Behaviour Prediction and Simulation of Partition-Based Biological Models – Some Open Issues of Lattice Paradigms Virginia Ecaterina Oltean, Radu Dobrescu, Loretta Ichim, Liliana Dobrica *Politehnica University of Bucharest, Faculty of Control and Computers, 77206 Bucharest, Romania, ( e-mail: ecaterina_oltean@yahoo.com, radud@isis.pub.ro, iloretta@yahoo.com, liliana@aii.pub.ro) Abstract: The state space partition problem for hybrid systems, closely related to prediction of systems’ behaviour, consists of finding the conditions under which a given quantised system is deterministic. Some biological processes present a hybrid nature, involving the interaction between the continuous evolution of internal cell variables, such as concentrations, and abrupt changes of discrete cells characteristics. In these classes of models, the basic modelling paradigm, implying a choice of the state space structure, is a lattice, with elements generally represented by single cells. This paper comparatively presents two approaches from the literature, both with a hybrid nature but involving two distinct lattice types. The description of their behaviour prediction mechanisms are overviewed and simulation of an abstract quasi-random invasion process, inspired from the second model of a brain tumor growth, is finally discussed, emphasising the difficulties in solving the general state space partition problem. Also a tumor-growth limiting model is proposed, as a dynamic nutrient inhibitor. Some aspects driving to a general lattice-based model of a growing process are finally discussed. Keywords: complex systems, hybrid systems, state space partition, hybrid automata, biological modelling, difference equation, simulation. 1. INTRODUCTION A large class of microscopic biological processes involve the interaction between the evolution of continuous variables, such as concentrations - conventionally described by differential laws -, and abrupt changes of discrete parameters, thus presenting a hybrid nature (Cassandras and Lafortune, 1999). The basic modelling paradigm in this case, implying a choice of the state space structure, is a lattice, with elements of the ordered set represented by single cells. The key point in the modelling effort is the estimation of systems’ evolution. In the partition-based hybrid paradigm, the continuous state space is partitioned into disjoint open regions, each of which defining a discrete state and a discrete event occurs whenever the continuous state trajectory crosses the boundary between two adjacent partition regions. Despite the fact that the differential or difference equations - which capture the continuous evolution - are deterministic, the resulting discrete state abstraction, observed only by the generated events sequences, may exhibit nondeterminstic behaviour. This is mainly caused by the fact that the initial continuous state is known only to reside within some initial partition region. In an effort to systematize this modelling problem, Lunze (2004) defines the state space partition problem for hybrid systems as finding the conditions under which a given quantised – i.e. discrete abstraction - system is deterministic. Thus, the choice of the state space structure of a hybrid dynamical model is closely related to the crucial problem of its behaviour prediction. This paper starts from two approaches in the biological modelling literature. The first one is dedicated to a biological protein regulatory network model, called in the sequel the Gosh-Tomlin protein-signalling model (Gosh and Tomlin, 2004). The other one describes a microscopic brain tumor growth process, based on discretisation of a nutrient diffusion equation, called in the sequel the Sander-Deisboeck tumor growth model (Sander and Deisboeck, 2002). These models involve two distinct lattice representations for cell-to-cell discrete interaction modelling. The continuous state variables are basically concentrations – of proteins in the first case and of nutrient, in the second, respectively – which, when hitting some limits, determine an abrupt change of the some cells parameters, thus a change of the systems discrete state. The paper is focussed on a comparative evaluation of the state space structures in both above-mentioned models, with emphasis on the implications on behaviour prediction. The goal of this discussion is the perspective of developing a general lattice-based growing model. A first attempt proposed in this direction is an abstract quasi-random lattice- based invasion process, inspired from the Sander-Deisboeck tumor growth model, given by a second-order nonlinear discrete-time nutrient equation. Also, the problem of modelling the influence of a growth inhibitor is introduced and a first version of an enriched model is proposed. The paper is organized as follows. Section 2 presents an overview of the state space partition problem of Lunze (2004) and a recall on the lattice concept. Section 3 introduces the lattice structure and the hybrid automata (Alur et al., 1995)