LIGAND BINDING - TUTORIAL #7 answer 7-1: protein partition function       The reaction scheme of a ligand replacement reaction is: PY + X <==> P + X + Y <==> PX + Y       The so-called partition function describes how many moles of PX and PY are in equilibrium with one mol of unliganded P (actually how many moles of verey species are in equilibrium with an arbitrarily chosen reference species; one might in principle take PX or PY as the reference species as well as P. The reference species is chosen for convenience, and in the present case we use P). We obtain:       [P] = reference species       [PX] = [P] x [X]/Kx       [PY] = [P] x [Y]/Ky where Kx is the dissociation equilibrium constant of ligand X (Kx = [P][X]/[PX]) and Ky is the dissociation equilibrium constant of ligand Y. Notice that in this reaction scheme the ligand X and Y are mutually exclusive and the triply liganded species PXY does not exist. We obtain the following binding polynomial:       [P]tot = [P] (1 + [X]/Kx + [Y]/Ky)       The fraction of X-bound ligand binding sites results:       [PX]/[P]tot = [X]/Kx / ((1 + [X]/Kx + [Y]/Ky) = [X]Ky / (KxKy + [Y]Kx +[X]Ky)       The concentration of ligand X required to saturate half the ligand binding sites (X50) results:       [PX]/[P]tot = 0.5 = X50Ky / (KxKy + [Y]Kx + X50Ky)       and we derive:       X50 = Kx (Ky + [Y]) / Ky       In a plot of X50 versus [Y], the above function is represented by a straight line with intercept Kx and slope Kx/Ky. The slope of the plot is often called the partition constant of the protein between the two ligands: Kp = Kx/Ky.       If the concentrations of X and Y are both high with respect to their respective dissociation constants ([X]>Kx and [Y]>Ky), we may neglect the contribution of the unliganded protein P to [P]tot and simplify:       [PX]/[P]tot = [X]Ky / ([Y]Kx +[X]Ky)       Under this approximation we obtain:       X50 = Kx [Y]/Ky       Notice that, because of the assumption we made ([Y]>Ky) the above equation necessarily implies X50 > Kx.       Warning: ligand Y is released during the binding of ligand X! It is necessary to check that the amount of Y released does not significantly affect [Y]free, or to calculate the change of [Y]free induced by the increasing concentrations of X. This control will be done at the end of the analysis.       How would you proceed for the analysis (click to select)? 1) Fit every binding curve of X independently, find a series of X50 values, then plot these as a function of Y and obtain Kx and Ky by linear regression. 2) Globally fit the equation developed above to all binding curves of X to obtain Kx and Ky. Back one step.