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))
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:
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.
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:
The diffusion-limited off-rate for the unimolecular reaction thus has the same form as the bimolecular one, albeit with a different diffusion coefficient.
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.