Topics of Interest

Methods of combinatonal enumeration [1] are used in chemistry to calculate numbers of positional, skeletal, configurational and stereo isomers, isomers of nonrigid molecules, isomerizations, and other reactions. They serve to derive reaction probabilities, spectroscopic signals, quantum chemical parameters, virial coefficients and other features [2] - [4]. The Pólya theorem allows one to calculate not only the total number of representatives of a class of compounds but also to subdivide them into groups with predefined properties of their molecular graphs.

Graphs can be enumerated by simple properties like vertex number, edge number, and vertex degree [5]. These values can be already interpreted as trivial topological indices. A method is discussed which allows the formation of new properties by combination of substructural properties. This is an extension of commonly used enumeration techniques. Thus, the molecular graphs of classes of acyclic compounds can be enumerated by topological indices such as number of paths of a defined length, connectivity indices and characteristic polynomial. The enumeration of path lengths in the molecular graphs of alkanes is demonstrated by example. Possible applications in the simulation of quantitative structure-property relations are discussed. A computer program allowing Pólya insertion in polynomials with several variables is under development.

Angenommen, uns interessieren die Molekülgraphen mit bestimmten Eigenschaften. Durch systematische Konstruktion aller entsprechenden Graphen läßt sich die gewünschte Information erhalten. Oft ist dieses Vorgehen jedoch nicht praktikabel. Die Kombinatorische Abzählungstheorie ermöglicht dann neben der Bestimmung der Gesamtzahl von Molekülgraphen vorgegebener Stoffklasse eine Aufteilung dieser Stoffklasse in Klassen von Graphen mit bestimmten Eigenschaften. Durch diese Klassifikation läßt sich eine genauere Übersicht über die betrachtete Stoffklasse erreichen. In der vorliegenden Arbeit wird gezeigt, wie die Bildung neuer Eigenschaftsparameter bei der Kombination von Eigenschaftsparametern im Abzählungsverfahren berücksichtigt werden kann. Damit lassen sich etwa über das Pólya-Verfahren Topologische Indizes, z.B. Konnektivitäts-Indizes und Charakteristisches Polynom, für Klassen nichtzyklischer Verbindungen ermitteln. Dies stellt eine Erweiterung bisher üblicher Abzählungsverfahren dar. Der Artikel gibt eine kurze Einführung in die Anwendung der gewichteten Form des Satzes von Pólya und beschreibt die rekursive Abzählung von Bäumen und Wurzelbäumen. Am Beispiel der Abzählung von Knotenzahl, Knotengrad und Spektrum aller Weglängen der Alkane wird das Rechenverfahren vorgestellt. Mögliche Anwendungen in der Quantitativen Struktur-Eigenschafts-Simulation (QSPR) werden kurz angedeutet.

This bibliography has been gathered during a practical training period at the Brandenburg University of Technology at Cottbus. Special emphasis has been laid on Petri nets applications in biochemistry for modelling, analysis, and simulation.

To the best of our knowledge, it was 1993 that Reddy et al. introduced Petri nets for the qualitative modelling of biochemical networks. Since that time, just a few papers appeared every year with similar approaches in order to model and/or analyse biochemical pathways, dealing with metabolism, gene regulation, or signal transduction, respectively. Only recently, there seems to be an increasing interest in that research topic, at least as far we can tell from the total number of published papers.
A quick search through the papers found reveals a quite large variety of Petri net extension, for qualitative modelling:

hierarchies,
colour
as well as for quantitative modelling and simulation:
time dependency: discrete, continuous, hybrid,
stochastics,
self-modification, in different variations, and
fuzzy Petri nets.

All these kinds of Petri nets are used on all levels of biological abstraction, for example for the modelling of the relationships between biochemical substances, cell components, cells, organs, individuals, species, organism classes, and populations.
In chemistry, applications seem to be restricted to the control and regulation of process technologies and installations. Just Kuroda et al. used Petri nets for modelling more complex reaction dynamics.
Also, many applications are conceivable in the field of stuff and energy flows, ecological systems, population dynamics and evolution. Suzuki introduces spatial hybrid Petri nets for that purpose.
In medicine, the majority of the published applications come from the field of management processes, for instance the passing on of patients, diagnosis preparation, display preparation, and therapy. But, there are also a few applications, modelling the development of some diseases.
A central topic of almost all investigated publications is the demonstration of a methodology how to use Petri nets for the chosen application area, often associated with the presentation of related software tools. Accordingly, almost all published applications seem to rely on demonstration examples only. Therefore, a major break through of a Petri net based technology for modelling and analysis in applications related to biology, chemistry, or medicine can´t be reported yet.

This paper demonstrates the first steps of a new integrating methodology to develop and analyse models of biological pathways in a systematic manner using well established Petri net technologies. The whole approach comprises step-wise modelling, animation, model validation as well as qualitative and quantitative analysis for behaviour prediction. In this paper, the first phase is addressed - how to develop and validate a qualitative model, which might be extended afterwards to a quantitative model.
The example used in this paper is devoted to apoptosis, the genetically programmed cell death. Apoptosis is an essential part of normal physiology for most metazoan species. Disturbances in the apoptotic process could lead to several diseases. The signal transduction pathway of apoptosis includes highly complex mechanisms to control and execute programmed cell death. This paper explains how to model and validate this pathway using qualitative Petri nets. The results provide a mathematically unique and valid model enabling the confirmation of known properties as well as new insights in this pathway.