## Introduction

A major aim of Systems Biology is the establishment of mathematical models of biological processes like signal transduction, metabolism, or gene regulation in order to gain insight in these nonlinear dynamical systems. As an initial step, an appropriate model structure has to be identified, i.e. the relevant molecular compounds and the nature and characteristic of their interactions. Then, the model's parameters like concentrations of compounds and rate constants are estimated from experimental data to calibrate the model. For this calibration step, an objective function assessing the goodness of fit can be optimized, e.g. the parameters are chosen to minimize deviations between measurements and model. A very efficient and flexible objective function for this purpose is the so-called *likelihood* which coincides with the least-squares criterion in typical Systems Biology applications.

An essential task of the modelling procedure is the assessment of uncertainty, e.g. by calculating confidence intervals for parameters and predictions. In the classical regression setting, this is typically accomplished by so-called *standard errors*, i.e. by propagating the measurement uncertainty using the *Gaussian law of error propagation* which is based on linearization of the model.

In Systems Biology, the models are typically *mechanistic*, i.e. the components of the models have counterparts in the biological process. Therefore, the mathematical models are typically nonlinear and more complex than in a regression setting. Frequently, ordinary differential equations are used to describe the dynamics of biochemical interactions. For such models, the likelihood is nonlinear and therefore confidence regions for model parameters can exhibit complex shapes. This renders classical approaches as rough approximations in the finite sample case. Sometimes, they are even infeasible, e.g. if structurally non-identifiable parameters are present.

In contrast, the *profile likelihood* approach [1, 2] results in confidence intervals which are invariant under parameter transformations [3] and therefore not affected by nonlinear distortions of the likelihood landscape. The profile likelihood is a one-dimensional representation of the likelihood indicating which values of a single parameter component are in statistical agreement with the available measurements. In the Systems Biology setting, the parameter profile likelihood has been proposed for the calculation of confidence intervals and in addition for the investigation of parameter identifiability [4, 5]. It is increasingly applied in recent years [6-12].

For the more general setting of a model prediction, a respective theoretical concept was established decades ago [13, 14]. However, the classical calculation of a prediction profile likelihood requires analytical formulas which are only available for trivial ODE models. To circumvent this hurdle, the prediction profile likelihood approach was presented in the context of differential equation models [15, 16]. Subsequently, this concept and its use for investigating practical non-observability were rephrased in [17], but without making reference to earlier literature.

The suggested calculation procedure in [15, 16] derives the prediction profiles likelihood either based on numerical constraint- or on penalized optimization. In these publications, it has been demonstrated by Monte-Carlo simulations, that the resulting confidence intervals have desired statistical properties like proper coverage. Moreover, the prediction profile likelihood has been utilized for a data-based observability analysis and for experimental design considerations. Within this concept, sampling of the parameter space is replaced by optimization which constitutes the most efficient way to numerically evaluate the parameter space.

In the following, the potential of likelihood profiles in Systems Biology is discussed and illustrated. For this purpose, a model of the EPO receptor is used [6].