Hybrid Model of Erythropoiesis
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Hybrid Model of Erythropoiesis P. Kurbatova • N. Eymard • V. Volpert
Received: 17 December 2012 / Accepted: 19 July 2013 Ó Springer Science+Business Media Dordrecht 2013
Abstract A hybrid model of cell dynamics is presented. It is illustrated by model examples and applied to study erythropoiesis (red blood cell production). In this approach, cells are considered as discrete objects while intra-cellular proteins and extra-cellular biochemical substances are described with continuous models. Spatial organization of erythropoiesis occurring in specific structures of the bone marrow, called erythroblastic island, is investigated. Keywords Hybrid model Erythropoiesis Erythroblastic island macrophage Regulatory mechanisms
1 Hybrid Models of Cell Dynamics Hybrid discrete-continuous models are widely used in the investigation of dynamics of cell populations in biological tissues and organisms that involve processes at different scales. In this approach biological cells are considered as discrete objects described either by cellular automata (Dormann and Deutsch 2002; Giverso et al. 2010; Jiang et al. 2005; McDougall et al. 2002; Scianna et al. 2009; Ste´phanou et al. 2005) or by various on-lattice or off-lattice models (Dillon et al. 2008; Jeon et al. 2010; Ramis-Conde et al. 2008) while intracellular and extracellular concentrations are described with continuous models, ordinary or partial differential equations. P. Kurbatova (&) Faculte´ de Me´decine Laennec, UMR 5558, Lyon, France e-mail: [email protected] N. Eymard V. Volpert Institut Camille Jordan, University Lyon 1, UMR 5208 CNRS, 69622 Villeurbanne, France e-mail: [email protected] V. Volpert e-mail: [email protected]
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In cellular automaton model each individual cell can be represented as a single site of lattice, as several connected lattice sites or the lattice site can be larger than an individual cell. A generalized cellular automaton approach is presented by the cellular Potts models (CPM). The CPM is a more sophisticated cellular automaton that describes individual cells, occupying multiple lattice sites, as extended objects of variable shapes. These models take into account surface energy of cell membrane. The CPM effective energy can control cell behaviors including cell adhesion, signalling, volume and surface area or even chemotaxis, elongation and haptotaxis (Giverso et al. 2010; Scianna et al. 2009). In each particular CA model, the rules which determine cell motion should be specified. It can be influenced by the interaction of cells with the elements of their immediate surrounding and by processes that involve cellular response to external signals like chemotaxis. The numerous models with gradient fields of chemical concentrations that govern motility of cells have been suggested. Cellular automaton have been used extensively to model a wide range of problems. Different stages of tumor development from initial avascular phase (Dormann and Deutsch 2002; Jiang et al. 2005) to invasion (
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