Abstract
We define a class of cellular interface problems (short: CIP) that mathematically model the exchange of molecules in a compartmentalised living cell. Defining and eventually solving such compartmental problems is important for several reasons. They are needed to understand the organisation of life itself, for example by exploring different ’origin of life’ hypothesis based on simple metabolic pathways and their necessary division into one or more compartments. In more complex forms investigating cellular interface problems is a way to understand cellular homeostasis of different types, for example ionic fluxes and their composition between all different cellular compartments. Understanding homeostasis and its collapse is important for many physiological medical applications. This class of models is also necessary to formulate efficiently and in detail complex signalling processes taking place in different cell types, with eukaryotic cells the most complex ones in terms of sophisticated compartmentalisation. We will compare such mathematical models of signalling pathways with rule-based models as formulated in membrane computing in a final discussion. The latter is a theory that investigates computer programmes with the help of biological concepts, like a subroutine exchanging data with the environment, in this case a programme with its global variables.
Keywords
- Markov Chain
- Interface Condition
- Membrane System
- Reaction Network
- Molecular Concentration
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
Aguda, B.D., Clarke, B.L.: Bistability in chemical reaction networks: theory and application to the peroxidase-oxidase reaction. J. Chem. Phys. 87, 3461–3470 (1987)
Babuška, I., Banerjee, U., Osborn, J.E.: Meshless and generalized finite element methods: a survey of some major results. In: [26], p. 120. Springer, Berlin (2003)
Babuška, I., Melenk, J.M.: The partition of unity method. Internat. J. Numer. Methods Engrg. 40(4), 727–758 (1997)
Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Crysl, P.: Meshless methods: An overview and recent developments. Computational Methods in Applied Mechanical Engineering 139, 3–47 (1996)
Bronnikova, T.V., Fed’kina, V.R., Schaffer, W.M., Olsen, L.F.: Period-doubling bifurcations in a detailed model of the peroxidase-oxidase reaction. J. Phys. Chem. 99, 9309–9312 (1995)
Clarke, B.L.: Stability of complex reaction networks. In: Prigogine, I., Rice, S. (eds.) Advan. Chem. Phys., vol. 43, pp. 1–216. Wiley, New York (1980)
Clarke, B.L., Jiang, W.: Method for deriving Hopf and saddle-node bifurcation hypersurfaces and application to a model of the Belousov-Zhabotinskii reaction. J. Chem. Phys. 99, 4464–4476 (1993)
Cornish-Bowden, A., Hofmeyer, J.-H.S.: The role of stoichiometric analysis in studies of metabolism: an example. J. Theor. Biol. 216, 179–191 (2002)
Craciun, G., Tang, Y., Feinberg, M.: Understanding bistability in complex enzyme-driven reaction networks. PNAS 30(103), 8697–8702 (2006)
Domijan, M., Kirkilionis, M.: Graph Theory and Qualitative Analysis of Reaction Networks. Networks and Heterogeneous Media 3, 95–322 (2008)
Domijan, M., Kirkilionis, M.: Bistability and Oscillations in Chemical Reaction Systems. Journal of Mathematical Biology (in press, 2008)
Eigel, M., George, E., Kirkilionis, M.: A Meshfree Partition of Unity Method for Diffusion Equations on Complex Domains. IMA Journal of Numerical Analysis (in press, 2008)
Eigel, M., Erwin, G., Kirkilionis, M.: The Partition of Unity Meshfree Method for Solving Transport-Reaction Equations on Complex Domains: Implementation and Applications in the Life Sciences. In: Griebel, M., Schweitzer, A. (eds.) Meshfree Methods for Partial Differential Equations IV. Lecture Notes in Computational Science and Engineering, vol. 65. Springer, Heidelberg (2009)
Eigel, M.: An adaptive mashfree method for reaction-diffusion processes on complex and nested domains. PhD thesis, University of Warwick (2008)
Field, R.J., Körös, E., Noyes, R.M.: Oscillations in chemical systems. 2. Thorough analysis of temporal oscillation in bromate-cerium-malonic acid system. J. Am. Chem. Soc. 94(25), 8649–8664 (1972)
Ferry, J.G., House, C.H.: The Stepwise Evolution of Early life Driven by Energy Conservation. Molecular Biology and Evolution 23(6), 1286–1292 (2006)
Field, R.J., Noyes, R.M.: Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60, 1877–1884 (1974)
Goldbeter, A., Dupont, G.: Allosteric regulation, cooperativity, and biochemical oscillations. Biophys. Chem. 37, 341–353 (1990)
Guckenheimer, J., Holmes, J.P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Applied Mathematics Sciences, vol. 42. Springer, Heidelberg (2002)
Heinrich, R., Schuster, S.: The Regulation of Cellular Processes. Chapman & Hall, Boca Raton (1996)
Hunt, K.L.C., Hunt, P.M., Ross, J.: Nonlinear Dynamics and Thermodynamics of Chemical Reactions Far From Equilibrium. Annu. Rev. Phys. Chem. 41, 409–439 (1990)
Ivanova, A.N.: Conditions for uniqueness of stationary state of kinetic systems, related, to structural scheme of reactions. Kinet. Katal. 20(4), 1019–1023 (1979)
Keener, J., Sneyd, J.: Mathematical Physiology. Springer, Heidelberg (1998)
Kirkilionis, M., et al. (eds.): Trends in Nonlinear Analysis. Springer, Heidelberg (2003)
Kirkilionis, M.: Reaction systems, graph theory and dynamical networks. In: Gauges, R., et al. (eds.) 5th Workshop on Computation of Biochemical Pathways and Genetic Networks, pp. 131–150. Logos-Verlag (2008)
Kirkilionis, M., Sbano, L.: An Averaging Principle for Combined Interaction Graphs. Part I: Connectivity and Applications to Genetic Switches. In: Advances in Complex Systems (2008); in revision. Also available as WMI Preprint 5/2008
Klonowski, W.: Simplifying principles for chemical and enzyme reaction kinetics. Biophys. Chem. 18, 73–87 (1983)
Krischer, K., Eiswirth, M., Ertl, G.: Oscillatory CO oxidation on Pt(110): Modeling of temporal self-organisation. J. Chem. Phys. 96, 9161–9172 (1992)
Kuznetsov, Y.: Elements of Applied Bifurcation Theory, 2nd edn. Applied Mathematical Sciences, p. 112. Springer, Heidelberg (1998)
Melenk, J.M., Babuška, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139(1-4), 289–314 (1996)
Misteli, T., Gunjan, A., Hock, R., Bustink, M., David, T.: Dynamic binding of histone H1 to chromatin in living cells. Nature 408, 877–881 (2000)
Paun, G.: Membrane Computing. An Introduction. Springer, Heidelberg (2002)
Ratto, G.M., Pizzorusso, T.: A kinase with a vision: Role of ERK in the synaptic plasticity of the visual cortex. Adv. Exp. Med. Biol. 557, 122–132 (2006)
Pavliotis, G.A., Stuart, A.M.: An Introduction to Multiscale Methods. Springer, Heidelberg (2008)
Perelson, A.S., Wallwork, D.: The arbitrary dynamic behavior of open chemical reaction systems. J. Chem. Phys. 66, 4390–4394 (1977)
Sbano, L., Kirkilionis, M.: Molecular Reactions Described as Infinite and Finite State Systems. Part I: Continuum Approximation. Warwick Preprint 05/2007
Sbano, L., Kirkilionis, M.: Molecular Reactions Described as Infinite and Finite State Systems Part Ii: Deterministic Dynamics and Examples. Warwick Preprint 07/2007
Sbano, L., Kirkilionis, M.: Multiscale Analysis of Reaction Networks. Theory in Biosciences 127, 107–123 (2008)
Schweitzer, M.A.: Efficient implementation and parallelization of meshfree and particle methods—the parallel multilevel partition of unity method, pp. 195–262. Springer, Berlin (2005)
Siegel, I.H.: Enzyme Kinetics. Wiley, Chichester (1975)
Selkov, E.E.: Self-oscillations in glycolysis. 1. A simple kinetic model. Eur. J. Biochem. 4, 79–86 (1968)
Slepchenko, B.M., Terasaki, M.: Cyclin aggregation and robustness of bio-switching. Mol. Biol. Cell. 14, 4695–4706 (2003)
Tyson, J.J., Chen, K., Novak, B.: Network dynamics and cell physiology. Nat. Rev. Mol. Cell Biol. 2, 908–916 (2001)
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Kirkilionis, M., Domijan, M., Eigel, M., George, E., Li, M., Sbano, L. (2009). A Definition of Cellular Interface Problems. In: Corne, D.W., Frisco, P., Păun, G., Rozenberg, G., Salomaa, A. (eds) Membrane Computing. WMC 2008. Lecture Notes in Computer Science, vol 5391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-95885-7_4
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