Communication: a simple analytical formula for the free energy of ligand-receptor-mediated interactions

Recently [P. Varilly, S. Angioletti-Uberti, B. M. Mognetti, and D. Frenkel, "A general theory of DNA-mediated and other valence-limited colloidal interactions," J. Chem. Phys. 137, 094108 (2012)], we presented a general theory for calculating 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 interaction free energy, which was previously obtainable only via a costly numerical thermodynamic integration. As a result, the computational effort to obtain potentials of interaction 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.