similarly, the ratio determined
by fitting an acid-base, two-state model to the pH-dependent binding
significantly correlated with the experimental efficacies for a variety of
ligands (link).
This is due to the ligand's ability to relatively favor the base form of the
receptor and thereby produce a shift in the original equilibrium to create more
of the base state of the receptor. This presents a general mechanism for
receptor activation within the confines of a plausible biophysical model.

THE TWO-STATE MODEL:

This is similar to most other two-state models with R and R* states corresponding to inactive and active receptor states except that this model relates the response to a fundamental equation for physical equilibrium, which can be solved for the net shift in the original equilibrium, RH,

Where R_{H} and R_{L} represent the amount
of unperturbed receptor existing in initial high and low affinity states
respectively, and D represents the concentration of the binding drug or ligand.
Download a prototype model for this equation in Excel (http://www.bio-balance.com/Prototype.htm).

With the dissociation constants, K_{DH} and K_{DL},
for the high and low affinity binding, this equation has been shown to
accurately model the dose-response behaviors for a wide variety of
drug-receptor systems (see **EXPERIMENTAL VERSUS CALCULATED DOSE-RESPONSE
CURVES** below).

**EXPERIMENTAL VERSUS CALCULATED DOSE-RESPONSE CURVES**

Consider a molecule that interconverts between two chemical
states *B* and *A*:

The Electrostatic Potential for the two-states of the
beta-2-Adrenergic Receptor (B2AR)

The base state with the cysteine as S- and the acid state with the cysteine as SH. The large blue region at the bottom of the receptor is the intracellular region which combiones with the G protein, while the smaller red region is on the extracellular side of the receptor and attracts the positively charged ligands. (Click on each B2AR image if you'd like a higher resolution image to download or display.)

Click here for more electrostatic potentials and graphics

** **

**Web Page: http://www.bio-balance.com
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