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**1 - 2**of**2**### Cycles and Communicating Classes in Membrane Systems and Molecular Dynamics

"... We are considering sequential membrane systems and molecular dynamics from the viewpoint of Markov chain theory. The configuration space of these systems (including the transitions) is a special kind of directed graph, called pseudo-lattice digraph, which is closely related to the stoichiometric mat ..."

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We are considering sequential membrane systems and molecular dynamics from the viewpoint of Markov chain theory. The configuration space of these systems (including the transitions) is a special kind of directed graph, called pseudo-lattice digraph, which is closely related to the stoichiometric matrix. Taking advantage of the monoidal structure of this space, we introduce the algebraic notion of precycle. A precycle leads to the identification of cycles by means of the concept of defect, which is a set of geometric constraints on configuration space. Two efficient algorithms to evaluate precycles and defects are given: one is an algorithm due to Contejean and Devie, the other is a novel branch-and-bound tree search procedure. Cycles partition configuration space into equivalence classes, called the communicating classes. The structure of the communicating classes in the free regime — where all rules are enabled — is analysed: testing for communication can be done efficiently. We show how to apply these ideas to a biological regulatory system. DOI of the journal version: 10.1016/j.tcs.2006.11.027 Key words: membrane systems, communicating classes, molecular dynamics, cycles in digraphs, vector addition systems.

### Reaction Cycles in Membrane Systems and Molecular Dynamics

"... Summary. We are considering molecular dynamics and (sequential) membrane systems from the viewpoint of Markov chain theory. The first step is to understand the structure of the configuration space, with respect to communicating classes. Instead of a reachability analysis by traditional methods, we u ..."

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Summary. We are considering molecular dynamics and (sequential) membrane systems from the viewpoint of Markov chain theory. The first step is to understand the structure of the configuration space, with respect to communicating classes. Instead of a reachability analysis by traditional methods, we use the explicit monoidal structure of this space with respect to rule applications. This leads to the notion of precycle, which is an element of the integer kernel of the stoichiometric matrix. The generators of the set of precycles can be effectively computed by an incremental algorithm due to Contejean and Devie. To arrive at a characterization of cycles, we introduce the notion of defect, which is a set of geometric constraints on a configuration to allow a precycle to be enabled, that is, be a cycle. An important open problem is the efficient calculation of the defects. We also discuss aspects of asymptotic behavior and connectivity, as well as give a biological example, showing the usefulness of the method for model checking. Corresponding author186 M. Muskulus et al.