Many biological proteins and enzymes can possess more than one binding site.
Usually, when a ligand  binds with a macromolecule , it can influence binding kinetics of other ligands  binding to the macromolecule.
A simplified mechanism can be formulated if the affinity of all binding sites can be considered independent of the number of ligands bound to the macromolecule. 
This is valid for macromolecules composed of more than one, mostly identical, subunits. It can be then assumed that each of these  subunits are identical, symmetric and that they possess only a single binding site. Then the concentration of bound ligands  becomes
In this case, , but comprises all partially saturated forms of the macromolecule:
where the saturation occurs stepwise
For the derivation of the general binding equation a saturation function  is defined as the quotient from the portion of bound ligand to the total
amount of the macromolecule:
K′ are so-called macroscopic or apparent dissociation constants and can result from multiple individual reactions. For example, if a macromolecule M has three binding sites, K′ describes a ligand being bound to any of the three binding sites. In this example, K′ describes two molecules being bound and K′ three molecules being bound to the macromolecule. The microscopic or individual dissociation constant describes the equilibrium of ligands binding to specific binding sites. Because we assume identical binding sites with no cooperativity, the microscopic dissociation constant must be equal for every binding site and can be abbreviated simply as K. In our example, K′ is the amalgamation of a ligand binding to either of the three possible binding sites (I, II and III), hence three microscopic dissociation constants and three distinct states of the ligand–macromolecule complex. For K′ there are six different microscopic dissociation constants (I–II, I–III, II–I, II–III, III–I, III–II) but only three distinct states (it does not matter whether you bind pocket I first and then II or II first and then I). For K′ there are three different dissociation constants — there are only three possibilities for which pocket is filled last (I, II or III) — and one state (I–II–III).
Even when the microscopic dissociation constant is the same for each individual binding event, the macroscopic outcome (K′, K′ and K′) is not equal. This can be understood intuitively for our example of three possible binding sites. K′ describes the reaction from one state (no ligand bound) to three states (one ligand bound to either of the three binding sides). The apparent K′ would therefore be three times smaller than the individual K. K′ describes the reaction from three states (one ligand bound) to three states (two ligands bound); therefore, K′ would be equal to K. K′ describes the reaction from three states (two ligands bound) to one state (three ligands bound); hence, the apparent dissociation constant K′ is three times bigger than the microscopic dissociation constant K.
The general relationship between both types of dissociation constants for n binding sites is
Hence, the ratio of bound ligand to macromolecules becomes
where  is the binomial coefficient.
Then the first equation is proved by applying the binomial rule