Structural Decomposition of Reactions of Graph-Like Objects

Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation. Introduction Compositional methods for the synthesis and analysis of computational systems remain a fruitful research topic with potential applications in practice. Though compositionality is most clearly exhibited in semantics for process calculi where structural operational semantics (SOS) can be found in its "pure" form, a slightly broader perspective is appropriate to make use of the fundamental ideas of SOS in interdisciplinary research. The first source of inspiration of the present paper is the κ-calculus [6], which is an influential modelling framework in Computational Systems Biology. The κ-calculus allows to give abstract, formal descriptions of biological systems that can be used to explain the reaction (rate) of complex systems, so-called complexes, in terms of the reaction (rate) of each of its subsystems, which are called partial complexes. Leaving quantitative aspects as a topic for future research, we concentrate on a specific sub-problem, namely the "purely structural" decomposition of reactions. In the κ-calculus, system states are composed of partial complexes and they have an intuitive, graphical representation. Hence, it is natural to investigate the decomposition of (reactions of) system states using concepts from graph transformation. In its simplest form, the idea of composition of graph transformations is by means of coproducts. Intuitively, the coproduct of two graphs models the assembly of two states put side by side and the two (sub-)states react independently of each other. A well-known, related theorem about graph transformations is the so-called Parallelism Theorem (see e.g. [5, Theorem 17]). A more general formalism of compositionality that is based on pushouts has been (re-)considered in In this paper, we shall remove the restriction to pushouts as a composition mechanism and generalize the results of [18] from pushouts to (pullback stable) colimits of arbitrary shape. This considerably enlarges the set of available gluing patterns. As a simple example, we can now equip each sub-state with several interfaces; this would be appropriate for the model of a cell in an organism that is in direct contact with each of its neighbouring cells with some part of its membrane; each area of contact would be modelled by a different interface. Content of the paper After reviewing some basic category theoretical concepts and the definition of adhesive categories in Section 1, we begin Section 2 with the "deconstruction" of models of system states; more precisely, we explain in Section 2.1 how suitably finite objects in adhesive categories arise as the colimit of a diagram of "atomic" objects, namely irreducible objects in the sense of The main problem, which is concerned with the decomposition of a "global" transformation into a family of "local" ones, is addressed in Section 3. We give a formal description of local decomposition problems, which consist of a given decomposition of a state (as a colimit of a certain shape) and a rule that describes a possible reaction of the state; to solve such a problem means to extend the decomposition of the state to a decomposition of the whole reaction (using colimits of the same shape). Section 3.1 presents a "global" solution, which first constructs the whole transformation "globally"; a "more local" solution of the problem is possible if we are given extra information that involve a generalization of the accommodations o

On functional module detection in metabolic networks

Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers (Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models

Generalised compositionality in graph transformation

We present a notion of composition applying both to graphs and to rules, based on graph and rule interfaces along which they are glued. The current paper generalises a previous result in two different ways. Firstly, rules do not have to form pullbacks with their interfaces; this enables graph passing between components, meaning that components may “learn” and “forget” subgraphs through communication with other components. Secondly, composition is no longer binary; instead, it can be repeated for an arbitrary number of components

On the Complexity of Reconstructing Chemical Reaction Networks

The analysis of the structure of chemical reaction networks is crucial for a better understanding of chemical processes. Such networks are well described as hypergraphs. However, due to the available methods, analyses regarding network properties are typically made on standard graphs derived from the full hypergraph description, e.g.\ on the so-called species and reaction graphs. However, a reconstruction of the underlying hypergraph from these graphs is not necessarily unique. In this paper, we address the problem of reconstructing a hypergraph from its species and reaction graph and show NP-completeness of the problem in its Boolean formulation. Furthermore we study the problem empirically on random and real world instances in order to investigate its computational limits in practice

Interactive Chemical Reactivity Exploration

Elucidating chemical reactivity in complex molecular assemblies of a few hundred atoms is, despite the remarkable progress in quantum chemistry, still a major challenge. Black-box search methods to find intermediates and transition-state structures might fail in such situations because of the high-dimensionality of the potential energy surface. Here, we propose the concept of interactive chemical reactivity exploration to effectively introduce the chemist's intuition into the search process. We employ a haptic pointer device with force-feedback to allow the operator the direct manipulation of structures in three dimensions along with simultaneous perception of the quantum mechanical response upon structure modification as forces. We elaborate on the details of how such an interactive exploration should proceed and which technical difficulties need to be overcome. All reactivity-exploration concepts developed for this purpose have been implemented in the Samson programming environment.Comment: 36 pages, 14 figure

Entropy production and coarse-graining in Markov processes

We study the large time fluctuations of entropy production in Markov processes. In particular, we consider the effect of a coarse-graining procedure which decimates {\em fast states} with respect to a given time threshold. Our results provide strong evidence that entropy production is not directly affected by this decimation, provided that it does not entirely remove loops carrying a net probability current. After the study of some examples of random walks on simple graphs, we apply our analysis to a network model for the kinesin cycle, which is an important biomolecular motor. A tentative general theory of these facts, based on Schnakenberg's network theory, is proposed.Comment: 18 pages, 13 figures, submitted for publicatio

Entropy production and coarse-graining in Markov processes

We study the large time fluctuations of entropy production in Markov processes. In particular, we consider the effect of a coarse-graining procedure which decimates {\em fast states} with respect to a given time threshold. Our results provide strong evidence that entropy production is not directly affected by this decimation, provided that it does not entirely remove loops carrying a net probability current. After the study of some examples of random walks on simple graphs, we apply our analysis to a network model for the kinesin cycle, which is an important biomolecular motor. A tentative general theory of these facts, based on Schnakenberg's network theory, is proposed.Comment: 18 pages, 13 figures, submitted for publicatio