Deriving ODEs from Reactions
For model simulation, SimBiology® derives ordinary differential equations (ODEs) from model reactions using mass-balance principles. The left-hand-side (LHS) of each ODE is the time-derivative of a model quantity and the right-hand-side (RHS) is defined using reaction fluxes that are derived from reaction rates and rate rules. In other words, SimBiology represents a system of ODEs as follows.
is an M-by-
1 vector containing
the rates of change for model quantities, S is an
v is an R-by-
1 flux vector.
M equals the total number of species, and
equals the total number of reactions in the model
During the conversion of model reactions into ODEs, SimBiology performs a dimensional
analysis to ensure each reaction flux has the dimension of
substance/time such as
mass/time. If the reaction rate has the dimension of
concentration/time, then SimBiology multiplies it by the
compartment volume to get the reaction flux. If the reaction rate has the dimension of
substance/time, then the flux is identical to the rate, and no
volume-correction is performed. If there are no units specified with the model, the
default dimension for a species (
concentration, and that for a flux is
substance/time. For such cases, the ODE is the flux divided by a
compartment volume to make the dimension of LHS and RHS consistent. See the following
figure for an illustration.
Suppose there is a reaction
x —> y, with the reaction
rate R1. The following figure explains the dimensional analysis
performed by SimBiology to make the dimensions of LHS and RHS of an ODE consistent.