Nondimensionalization is more science than art

TL;DR

This paper presents a systematic, algorithmic approach to nondimensionalization of biological models using rational invariants, making the process more rigorous and accessible than traditional artful methods.

Contribution

The authors develop an algorithm extending differential algebra and invariant theory to automate nondimensionalization for models described by rational ODEs, including initial conditions and invariants.

Findings

Algorithm successfully nondimensionalizes various biological models.

Extension includes initial conditions and user-selected invariants.

Validated on Michaelis-Menten equations as a benchmark.

Abstract

When faced with a mathematical model, often the first step is to reduce the complexity of the model by turning variables and parameters into dimensionless quantities. This process is often performed by hand, relying on a skill practiced over many years, and attempted for small models. Nondimensionalization is often considered an art, as there is no formal method accessible to applied scientists. Here we show how to systematically perform nondimensionalization for arbitrarily sized models described by rational first order ordinary differential equations. We translate and extend an existing approach for computing rational invariants of the maximal scaling symmetry, which combines ideas from differential algebra, invariant theory and linear algebra, to the setting arising in biological models. The modeler inputs the system of equations and our implemented algorithm outputs the nondimensional quantities for the corresponding nondimensionalized model. We extend the algorithm to include initial conditions, and the modeler's choice of invariants, thereby including a larger class of nondimensionalizations. We further prove that any dimensionally consistent change of variables preserves the dimension of the maximal scaling symmetry. We showcase the framework on various models, including the classical Michaelis-Menten equations, which serves as a benchmark for asking and answering specific modeling questions.