Nucleo-cytoplasmic shuttling of APC can maximize β-catenin/TCF concentration

β-catenin is the key player of the canonical Wnt pathway. Its activity is mainly regulated via protein degradation. In the nucleus, its interaction with TCF initiates target gene expression. Although the functional relevance is unclear, it has been shown that β-catenin antagonists are also capable of nucleo-cytoplasmic shuttling. The focus of our systems biology analysis lies on the β-catenin subcellular distribution regulated by the antagonist and scaffolding protein APC. We address the following questions: Can the concentration of the transcription factor complex [β-catenin/TCF], which is considered as the output of the pathway, be maximized by APC nucleo-cytoplasmic shuttling and how is retention of β-catenin by APC influencing this output? We established a mathematical model based on experimental findings to examine the influence of nucleo-cytoplasmic shuttling of APC and retention of β-catenin by APC on the output of the pathway. The model is based on ordinary differential equations and includes protein shuttling between the two compartments nucleus and cytoplasm as well as protein complex formation in each compartment. We discuss how the steady state concentration of [β-catenin/TCF] is influenced by APC shuttling and retention. The analysis of the model shows that the breakdown of β-catenin cytoplasmic retention induced by APC shuttling can enhance nuclear accumulation of β-catenin and hence maximize the output of the pathway. Using mathematical modelling, we demonstrate that in certain parameter ranges, the steady state concentration of [β-catenin/TCF] benefits from APC shuttling. The inhibitory effect of APC is alleviated due to shuttling of APC. Surprisingly, our study therefore indicates that the nucleo-cytoplasmic shuttling of APC can have a beneficial effect on the output of the pathway in steady state, although APC is in general a β-catenin antagonizing protein. Furthermore, we show that saturated protein translocation can under certain conditions be modelled by pure diffusion. A difference in the shuttling rate constants of sufficient orders of magnitude leads to an accumulation in either compartment, which corresponds to saturation in translocation.