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Subsections
Continuous logic
Continuous logic is used to refer to models of regulatory systems in which the activation of a given gene is again considered to be Boolean, but the analytical treatment of the models is more similar to that used in continuous models than previous approaches. Furthermore the system states that are measured are generally qualitative in nature (in comparison to true continuous approaches, where system states are frequently quantitative). Such continuous logic models have been used both for ensemble approaches to determining classes of behaviour as well as qualitative simulation approaches for modelling the behaviour of particular systems.
Glass networks use piecewise linear differential equations (PLDEs) to describe the switching of gene states in continuous time [49]. This methodology has the advantage of rendering systems amenable to analysis, while still allowing complex periodic and chaotic dynamic patterns. The motivating question for this formalism is: ``Given a network with a certain logical structure, what are the possible dynamics that can be found in this network?'' [40].
Assumptions
In order to simplify mathematical analysis, nonlinearities in the updating function are eliminated by replacing continuous sigmoidal functions with discontinuous step functions. The rate equations that result from this approximation are in the form of piecewise linear differential equations. The n-dimensional phase space of a model may therefore be pictured as being divided by threshold hyperplanes into volumes corresponding to qualitative states of the system (spaces in which the system behaves in a qualitatively distinct way) (see Figure 9). Transitions between neighbouring qualitative states occurs whenever a solution starting in one region ends in another region. These systems have two types of steady states: regular steady states, lying within a volume and singular steady states, lying on one or more threshold planes between volumes.
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One of the primary advantages of using differential equations to model updating functions is that time may incorporated in a continuous fashion. A disadvantage of this approach is that analytical methods frequently scale poorly, limiting analysis to small systems, the use of unrealistic simplifications, or the use of numerical simulation, which typically requires the introduction of some form of temporal discretization.
A common feature of these formalisms is their motivations is frequently to render the dynamics of complex regulatory systems tractable to mathematical analysis. As a result, they omit many complicating features of real biological systems, including time delays, spatial structure, sigmoidal activation and regulatory control of decay rates. A framework that allows regulatory mechanisms to be described more comprehensively has been proposed by Mestl and colleagues [85], however, this additional complexity limits the application of these techniques to relatively small systems.
Applications
Early usage of this formalism was restricted to analytical techniques [39] and ensemble approaches [49]. The mathematical complexity of the approach frequently limited application to only very small systems (two or three interacting genes). Recently, a streamlined qualitative simulation technique based on PLDE models has been proposed [34,33] that is extendable to large systems and has been used to model sporulation in B. subtilis [32].
Next: Further reading Up: Logical models Previous: Generalised logic Nic Geard 2004-05-06