Differential Equations of Multiple Coupled Compartments (up to four):

In biological, biophysical, biochemical and pure chemical Systems many different but somehow connected compartments are in exchange with each other and cannot be observed or should not be described separately. The problem of three or four coupled equilibrium reactions with first order rate constants for forward and backward reactions represent a very general scenario for many real problems in physics, chemistry and biology (see Fig.1). Therefore a mathematical description via differential equations (DES) is a necessary basis for understanding observed data or modelling such systems (see Eq.1).

 In our example the uptake process of a substance trough gills into the blood of a fish is a process through 3 compartments in general. In detail, at each of these compartment boundaries there is a system of at least 4 different phases, which have to be treated as four coupled compartments (see fig 2).

 The basic knowledge of solving such DES has been known for more than 200 years. Various numerical methods have been developed as well and lead to excellent result. However, there are situations where an analytical solution is desirable; e.g. backward fitting of parameters or reduction of computational efforts. But we were not able to find an analytical solution for 4 coupled compartments in the published literature. Only advice on how to solve this problem in general can be found. (We would be grateful for any information on a published analytical solution).

Here we have taken the effort to solve the DES that describes the concentration-time profiles for all compartments based on known starting conditions. We validated the results against numerical methods for each of the 4 compartments.

Larisch, W., Goss, K.-U. (2017): Calculating the first-order kinetics of three coupled, reversible processes
SAR QSAR Environ. Res. 28 (8), 651 - 659.