We shall discuss reaction diffusion systems arising in the study of biological systems where concentration fluctuations   play an important role.  This includes the study of cell signaling, cell polarization, and directed cell motility through detection  of shallow concentration gradients.  To study these systems  we shall formulate stochastic partial differential equations to  model intrinsic fluctuations in the concentration fields associated with these processes.  A number of challenges arise in  approximating the solutions of such SPDE's. Further issues  arise when considering non-periodic domains with complex geometries which arise naturally in modeling  biological systems.  We shall also discuss an approach to obtain  spatially adaptive numerical discretization of SPDE's, which specifically address how to discretize stochastic terms at coarse-refined  interfaces.  In the spirit of the seminar we shall try to make the talk accessible to a wide audience with ample time for  informal discussion.