Karl-Peter Hadeler

PAProC: a prediction algorithm for proteasomal cleavages available on the WWW.

Abstract. The first version of PAProC (Prediction Algorithm for Proteasomal Cleavages) is now available to the general public. PAProC is a prediction tool for cleavages by human and yeast proteasomes, based on experimental cleavage data. It will be particularly useful for immunologists working on antigen processing and the prediction of major histocompatibility complex classxa0I molecule (MHCxa0I) ligands and cytotoxic T-lymphocyte (CTL) epitopes. Likewise, in cases in which proteasomal protein degradation has been indicated in disease, PAProC can be used to assess the general cleavability of disease-linked proteins. On its web site (http://www.paproc.de), background information and hyperlinks are provided for the user (e.g., to SYFPEITHI, the database for the prediction of MHCxa0I ligands).

Cellular Automata : Analysis and Applications

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Dynamical Systems of Population Dynamics

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Mathematische Demographie und Epidemiologie

After a short review of the the role of population dynamics in the formation of the theory of dynamical systems some recent developments are described, centered around work within the DANSE project: Reaction transport equations which are refinements of reaction diffusion equations, showing travelling front solutions and pattern formation, with applications to chemotaxis and the formation of polarized groups of animals or cells. Populations structured by age are modeled by conservation laws and by delay equations, the thorough investigation of the close connection between these two classes of systems leads to a solid justification of the latter as population models. Particular attention is given to models for the spread of infectious diseases: Models describing outbreaks in closed populations, optimal vaccination and control policies, the conditions for backward bifurcations and (unwanted) hysteresis phenomena in public health. Epidemic spread in space (travelling front problems), simplification of complex models to modulation equations. Modeling the behavior of single individuals connects dynamical systems to stochastic processes: Synchronous and asynchronous cellular automata, interacting particle systems, concrete applications are ring vaccination (in animal husbandry) and contact tracing (in human populations).

Cellular Automata: Basic Definitions

Die Demographie als Bevolkerungswissenschaft beschaftigt sich mit den Regeln, nach denen sich menschliche Populationen entwickeln, mit ihrem Altersaufbau, der Zahl der Geburten und Todesfalle und sozialen Charakteristika. Im Vordergrund steht dabei nicht so sehr das Schicksal des Individuums als die Erklarung vergangener Entwicklungen und die Prognose. Die Prognose kann nur unter der Pramisse rebus sic stantibus erfolgen. Die Erfahrung zeigt, das nicht nur Emigration und Immigration, sondern auch relativ geringe politische oder okonomische Schwankungen etwa das Reproduktionsverhalten entscheidend beeinflussen konnen.

Applications in Various Areas

What is a cellular automaton? This question is not too easy to answer. In the present approach, we define cellular automata in a very narrow sense: the automaton is deterministic and it has a high degree of symmetry. This narrow definition allows us to develop a relatively rich theory. In applications, however, these assumptions are quite often relaxed. Many mathematical systems describe the behavior of single particles, molecules or cells. Such small entities, described on a small spatial scale, do not follow strict deterministic laws but are subject to stochastic variation. Hence it is quite natural to generalize the concept of a cellular automaton in such a way that the local rule depends on random variables.

Language Classification of Kůrka

There are many examples where cellular automata contribute to the understanding of scientific phenomena. In the following, we briefly sketch three of these applications to demonstrate the flexibility of cellular automata as a modeling approach. All these models allow for a specific analysis of their dynamics, at least in some heuristic way.

Lotka–Volterra and Replicator Systems

The classifications considered so far focused on the long term dynamics (attractors) respectively the complexity of the dynamics (Lyapunov stability). Culik and Hurd [38] and Kůrka [111] turned their attention to the “automaton” aspect of cellular automata. They developed a classification based on the theory of finite automata resp. formal grammars.

Chaos and Lyapunov Stability

Nevertheless, there are few mathematical results that apply to the whole class of Lotka–Volterra systems and it is not known, in general, which dynamics can occur within this class.

Besicovitch and Weyl Topologies

In the preceding section we introduced a classification of cellular automata based on attractors, their number and structure. In the present section we focus on the complexity of the dynamics. The two aspects are not independent, but differ slightly. We start with Devaney’s definition of chaos, and relate this definition to the Hurley classification. Thereafter we investigate a class of cellular automata that induce chaotic dynamics: permutive cellular automata. In the last part of this section we focus on one special property of chaotic dynamics: sensitive dependency on initial conditions. It is possible to recognize complex dynamics by inspecting the fate of two neighboring points.