The Swelling of Mitochondria: In Vivo

This chapter deals with the swelling process in a living organism, so that in this case we do not have a controlled environment as we had in the test tube. Here there are three main factors that differ a great deal from the in vitro case, which was considered in Chap. 3 , that is: Mitochondria are not uniformly distributed. The induced calcium source is very localized. The cell is not a closed system. This chapter consists of two subsections. In Sect. 5.4, we formulate mitochondria swelling scenario in vivo as a PDE-ODE system with Robin boundary conditions, which in turn reflects the fact that calcium ions can enter and leave the cell across the permeable membrane. In this section we prove the well posedness and study large-time asymptotics of solutions of the corresponding initial boundary value PDE-ODE problem. Moreover, in Sect. 5.4 it is shown that partial and complete swelling behavior in vivo depends on the balance of concentration as well as on the threshold parameter of initiation for swelling. In Sect. 5.5 we illustrate the results of Sect. 5.4 with numerical simulations. Note that in the previous chapters mathematical modeling of swelling of mitochondria and their analysis in vitro, in vivo, and in silico were governed by the concept of standard diffusion for calcium concentrations. The use of the standard Laplacian to describe diffusion processes is generally accepted for mathematical idealization and indeed can explain some aspects of mitochondria swelling scenarios observed in experiments. However, from the viewpoint of applications it is also desirable to analyze the effect of degenerate diffusion, because there is no species which can diffuse infinitely fast (a property of the standard diffusion).