Geomcell, towards modeling the muscle cell dynamic geometry
Július Parulek1,2*, Michal
Remiš2, Miloš Šrámek2 and Ivan
Zahradník1
1Institute
of Molecular Physiology and Genetics, Slovak
2Faculty of Mathematics, Physics and
Informatics,
*julius.parulek@savba.sk
In geomcell (Parulek et al., 2009), we exploit techniques of implicit surfaces, defined by implicit functions, to materialize a static model of muscle cells. Here we introduce a next step in the geomcell development process that is represented by an addition of a physical layer to the myofibrillar system in order to represent their dynamics. Myofibrils, the contractile fibres, are thin and long cylindrical objects, divided longitudinally into Z, I, and A bands, giving rise to the striated pattern of this type of muscle cells under a microscope. In the model, they are defined by a set of transversal cross-sectional graphs (c-graph) in a system of parallel planes that are divided into a set of closed polygons. C-graphs are prepared according to real electron microscopic (EM) images of transversal cell sections. The resultant implicit function f of a single myofibril is obtained by interpolation between the neighbouring c-graphs. To simulate myofibrillar dynamics (contraction), we add a special offset function g to a myofibrillar function f, i.e. f(x)+g(x,d), where x is a space and d is a deformation parameter. The offset function g adjusts the values of f by distributing the deformation along the longitudinal myofibril axis. The parameter x controls the position along the myofibrillar longitudinal axis and defines the general shape of deformation. The parameter d controls the size of deformation, i.e. the amount a myofibrillar surface is “pushed out”, and implies the actual shortening (Fig. 1). As was observed on EM images, the surface deformation and the contraction (shortening) occur in the I-Band regions, whereas the remaining regions of the myofibril stay not deformed.
Fig 1. Deformation of the sarcomere. (Left) Contraction of a myofibril. The I-Band area is shortened and “pushed
out” according to the deformation term g given in the graph. (Right) The deformation graph. Different colors represents plots of
the deformation term g(x,d):
d=0.2 (green), d=0.6 (blue) and d=1 (red).
References
Parulek J, Šrámek M, Červeňanský
M, Novotová M and Zahradník I. A cell architecture modeling system based on quantitative ultrastructural
characteristics. In: V. Maly (ed.), Systems Biology, Methods in Molecular
Biology 500: 289–313, Springer-Verlag, Berlin
Heidelberg 2009, ISBN 978-1-934115-64-0