Multiscale Modelling: A Mobile Membrane Approach

Proceedings of 5th Workshop on Membrane Computing and Biologically Inspired Process Calculi (MeCBIC 2011) Pages 113–114, 2011. c⃝F. Buti, M. Callisto De Donato, F. Corradini, E. Merelli and L. Tesei All the rights to the paper remain with the authors. Multiscale Modelling: A Mobile Membrane Approach (Work in Progress) Federico Buti, Massimo Callisto De Donato, Flavio Corradini, Emanuela Merelli, Luca Tesei School of Science and Technology, Computer Science Division University of Camerino, Camerino, Italy {name.surname}@unicam.it Nowadays, multiscale modelling is recognized as the most suitable way to study biological processes. Indeed, almost every phenomenon in nature exhibits a multiscale behaviour, i.e., it is the outcome of interactions that occur at different spatial and temporal scales. Although several ways to provide “multilayer” models have been proposed, only Complex Automata naturally embed spatial infor- mation and realize the multiscale approach with well-established inter-scale integration schemas. Recently, such approach has been restated in terms of Spatial P systems - a variant of P systems with a more geometric concept of space. In this work we discuss how mobile membranes, a variant of membrane systems inspired by the bio- logical movements of endocytosis and exocytosis, can be efficaciously exploited to define a uniform multiscale coupling scheme relying only on the features of the formalism itself. 1 Introduction Many biological phenomena are characterized by interactions involving different spatial and temporal scales simultaneously. At each scale, different structures come into play. Consequently, several compu- tational methodologies have been developed for modelling biological processes with different degrees of resolution. Among them, multiscale modelling rose up as the most suitable one. In such approach, several models at different spatial and temporal scales, are coupled together. How these models are homogenized, i.e, integrated is a key aspect to guarantee that the overall model is faithful to the real phenomenon. Recently, complex Automata (CxA) [4] were introduced as a robust multiscale modelling solution which takes in account coupling issues. In CxA the multiscaling is realised on uniform components, the Cellular Automata, (CA) by different and well-established integration schemas. Any CxA consists of a finite grid of CA cells, where each cell has an associated state taken from a finite set of different states. In [2], authors rephrased the CxA approach in term of Spatial P Systems (SPs) [1], a variant of P Systems enriched with a notion of space. Similarly to CxA, SPs represents space as a geometric 2D grid-based set of cells that contain objects as in classical P Systems. As a case study, authors provided a uniform multiscale model of bone remodelling, a common multiscale phenomena, as an aggregation of single SP models at different scales, coupled through functions external to the paradigm. Following their approach, we discuss how mobile membranes[3] can be exploited to define a multiscale coupling scheme which relies only on the operations provided from the mobile membrane paradigm itself. 2 Bone remodelling with mobile membranes Bone remodelling is a biological phenomenon in which mature bone tissue is removed from the skeleton (resorption) and new bone tissue is formed (formation); such process is multiscale since macroscopic arXiv:1108.3434v1 [cs.FL] 17 Aug 2011 Model BMUi Cui Ti Vi Cu1 BMU1 T1 V1 Cu3 BMU3 T3 V3 Cu2 BMU2 T2 V2 Cun BMU2 Tn Vn (a) MODEL BMUi Mi MBn MBn-1 MBn-1 MBi3 MBi2 MBi1 MBi6 MBi5 MBi4 (b) BMU Figure 1: Multiscale model for bone remodelling. Figure (a) represents the whole model containing the coupling membranes CU (red). Figure (b) shows a detailed BMU (cell scale) membrane. behaviours, e.g. mechanical stimuli at the tissutal level and microstructure (cell scale actors - e.g. hor- mones, receptors etc.) strongly influence each other. Figure 1(a) shows the proposed model based on mobile membranes. As in [2], we consider two resolution levels: the tissutal and the cellular ones. Tis- sue level is divided into a series of membranes Ti (cyan), each representing a portion of bone. Each membrane contains, at any moment, a number of objects c proportional to the mineralisation value (ex- pressed as a density in a certain interval) of the tissue. At the cellular scale we consider a series of BMUi membranes (green), i.e. “Bone Multicellular unit” membranes representing the cellular sites in which remodelling occurs1. The integration of the two scales is realised through a coupling membrane CUi (red) and a simple membrane Vi (violet), that moves between the tissutal and cellular membranes carrying the coupling information. No external function is needed since the paradigm provide the right means to handle the coupling. 3 Future work We sketched a possible application of Mobile Membranes to the definition of a multiscale coupling scheme which, differently from [

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