On the Origins and Rarity of Locally but Not Globally Identifiable Parameters in Biological Modeling

Abstract

Structural identifiability determines the possibility of estimating the parameters of a model by observing its output in an ideal experiment. If a parameter is structurally <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">locally</i> but not globally identifiable (SLNGI), its true value cannot be uniquely inferred because several equivalent solutions exist. In biological modeling it is sometimes assumed that local identifiability entails global identifiability, which is convenient because local identifiability tests are typically less computationally demanding than global tests. However, this assumption has never been investigated beyond demonstrating the existence of counter-examples. To clarify this matter, in this paper we began by asking <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">how often</i> a structurally locally identifiable parameter is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">not</i> globally identifiable in systems biology. To answer this question empirically we assembled a collection of 102 mathematical models from the literature, with a total of 763 parameters. We analysed their identifiability, determining that approximately 5% of the parameters are SLNGI. Next we investigated how the SLNGI parameters arise, tracing their origin to particular features of the model equations. Finally, we investigated the possibility of obtaining false estimates. Some of the solutions that are mathematically equivalent to the true one involved parameters and/or initial conditions with negative values, which are not biologically meaningful. In other cases the true solution and the equivalent one were in the same range. These results provide insight about a previously unexplored hypothesis, and suggest that in most (albeit not all) systems biology applications it suffices to test for structural local identifiability.