In the steady-state approximation, we demand that the time derivative of the intermediate complex vanishes, thus assuming the intermediate complex population changes slowly. The effective rate thus obtained is:
kon,eff=kD+qkD-+q=kD+1+kD-/q-1
(11)
We may compare this rate to the rate obtained via SSS theory for both the bimolecular and unimolecular cases. The diffusion-limited rate for the freely diffusing particle (the bimolecular interaction) is (see equation (8))
kDbi+=4Dbia.
(12)
For the polymer end-to-end (unimolecular) contact formation, we use the appropriate diffusion-limited on-rate (see equation (9))
kDuni+=4Dunia2L2/3-32 .
(13)
The formulation of an appropriate diffusion-limited off-rate, in both types of reactions, requires further discussion. We follow the considerations laid out by [Lapidus, Eaton & Hofrichter (2000)] and [Wang & Davidson (1966)]. We find the expression for the equilibrium constant for the diffusion-limited reaction, and use it to express the diffusion off-rate:
kD-=kD+/Ka,D.
(14)
For two well-mixed particles A and B, the concentration of freely-diffusing particles within radius rAB of each other is:
nAB,bi=43rAB3AB
Where concentrations of species are in square brackets. One may treat the expression 43rAB3A as the fraction of ligand B that is within the radius. Multiplying it by the concentration B produces the concentration of overlapping molecules (in an interaction-less world). Since diffusion is stochastic in nature, we assume that this is the concentration of encounter complexes nAB,bi=AB. The equilibrium constant is thus the volume defined by the reaction radius (sum of radii for the reacting partners), independent of the diffusion coefficient.
Ka,Dbi=43rAB3
(15)
From (14) we obtain:
kDbi-AB=3Dbi/rAB2.
(16)
The same notion applies for diffusion between the ends of a polymer - only now the end-to-end distance probability is not uniform. The Jacobson-Stockmayer factor S (equation (10)) measures exactly the probability to occupy a small volume vs, which is Pvs=vsS-1. Substituting 43rAB3 for vs, we obtain the fraction of ligand B that is within this radius. As this is the ratio between the amount of ligand that is at the encounter complex to the amount of non-encountered ligand, this expression plays the role of the equilibrium constant of the unimolecular reaction:
Ka,DuniAB=43rAB3S-1.
(17)
The diffusion-limited off-rate for the unimolecular reaction thus has the same form as the bimolecular one, albeit with a different diffusion coefficient.
kDuni-AB=3Duni/rAB2
(18)
By substituting equations (12) and (16) into equation (11), we get the effective bimolecular on-rate:
kon,eff,bi=4Dbia1+3Dbi/qa2-1
Similarly, for the effective unimolecular on-rate, we substitute equations (13) and (18) into equation (11):
kon,eff,uni=4Dunia2L2/33/21+3Duni/qa2-1
Both results coincide with the terms of the equivalent order in the SSS solution (cf. equations (8) and (9)) if q=3/a=4a243a3.