  Recently \1, we presented a general theory for calculat- ing the strength and
properties of colloidal interactions mediated by ligand-receptor bonds (such as
those that bind DNA-coated colloids). In this communication, we derive a
surprisingly simple analytical form for the inter- action free energy, which
was previously obtainable only via a costly numerical thermodynamic
integration. As a result, the computational effort to obtain potentials of in-
teraction is significantly reduced. Moreover, we can gain insight from this
analytic expression for the free energy in limiting cases. In particular, the
connection of our general theory to other previous specialised approaches is
now made transparent. This important simplification will significantly broaden
the scope of our theory.
