Graphics for the net shift

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 K₁ and K₂ 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.

 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