Abstract
We show that in the context of the Iyer et al. 67-variable cardiac myocycte model (IMW), it is possible to replace the detailed 13-state probabilistic model of the sodium channel dynamics with a much simpler Hodgkin-Huxley (HH)-like two-state sodium channel model, while only incurring a bounded approximation error. The technical basis for this result is the construction of an approximate bisimulation between the HH and IMW sodium channel models, both of which are input-controlled (voltage in this case) CTMCs.
The construction of the appropriate approximate bisimulation, as well as the overall result regarding the behavior of this modified IMW model, involves: (1) Identification of the voltage-dependent parameters of the m and h gates in the HH-type channel via a two-step fitting process, carried out over more than 22,000 representative observational traces of the IMW channel. (2) Proving that the distance between observations of the two channels is bounded. (3) Exploring the sensitivity of the overall IMW model to the HH-type sodium-channel approximation. Our extensive simulation results experimentally validate our findings, for varying IMW-type input stimuli.
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References
Bartocci, E., Cherry, E., Glimm, J., Grosu, R., Smolka, S.A., Fenton, F.: Toward real-time simulation of cardiac dynamics. In: Proceedings of the 9th International Conference on Computational Methods in Systems Biology, CMSB 2011, pp. 103–112. ACM (2011)
Bauer, F.L., Fike, C.T.: Norms and Exclusion Theorems. Numerische Matematik (1960)
Boyd, S.: EE 263: Introduction to Linear Dynamical Systems, lecture notes. In: Stanford Engineering Everywhere, SEE (2010)
Bueno-Orovio, A., Cherry, E.M., Fenton, F.H.: Minimal model for human ventricular action potentials in tissue. J. of Theor. Biology 253(3), 544–560 (2008)
Cherry, E.M., Fenton, F.H.: Visualization of spiral and scroll waves in simulated and experimental cardiac tissue. New Journal of Physics 10, 125016 (2008)
Chiavazzo, E., Gorban, A.N., Karlin, I.V.: Comparisons of invariant manifolds for model reduction in chemical kinetics. Comm. Comp. Phys. 2, 964–992 (2007)
Epstein, I.R., Pojman, J.A.: An Introduction to Nonlinear Chemical Dynamics. Oxford University Press, London (1998)
Fenton, F., Karma, A.: Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos 8(1), 20–47 (1998)
Fenton, F.H., Cherry, E.M.: Models of cardiac cell. Scholarpedia 3, 1868 (2008)
Fink, M., Noble, D.: Markov models for ion channels: Versatility versus identifiability and speed. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367(1896), 2161–2179 (2009)
Fisher, J., Piterman, N., Vardi, M.Y.: The Only Way Is Up. In: Butler, M., Schulte, W. (eds.) FM 2011. LNCS, vol. 6664, pp. 3–11. Springer, Heidelberg (2011)
Girard, A.: Controller synthesis for safety and reachability via approximate bisimulation. Automatica 48, 947–953 (2012)
Girard, A., Pappas, G.J.: Approximate bisimulations for nonlinear dynamical systems. In: Proc. of CDC 2005, The 44th Int. Conf. on Decision and Control, Seville, Spain. IEEE (December 2005)
Girard, A., Pappas, G.J.: Approximate bisimulation relations for constrained linear systems. Automatica 43, 1307–1317 (2007)
Girard, A., Pappas, G.J.: Approximation metrics for discrete and continuous systems. IEEE Transactions on Automatic Control 52(5), 782–798 (2007)
Gorban, A.N., Karlin, I.V.: Method of invariant manifold for chemical kinetics. Chem. Eng. Sci. 58, 4751–4768 (2003)
Gorban, A.N., Kazantzis, N., Kevrekidis, I.G., Ottinger, H.C., Theodoropoulos, C.: Model reduction and coarse-graining approaches for multiscale phenomena. Springer (2006)
Grosu, R., Batt, G., Fenton, F.H., Glimm, J., Le Guernic, C., Smolka, S.A., Bartocci, E.: From Cardiac Cells to Genetic Regulatory Networks. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 396–411. Springer, Heidelberg (2011)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology 117, 500–544 (1952)
Iyer, V., Mazhari, R., Winslow, R.L.: A computational model of the human left-ventricular epicardial myocytes. Biophysical Journal 87(3), 1507–1525 (2004)
Jahnke, T., Huisinga, W.: Solving the chemical master equation for monomolecular reaction systems analytically. Journal of Mathematical Biology 54, 1–26 (2007)
Fisher, J., Harel, D., Henzinger, T.A.: Biology as reactivity. Communications of the ACM 54(10), 72–82 (2011)
Keener, J.: Invariant manifold reductions for markovian ion channel dynamics. Journal of Mathematical Biology 58(3), 447–457 (2009)
Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Springer (1996)
Kienker, P.: Equivalence of aggregated markov models of ion-channel gating. Proceedings of the Royal Society of London. B. Biological Sciences 236(1284), 269–309 (1989)
Kuo, C.-C., Bean, B.P.: Na channels must deactivate to recover from inactivation. Neuron 12, 819–829 (1994)
Lee, E., Varaiya, P.: Structure and Interpretation of Signals and Systems. Pearson Education (2003)
Irvine, L.A., Saleet Jafri, M., Winslow, R.L.: Cardiac sodium channel markov model with tempretature dependence and recovery from inactivation. Biophysical Journal 76, 1868–1885 (1999)
Luo, C.H., Rudy, Y.: A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circulation Research 74(6), 1071–1096 (1994)
MATLAB. Choosing a solver, http://www.mathworks.com/help/toolbox/optim
MATLAB. Curve fitting toolbox, http://www.mathworks.com/products/curvefitting
MATLAB. Nonlinear numerical methods, http://www.mathworks.com/help/techdoc/ref/f16-5872.html
MATLAB. Optimization toolbox, http://www.mathworks.com/help/toolbox/optim
Murray, J.D.: Mathematical Biology. Springer (1990)
Myers, C.J.: Engineering Genetic Circuits. CRC Press (2010)
National Science Foundation (NSF). Computational Modeling and Analysis of Complex Systems (CMACS), http://cmacs.cmu.edu
Noble, D.: A modification of the Hodgkin-Huxley equations applicable to purkinje fibre action and pace-maker potentials. J. Physiol. 160, 317–352 (1962)
Radulescu, O., Gorban, A.N., Zinovyev, A., Lilienbaum, A.: Robust simplifications of multiscale biochemical networks. BMC Systems Biology 2(1), 86 (2008)
Smith, N., Crampin, E.: Development of models of active ion transport for whole-cell modelling: Cardiac sodium–potassium pump as a case study. Progress in Biophysics and Molecular Biology 85(2-3), 387–405 (2004), Modelling Cellular and Tissue Function
ten Tusscher, K.H., Noble, D., Noble, P.J., Panfilov, A.V.: A model for human ventricular tissue. American Journal of Physiology 286, H1573–H1589 (2004)
Wang, C., Beyerlein, P., Pospisil, H., Krause, A., Nugent, C., Dubitzk, W.: An efficient method for modeling kinetic behavior of channel proteins in cardiomyocytes. IEEE/ACM Trans. on Computational Biology and Bioinformatics 9(1), 40–51 (2012)
Whiteley, J.P.: Model reduction using a posteriori analysis. Mathematical Biosciences 225(1), 44–52 (2010)
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Murthy, A. et al. (2012). Approximate Bisimulations for Sodium Channel Dynamics. In: Gilbert, D., Heiner, M. (eds) Computational Methods in Systems Biology. CMSB 2012. Lecture Notes in Computer Science(), vol 7605. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33636-2_16
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DOI: https://doi.org/10.1007/978-3-642-33636-2_16
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