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...networks of interactions that cannot be easily understood without resorting to formal modelling and automated analysis. The generalized logical formalism initially proposed by René Thomas in the 70s =-=[25, 26, 27]-=-, is a discrete modelling formalism that has proved to be an effective way to capture regulation processes and analyze them. It has been successfully applied to the study of a variety of regulatory ne...

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...networks of interactions that cannot be easily understood without resorting to formal modelling and automated analysis. The generalized logical formalism initially proposed by René Thomas in the 70s =-=[25, 26, 27]-=-, is a discrete modelling formalism that has proved to be an effective way to capture regulation processes and analyze them. It has been successfully applied to the study of a variety of regulatory ne...

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...networks of interactions that cannot be easily understood without resorting to formal modelling and automated analysis. The generalized logical formalism initially proposed by René Thomas in the 70s =-=[25, 26, 27]-=-, is a discrete modelling formalism that has proved to be an effective way to capture regulation processes and analyze them. It has been successfully applied to the study of a variety of regulatory ne...

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...into account the neighboring relation δ . 2.3 Petri net semantics In order to analyze regulatory bundles, a Petri net semantics has been defined [5]. For this purpose, a colored variant of Petri nets =-=[13]-=- has been used in which: • places model containers of tokens, the latter being arbitrary values, e.g., pairs of integers as in the following; • transitions model activities that consume and produce to...

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...tic regulation on a complex dynamic geometry. Various formalism have been proposed to face these needs relying on various explicit representations of space [24]: molecular dynamics, spatial Gillespie =-=[23]-=-, partial differential equations (PDE), lattice based methods (e.g., cellular automata), etc. See Fig 13 for a brief taxonomy of space representation in biological modelling. In this context, we want ...

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...patial evolutions, enumerations of the possible evolutions, graphical rendering of spatial information. To do so, we resort to topological collections. Topological collections have been introduced in =-=[9]-=- to describe arbitrary complex spatial structures that appear in biological systems [10]. They have been used to represents states of dynamical systems with a time varying structure [7, 12]. We will u...

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...ted degrees. 3.1 Placing modules on grids Simple topological collections can be defined as group based fields (GBF), that can be considered as associative arrays whose indexes are elements in a group =-=[8]-=-. The latter is defined by a finite presentation: a set of generators together with some constraints on their combinations. Thus a GBF can be pictured as a labelled graph where the underlying graph is...

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...ial information. To do so, we resort to topological collections. Topological collections have been introduced in [9] to describe arbitrary complex spatial structures that appear in biological systems =-=[10]-=-. They have been used to represents states of dynamical systems with a time varying structure [7, 12]. We will use them in this paper as a unifying framework to represent arbitrary neighborhood relati...

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... introduced in [9] to describe arbitrary complex spatial structures that appear in biological systems [10]. They have been used to represents states of dynamical systems with a time varying structure =-=[7, 12]-=-. We will use them in this paper as a unifying framework to represent arbitrary neighborhood relationships. In this context, a neighborhood relationship between modules is represented by a labelled gr...

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... introduced in [9] to describe arbitrary complex spatial structures that appear in biological systems [10]. They have been used to represents states of dynamical systems with a time varying structure =-=[7, 12]-=-. We will use them in this paper as a unifying framework to represent arbitrary neighborhood relationships. In this context, a neighborhood relationship between modules is represented by a labelled gr...

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...ay, which allows to explore all the possible evolutions of the spatial structure. 4.4 Implementation and applications Spatialized regulatory bundles have been implemented as a prototype, using SNAKES =-=[18, 17]-=- for the Petri net part, and MGS [16] for the topological collections part. SNAKES is a general purpose Petri net library implemented in Python, that is particularly suitable for the quick development...

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... an effective way to capture regulation processes and analyze them. It has been successfully applied to the study of a variety of regulatory networks comprising relatively large numbers of components =-=[20, 21]-=-. This formalism however does not provide any modelling device to specifically address the question of multi-cellular systems, where the regulatory networks of cells can influence each other in a way ...

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...o produce a graphical rendering of an n-BDG. Here also, MGS simplifies the problem by proposing graph layout algorithms that have been already proved relevant to represent graphically cells positions =-=[19, 22]-=-. In an n-BDG, a notion of distance can be defined as the length of the shortest path between two nodes. If such a path does not exist, i.e., if the graph is not connected, the distance can be assumed...

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... in a disconnected tissue, which may be undesirable too. However, the simplicity of grids is well adapted to model, e.g., accretive growth that occurs at the borders of a tissue like leaves in plants =-=[11]-=-. Notice also that the graphical rendering of GBF is trivial to obtain because the grids are predefined. 3.2 A generalization of grids In order to overcome the limitations of grids in the presence of ...

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...o produce a graphical rendering of an n-BDG. Here also, MGS simplifies the problem by proposing graph layout algorithms that have been already proved relevant to represent graphically cells positions =-=[19, 22]-=-. In an n-BDG, a notion of distance can be defined as the length of the shortest path between two nodes. If such a path does not exist, i.e., if the graph is not connected, the distance can be assumed...

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...cell signaling, mechanical stresses and genetic regulation on a complex dynamic geometry. Various formalism have been proposed to face these needs relying on various explicit representations of space =-=[24]-=-: molecular dynamics, spatial Gillespie [23], partial differential equations (PDE), lattice based methods (e.g., cellular automata), etc. See Fig 13 for a brief taxonomy of space representation in bio...

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... is an arc from s toward s′. Such a graph defines a transition system that is suitable to perform model-checking of various reachability- or causality-related properties. For instance, consider state =-=[1,0,0]-=- of the previous network (indexed as [xA,xB,xC]). Only component B may evolve, because KB(xA) 7→ 1, KA(xC) 7→ 1 and KC(xB) 7→ 0 in state [1,0,0]; this yields a new state [1,1,0]. The state space reach...

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...fiers); another possibility is to resort to abstraction, for instance by applying to module identifiers a state space reduction technique originally designed for systems with dynamic process creation =-=[14, 15]-=-. Finally, we intend to study an alternative approach for spatial information in order to address systems such as blood-cells populations. The idea is to implement the spatial relation as a purely sto...

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...igration, division or apoptosis has been added. This prototype implementation has been applied to the study of two simple biology-inspired examples, allowing to test various features of our framework =-=[3]-=-: 1. A system with transcription signal cascading resulting in a cell division allowed to observe the alternation of growth phases and divisions, which can be interpreted as a mutual influence between...

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...information should be compatible with existing graphical rendering techniques. Contributions. Our starting point is a modelling approach developed in [2, 6] and extended later to introduce modularity =-=[5, 4]-=-. This approach itself is based on logical regulatory networks [25, 26, 27] that allow for modelling regulation within a biological system. A Petri net semantics is provided in order to analyze the va...

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...information should be compatible with existing graphical rendering techniques. Contributions. Our starting point is a modelling approach developed in [2, 6] and extended later to introduce modularity =-=[5, 4]-=-. This approach itself is based on logical regulatory networks [25, 26, 27] that allow for modelling regulation within a biological system. A Petri net semantics is provided in order to analyze the va...

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...fiers); another possibility is to resort to abstraction, for instance by applying to module identifiers a state space reduction technique originally designed for systems with dynamic process creation =-=[14, 15]-=-. Finally, we intend to study an alternative approach for spatial information in order to address systems such as blood-cells populations. The idea is to implement the spatial relation as a purely sto...

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...ll be computed dynamically with respect to some lower-level neighboring information. Topological collections of various kinds have been implemented in an experimental programming languages called MGS =-=[16]-=-. This language is used here to implement a database that records the neighborhood relationships and that can be queried and updated efficiently. 3 Representing spatial information In this section, we...

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...ay, which allows to explore all the possible evolutions of the spatial structure. 4.4 Implementation and applications Spatialized regulatory bundles have been implemented as a prototype, using SNAKES =-=[18, 17]-=- for the Petri net part, and MGS [16] for the topological collections part. SNAKES is a general purpose Petri net library implemented in Python, that is particularly suitable for the quick development...

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... an effective way to capture regulation processes and analyze them. It has been successfully applied to the study of a variety of regulatory networks comprising relatively large numbers of components =-=[20, 21]-=-. This formalism however does not provide any modelling device to specifically address the question of multi-cellular systems, where the regulatory networks of cells can influence each other in a way ...