# Dynamical compensation and structural identifiability: analysis, implications, and reconciliation

## TL;DR

This paper reveals that dynamical compensation in biological systems is equivalent to structural unidentifiability, discusses its implications, and proposes alternative definitions to reconcile dynamical compensation with model identifiability.

## Contribution

It clarifies the relationship between dynamical compensation and structural identifiability, and offers methods to achieve both in biological models.

## Key findings

- Dynamical compensation is equivalent to lack of structural identifiability.
- Unidentifiable models can produce misleading biological insights.
- Alternative definitions can ensure dynamical compensation without sacrificing identifiability.

## Abstract

The concept of dynamical compensation has been recently introduced to describe the ability of a biological system to keep its output dynamics unchanged in the face of varying parameters. Here we show that, according to its original definition, dynamical compensation is equivalent to lack of structural identifiability. This is relevant if model parameters need to be estimated, which is often the case in biological modelling. This realization prompts us to warn that care should we taken when using an unidentifiable model to extract biological insight: the estimated values of structurally unidentifiable parameters are meaningless, and model predictions about unmeasured state variables can be wrong. Taking this into account, we explore alternative definitions of dynamical compensation that do not necessarily imply structural unidentifiability. Accordingly, we show different ways in which a model can be made identifiable while exhibiting dynamical compensation. Our analyses enable the use of the new concept of dynamical compensation in the context of parameter identification, and reconcile it with the desirable property of structural identifiability.