Higher-order Lie symmetries in identifiability and predictability analysis of dynamic models

Parameter estimation in ordinary differential equations (ODEs) has manifold applications not only in physics but also in the life sciences. When estimating the ODE parameters from experimentally observed data, the modeler is frequently concerned with the question of parameter identifiability. The source of parameter nonidentifiability is tightly related to Lie group symmetries. In the present work, we establish a direct search algorithm for the determination of admitted Lie group symmetries. We clarify the relationship between admitted symmetries and parameter nonidentifiability. The proposed algorithm is applied to illustrative toy models as well as a data-based ODE model of the $\text{NF}\kappa \text{B}$ signaling pathway. We find that besides translations and scaling transformations also higher-order transformations play a role. Enabled by the knowledge about the explicit underlying symmetry transformations, we show how models with nonidentifiable parameters can still be employed to make reliable predictions.

Figure 1

Reaction scheme of the $\text{NF}\kappa \text{B}$ signaling pathway. $\text{NF}\kappa \text{B}$ is bound in a complex with $\text{I}\kappa \text{B}\alpha$ in the cytoplasm. Complex phosphorylation by pIKK leads to complex dissociation, $\text{I}\kappa \text{B}\alpha$ degradation, and $\text{NF}\kappa \text{B}$ shuttling to the nucleus. The associated reactions are indicated by arrows labeled by reaction flow expressions.

Figure 2

Different time courses of pIKK. (a) pIKK activation as expected from the literature. (b) Transformation (20) applied to the expected activation for different values of $\epsilon$ (renormalized to a peak height of 1).