Spatial Calculus of Looping Sequences

Roberto Barbuti, Andrea Maggiolo-Schettini, Paolo Milazzo, Giovanni Pardini
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
This paper presents Spatial CLS, an extension of the Calculus of Looping Sequences (CLS) with spatial features. Spatial CLS allows keeping track of the position of biological elements in a continuous space (2D or 3D) as time passes. The movement of elements in the space can be precisely described, and elements can interact when constraints on their positions are satisfied such as, for example, if two elements are close enough. As for CLS, membranes and elements inside them can be directly
more &raquo; ... d in the syntax. Spatial CLS allows describing the space occupied by elements and membranes. The space occupied by different objects is always kept disjoint. The validity of this constraint is ensured at all times by the semantics of the calculus. In order to model specific behaviors, the modeler can provide an algorithm to rearrange the position of objects in case of a space conflict. Being an extension of CLS, Spatial CLS provides a simple and powerful syntax, based on rewrite rules, for describing the possible reactions among elements of a system. Moreover, rewrite rules are endowed with a stochastic reaction rate parameter. The aim of Spatial CLS is to enable a more accurate description of those biological processes whose behavior depends on the exact position of the elements. As example applications of the calculus, we present a model of cell proliferation, and a model of the quorum sensing process in Pseudomonas aeruginosa. , and the P System Modeling Framework [20], Psim [21, 22] , and P-Lingua [23-25] for P systems. Recently, in order to model the protein chemistry of biological cells for phenomena where spatial effects are important, particle-based simulators have been developed [26, 27] and extensions of the π -calculus with spatial features have been proposed [28] [29] [30] . Other approaches dealing with spatial aspects, that have been used for describing biological systems, include cellular automata [31] [32] [33] , and models based on differential equations, such as those describing the reaction diffusion process and spatial pattern formation [34, 35] . The Calculus of Looping Sequences (CLS) [12] [13] [14] allows the modeling of biological systems and of their evolution. It is based on term rewriting, hence a CLS model is composed of a term, which describes the biological system, and a set of rewrite rules, modeling its evolution. Two kinds of structures are provided by the calculus: sequences, used to represent simple entities of biological systems, such as proteins and DNA strands, and looping sequences that can be used to model more complex structures such as membranes. In this paper we present the Spatial Calculus of Looping Sequences (Spatial CLS), which extends CLS by allowing spatial information to be associated with CLS structures when this information is relevant for determining the system behavior. In Spatial CLS, all structures are embedded in an Euclidean space, which may be either 2D or 3D according to the needs of the modeler. Structures are associated with a precise position in space, and their movement can be precisely described. The interaction can be constrained to the position of the involved elements. As an example, we can impose that a protein can enter a cell membrane only if it is within a certain distance from the membrane itself. Both deterministic and stochastic motions of elements can be described. An important feature of the Spatial CLS is the description of the space occupied by the elements, with the constraint that there cannot be any conflict between the space occupied by different elements. In particular, an "exclusion space" can be associated with elements, which is either circular or spherical, according to the dimension of the considered space (2D/3D). The semantics ensures that no space conflicts between elements arise during the evolution of the system. The calculus allows the use of different strategies to rearrange the elements, in case of a space conflict. The semantics prescribes that, if no valid arrangement can be found, then the event that would cause the conflict cannot occur. Finally, rewrite rules are endowed with kinetic parameters describing their stochastic application rate. The aim of Spatial CLS is to enable a more accurate description of those biological processes whose behavior depends on the exact position of the elements. This high level of accuracy is especially useful for cell biology, where there can be a high degree of spatial organization and molecular species may be distributed not uniformly in the space [36] . Such descriptions can then be used to simulate the systems, so as to obtain a faithful representation of their evolution. Handling spatial information in a simulator may have a high computational cost. However, Spatial CLS allows specifying spatial information only for those elements for which such information is relevant, thus enabling the modeler to mix descriptions at different levels of abstraction. As example applications of the calculus we present a model of cell proliferation, as it happens during the development of a biological tissue, and a model of the quorum sensing process in P. aeruginosa. In the case of cell proliferation, we formalize an algorithm for the rearrangement of the objects in the system, which tries to resolve space conflicts by simulating the movement of elements as if they push each other when their exclusion spaces overlap. We show the results of simulation of the two models. The paper is structured as follows. In Section 2 we recall the definition of the variant of the Calculus of Looping Sequences used as the basis for the extension into the Spatial CLS. Then, in Section 3, we introduce the syntax of Spatial CLS and, in Section 4 we present the formal definition of the semantics. Section 5 provides a definition of the Arrange algorithm, used by the semantics to perform a rearrangement of elements in case of space conflicts. Finally, in Section 6, we show some examples of applying the features of the calculus to biological modeling, and present the complete models of cell proliferation and the quorum sensing process, along with the results of their simulations.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1016/j.tcs.2011.01.020</a> <a target="_blank" rel="external noopener" href="">fatcat:pv3avqb5yncovfybqacz2ctysi</a> </span>
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