Membrane Computing was introduced as a computational paradigm in Natural Computing. The models introduced, called Membrane (or P) Systems, provide a coherent platform to describe and study living cells as computational systems. Membrane Systems have been investigated for their computational aspects and employed to model problems in other fields, like: Computer Science, Linguistics, Biology, Economy, Computer Graphics, Robotics, etc. Their inherent parallelism, heterogeneity and intrinsic versatilityallow them to model a broad range of processes and phenomena, being also an efficient means to solve and analyze problems in a novel way.

Membrane Computing has been used to model biological systems, becoming with time a thorough modeling paradigm comparable, in its modeling and predicting capabilities, to more established models in this area. This book is the result of the need to collect, in an organic way, different facets of this paradigm.

The chapters of this book, together with the web pages accompanying them, present different applications of Membrane Systems to Biology. Deterministic, non-deterministic and stochastic systems paired with different algorithms and methodologies show the full potential of this framework.

The book is addressed to researchers interested in applications of discrete biological models and the interplay between Membrane Systems and other approaches to analyze complex systems."

This volume contains a selection of papers presented at the 9th Workshop on Membrane Computing, WMC9, which took place in Edinburgh, UK, during July 28-31,2008. The ?rst three workshopson membrane computing were or- nized in Curtea de Arge, s, Romania - they took place in August 2000 (with the proceedings published in Lecture Notes in Computer Science, volume 2235), in August 2001 (with a selection of papers published as a special issue of Fun- menta Informaticae, volume 49, numbers 1-3, 2002), and in August 2002 (with the proceedings published in Lecture Notes in Computer Science, volume 2597). The next ?ve workshops were organized in Tarragona, Spain, in July 2003, in Milan, Italy, in June 2004, in Vienna, Austria, in July 2005, in Leiden, The Netherlands, in July 2006, and in Thessaloniki, Greece, in June 2007, with the proceedings published as volumes 2933, 3365, 3850, 4361, and 4860 of Lecture Notes in Computer Science."

Membrane systems are a new class of distributed and parallel model of computation inspired by the subdivision of living cells into compartments delimited by membranes. Their hierarchical internal structure, their locality of interactions, their inherent parallelism and also their capacity to create new compartments, represent the distinguishing hallmarks of membrane systems. Membrane computing, the study of membrane systems, is a fascinating and fast growing area of research. The main streams of current investigations in Membrane Computing concern theoretical computer science and the modelling of complex systems. In this monograph Pierluigi Frisco considers the former trend: he presents an in-depth study of the formal language and computational complexity aspects of the most widely investigated models of membrane systems. This study gives a comprehensive understanding of the computational power of the models considered, shows different proof techniques used for such study, and introduces links highlighting the similarities and differences between the their computational power. These models cover a broad range of features, giving a grasp of the enormous flexibility of the framework offered by membrane systems. Aimed at graduates and researchers in the field, who can use it as a reference text, and to people with an initial interest in Membrane Computing, who can use it as a clear and up to date starting point for Membrane Computing.