Dimensionality Reduction of Bistable Biological Systems
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Dimensionality Reduction of Bistable Biological Systems A. Zakharova · Z. Nikoloski · A. Koseska
Received: 25 July 2012 / Accepted: 4 January 2013 / Published online: 8 February 2013 © Society for Mathematical Biology 2013
Abstract Time hierarchies, arising as a result of interactions between system’s components, represent a ubiquitous property of dynamical biological systems. In addition, biological systems have been attributed switch-like properties modulating the response to various stimuli across different organisms and environmental conditions. Therefore, establishing the interplay between these features of system dynamics renders itself a challenging question of practical interest in biology. Existing methods are suitable for systems with one stable steady state employed as a well-defined reference. In such systems, the characterization of the time hierarchies has already been used for determining the components that contribute to the dynamics of biological systems. However, the application of these methods to bistable nonlinear systems is
Electronic supplementary material The online version of this article (doi:10.1007/s11538-013-9807-8) contains supplementary material, which is available to authorized users. A. Zakharova () · A. Koseska Center for Dynamics of Complex Systems, University of Potsdam, Campus Golm, Karl-Liebknecht-Str. 24, 14476 Potsdam, Germany e-mail: [email protected] A. Zakharova Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany Z. Nikoloski Institute for Biochemistry and Biology, University of Potsdam, Campus Golm Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany Z. Nikoloski Max Planck Institute for Molecular Plant Physiology, Am Mühlenberg 1, 14476 Potsdam, Germany e-mail: [email protected] A. Koseska Institute of Physics, Humboldt University, Newtonstr. 15, 12489 Berlin, Germany
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impeded due to their inherent dependence on the reference state, which in this case is no longer unique. Here, we extend the applicability of the reference-state analysis by proposing, analyzing, and applying a novel method, which allows investigation of the time hierarchies in systems exhibiting bistability. The proposed method is in turn used in identifying the components, other than reactions, which determine the systemic dynamical properties. We demonstrate that in biological systems of varying levels of complexity and spanning different biological levels, the method can be effectively employed for model simplification while ensuring preservation of qualitative dynamical properties (i.e., bistability). Finally, by establishing a connection between techniques from nonlinear dynamics and multivariate statistics, the proposed approach provides the basis for extending reference-based analysis to bistable systems. Keywords Bistability · Time-scales hierarchy · Similarity transformation · Canonical correlation analysis · Dimensionality reduction
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