# American Institute of Mathematical Sciences

August  2017, 37(7): 4131-4158. doi: 10.3934/dcds.2017176

## Mathematical analysis of an in vivo model of mitochondrial swelling

 1 Institute for Computational Biology, Helmholtz Zentrum München, Ingolstäder Landstr. 1, 85764 Neuherberg, Germany 2 Department of Applied Phsyics, School of Science and Engineering, Waseda University 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-855, Japan 3 Department of Mathematics and Statistics, University of Guelph, Guelph ON, N1G2W1, Canada

* Corresponding author: Messoud Efendiev

Received  October 2016 Revised  February 2017 Published  April 2017

Fund Project: M.O. is partly supported by the Grant-in-Aid for Scientific Research #15K13451, the Ministry of Education, Culture, Sports, Science, and Technology, Japan; H.J.E. is partly supported by the Natrural Science and Engineering Researc Council of Canada through a Discovery Grant.

We analyze the effect of Robin boundary conditions in a mathematical model for a mitochondria swelling in a living organism. This is a coupled PDE/ODE model for the dependent variables calcium ion contration and three fractions of mitochondria that are distinguished by their state of swelling activity. The model assumes that the boundary is a permeable 'membrane', through which calcium ions can both enter or leave the cell. Under biologically relevant assumptions on the data, we prove the well-posedness of solutions of the model and study the asymptotic behavior of its solutions. We augment the analysis of the model with computer simulations that illustrate the theoretically obtained results.

Citation: Messoud Efendiev, Mitsuharu Ôtani, Hermann J. Eberl. Mathematical analysis of an in vivo model of mitochondrial swelling. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4131-4158. doi: 10.3934/dcds.2017176
##### References:
 [1] S. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal., 41 (2003), 306-324.  doi: 10.1137/S0036142902401311. [2] H. Brézis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans un Espace de Hilbert, North Holland, Amsterdam, The Netherlands, 1973. [3] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello), Academic Press, (1971), 101-179. [4] M. A. Efendiev, M. Ôtani and H. J. Eberl, A coupled PDE/ODE model of mitochondrial swelling: Large-time behavior of homogeneous Dirichlet problem, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 1-13.  doi: 10.1166/jcsmd.2015.1070. [5] S. Eisenhofer, A coupled system of ordinary and partial differential equations modeling the swelling of mitochondria, PhD Dissertation, TU Munich, 2013. [6] S. Eisenhofer, M. A. Efendiev, M. Ôtani, S. Schulz and H. Zischka, On a ODE-PDE coupling model of the mitochondrial swelling process, Discrete and Continuous Dynamical Syst. Ser. B, 20 (2015), 1031-1057.  doi: 10.3934/dcdsb.2015.20.1031. [7] S. Eisenhofer, F. Toókos, B. A. Hense, S. Schulz, F. Filbir and H. Zischka, A mathematical model of mitochondrial swelling BMC Research Notes, 3 (2010), p67. doi: 10.1186/1756-0500-3-67. [8] G. Kroemer, L. Galluzzi and C. Brenner, Mitochondrial membrane permeabilization in cell death, Physiological Reviews, 87 (2007), 99-163. [9] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46 (1982), 268-299.  doi: 10.1016/0022-0396(82)90119-X. [10] V. Petronilli, C. Cola, S. Massari, R. Colonna and P. Bernardi, Physiological effectors modify voltage sensing by the cyclosporin A-sensitive permeability transition pore of mitochondria, Journal of Biological Chemistry, 268 (1993), 21939-21945. [11] R. Rizzuto and T. Pozzan, Microdomains of intracellular $\textrm{Ca}^{2+}$: Molecular determinants and functional consequences, Physiological Reviews, 86 (2006), 369-408. [12] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [13] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, J. A. Barth, 1995. [14] H. Zischka, N. Larochette, F. Hoffmann, D. Hamöoller, N. Jägemann, J. Lichtmannegger, L. Jennen, J. Müller-Höcker, F. Roggel, M. Göttlicher, A. M. Vollmar and G. Kroemer, Electrophoretic analysis of the mitochondrial outer membrane rupture induced by permeability transition, Analytical Chemistry, 80 (2008), 5051-5058.  doi: 10.1021/ac800173r.

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##### References:
 [1] S. Brenner, Poincaré-Friedrichs inequalities for piecewise $H^1$ functions, SIAM J. Numer. Anal., 41 (2003), 306-324.  doi: 10.1137/S0036142902401311. [2] H. Brézis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans un Espace de Hilbert, North Holland, Amsterdam, The Netherlands, 1973. [3] H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello), Academic Press, (1971), 101-179. [4] M. A. Efendiev, M. Ôtani and H. J. Eberl, A coupled PDE/ODE model of mitochondrial swelling: Large-time behavior of homogeneous Dirichlet problem, Journal of Coupled Systems and Multiscale Dynamics, 3 (2015), 1-13.  doi: 10.1166/jcsmd.2015.1070. [5] S. Eisenhofer, A coupled system of ordinary and partial differential equations modeling the swelling of mitochondria, PhD Dissertation, TU Munich, 2013. [6] S. Eisenhofer, M. A. Efendiev, M. Ôtani, S. Schulz and H. Zischka, On a ODE-PDE coupling model of the mitochondrial swelling process, Discrete and Continuous Dynamical Syst. Ser. B, 20 (2015), 1031-1057.  doi: 10.3934/dcdsb.2015.20.1031. [7] S. Eisenhofer, F. Toókos, B. A. Hense, S. Schulz, F. Filbir and H. Zischka, A mathematical model of mitochondrial swelling BMC Research Notes, 3 (2010), p67. doi: 10.1186/1756-0500-3-67. [8] G. Kroemer, L. Galluzzi and C. Brenner, Mitochondrial membrane permeabilization in cell death, Physiological Reviews, 87 (2007), 99-163. [9] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46 (1982), 268-299.  doi: 10.1016/0022-0396(82)90119-X. [10] V. Petronilli, C. Cola, S. Massari, R. Colonna and P. Bernardi, Physiological effectors modify voltage sensing by the cyclosporin A-sensitive permeability transition pore of mitochondria, Journal of Biological Chemistry, 268 (1993), 21939-21945. [11] R. Rizzuto and T. Pozzan, Microdomains of intracellular $\textrm{Ca}^{2+}$: Molecular determinants and functional consequences, Physiological Reviews, 86 (2006), 369-408. [12] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [13] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, J. A. Barth, 1995. [14] H. Zischka, N. Larochette, F. Hoffmann, D. Hamöoller, N. Jägemann, J. Lichtmannegger, L. Jennen, J. Müller-Höcker, F. Roggel, M. Göttlicher, A. M. Vollmar and G. Kroemer, Electrophoretic analysis of the mitochondrial outer membrane rupture induced by permeability transition, Analytical Chemistry, 80 (2008), 5051-5058.  doi: 10.1021/ac800173r.
Model simulation with $\alpha=10<C^-$: Shown are $u, N_1, N_2, N_3$ for selected times.
Model simulation with $\alpha=10<C^-$: Shown is $N_1$ for selected times.
Simulation to illustrate partial swelling in Theorem 5.2, using initial data (ref{T2init:eq}): shown is the minimum value of $N_2$ as a function of time for different base calcium ion concentrations $u_{base}$ (top left), along with the steady state distributions for $N_1$ (top right), $N_2$ (bottom left), and $N_3$ (bottom right) in the case $u_{base}=100$.
Mitochondria populations $N_1$ and $N_2$ as a function of time in three points of the domain on a line through the center point: A (close to the boundary), B (half way between boundary and center), C (in the center), for six different values of the external calcium ion concentration $\alpha$.
Default parameter values, cf also [5]
 parameter symbol value remark lower (initiation) swelling threshold $C^-$ 20 (varied) upper (maximum) swelling threshold $C^+$ 200 maximum transition rate for $N_1\rightarrow N_2$ $f^\ast$ 1 maximum transition rate for $N_2\rightarrow N_3$ $g^\ast$ 1 diffusion coefficient $d_1$ 0.2 (varied) feedback parameter $d_2$ 30
 parameter symbol value remark lower (initiation) swelling threshold $C^-$ 20 (varied) upper (maximum) swelling threshold $C^+$ 200 maximum transition rate for $N_1\rightarrow N_2$ $f^\ast$ 1 maximum transition rate for $N_2\rightarrow N_3$ $g^\ast$ 1 diffusion coefficient $d_1$ 0.2 (varied) feedback parameter $d_2$ 30
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