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June  2015, 20(4): 1031-1057. doi: 10.3934/dcdsb.2015.20.1031

## On an ODE-PDE coupling model of the mitochondrial swelling process

 1 Institute of Computational Biology, Helmholtz Center Munich, Ingolstädter, Landstrasse 1, 85764 Neuherberg, Germany 2 Helmholtz Zentrum München, Institute of Computational Biology, Ingolstädter Landstrasse1, D-85764 Neuherberg, 3 Department of Applied Physics, Waseda University, 3-4-1, Okubo, Tokyo, 169-8555 4 Institute of Molecular Toxicology and Pharmscology, Helmholtz Center Munich, Ingolstädter, Landstrasse 1, 85764 Neuherberg, Germany, Germany

Received  June 2013 Revised  August 2014 Published  February 2015

Mitochondrial swelling has huge impact to multicellular organisms since it triggers apoptosis, the programmed cell death. In this paper we present a new mathematical model of this phenomenon. As a novelty it includes spatial effects, which are of great importance for the in vivo process. Our model considers three mitochondrial subpopulations varying in the degree of swelling. The evolution of these groups is dependent on the present calcium concentration and is described by a system of ODEs, whereas the calcium propagation is modeled by a reaction-diffusion equation taking into account spatial effects. We analyze the derived model with respect to existence and long-time behavior of solutions and obtain a complete mathematical classification of the swelling process.
Citation: Sabine Eisenhofer, Messoud A. Efendiev, Mitsuharu Ôtani, Sabine Schulz, Hans Zischka. On an ODE-PDE coupling model of the mitochondrial swelling process. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1031-1057. doi: 10.3934/dcdsb.2015.20.1031
##### References:
 [1] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, 5th edition, Garland Science, 2007. Google Scholar [2] M. A. Aon, S. Cortassa, E. Marban and B. O'Rourke, Synchronized whole cell oscillations in mitochondrial metabolism triggered by a local release of reactive oxygen species in cardiac myocytes, Journal of Biological Chemistry, 278 (2003), 44735-44744. doi: 10.1074/jbc.M302673200.  Google Scholar [3] M. A. Aon, S. Cortassa and B. O'Rourke, Percolation and criticality in a mitochondrial network, Proceedings of the National Academy of Sciences of USA, 101 (2004), 4447-4452. doi: 10.1073/pnas.0307156101.  Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations, North Holland, 1992.  Google Scholar [5] S. Baranov, I. Stavrovskaya, A. Brown, A. Tyryshkin and B. Kristal, Kinetic model for $Ca^{2+}$-induced permeability transition in energized liver mitochondria discriminates between inhibitor mechanisms, Journal of Biological Chemistry, 283 (2008), 665-676. Google Scholar [6] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, vol. 5, North Holland, 1973. Google Scholar [7] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar [8] G. Calamita, D. Ferri, P. Gena, G. Liquori, A. Cavalier, D. Thomas and M. Svelto, The inner mitochondrial membrane has aquaporin-8 water channels and is highly permeable to water, Journal of Biological Chemistry, 280 (2005), 17149-17153. doi: 10.1074/jbc.C400595200.  Google Scholar [9] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, 2002.  Google Scholar [10] S. Eisenhofer, A Coupled System of Ordinary and Partial Differential Equations Modeling the Swelling of Mitochondria,, PhD Thesis, ().   Google Scholar [11] 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.  Google Scholar [12] D. Gilbarg and T. N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, 1983. Google Scholar [13] D. Green and G. Kroemer, The pathophysiology of mitochondrial cell death, Science, 305 (2004), 626-629. doi: 10.1126/science.1099320.  Google Scholar [14] D. Hunter, R. Haworth and J. Southard, Relationship between configuration, function, and permeability in calcium-treated mitochondria, Journal of Biological Chemistry, 251 (1976), 5069-5077. Google Scholar [15] G. Kroemer, L. Galluzzi and C. Brenner, Mitochondrial membrane permeabilization in cell death, Physiological Reviews, 87 (2007), 99-163. doi: 10.1152/physrev.00013.2006.  Google Scholar [16] G. Leoni and M. Morini, Necessary and sufficient conditions for the chain rule in $W_{loc}^{1,1}(\mathbbR^n ; \mathbbR^d)$ and $BV_{loc}(\mathbbR^n ; \mathbbR^d )$, J. Eur. Math. Soc., 9 (2007), 219-252. doi: 10.4171/JEMS/78.  Google Scholar [17] S. Massari, Kinetic analysis of the mitochondrial permeability transition, Journal of Biological Chemistry, 271 (1996), 31942-31948. Google Scholar [18] S. Naghdi, M. Waldeck-Weiermair, I. Fertschai, M. Poteser, W. Graier and R. Malli, Mitochondrial $Ca^{2+}$ uptake and not mitochondrial motility is required for STIM1-Orai1-dependent store-operated $Ca^{2+}$ entry, Journal of Cell Science, 123 (2010), 2553-2564. Google Scholar [19] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, Journal of Differential Equations, 46 (1982), 268-299. doi: 10.1016/0022-0396(82)90119-X.  Google Scholar [20] P. Petit, M. Goubern, P. Diolez, S. Susin, N. Zamzami and G. Kroemer, Disruption of the outer mitochondrial membrane as a result of large amplitude swelling: The impact of irreversible permeability transition, FEBS letters, 426 (1998), 111-116. doi: 10.1016/S0014-5793(98)00318-4.  Google Scholar [21] 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. Google Scholar [22] A. Pokhilko, F. Ataullakhanov and E. Holmuhamedov, Mathematical model of mitochondrial ionic homeostasis: Three modes of $Ca^{2+}$ transport, Journal of Theoretical Biology, 243 (2006), 152-169. doi: 10.1016/j.jtbi.2006.05.025.  Google Scholar [23] R. Rizzuto, S. Marchi, M. Bonora, P. Aguiari, A. Bononi, D. De Stefani, C. Giorgi, S. Leo, A. Rimessi, R. Siviero, E. Zecchini and P. Pinton, $Ca^{2+}$ transfer from the ER to mitochondria: When, how and why, Biochimica et Biophysica Acta (BBA)-Bioenergetics, 1787 (2009), 1342-1351. Google Scholar [24] R. Rizzuto and T. Pozzan, Microdomains of intracellular $Ca^{2+}$: Molecular determinants and functional consequences, Physiological Reviews, 86 (2006), 369-408. Google Scholar [25] V. Selivanov, F. Ichas, E. Holmuhamedov, L. Jouaville, Y. Evtodienko and J. Mazat, A model of mitochondrial $Ca^{2+}$-induced $Ca^{2+}$ release simulating the $Ca^{2+}$ oscillations and spikes generated by mitochondria, Biophysical Chemistry, 72 (1998), 111-121. Google Scholar [26] H. Zischka, N. Larochette, F. Hoffmann, D. Hamöller, 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. Google Scholar

show all references

##### References:
 [1] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, 5th edition, Garland Science, 2007. Google Scholar [2] M. A. Aon, S. Cortassa, E. Marban and B. O'Rourke, Synchronized whole cell oscillations in mitochondrial metabolism triggered by a local release of reactive oxygen species in cardiac myocytes, Journal of Biological Chemistry, 278 (2003), 44735-44744. doi: 10.1074/jbc.M302673200.  Google Scholar [3] M. A. Aon, S. Cortassa and B. O'Rourke, Percolation and criticality in a mitochondrial network, Proceedings of the National Academy of Sciences of USA, 101 (2004), 4447-4452. doi: 10.1073/pnas.0307156101.  Google Scholar [4] A. Babin and M. Vishik, Attractors of Evolution Equations, North Holland, 1992.  Google Scholar [5] S. Baranov, I. Stavrovskaya, A. Brown, A. Tyryshkin and B. Kristal, Kinetic model for $Ca^{2+}$-induced permeability transition in energized liver mitochondria discriminates between inhibitor mechanisms, Journal of Biological Chemistry, 283 (2008), 665-676. Google Scholar [6] H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert, vol. 5, North Holland, 1973. Google Scholar [7] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar [8] G. Calamita, D. Ferri, P. Gena, G. Liquori, A. Cavalier, D. Thomas and M. Svelto, The inner mitochondrial membrane has aquaporin-8 water channels and is highly permeable to water, Journal of Biological Chemistry, 280 (2005), 17149-17153. doi: 10.1074/jbc.C400595200.  Google Scholar [9] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, 2002.  Google Scholar [10] S. Eisenhofer, A Coupled System of Ordinary and Partial Differential Equations Modeling the Swelling of Mitochondria,, PhD Thesis, ().   Google Scholar [11] 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.  Google Scholar [12] D. Gilbarg and T. N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, 1983. Google Scholar [13] D. Green and G. Kroemer, The pathophysiology of mitochondrial cell death, Science, 305 (2004), 626-629. doi: 10.1126/science.1099320.  Google Scholar [14] D. Hunter, R. Haworth and J. Southard, Relationship between configuration, function, and permeability in calcium-treated mitochondria, Journal of Biological Chemistry, 251 (1976), 5069-5077. Google Scholar [15] G. Kroemer, L. Galluzzi and C. Brenner, Mitochondrial membrane permeabilization in cell death, Physiological Reviews, 87 (2007), 99-163. doi: 10.1152/physrev.00013.2006.  Google Scholar [16] G. Leoni and M. Morini, Necessary and sufficient conditions for the chain rule in $W_{loc}^{1,1}(\mathbbR^n ; \mathbbR^d)$ and $BV_{loc}(\mathbbR^n ; \mathbbR^d )$, J. Eur. Math. Soc., 9 (2007), 219-252. doi: 10.4171/JEMS/78.  Google Scholar [17] S. Massari, Kinetic analysis of the mitochondrial permeability transition, Journal of Biological Chemistry, 271 (1996), 31942-31948. Google Scholar [18] S. Naghdi, M. Waldeck-Weiermair, I. Fertschai, M. Poteser, W. Graier and R. Malli, Mitochondrial $Ca^{2+}$ uptake and not mitochondrial motility is required for STIM1-Orai1-dependent store-operated $Ca^{2+}$ entry, Journal of Cell Science, 123 (2010), 2553-2564. Google Scholar [19] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, Journal of Differential Equations, 46 (1982), 268-299. doi: 10.1016/0022-0396(82)90119-X.  Google Scholar [20] P. Petit, M. Goubern, P. Diolez, S. Susin, N. Zamzami and G. Kroemer, Disruption of the outer mitochondrial membrane as a result of large amplitude swelling: The impact of irreversible permeability transition, FEBS letters, 426 (1998), 111-116. doi: 10.1016/S0014-5793(98)00318-4.  Google Scholar [21] 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. Google Scholar [22] A. Pokhilko, F. Ataullakhanov and E. Holmuhamedov, Mathematical model of mitochondrial ionic homeostasis: Three modes of $Ca^{2+}$ transport, Journal of Theoretical Biology, 243 (2006), 152-169. doi: 10.1016/j.jtbi.2006.05.025.  Google Scholar [23] R. Rizzuto, S. Marchi, M. Bonora, P. Aguiari, A. Bononi, D. De Stefani, C. Giorgi, S. Leo, A. Rimessi, R. Siviero, E. Zecchini and P. Pinton, $Ca^{2+}$ transfer from the ER to mitochondria: When, how and why, Biochimica et Biophysica Acta (BBA)-Bioenergetics, 1787 (2009), 1342-1351. Google Scholar [24] R. Rizzuto and T. Pozzan, Microdomains of intracellular $Ca^{2+}$: Molecular determinants and functional consequences, Physiological Reviews, 86 (2006), 369-408. Google Scholar [25] V. Selivanov, F. Ichas, E. Holmuhamedov, L. Jouaville, Y. Evtodienko and J. Mazat, A model of mitochondrial $Ca^{2+}$-induced $Ca^{2+}$ release simulating the $Ca^{2+}$ oscillations and spikes generated by mitochondria, Biophysical Chemistry, 72 (1998), 111-121. Google Scholar [26] H. Zischka, N. Larochette, F. Hoffmann, D. Hamöller, 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. Google Scholar
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