Md. Ariful Islam
ABSTRACT
By appealing to the small-gain theorem of one of the authors (Girard), we show that the 13-variable sodium-channel component of the 67-variable IMW cardiac-cell model (Iyer-Mazhari-Winslow) can be replaced by an approximately bi-similar, 2-variable HH-type (Hodgkin-Huxley) abstraction. We show that this substitution of (approximately) equals for equals is safe in the sense that the approximation error between sodium-channel models is not amplified by the feedback-loop context in which it is placed. To prove this feedback-compositionality result, we exhibit quadratic-polynomial, exponentially decaying bisimulation functions between the IMW and HH-type sodium channels, and also for the IMW-based context in which these sodium-channel models are placed. These functions allow us to quantify the overall error introduced by the sodium-channel abstraction and subsequent substitution in the IMW model. To automate computation of the bisimulation functions, we employ the SOSTOOLS optimization toolbox. Our experimental results validate our analytical findings. To the best of our knowledge, this is the first application of δ-bisimilar, feedback-assisting, compositional reasoning in biological systems.
- M. Antoniotti, C. Piazza, A. Policriti, M. Simeoni, and B. Mishra. Taming the complexity of biochemical models through bisimulation and collapsing: Theory and practice. Theoretical Computer Science, 325:2004, 2004. Google Scholar
Digital Library
- R. Barbuti, A. Maggiolo-Schettini, P. Milazzo, and A. Troina. Bisimulation congruences in the calculus of looping sequences. In International Colloquium on Theoretical Aspects of Computing (ICTAC'06), LNCS 4281, pages 93--107. Springer, 2006. Google ScholarDigital Library
- M. Curti, P. Degano, C. Priami, and C. T. Baldari. Modelling biochemical pathways through enhanced pi-calculus. Theoretical Computer Science, 325(1):111--140, 2004. Google ScholarDigital Library
- J. Fisher, D. Harel, and T. A. Henzinger. Biology as reactivity. Communications of the ACM, 54(10):72--82, October 2011. Google ScholarDigital Library
- J. Fisher and N. Piterman. The executable pathway to biological networks. Briefings in Functional Genomics, 9(1):79--92, Jan. 2010.Google ScholarCross Ref
- J. Fisher, N. Piterman, and M. Y. Vardi. The only way is up. In Proceedings of the 17th International Conference on Formal methods, FM'11, pages 3--11, Berlin, Heidelberg, 2011. Springer-Verlag. Google ScholarDigital Library
- A. Girard. A composition theorem for bisimulation functions. Pre-print, 2007. arXiv:1304.5153.Google Scholar
- A. Girard and G. J. Pappas. Approximate bisimulations for nonlinear dynamical systems. In Proceedings of 44th IEEE Conference on Decision and Control, Serville, Spain, December 2005.Google ScholarCross Ref
- R. Grosu, G. Batt, F. H. Fenton, J. Glimm, C. L. Guernic, S. A. Smolka, and E. Bartocci. From cardiac cells to genetic regulatory networks. In Proceedings of the 23rd International Conference on Computer Aided Verification, pages 396--411. Springer, 2011. Google ScholarDigital Library
- A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117:500--544, 1952.Google ScholarCross Ref
- M. A. Islam. Supplement to compositionality results for cardiac cell dynamics. www.cs.sunysb.edu/~amurthy/hscc14_supp.htm, also available on arxiv.org, 2013.Google Scholar
- V. Iyer, R. Mazhari, and R. L. Winslow. A computational model of the human left-ventricular epicardial myocytes. Biophysical Journal, 87(3):1507--1525, 2004.Google ScholarCross Ref
- J. Kapinski, A. Donzé, F. Lerda, H. Maka, S. Wagner, and B. H. Krogh. Control software model checking using bisimulation functions for nonlinear systems. In 47th IEEE Conference on Decision and Control (CDC), pages 4024--4029, 2008.Google ScholarCross Ref
- J. P. Keener. Invariant manifold reductions for Markovian ion channel dynamics. Journal of Mathematical Biology, 58(3):447--457, July 2009.Google ScholarCross Ref
- Md. A. Islam, A. Murthy, E. Bartocci, S. A. Smolka, and R. Grosu. Compositionality results for cardiac cell dynamics. In Proceedings of CMSB'13, the 11th Conference on Computational Methods in Systems Biology, LNCS, Klosterneuburg, Austria, Sept. 2013. Springer.Google Scholar
- A. Murthy, Md. A. Islam, E. Bartocci, E. Cherry, F. H. Fenton, J. Glimm, S. A. Smolka, and R. Grosu. Approximate bisimulations for sodium channel dynamics. In Proceedings of CMSB'12, the 10th Conference on Computational Methods in Systems Biology, LNCS, London, U.K., October 2012. Springer. Google ScholarDigital Library
- D. Noble. A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pace-maker potentials. Journal of Physiology, 160(2):317--352, 1962.Google ScholarCross Ref
- S. Prajna, A. Papachristodoulou, P. Seiler, and P. A. Parrilo. SOSTOOLS: Sum of squares optimization toolbox for MATLAB, 2004.Google Scholar
- A. Regev, E. M. Panina, W. Silverman, L. Cardelli, and E. Shapiro. Bioambients: An abstraction for biological compartments. Theoretical Computer Science, 325(1):141--167, Sept. 2004. Google ScholarDigital Library
- E. D. Sontag. Input to state stability: Basic concepts and results. In Nonlinear and Optimal Control Theory, pages 163--220. Springer, 2006.Google Scholar
Index Terms
- Compositionality results for cardiac cell dynamics
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