APPENDIX - Derivation of the Method

Given a mathematical ratio:

Then the expression: can be changed to:

Solving for gives:

(1)

Where represents a
specific addition to *a* and a
corresponding subtraction from *b*
that maintains the equality for the two ratios. This is a fundamental equation for
equilibrium. (Return to
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We may also broaden *x*
and *y* to include functions of *x* such as *f(x)* and *g(x)* then,

(2)

This expression compares the relative effects of the two
functions *f(x)* and *g(x)* in place of
*x* and *y* on *a*
and *b* by determining over their
common domain *x*.

Now, instead of the mathematical ratio consider a
molecule that interconverts between two chemical states *B* and *A*:

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

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

The initial binding of *S*
will be determined by the initial equilibrium concentrations of *B* and *A*
and the affinity constants *K*_{1}
and *K*_{2} that *S* has for *A*
and *B* respectively. Under the
constraints imposed by these initial conditions the amounts of *SB* and *SA*
can be described by the Langmuir binding expressions:

and

These functions determine the amount of the initial binding
of *S* to *B* and *A*.

If then the binding
of *S* to *A* and *B*
will depend on the initial relative 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 the initial ratio of [*A*]/[*B*]. This relatively unequal binding will
necessarily 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 stress one
side of the equilibrium (given ), which would then be compensated by a shift toward the
stressed side to restore the equilibrium. This shift would necessitate the
transfer of some amount of one side of the equilibrium to the relatively
stressed side of the equilibrium.

As an aside, if we didn't know 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* states would increase by Le
Chatelier's principle because the total *B* states would include the free *B* plus *SB* plus the amount
shifted from *A* to relieve the
stress on the original equilibrium. Therefore the total amount of the *B* states 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 states 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 *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 (2)
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* will or will not perturb 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)

where the chemical states have replaced the mathematical
terms in (2), then the ratio would approach zero as more of *S* binds to *A.* This reflects 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 for this particular example and isn't meant to imply
that can never be
negative). Certainly this makes
sense for the straightforward case where *S* binds only to *A* and not
to *B*.

Therefore our thinking must be correct in order to
understand the true nature and direction of the 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 states of *A* by including the formation of a new potential state *SA*. Therefore this increase in the potential chemical
states should be accounted for by the addition of *SA* and *SB*
to *A* and *B* rather than their subtraction. In this sense, the
ratio on the left side of equation (4) represents a probalistic sum of the
potential chemical states of *A*
compared to *B* in the presence of *S*. Any increase in the formation of *SA* would be considered as an increase in the potential
reservoir for the *A* thereby
increasing the probability that we would find more *A* 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]
(link). 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. 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 base states of the receptor. This presents a general mechanism for
receptor activation within the confines of a plausible biophysical model.

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[1] Rubenstein,
L. and Lanzara, R. *Activation of G Protein-Coupled Receptors Entails
Cysteine Modulation of Agonist Binding*, J.
Molecular Structure (Theochem) 430/1-3: 57-71 (1998).