Mathematical Models

Models are limited or simplified representations of real systems and they are used to study the complexity of their real counterparts. Today, the area of modelling biological systems is dominated by approaches referred to as classic methods, such as mathematical modelling [4].

Quantitative analysis of the dynamics of biological networks often requires differential equations [6], which determine the rates of the flows in the system under various conditions. Most of biological systems are very complex and show nonlinear behaviours, which makes it hard to solve them analytically. While universal mathematical software packages with numerical equation-solving capabilities exist, specific tools have been developed in the field of metabolic networks, including Gepasi [7], WinSAAM [4] and DBSolve [8].

In addition, tools for modelling biochemical systems using Petri nets (PN) [9, 10] and stochastic simulation [28, 29] are available. An advantage on Petri nets is that equations are visualized graphically so that the user can construct and modify them very easily. PN are bipartite graphs built from four types of elements: places, transitions, arcs, and tokens. Each equation is directly modelled using a transition which is graphically represented by a vertical or horizontal bar. Places are circles containing information about concentrations of substances represented by tokens (dots). Arcs are directed edges indicating the flow of materials consumed and produced by each transition. Actually, a transition will consume tokens from places connected as input and produce tokens at places connected as output. To define reaction rates, transitions can be associated with equations which are computed in a continous or discrete manner at runtime. Furthermore, stoichiometric coefficients can be assigned to arcs to determine the number of tokens processes in each transition. The configuration of tokens represents the state of the system.

In the early stages of the modelling process, the modeller analyzes the mechanism of the real system under study. As mentioned in Section 11.1, these mechanisms can describe classes of processes. Figure 11.3 shows a simplified example of a Petri-net model for the interaction of receptors with G protein, based on the example shown in Figure 11.1. The displayed network represents a pattern, which is applicable to every receptor of its type, e.g., the activation of adenylate cyclase or of phos-pholipase C. For each application, the identities of places and transitions have to be substituted by the objects ofthe real system. The configuration ofthe system is given mythe initial number of tokens. In the example signal, receptor, G protein, and effector are present which one element each.

Representing sets of equations by Petri nets makes it clear that biological systems can be interpreted as special types of graphs. Graph theory has been applied to metabolic networks in the past [11, 12]; in principle, it enables us to analyse the complexity of biological networks. For many problems, the total state of systems, details of the system, and the kinetics of the system processes are unknown. In addition, detailed information can be hidden within large datasets, which have to be mined. Additionally, biological models can be very complex and need to be verified to avoid inconsistencies, which are not easily detectable. However, an advantage of graph theory is that previous work in this broad area has provided a robust library of directly applicable methods for calculating qualitative information about the effective range of agents.

Fig. 11.3 The mechanism of G-protein interacting with a receptor, visualized as a Petri net.
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