answer 6-5: ligand solubility < Kd>/d> If the ligand's solubility equals or is lower than Kd, fitting of the signal yields unreliable estimates of the upper asymptote of the ligand binding isotherm and, consequently, of the fraction of liganded sites and Kd. In this case an important distinction is in order because our signal may either: 1) monitor the absolute concentration of [P] or [PX]. For example, if you use a radioactively labelled ligand and measure the radioactivity in the precipitated protein, you have a quantitative, absolute measure of PX. In this case, and provided that you have also a good absolute extimate of [P]tot, you can replace the signal with an estimate of the fraction of bound sites, and fit that to the ligand binding function, as we did in tutorial #1. Your estimate of the Kd will be reliable even if you could explore only a very small fraction of the ligand binding isotherm. 2) Monitor only the relative amount of [P] and [PX], i.e. the fraction of liganded and unliganded sites. For example if you use absorbance of fluorescence changes and you don't have independent estimates of extinction coefficients or fluorescence yields, you rely on the high and low asymptotes to correlate the total amplitude of the transition to [P]tot and to recalculate the absolute concentrations of [P] and [PX] (if necessary). Case 1 is non-problematic: if you can measure absolute values for [PX] and [P]tot you can derive a reliable estimate of Kd even from analysis of a small fraction of the ligand binding isotherm. Case 2 is problematic. You should devise some method to either: (i) increase the ligand affinity; or (ii) obtain absolute estimates of [PX]. Some suggestions are given below. One can record ligand binding isotherms over a wide range of temperatures: this has two effects because it changes both the solubility of the ligand and its affinity for the protein, depending on the ΔH of the reaction. The results should then be fitted globally for the Kd at a reference temperature and the ΔH. Other experimental conditions may be subject to a similar exploration e.g. pH and ionic strength. With some luck one may find a condition where the affinity of the protein for the ligand and the solubility of the ligand allow he/she to record a complete ligand binding isotherm, plus several incomplete ones whose interpretation takes advantage of the global fitting procedure. One may titrate the ligand with the protein, rather than the opposite way round, assuming that the solubility of the protein, and its concentration are high enough to saturate the ligand. One may determine the extinction coefficient of PX on a similar protein with higher affinity for the same ligand. This method is applicable if the ligand binds to a protein bound cofactor common to many related proteins. To conclude the series of tutorials #4, #5, and #6 we summarize: 1) a ligand is said to have high affinity if its Kd is lower than the minimum protein concentration the sensitivity of our instruments allows us to use. THe analysis of the ligand binding isotherm requires to take into account that the free ligand concentration is significantly lower than the total ligand concentration. 2) A ligand is said to have very high affinity if its Kd is much lower than the minimum protein concentration the sensitivity of our instruments allows us to use. The free ligand concentration is so much lower than the total ligand concentration that no mathematical analysis can estimate it reliably; thus Kd can be determined only if we directly measure [X]free. This condition, however, allows us to estimate the ligand binding stoichiometry. 3) A ligand is said to have low affinity if its Kd is higher than the maximum ligand concentration we can practically achieve because of its solubility or other problems (e.g. ionic strength, deleterious effects on protein stability). Thus, the concepts of high and low affinity relate to practical problems we cencounter on specific systems rather than on absolute values of Kd. This tutorial is complete. |