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Subsections
Generalised logic
The generalised logic formalism for modelling GRNs has been developed by René Thomas and a number of colleagues over the past three decades [127]. While its origins lie in similar areas to the Boolean models described above, it is distinguished by several features: it is inherently asynchronous, it allows variables to take multiple logical values and it allows for a more sophisticated definition of logical interactions, involving multiple thresholds and parameters. Generalised logic is also motivated by a different set of questions. While Kauffman's networks were developed to investigate the theoretical properties of an entire class of networks, generalised logic tends to focus on models of actual systems. It provides a set of tools with which to characterise and analyse networks derived either from known interactions or from measured patterns of gene expression in terms of their dynamic steady states.
Assumptions
Although the initial version of the generalized logic formalism described the state of a gene in a Boolean fashion [123], later iterations introduce the possibility of state variables assuming more than two levels [133,124]. The argument for multivariate logic is that when a particular element acts in more than one context, it cannot necessarily be assumed that the thresholds required for each of these actions to occur is going to be equal. For example product X may have an effect on gene Y when it reaches concentration c1 and also have a further effect on gene Z at concentration c2 (see Figure - multiple thresholds).
The generalized logical formalism also allows for a considerably more sophisticated form of logical updating than the Boolean rules used in RBNs. The first refinement is the introduction of logical parameters, which allow for weighted gene interactions [113]. The argument for allowing this complication is that genes may be expressed to different extents in different circumstances and therefore may affect the expression of another gene to varying degrees. The second refinement concerns the possibility that some steady states of a system, particularly unstable ones, may be located at the threshold values [113]. This issue is dealt with by introducing logical values for the thresholds, as well as for expression levels below and above thresholds.
Unlike RBNs, in which time is measured discretely, the generalized logic formalism uses continuous time, allowing for asynchronous updating of elements [124]. It is important to note that the form of asynchrony used here is deterministic in its ordering of element updating. Instead of a set of deterministic state transition rules, a generalized logic model defines a set of functions mapping current states to their image, or the state towards which a system would tend to move if all variable updates were carried out. This transition is enhanced by the inclusion of two time delays, one describing the period between a gene switching on and its product reaching functional levels and the other describing the period between a gene swtiching off and its product dropping below functional levels (see figure - time delays). The use of asynchrony produces to systems containing more complex sets of periodic attractors than standard synchronous networks and the dynamics of such systems tend to be closer to equivalent differential models.
Model Description
The first stage in building a logical description of a system is to specify the graph of positive and negative interactions between logical elements. From this diagram, logical equations, and a corresponding image table may be inferred. It is important to note that, unlike the Boolean network approach described above, the image table does not show deterministic transitions. Whereas the standard Boolean network assumed synchronous updating of all elements, the generalised logic formalism is inherently asynchronous. Therefore, in a transition involving the change of state of two genes, the probability of both genes being updated simulataneously is infinitesimally small. Therefore, one of the two possible transitions will occur first, dependant on the time delay for that element, and determine the next state. Carrying out this process for all states results in a transition graph, from which steady states and cycles can be identified (see Figure 8). The path that will actually be taken from this graph can be determined by considering the time delays of each transition. A more thorough description of the model, including more advanced elements such as logical parameters and multi-valued logical variables is given in [127].
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Theoretical results and Applications
Once a model has been built, the set of logical equations can be analysed to determine the logical steady states of the system, analogous to attractors in Boolean systems. A state space can then be constructed in which each state corresponds to a qualitative behaviour of the system.
The ease with which qualitative analyses can be carried out has been increased by theorems which allow the identification of steady states of a system by considering the characteristic state of the circuits that make up a system [114,126]. A circuit is a complete loop in the system's interaction graph. The characteristic state of a circuit is the intersection of the thresholds beyond which each variable in the circuit is active. The properties of these circuits will determine which of them are functional in given conditions, and from this knowledge, the steady states of the system can be determined. While the number of logical states grows rapidly with the size of the system, the number of circuits increases much more slowly, therefore the ability to derive steady states from characteristic states greatly improves the scalability of this type of analysis.
A feedback circuit can also be described as positive or negative, depending on whether it contains an even or odd number of inhibitory interactions respectively. Negative circuits generate homeostasis, while positive circuits are involved in multistationarity, and hence differentiation. Circuits may interact to produce multistationarity in a number of different conditions [128].
The generalised logic formalism has been applied to the analysis of a number of real genetic systems, including phage-
[122], dorso-ventral patterning in Drosophila [106] and flower morphogenesis in Arabidopsis thaliana [84].
Strengths and limitations
The generalized logic formalism is a powerful method for analysing networks whose interactions are well known. It enables the possible qualitative behaviours of a system to be determined in a rigorous and scalable fashion. The use of logical values corresponding to functional thresholds removes the necessity of having to set the values of large numbers of real parameters. The process is amenable to being automated by a computer and it has been demonstrated to be effective for the induction of gene networks from expression data. The explicit inclusion of time delays leads to a considerably more accurate picture of biological systems than synchronous Boolean networks and a number of theoretical insights into necessary conditions for multistationarity have been shown.
One of the primary limitations of this approach is that, because it has been designed for the detailed analysis of relatively small systems consisting of well characterized interactions, its scalability is limited. It is less suited to the exploration of different classes of behaviour and of large, less well-known systems. Furthermore, phenomena such as cyclic behaviour in generalized logic models are quite sensitive to those parameters which do require specification, such as time delays.
Next: Continuous logic Up: Logical models Previous: Boolean networks Nic Geard 2004-05-06