Many systems in the engineering and biology involve moving interfaces or boundaries. Front tracking method is one of the most accurate and efficient computational approaches for studying such systems. A main challenge of developing front tracking algorithms is to capture the interface topological changes. In this talk I shall introduce an improved three-dimensional front tracking method and consider an application for turbulent mixing driven by Rayleigh-Taylor instability, which shows an excellent agreement with the experiments. For the second part of the talk, I will present a computational analysis of cell signaling in biology and medicine. Scaffold, a class of proteins, plays many important roles in signal transduction. Through studying various models of scaffold, I will show novel regulations induced by its spatial location and switch-like responses due to scaffold. To efficiently compute the models, we introduce a new fast numerical algorithm incorporated with adaptive mesh refinement for solving the stiff systems with spatial dynamics.