Scale dependent association rate constants

When small subvolumes are needed in 3D, or when running 2D simulations, it is often necessary to use scale dependent association and dissociation rates constants [Fange et al. 2010]. In 2D simulations it is, in fact, a requirement to use the scale dependent reaction rates. In 3D simulations the scale dependent rate constant, below, will converge to the macroscopic association rate constant when the subvolumes are large and/or the reactions are not diffusion controlled. In the latter case, the scale dependent reaction rate constants can be turned off.

The scale dependent association rates are calculated according to

Equation 6.1. 

Here k is the microscopic association rate constant on the reaction radius (see Figure 3.13, “ The XML definition of AssociationRateConst. ”), α is the degree of diffusion control (see Figure 3.15, “ The XML definition of DegreeOfDiffusionControl. ”) and where ρ is the reaction radius (see Figure 3.14, “ The XML definition of ReactionRadius. ”), and h thickness of a concentric shell centred around one of the reactants. When the other reactant is within the shell, a reaction can occur.

Since MesoRD does not use spherical shells around molecules, rather a cartesian grid, h needs to chosen in the particular way described below. Since a molecule can reside anywhere within a subvolume, the region in which a molecule can react with its reaction partner should not only include the subvolume in which the molecule resides, but also its nearest neighbours. By requiring that the reaction volumes for the spherical case and the cartesian case should be the same, h is chosen to fulfil the relation

Equation 6.2. 

where l is the subvolume size.

The reaction rates must be calculated in agreement with the fact that reactions can occur within the center subvolume and its nearest neighbours [16] . In MesoRD this a achieved as follows. Using the example above where A and B react for form C, the reaction rate is calculated as

Equation 6.3. 

where N is the number of neighbouring subvolumes to subvolume i, and and are the diffusion rates of A and B respectively. Each combination of A and B molecules in the neighbours and center subvolume is, however, sampled independently such that only neighbours which actually have a molecule are selected for a reactive event. Upon an association reaction event the product molecule is always placed in the centre subvolume, i. Upon a dissociation event, the two products are distributed uniformly among the neighbours and the centre subvolume. Here the probablity of placing reactant A in the center subvolume is and the probability of placing reactant B in the center subvolume is . The fractions of diffusion-rates defining the probability of selecting a specific reactant from the center subvolume were introduced in MesoRD version 1.1. Previous release (1.0) assumed the same diffusion-rates for both reactants.

[16] Reactions can occur in neighbouring subvolumes because diffusion events includes mixing time, and diffusion limited reactions does not always have time to mix. Furthermore, in a discretised system the resolution is always twice as large as discretisation length.