An algorithm for simulating reaction-diffusion systems on complex geometries is presented, giving insight into how the interplay of cell geometry and biochemistry can control polarity in living cells.
Cell polarity is critical to the functioning of living organisms1. It controls the correct positioning of cells in the tissue by, among other things, determining the direction of cell migration. From a molecular perspective, cell polarity results from the accumulation of certain determinants at certain locations on the cell membrane. Such localization can trigger molecular events that break the symmetry of cells and define the preferred orientation axis (e.g., the cell front and cell back during migration).2. In most cases, the relevant reactions driving polarity take place at the membrane3. Diffusion at the cell membrane is much slower than in the bulk cytoplasm. Therefore, the main cytoplasm can often be viewed as a large reservoir from which proteins are transferred to the membrane via binding/debinding reactions. Consequently, the rate-limiting processes that can drive polarization are membrane-bound. Several lines of experimentation have shown that cell polarity can be formulated as a reaction-diffusion problem, in which nonlinear biochemical interactions coupled with molecular diffusion lead to inhomogeneities in the spatial localization of key polarity determinants4. The establishment of cell polarity can therefore be described as a problem of symmetry breaking. In the simulations of existing mathematical polarity models, one usually assumes homogeneous, uniform initial conditions and observes how the dynamics produce non-uniform steady-state conditions. For practical reasons, most available cell polarity simulations are limited to overly simple geometries, namely spheres and cubes. However, the geometry of cells can be complex, and geometric features such as Gaussian curvature can affect reaction-diffusion dynamics. There is therefore a need for fast and effective numerical methods for simulating partial differential equations in complex geometries. To fill this need and elucidate how cell geometry can affect polarization, in natural informaticsMiller et al5. provide an effective numerical method for simulating continuous models on complex geometries.