The MEK/ERK signalling pathway is involved in cell division, cell specialisation, survival and cell death [76]. Here we study a polynomial dynamical system describing the dynamics of MEK/ERK proposed by Yeung et al. [94] with their experimental setup, data and known biological information. The experimental dataset is a time-course of ERK measurements in different phosphorylation states following activation of either wild-type MEK or MEK mutations associated with cancer or developmental defects. We demonstrate how methods from computational algebraic geometry, differential algebra, Bayesian statistics and computational algebraic topology can inform the model reduction, identification and parameter inference of MEK variants, respectively. Throughout, we show how this algebraic viewpoint offers a rigorous and systematic analysis of such models.