In many cellular systems, stresses generated by the cytoskeleton are regulated by diffuse chemical signals. Simultaneously, these stresses may drive the flow of intracellular fluid, which in-turn advects said chemical signals. The mechano-chemical coupling in these complex materials can often lead to complex pattern formation, including various standing, traveling and rotating waves. In this work, we present one example of pattern formation in the plasmodium of the slime mold Physarum polycephalum, where the relevant instability determines the geometry of the cell. We then present a framework to study and simulate mechano-chemical pattern formation in moving geometries using the Immersed Boundary Method (IBM). The IBM is widely used in the simulation of systems involving fluid-structure interaction. However, the study of chemical systems often necessitates the ability to impose Neumann or Robin boundary conditions at irregular locations on the Eulerian grid. To address this, we present a method for imposing Neumann and Robin boundary conditions within an IBM framework. The method requires solving a larger, augmented linear system. However, this sytem is well-conditioned and can be solved with a small number of iterations of a standard Kylov method without preconditioning. This method is then used to simulate our system of interest, and determine the cell geometries which result from mechano-chemical instabilities.