One common mathematical interpretation of a biochemical model in COPASI is to represent it as a system of ordinary differential equations (ODEs). In this framework, each variable of the ODE system corresponds to the particle number of a species in the model.
To construct the right-hand side (RHS) of these differential equations, COPASI follows several steps:
Convert Particle Numbers to Concentrations: The particle numbers are first converted to concentrations. This takes into account the units of substance amount and the compartment volume.
For single compartment reactions, the result of the kinetic function is multiplied by the compartment volume to obtain the amount rate. For multi-compartment reactions, the output is already considered to be in amount units (e.g., moles) per unit time.
Convert to Particle Number Rates: The resulting amount per unit time is multiplied by an appropriate conversion factor to yield particle numbers per unit time.
COPASI also automatically analyzes the model to identify conserved values: these are linear combinations of particle numbers that remain constant throughout the system’s time evolution. Each conservation relation can be used to eliminate one variable, thereby reducing the system to a smaller set of independent variables. The remaining variables, called dependent variables, are expressed as linear combinations of the independent ones.
COPASI performs this model reduction transparently, while also indicating in the graphical interface which species are treated as independent and which as dependent. Technically, identifying conservation relations involves finding rows in the stoichiometry matrix that are linear combinations of other rows. To achieve this, COPASI uses Householder QR factorization (Vallabhajosyula06).