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User Manual | The Model in COPASI | Deterministic Interpretation of the Model

Deterministic Interpretation of the Model

One possible mathematical interpretation of the model is to convert it into a set of ordinary differential equations. The variables of the equation are the particle numbers of the species in the model. The right hand side of the differential equation are constructed as follows: The particle numbers are converted to concentrations taking into account the unit for amounts of substance and dividing by compartment volume. These concentrations are used to calculate the reaction fluxes. The kinetic functions, as they are defined in COPASI give as result a value that is a concentration rate (for single compartment reactions) or an amount of substance rate (for multi-compartment reactions), respectively. So for single compartment reactions the value of the kinetic function is multiplied by the compartment volume; kinetics for multi-compartment reactions are assumed to already be expressed in units of amount of substance (e.g. moles) per time. The resulting value is then multiplied by a factor to convert amount of substance per time to particle numbers per time. Linear combinations of these values, using the stoichiometries as coefficients, result in particle number rates for all species. These form the right hand side of the differential equations.

COPASI automatically performs an analysis of the model by which conserved values are found. The conserved values COPASI is looking for are linear combinations of particle numbers that do not vary during the time evolution of the system. Each conservation relation can be used to eliminate one variable of the system, leading to a reduced system with a smaller number of variables. These variables are called the independent variables of the system; the dependent variables are defined as linear combinations of independent variables. COPASI handles this model reduction transparently, but it is displayed in the GUI which species are treated as independent or dependent variables. Technically finding the conservation relation means finding rows in the stoichiometry matrix that can be expressed as linear combinations of other rows. COPASI uses Householder QR factorization [Vallabhajosyula06] to do this.