Nanomedicine, Volume I: Basic Capabilities
© 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, cligand (molecules/nm3).10 For simple processes, the equilibrium constant Kd, taken in the direction of dissociation, is:
where kd is the dissociation rate constant (sec-1) and ka is the association rate constant (nm3/molecule-sec). The ka 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 Kd by a factor of 3000, due to kd increasing by a factor of 1600 but ka decreasing only by a factor of 1.7. Hence it is the rate constant of dissociation, kd, 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:
Observed half-lives range from <0.1 microsec (kd ~ 107 sec-1) for the enzyme catalase to a few months (kd ~ 10-7 sec-1) for enzyme inhibitors such as the Kunitz inhibitor of trypsin406 and for avidin-biotin binding.407 The smaller the kd (or the Kd), the greater the affinity and so the more firmly the receptor grasps the ligand.
The probability Poccupied that a receptor will be occupied10 is given by:
where Punoccupied = 1 - Poccupied. To ensure Poccupied = 99% receptor occupancy, Kd must ~ cligand / 100. For target molecules present at the 10-3 - 10-11 gm/cm3 concentrations typically found in human blood (Appendix B), cligand = 3 x 10-3 molecules/nm3 for glucose to cligand ~ 10-11 molecules/nm3 for female serum testosterone, giving a range of Kd ~ 10-4 - 10-13 molecules/nm3 to achieve 99% occupancy.
How much binding energy per receptor will this require? The free energy of dissociation DGd of a ligand-receptor complex is related to its equilibrium dissociation constant Kd by:
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 Kd gives a range for DGd 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 DGs = 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 Kd, the receptor design must provide a free energy of binding of at least:
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