Validation of the model
The model was tested in a number of ways. The first test was performed by simulating an enzymatic reaction: A + E → B + E. 1000 molecules of A were simulated with 10 molecules of E and the effective reaction rate was measured by fitting the change in the concentration of A over time. We ran 4 sets of simulations with the following parameters for the diffusion rate d of the chemical objects and timestep Δt: 1) d = 1, Δt = 0.01; 2) d = 1, Δt = 0.001; 3) d = 5, Δt = 0.01; 4) d = 5, Δt = 0.001. For each set of d and Δt, the reaction rate k was successively set to 106.0, 106.2, 106.4 and 106.6 M-1s-1. For comparison purposes, these 4 sets of runs were performed using both the present model and the Andrews and Bray model. The corrected binding radius used in the Andrews and Bray approach was calculated using the code provided by the authors.
As has been pointed out by Andrews and Bray [17], in such simulations the measured rate of the reaction varies with time. This is due to the fact that the simulation starts, the local concentration gradient is not yet established, and the initial reaction rate is, therefore, higher than the desired rate as the local concentration gradient is not yet established. Subsequently, the system tends towards a steady state, and the reaction rate is correctly predicted by both methods with 99% accuracy when using the correction term. The two methods were statistically indistinguishable over the four sets of runs (slope and intercept of kmeasured = f (kdesired) were identical with p > 0.55).
The model was also tested for reactions at low numbers of reactants (nA = 10) where the effective rate of the reaction becomes subject to significant stochastic fluctuations. 10000 runs were performed using both the present and corrected Smoluchowski approaches; the reaction rate was determined for each run. Again four sets of runs were performed using the same diffusion constants and timesteps as described above. The reaction rate used was 106 M-1s-1. The distribution of the reaction rates at low concentrations produced by the present and corrected Smoluchowski approaches were compared and found to be indistinguishable (p > 0.2 on t-test).
Finally, the present and corrected Smoluchowski approaches were also compared in a situation containing a concentration gradient. The concentration gradient was produced by a point source of a molecule A (k = 2 s-1) which reacts with an enzyme E ([E] = 40 nM) with kE = 106 M-1s-1. The diffusion constants and timestep parameters where again varied as previously. The gradient generated were found to be identical (p > 0.9 on U-test). All statistical analyses were performed using the R package [30].