## Introduction

Molecular and cell biologists frequently question why we need mathematical models to understand biological processes. However, owing to the complexity of biological systems, it is evident that a higher level of abstraction is required to decode the ‘language’ of cells and to gain insights into how signals from the environment are integrated and how decisions for life, death, proliferation or differentiation are regulated. In recent years, much progress has been made in the qualitative analyses of biological systems. Yet, to advance the understanding of complex diseases and the process of drug discovery (including efficacy, safety and consequently the outcome in patients), computational tools are essential that are able to integrate the plethora of experimentally observed information and facilitate the prediction of cellular responses. The reliability of these predictions critically depends on the availability of quantitative data to capture cellular events over time with sufficient quality to calibrate the mathematical models.

Cellular signal transduction pathways process extracellular signals that are received by cell surface receptors; these receptors are activated and translate this information via signalling networks to cellular responses. As the ‘omics’ technologies facilitated the identification of key components of signalling pathways in high-throughput systems, the current focus is the investigation of the connectivity, crosstalk and dynamics of these networks. From systems-based approaches, we have learned that temporal dynamics [1, 2], spatial distribution [3, 4] and cell-to-cell variability [5–7] are key systems properties that lead to context-specific cellular responses. These insights serve as inspiration to further investigate the emergent properties of signalling pathways and how they are quantitatively linked to decisions concerning cell fate.

In this review we provide an overview of modelling concepts describing biological systems, in particular cancer signalling pathways, to demonstrate the power of modelling approaches in addressing urgent biological questions. We do not intend to give a theoretical introduction to modelling strategies in general as this has been previously provided elsewhere [8–10]. Rather, we emphasize that when using mathematical models to predict cellular behaviour, certain requirements concerning the modelling strategy have to be accomplished to facilitate the prediction of cellular behaviour and a deeper understanding of the biological system. Therefore, a summary of modelling signalling pathways using ordinary differential equations (ODEs) is provided. Because the identifiability of parameters in these models is one prerequisite for achieving models with high predictive power, we introduce the theoretical background on identifiability in ODE-based models.

A signal transduction network that has been intensively studied by systems-based approaches is the erythropoietin receptor (EpoR) signalling system. Therefore, we highlight the most recent advances in this field and demonstrate how mathematical modelling enabled the identification of key system properties. Furthermore, we discuss how systems-based approaches can be employed to address complex questions in pharmacology. In particular, advances in systems biology approaches that investigate the risk of Epo treatment in patients with cancer are presented.