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 Graphics page - link )

 

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 K₁ and K₂ 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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