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Scatchard's equation

The preceding analysis presupposes that the total concentration of sites, [So] = n[mmtot], is known beforehand. In most cases, however, the number of sites per macromolecule, n, is not known; only the concentration of macromolecules, [mmtot], can be specified. During the "titration" of mm by addition to the solution of varying concentrations [Lo] of ligand, the concentration of free ligand, [L], is determined. In this instance, Fig 11.9 is useless because [So] is unknown. To analyze such a data set, the following measure of ligand binding is defined:

[Lo ] - [L] _ concentration of bound ligand [mm tot ] concentration of macromolecule

With this experimentally-determinable quantity, Eqs (11.14) -yield (see Problem 11.3b):

(11.16) can be manipulated to

In this form, a plot of v/[L] against v should yield a straight line with a slope of 1/K and an intercept of n/K. From such data treatment, both the equilibrium constant for ligand binding, K, and the number of sites per macromolecule, n, can be determined.

### Site occupancy

It is often important to know the fraction of the macromolecules that bind 0,1, .. ..n ligands. This cannot be had from Scatchard's equation, but a modest extension of the model provides this information. The notation for partially-saturated macromolecules is shown in Fig. 11.10 for n = 4 (4 binding sites per macromolecule). The objective is to determine the

Fig. 11.10 Combinations of ligands (circles) bound to sites on a macromolecule with 4 identical, independent sites v

Fig. 11.10 Combinations of ligands (circles) bound to sites on a macromolecule with 4 identical, independent sites concentrations of each of the five species, designated as [mm*Lk], where 0 £ k £ 4. These concentrations must satisfy two conservation equations. The first is for macromolecules:

[mmtot] = [mm] + [mm*L] + [mm*L2] + [mm*L3] + [mm*Lj] (11.23a)

and the second is for bound ligands:

[Lo] - [L] = [mm*L] + 2[mm*L2] + 3[mm*L3] + 4[mm*L4] (11.24a)

If fk = [mm*Lk]/[mmtot] is the fraction of the macromolecules with k bound ligands, with Eq (11.21) the above equations become:

The most straightforward way of obtaining these fractions is by the Monte-Carlo technique. Specified are M macromolecules with 4 sites on each and N bound ligands These quantities are related by v = N/4M (the denominator is the total number of sites). The flow chart for this calculation is shown in Fig. 11.11. The left-hand side (separated from the right-hand side by the vertical dashed line) is the actual Monte Carlo calculation. The heart of the computation is the site-occupancy matrix, kocc(i,m), which is zero if the ith site empty and one if it contains a bound ligand. Ligands are "thrown" at the macromolecules until all N have found an empty site. The right hand side of Fig 11.11 is the bookkeeping necessary to produce the concentrations of the macromolecules with 0 ,.. ..4 sites occupied. This computation is repeated many times in order to obtain good statistics.

Figure 11.12 shows the result of this computation for the complete range of site saturation. The ordinates are the fractions f0,..f4 multiplied by 100. Each intermediate states (mm*L1, mm*L2 and mm*L3) rises to a maximum and then falls to zero. As the fractional saturation approaches unity, all macromolecules have ligands bound to all 4 of their sites.

Note that this computation and Fig. 11.12 are independent of the ligand binding equilibrium constant K, the ligand concentration [Lo] and the macromolecule concentration [mmtot]. These parameters determine the value of v, however.

### 11.6.2 Dual independent binding sites

If the macromolecule possesses two sets of independent sites, at concentrations Ro and Qo, with nR and nQ per molecule, the site concentrations are related to the concentration [mmtot] of the macromolecule by:

There are now two equilibrium expressions, one for each site type:

average each [rnrn*Lh] over J iteratioi

Monte Carlo calculation of ligand binding to macromolecules [site-occupancy flag matrix kocc ( i,m)

store [rnrn*Lk]

i = site on rn (i-a»4) m = macromolecule (m = 1~>M) n = ligand { n = 1 -~*N) | [rnm'LJ = percent of macromolecuies binding k ligands (k = 0-^-4) kocc < i,m) = site-occupancy matrix ( 0 = empty; 1 = binding a ligand)

m = 0 |

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