Distilling identifiable and interpretable dynamic models from biological data

Abstract

Mechanistic dynamical models allow us to study the behavior of complex biological systems. They can provide an objective and quantitative understanding that would be difficult to achieve through other means. However, the systematic development of these models is a non-trivial exercise and an open problem in computational biology. Currently, many research efforts are focused on model discovery, i.e. automating the development of interpretable models from data. One of the main frameworks is sparse regression, where the sparse identification of nonlinear dynamics (SINDy) algorithm and its variants have enjoyed great success. SINDy-PI is an extension which allows the discovery of rational nonlinear terms, thus enabling the identification of kinetic functions common in biochemical networks, such as Michaelis-Menten. SINDy-PI also pays special attention to the recovery of parsimonious models (Occams razor). Here we focus on biological models composed of sets of deterministic nonlinear ordinary differential equations. We present a methodology that, combined with SINDy-PI, allows the automatic discovery of structurally identifiable and observable models which are also mechanistically interpretable. The lack of structural identifiability and observability makes it impossible to uniquely infer parameter and state variables, which can compromise the usefulness of a model by distorting its mechanistic significance and hampering its ability to produce biological insights. We illustrate the performance of our method with six case studies. We find that, despite enforcing sparsity, SINDy-PI sometimes yields models that are unidentifiable. In these cases we show how our method transforms their equations in order to obtain a structurally identifiable and observable model which is also interpretable. Author summaryDynamical models provide a quantitative understanding of complex biological systems. Since their development is far from trivial, in recent years many research efforts focus on obtaining these models automatically from data. One of the most effective approaches is based on implicit sparse regression. This technique is able to infer biochemical networks with kinetic functions containing rational nonlinear terms. However, as we show here, one limitation is that it may yield models that are unidentifiable. These features may lead to inaccurate mechanistic interpretations and wrong biological insights. To overcome this limitation, we propose an integrated methodology that applies additional procedures in order to ensure that the discovered models are structurally identifiable, observable, and interpretable. We demonstrate our method with six challenging case studies of increasing model complexity.