Universality in Multiparameter Fitting: Sloppy Models

Abstract

In order to understand a variety of physical phenomena (such as signaling networks in molecular biology or crystal structures in condensed matter physics), scientists often develop models with many unknown or tunable parameters. Such multi-parameter models and systems are often sloppy. For practical purposes their behavior depends only on a few stiffly constrained combinations of parameters; other directions in parameter space can change by orders of magnitude without significantly changing the behavior. We develop the theoretical basis of sloppiness and argue that there is in fact a new universality class to which these models belong.

We begin by defining sloppiness (an exponentially large range of sensitivity to different combinations of parameters, with a roughly uniform distribution of sensitivities between the extremes). We then document sloppiness in a variety of models from different scientific fields. Several mathematically well-defined classes of models, some sloppy and some not sloppy, are then analyzed to understand the origins of sloppiness. Drawing connections to the field of random matrix theory, we derive an ensemble of sloppy models. The heart of sloppiness in this ensemble is shown to be the Vandermonde matrix. By demonstrating the novel statistical properties of this ensemble we argue that it constitutes a new universality class. Inspired by the properties of this Vandermonde ensemble we develop new tools for analyzing complex, real-world models with many parameters.

In the final section we focus on a particular complex, real-world model with many parameters. We formulate and analyze a mathematical description of the quorum sensing network in the bacterium Agrobacterium tumefaciens. This network allows Agrobacterium to regulate gene expression in accordance with its population density. The mathematical description includes twenty four unknown parameters quantifying the biochemical interactions. While not complete, the model provides insight into the quorum sensing process and we suggest ways of coupling the model with experiments in the future.