Algorithmic Non-dimensionalisation of Polynomial Dynamical Systems with CRNNondim

Abstract

Although non-dimensionalisation is a fundamental step in modelling, it is rarely obvious, especially for large dynamical systems, how to systematically construct optimal non-dimensionalisations. In this work, we develop a general mathematical framework for non-dimensionalising polynomial ODEs by interpreting them as chemical reaction networks (CRNs) and characterise the space of admissible scalings. Within this viewpoint, admissible scalings of time and species are encoded by Laurent monomials, and we derive exact formulae for the minimal number of multiplicative dimensionless parameters required by the reaction-channel coefficients and by the full initial value problem, together with characterisations of the scale assignments that attain structural lower bounds. These counts are expressed in terms of structural properties of the underlying CRN, and we show that the full initial value problem requires no more parameters than the corresponding Buckingham count. These results lead to constructive algorithms which we analyse rigorously and implement in the open-source Python package CRNNondim, available at https://github.com/georgio-hawi/crnnondim . Given a polynomial system or CRN, CRNNondim determines species and time scalings that bring variables as close to unity as possible while minimising the number of free parameters, and among the admissible scalings it chooses free parameters that result in the corresponding dimensionless system being sparse and simple. We apply CRNNondim to numerous well-known biological and biochemical models, yielding simpler dimensionless systems with fewer free parameters, and interpreting the parameters that govern their dynamics.