© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

3.5.2 Ligand-Receptor Affinity

For a ligand binding to a receptor in a solvent, there will be a characteristic frequency with which existing ligand-receptor complexes dissociate as a result of thermal excitation, and a characteristic frequency with which empty receptors bind ligands as a result of Brownian encounters, forming new complexes, with the frequency of binding proportional to the concentration of the ligand in solution, c_ligand (molecules/nm³).¹⁰ For simple processes, the equilibrium constant K_d, taken in the direction of dissociation, is:

{Eqn. 3.22}

where k_d is the dissociation rate constant (sec^-1) and k_a is the association rate constant (nm³/molecule-sec). The k_a rate constant reflects mainly the molecular weight of the ligand, and thus varies little among antibody, enzyme, or other receptor systems. For example, Delaage⁴⁰¹ notes that changing the solution pH for growth hormone from 7.5 to 4.0 increases K_d by a factor of 3000, due to k_d increasing by a factor of 1600 but k_a decreasing only by a factor of 1.7. Hence it is the rate constant of dissociation, k_d, which accounts for the vast bulk of affinity in receptor systems.

Thus receptor affinity is usually taken as the inverse of the dissociation rate constant, which may be placed in the context of the half-life of the ligand-receptor complex approximated⁴⁰¹ by:

{Eqn. 3.23}

Observed half-lives range from <0.1 microsec (k_d ~ 10⁷ sec^-1) for the enzyme catalase to a few months (k_d ~ 10^-7 sec^-1) for enzyme inhibitors such as the Kunitz inhibitor of trypsin⁴⁰⁶ and for avidin-biotin binding.⁴⁰⁷ The smaller the k_d (or the K_d), the greater the affinity and so the more firmly the receptor grasps the ligand.

The probability P_occupied that a receptor will be occupied¹⁰ is given by:

{Eqn. 3.24}

where P_unoccupied = 1 - P_occupied. To ensure P_occupied = 99% receptor occupancy, K_d must ~ c_ligand / 100. For target molecules present at the 10^-3 - 10^-11 gm/cm³ concentrations typically found in human blood (Appendix B), c_ligand = 3 x 10^-3 molecules/nm³ for glucose to c_ligand ~ 10^-11 molecules/nm³ for female serum testosterone, giving a range of K_d ~ 10^-4 - 10^-13 molecules/nm³ to achieve 99% occupancy.

How much binding energy per receptor will this require? The free energy of dissociation DG_d of a ligand-receptor complex is related to its equilibrium dissociation constant K_d by:

{Eqn. 3.25}

which refers to a standard reference state where all chemical species are 1 M (i.e., K₀ ~ 0.6 molecules/nm³) and attributes a free energy of zero to a complex with a dissociation constant of 1 M.⁴⁰²

For T = 310 K, the range of required K_d gives a range for DG_d of 39.4 zJ for glucose (at typical serum concentrations) to 128 zJ for female serum testosterone.

However, when ligand and receptor associate there is a loss of three degrees of freedom in each of translational and rotational entropy, which may be estimated using the classical Sackur-Tetrode equations, giving an entropic free energy range (for translation and rotation combined) of DG_s = 80 zJ for very small molecules (MW ~10 daltons), to 120 zJ (MW ~10² daltons), 200 zJ (MW ~10⁴ daltons), and 280 zJ for large molecules (MW ~10⁶ daltons).⁴²⁷

Thus to form a ligand-receptor complex with a dissociation constant K_d, the receptor design must provide a free energy of binding of at least:

{Eqn. 3.26}

or 120-410 zJ/molecule for designed receptors achieving 99% occupancy operating over the likely range of physiological concentrations and temperatures. This is consistent with Drexler's estimate of 161 zJ binding energy required to ensure reliable receptor occupancy for small plentiful molecules.¹⁰

Last updated on 7 February 2003