We consider moving boundary problems for biophysics and introduce a new computational framework to handle the complexity of the bulk-surface PDEs. In our framework, interpretability is maintained by adapting the fast, generalizable and accurate structure preservation scheme in [Q. Cheng and J. Shen, \textit{Computer Methods in Applied Mechanics and Engineering}, 391 (2022)]. We show that mesh distortion is mitigated by adopting the pioneering work of [B. Duan and B. Li, \textit{SIAM J. Sci. Comput.}, 46 (2024)], which is tied to an Arbitrary Lagrangian Eulerian (ALE) framework. We test our algorithms accuracy on moving surfaces with boundary for the following PDEs: advection-diffusion-reaction equations, phase-field models of Cahn-Hilliard type, and Helfrich energy gradient flows. We performed convergence studies for all the schemes introduced to demonstrate accuracy. We use a staggered approach to achieve coupling and further verify the convergence of this coupling using numerical experiments. Finally, we demonstrate broad applicability of our work by simulating state-of-the-art tests of biophysical models that involve membrane deformation.

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