Modeling Morphogenesis in Multicellular Structures with Cell Complexes and L-systems
We consider computational modeling of biological systems that consist of discrete components arranged into linear structures. As time advances, these components may process information, communicate and divide. We show that: (1) the topological notion of c
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Abstract We consider computational modeling of biological systems that consist of discrete components arranged into linear structures. As time advances, these components may process information, communicate and divide. We show that: (1) the topological notion of cell complexes provides a useful framework for simulating information processing and flow between components; (2) an indexfree notation exploiting topological adjacencies in the structure is needed to conveniently model structures in which the number of components changes (for example, due to cell division); and (3) Lindenmayer systems operating on cell complexes combine the above elements in the case of linear structures. These observations provide guidance for constructing L-systems and explain their modeling power. Lsystems operating on cell complexes are illustrated by revisiting models of heterocyst formation in Anabaena and by presenting a simple model of leaf development focused on the morphogenetic role of the leaf margin.
1 Introduction There is a feedback between mathematics and studies of nature. On one hand, mathematical concepts—even though they may eventually be formalized in an axiomatic way—are often inspired and motivated by studies of nature. On the other hand, they facilitate these studies by providing proper mathematical tools (Fig. 1). In this context, we consider computational methods needed to model the development of multicellular structures, in particular plants. We show that these methods are not merely a new application of partial differential equations, traditionally used to model spatio-temporal phenomena in mathematical physics. Instead, developmental modeling of multicellular structures requires an integration of
P. Prusinkiewicz (*) • B. Lane Department of Computer Science, University of Calgary, Calgary, AB T2N 1N4, Canada e-mail: [email protected] V. Capasso et al. (eds.), Pattern Formation in Morphogenesis, Springer Proceedings in Mathematics 15, DOI 10.1007/978-3-642-20164-6_12, # Springer-Verlag Berlin Heidelberg 2013
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Fig. 1 A conceptual model of relations between natural science and mathematics
inspiration
studies of nature
mathematical concepts
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tools rooted in different branches of mathematics and computer science. This combination includes L-systems [1], ordinary differential equations, and the topological notion of cell complexes [2]. The structures we consider are the spatial arrangements of discrete components that process information and communicate. These structures are dynamic, which means that not only the state of the components, but also their number can change over time. The development is symplastic: the neighborhood relations can only be changed as a result of the addition or removal of components (in contrast to animal cells, plant cells do not move with respect to each other). We limit our examples to linear structures consisting of sequences of cells, although similar problems occur in the modeling of branching plant structures at the large
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