## 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:

$\stackrel{˙}{x}=S\cdot v$

$\stackrel{˙}{x}$ is an M-by-1 vector containing the rates of change for model quantities, S is an M-by-R stoichiometry matrix, v is an R-by-1 flux vector. M equals the total number of species, and R 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 amount/time or 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 (DefaultSpeciesDimension) is 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.