Accurate prediction of deformation and collapse of the upper airway during breathing is required for effective and personalised treatment of obstructive sleep apnoea (OSA). While numerical modelling techniques such as fluid–structure interaction (FSI) are promising, an outstanding challenge is to accurately predict the deformation of the airway during breathing and thus the occurrence of OSA. These difficulties arise because the effective stiffness of the soft tissue in the human upper airway varies due to neuromuscular effects on the stiffness of the underlying muscles. In addition, both the elasticity and anisotropy of the soft tissues along the upper airway are poorly characterised. Finally, gravitational effects on anatomic features are yet to be considered. In this study, a validated FSI technique is introduced that allows prediction of the extent and position of the major deformation in the upper airway. This technique is used to analyse the behaviour of the upper airway in the two most common sleeping positions and for a range of effective tissue stiffnesses. The results demonstrate that sleeping position, gravity and soft tissue stiffness (used here as a proxy for neuromuscular effects) are the main factors that affect upper airway collapse. Therefore, this study provides new insights into the mechanisms of OSA and a new methodology that significantly advances the patient-specific treatment of OSA.