Mathematical Analysis of Vitro Models

This chapter is devoted to the mathematical analysis of the model obtained in Sect. 3.3 and consists of four subsections. In Sect. 4.1 we verify the obtained model and prove the well posedness of the PDE-ODE systems corresponding to it. Having ordinary differential equations describing the evolution in time of three subpopulations of mitochondria indicates that we assume the mitochondria in the test tube, as well as within cells, do not move in any direction and hence the spatial effects are only introduced by the calcium evolution obeying a partial differential equation (PDE), namely a reaction-diffusion equation. We study the longtime dynamics of solutions of the PDE-ODE coupling. We obtain the complete classifications of the limiting profile of solutions and study partial and complete classification scenarios depending on the given data. Note that these scenarios, namely partial and complete swelling scenarios, have been observed in experiments. Section 4.3 deals with the numerical simulation (in silico) of PDE-ODE systems. One of the remarkable results here is the clearly visible spreading calcium wave. If we compare the dynamics with those of simple diffusion without any positive feedback, the numerical results show that the resulting calcium evolution induced by mitochondria swelling is indeed completely different. Our numerical simulations show that a small change in the initial distribution of calcium is enough to shift the behavior from partial to complete swelling behavior. In Sect. 4.4 we continue our analysis of a coupled PDE-ODE model of calcium induced mitochondria swelling in vitro. More precisely, we study the longtime dynamics of solutions of PDE-ODE systems under homogeneous Dirichlet boundary conditions. Note that, biologically, this kind of boundary condition appears if we put some chemical material on the test-tube wall that binds calcium ions and hence removes it as a swelling inducer. We especially emphasize that the analytical machinery that was developed in Sect. 4.1 is not applicable under homogeneous Dirichlet boundary conditions and therefore must be extended. In this section we show that the calcium ion concentration will tend to zero and that, in general, complete swelling will not occur as time goes to infinity. This distinguishes the situation under Dirichlet boundary conditions from the situation under Neumann boundary conditions that were analyzed in Sect. 4.1. In Sect. 4.5, we carry out numerical simulations validating the analytical results of Sect. 4.4.