Patterning of the MinD cell division protein in cells of arbitrary shape can be predicted using a heuristic dispersion relation

Many important cellular processes require the accurate positioning of subcellular structures. Underpinning many of these are protein systems that spontaneously generate spatiotemporal patterns. In some cases, these systems can be described by non-linear reaction-diffusion equations, however, a full description of such equations is rarely available. A well-studied patterning system is the Min protein system that underpins the positioning of the FtsZ contractile ring during cell division in Escherichia coli. Using a coordinate-free linear stability analysis, the reaction terms can be separated from the geometry of a cell. The reaction terms produce a dispersion relation that can be used to predict patterning on any cell shape and size. Applying linear stability analysis to an accurate mathematical model of the Min system shows that while it correctly predicts the onset of patterning, the dispersion relation fails to predict oscillations and quantitative mode transitions. However, we show that data from full solutions of the Min model can be used to generate a heuristic dispersion relation. We show that this heuristic dispersion relation can be used to approximate the Min protein patterning obtained by full simulations of the non-linear reaction-diffusion equations. Moreover, it also predicts Min patterning obtained from experiments where the shapes of E. coli cells have been deformed into rectangles or arbitrary shapes. Using this procedure, it should be possible to generate heuristic dispersion relations from protein patterning data or simulations for any patterning process and subsequently use these to predict patterning for arbitrary cell shapes.