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# Refining Cheng-Prusoff equation

A generalisation of the Cheng-Prusoff formula relating affinity constant and the concentration of inhibitor giving 50% inhibition has been derived for the case when concentrations of ligands are not in a great excess. It shows that methods ensuring precise solving binding equations (including computer approaches and equations derived in the present work) can not be used if these "Cheng-Prusoff conditions" of ligands' excess are not observed. The trouble is that beyond these conditions analysis becomes unreliable: the error contained in estimations of necessary parameters appears abruptly multiplied in the calculated affinity constant.

first submitted to J.Theor.Biol. in March 95; also rejected from Biochem.Journal; no response from Trends in Pharm.Sci.; published in Internet in July 96

Short demonstration: an amazingly simplified Cheng-Prusoff eqn.

Derivation of all equations in the article is based on a single mathematical trick of introducing new variable f - the fraction of free binding sites in the absence of inhibitor. That is, f equals to approx. unity when the concn. of labeled ligand is small; and it approaches zero when this concn. is big and labeled ligand (tracer) almost completely saturates binding sites.

As a demonstration of the amazing effect of this substitution let's use it to rewrite famous Cheng-Prusoff equations:

I50=Kd (1+ T0/Kd*)          
where I50 is the concn. of inhibitor giving 50% inhibition; Kd - dissociation constant of inhibitor-binder reaction; Kd*- dissociation const. of tracer-binder reaction and T0 is the total added concn. of the tracer.

So, the new variable fraction f of free binding sites is defined as:  f=(S0-Bmax*)/S0
where S0 is the total concentration of binding sites and Bmaxis the bound concn. of tracer in the absence of inhibitor (i.e. the maximal signal on inhibition curve)
Then, in the absence of inhibitor and under the Cheng-Prusoff conditions the mass action low is written as:
Bmax* =T0(S0-Bmax*)/Kd*
Now we replace Bmax* with new variable fthat is  Bmax*=(1-f) S0 and (S0-Bmax*)=f S0
(1-f)S0=T0f S0/Kd*
after obvious rearrangements:
1-1/f=T0/Kd*
1/f=1+T0/Kd*
comparing this with eqn.  we see that a much simpler form of Cheng-Prusoff equation is just:

I50=Kd/f              

This eqution is mathematically equivalent to Cheng-Prusoff, i.e. it was derived with the same assumptions and it is valid inder the same conditions. But this form has substantial practical advantages:

1. Physical meaning of this equation is much more clear: I50 deviates from affinity diss. constant if the tracer saturates considerable fraction of binding sites.

2. Procedure of rigorous using eqn.  remains essentially the same -- as a rule, we still should conduct both inhibition and saturation experiments. But now we may just ESTIMATE one parameter instead of MEASURING two. S0 is the total concentration of binding sites (the same in inhibition and in saturation experiments, of course); T0 is the concentration of tracer used in the inhibition experiment. Note, that  signal Bmax* on inhibition curve should not ideally coincide (as in this picture) with signal produced by T0 concentration of tracer on saturation curve because of the larger inter-experimental measurement error.
Fraction f should be understood on this picture as measured in units relative to S0;  that is, here it is equal approximately to 0.5

3. Eqn.  also opens an opportunity to estimate fraction f in alternative experiment when obtaining saturation curve is impossible. This situation occurs in a not too important immunological problem of measuring "true" affinity constants, for which monstrous method proposed by Friguet is now most widely used. Read about it in the full text version.

The article itself derives a generalization of Eqn. valid beyond Cheng-Prusoff conditions which is just:

I50=S0 + Kd/f             

where S0 is total concn. of binding sites.

Unfortunately, this equation proves that it can not be used practicaly beyond "Cheng-Prusoff conditions": To get Kd we have to subtract S0 from I50 Under "Cheng-Prusoff conditions" S0 is negligibly small. If it is not small, the measurement errors contained in values for I50 and S0 appear abruptly muliplied in I50-S0. If S0 and I50 values become closer measurement errors may even lead to negative estimate for Kd

My works are more frequently plagiarized than cited properly. Therefore I expect that equations  and  must have been stolen as well. I would greatly appreciate information about publications plagiarizing these equations. If you know such instance please write me to dyuryev@yandex.ru