Phase field approach for modeling intracellular dynamics

We introduce a phase field approach for diffusion inside and outside a closed cell with damping and with source terms at the interface. The method is compared to exact solutions (where possible) and the more traditional finite element method. It is shown to be very accurate, easy to implement and computationally inexpensive. We apply our method to a recently introduced model for chemotaxis by Rappel et al. [Biophys. In dealing with free boundary problems, the so called phase field approach [2] appears as a method of choice. It has successfully been applied to various problems ranging from dendritic solidification [3], viscous fingering [4] and crack propagation [5]. In the spirit of time-dependent Ginzburg-Landau models, the method avoids the tracking of the interface by introducing an auxiliary field that locates the interface and whose dynamics is coupled to the other physical fields through an appropriate set of partial differential equations. In comparison to the more traditional boundary integral methods, the method is much simpler to implement numerically. In this Brief Report we introduce a phase field model for intracellular dynamics i.e. diffusion inside and outside a stationary, closed domain with source terms at the interface. We apply the method to a recently introduced model for the response of a Dictyostelium amoeba [6] following stimulation with the chemoattractant cAMP [1]. In [1], due to the need to use a finite element method the numerical implementation of the model was limited to two space dimensions and the cells were treated as disks. As we will see below, the phase field method is capable of faithfully capturing no-flux boundary conditions. Thus, it becomes feasible to investigate more realistic cell shapes in three dimensions. Before we introduce our approach, we like to point out some possible extensions of our methodology. The phase field approach can be easily modified to include problems where the domain boundary is not stationary. For example, force generation on cell membranes, leading to shape changes, can be incorporated within the phase field approach in a straightforward manner. This would require adding an additional equation for the phase field, but does not require the explicit calculation of a boundary. We should stress that attempting to model this type of problem using conventional techniques, with explicit boundary tracking, become quite cumbersome. Let us first introduce the most salient ingredients of our method. Our purpose is to describe the situation where some chemoattractant diffuses …