State trajectory of the Belousov-Zhabotinsky reaction

Anna Zhyrova, Dalibor Štys, Tomáš Náhlík

Faculty of Fisheries and Protection of Waters, School of Complex Systems, University of South Bohemia, Zámek 136, 373 33 Nové Hrady, Czech Republic

All biological systems ranging from organized herds to colonies of individual cells represent the self-organizing (SO) system. Studying and modeling of these systems will allow us to predict their future behavior and interaction with other systems. The aim of our investigation was to develop a method of analysis for SO systems. As a basic model of intracellular (or intra-organelles) pattern formation was chosen Belousov-Zhabotinsky reaction.

Data were received by non-invasive methods (photographing the surface of the reaction). For maximization information gain all images were processed by the method developed on our institute which is called information entropy. It is based on Rényi entropy equation.

Where α is called Rényi entropy coefficient. The colour channels and different Rényi entropy coefficients may be combined to best discriminate individual states.

For the further processing of data we used principal component analysis (PCA) provided by the Unscrambler X 10.1. We are using results of PCA for construction of state trajectory of reaction. For dividing obtained data into several clusters we use cluster analysis provides by Unscrambler. The cluster analysis is based on k-mean clustering method.

Each cluster of the trajectory represents an event or subset of the states of the reaction. Size and position of clusters depends on the trajectory and on number of clusters. We proposed seven clusters now as a first estimation, satisfied by the results: clusters are well separated; images in transitions between clusters show some changes. The loadings of the first third principal components (PC-1, PC-2 and PC-3) on the all experiments data were compared, which indicates that the using method well describes the main characteristic of the system parameters. This allows us to assume that the method can be applied to the analysis of more complex systems such as cells.