In this paper, we aim at modelling and analyzing the regulation processes in multi-cellular biological systems, in particular tissues. The modelling framework is based on interconnected logical regulatory networks a la Rene Thomas equipped with information about their spatial relationships. The semantics of such models is expressed through colored Petri nets to implement regulation rules, combined with topological collections to implement the spatial information. Some constraints are put on the the representation of spatial information in order to preserve the possibility of an enumerative and exhaustive state space exploration. This paper presents the modelling framework, its semantics, as well as a prototype implementation that allowed preliminary experimentation on some applications.
Regulation processes are the corner stone to understand many aspects of biological systems. Regulations occur at many levels: transcription and translation of the genetic material, protein modifications, etc. They define complex 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 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 that is dependent on the spatial relationships between the cells. This paper is a step toward providing a modelling framework for the regulation in multicellular systems, in particular tissues, taking into account cells migration, division and apoptosis (death). This framework will be applied to the analysis of systems such as developmental processes, invasive cancers, plant growth, etc.
In such a modelling framework, we would like to preserve the ability to perform modelchecking based analysis in order to be able to assess causality-related properties and observe rare events, which usually cannot be obtained through simulation. This constrains the possible choices for modelling spatial information. In particular, their should be a finite number of possible spatial evolutions from a given configuration and they should be enumerable. Moreover, each spatial configuration should be represented in a normalized form, allowing the recognition of two identical configurations. For instance, floating-point positions are not a possible solution, instead, we shall use solutions based on discrete representations. Finally, we would like to be able to obtain easily a graphical representation of a cell population, i.e., the chosen representation of spatial 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 various properties of so modelled systems. In [5,4], modularity allows an easier modelling of multi-cellular systems, each cell being represented by a module. However, the spatial relationship between the modules within a model is represented in a very abstract way, with no link to any kind of geometrical or topological information.
Our main contribution in this paper is twofold. First, we propose a way to specify such information in a very flexible way, without significantly increasing the complexity of the original framework. Second, we define devices to model efficiently the spatial transformations related to the apoptosis, migration and division processes. The former goal is achieved by decoupling the regulation rules in the model from the spatial information and transformations. Both aspects being linked through a standardized interface, allowing the modeler for using various approaches to spatial representation. The latter goal is achieved by a careful design of this interface in order to enable a smooth bi-directional communication between the two parts of a model.
Then, we present two approaches to the modelling of spatial information. One is based on predefined grids and another is based on bounded-degree graphs. Both methods have pros and cons, depending on the application domain of interest. Finally, our framework is given a Petri net semantics that is implemented in a prototype, allowing to prove the feasibility of the approach and to run preliminary experiments on simple applications.
Notice that the approach presented in this paper is applicable only if one sort of module is considered at the same time, i.e., when all the cells in a tissue are of the same kind. The extension to take into account multi-sorted systems is quite straightforward, but the resulting notations are much more complex. Our prototype implementation actually does not have any such limitation. However, for this paper, an intuitive and simpler presentation has been preferred.
Outlines. The next section introduces the background of our work, in particular the modelling framework we start from. Our contributions are then presented in sections 3 and 4, the former being dedicated to the modelling of spatial information while the latter presents the extended modelling framework. The paper ends with a conclusion and a discussion about future works.
A logical regulatory network is usually depicted as a graph wh
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