A modelling framework describing the enzyme regulation of membrane lipids underlying gradient perception in Dictyostelium cells II: input-output analysis

Spatial sensing in Dictyostelium involves localization of the phosphoinositide lipids PI(3,4,5)P3 and PI(3,4)P2 at the leading edge of the cell in response to an external gradient. We have previously proposed a modelling framework describing the regulation of these lipids by the enzymes PI3K and PTEN. In this paper we analyse this regulation from an input-output perspective. When the inputs are homogeneous, we obtain explicit analytical expressions for the lipid concentrations as a function of enzyme concentrations and model parameters. We also show that the system can be cast as an open-loop bilinear control system, and employ control engineering tools to show that a local three-dimensional region in the four-dimensional phase space can be accessed by temporally varying either or both enzyme concentrations. For spatially graded enzyme profiles, we show that diffusion limits the extent to which lipid profiles can be manipulated by enzymes. However, we also demonstrate that for certain ranges of network parameters, increasing lipid diffusion can lead to an increase in steady-state leading-edge concentrations of PI(3,4,5)P3 or PI(3,4)P2, even though all lipid diffusion coefficients are equal. Finally, in order to determine the extent to which lipid profiles can be regulated by the enzymes, we formulate and solve inverse problems, where we determine the enzyme profiles required to realize particular lipid profiles at steady state.