Algebra, Geometry and Topology of ERK Kinetics

Abstract

The MEK/ERK signalling pathway is involved in cell division, cell specialisation, survival and cell death (Shaul and Seger in Biochim Biophys Acta (BBA)-Mol Cell Res 1773(8):1213–1226, 2007). Here we study a polynomial dynamical system describing the dynamics of MEK/ERK proposed by Yeung et al. (Curr Biol 2019, https://doi.org/10.1016/j.cub.2019.12.052) 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.

Phase plane dynamics of ERK phosphorylation

“…Next, we use a steady-state approximation for complexes, which is valid when ( 16 , 21 ), and rescale the problem as follows: . This leads to the following reduced model: …”

“…The simplifications include a coarse-grained description of the ERK/MEK association and ATP/ADP exchange. Even with these simplifications, there are six rate constants that prove hard to estimate with high confidence based on the measured time courses ( 16 ). This difficulty in solving the inverse kinetics problem is a generic feature of multiparameter dynamic models ( 17 ).…”

Simulating BRAFV600E-MEK-ERK signalling dynamics in response to vertical inhibition treatment strategies

“…Marsh et al used model reduction techniques, including those based on algebraic topology, to reduce an ERK reaction network model. Such methods are useful as mathematical models of intracellular pathway dynamics can become large, which may complicate model analysis and parameter estimation 59 . Moreover, the experimental and computational modelling studies by Schoeberl et al 28 and Fujioka et al 60 have made important contributions to the estimation and reporting of parameter values that pertain to MAPK-cascade-related rate constants and kinase concentrations in human cells.…”

Simulating BRAFV600E-MEK-ERK signalling dynamics in response to vertical inhibition treatment strategies

“…Such methods are useful as mathematical models of intracellular pathway dynamics can become large, which may complicate model analysis and parameter estimation. 59 Moreover, the experimental and computational modelling studies by Schoeberl et al 28 and Fujioka et al 60 have made important contributions to the estimation and reporting of parameter values that pertain to MAPK-cascade related rate constants and kinase concentrations in human cells.…”