Another fundamental area of research is the development of mathematical models of signalling networks. These models are important as they allow sim ulating the transmission of signals in such networks, from the environmental stimuli to the cell responses.
Several approaches have been proposed for the inference of large regulatory networks from gene expression data. For example, several authors have used discrete models of Boolean networks (Rangel et al. 2004); Bayesian networks (Friedman 2003; Husmeier 2003); continuous models of neural networks (van Someren et al. 2002); differential equations (Kobayashi et al. 2002). An in-depth discussion of these methods goes beyond the scope of this review. We will instead focus on those approaches used to model gene networks in Arabidopsis. Strikingly, the models are still very limited, highlighting the need for further developments.
Several authors have started adapting Boolean language to represent and analyze interactions between pathways (Genoud and Metraux 1999; Genoud et al. 2001, 2002; Devoto et al. 2005). With this language the quantitative features of a molecular interaction may be described discontinuously by several qualitative steps. Using Boolean gates, signalling processes may be represented more accurately as network-like structures than with linear sequences of events in intuitive formalism. Interfering input signals reach a Boolean gate through switches with a molecular identity, generating an output signal that results from the combination of all inputs going through the gate (Arkin and Ross 1994). By using digital simulation programs, as described by Genoud et al. (2001), it is possible to predict the outputs of the logical gates by activating or inactivating input signals. Devoto and Turner (2005) represented using Boolean gates, the integration of the JA pathway with the SA, ET and light signalling pathways inferring the existence of multiple interferences and intersections from genetic and transcription profiling information. In addition, they have used simple Boolean language (one logical operator per molecule, or per complex of molecules) to identify groups of genes whose expression is differentially regulated by MeJA and/or wounding (Devoto et al. 2005). Further implementation on the digital simulation has been described by Trevino Santa Cruz et al. (2005) where it has been suggested in order to overcome the simplifications of digital networks by integrating noise and clock signals on a digital simulator in order account for the existence of signalling background and circadian rhythms in biological systems. In the first case a digital signal could be added to a source of random signal in order to simulate biological noise. The activation of an oscillating clock-like signal could be placed under the control of a simple ON/OFF switch through an AND operator.
One of the most studied abiotic stresses is water deprivation. During drought, the plant hormone ABA inhibits stomatal opening and promotes stomatal closure, thereby promoting water conservation. Abscisic acid signal transduction in guard cells is therefore one of the best characterized signalling systems in plants. Li et al. (2006) have formalized the large amount of information that has been gathered on ABA induction of stomatal closure from individual experiments and used this information to reconstruct the ABA signalling network. An advantage of the method used here over other methods such as those used in Science's Signal Transduction Knowledge Environment (STKE) connection maps (Assmann 2004) is the inclusion of intermediate nodes when direct physical interactions between two components have not been demonstrated. In this model, the dynamics of state changes are governed by Boolean rules providing the state transition of each node given the state of its regulators (upstream nodes). The model obtained sums up the regulation of more than 40 identified network components, and it is in agreement with previous experimental results. By simulating gene disruptions and pharmacological interventions, the robustness of the network against perturbations was also assessed. Simulations of stomatal response as derived from the proposed model provide an efficient tool for the identification of candidate manipulations that have the best chance of conferring increased drought stress tolerance and for the prioritization of future experimental analyses.
Recently Wang et al. (2006) have inferred a network in Arabidopsis based on 35 links generated from stress response datasets in shoots. GNR (Gene Network Reconstruction tool; http://zhangorup.aporc.org/bioinfo/grninfer/, http://digbio.missouri.edu/grninfer/ and http://intelligent.eic.osaka-sandai. ac.jp) is based on linear programming and differential equations aimed to reconstruct gene network using multiple datasets from different sources without normalization among the datasets. One of the main limitations of gene expression datasets consists of relatively few time points with respect to a large number of genes (generally in thousands). In addition to the dimensionality problem of the data, another problem is that the derived gene networks often have heavily connected gene regulatory associations among nodes. This method provides a general scaffold to analyze microarray data by fully exploiting all available microarray data for a given species, so as to improve the problem of dimensionality or data scarceness. An assumption for the proposed method is that the structure of the regulatory network is stationary, and does not "rewire" under the environmental conditions for those different datasets. Nine whole-genome Affymetrix chips microarray datasets related to the stress responses, each with six or more time points and each for root and shoot experiments (ATGenExpress database, TAIR, http://www.arabidopsis.org/) were used to test this method.
Despite that the main focus of this review is to examine latest progresses in modelling stress regulatory networks, it is worth highlighting the impact that spatial structures have on gene expression dynamics and the advancements made in growth modelling. Artificial life simulations provide a basis for evaluating methods to reconstruct regulatory networks based on gene expression measurements. The effects of spatial growth on gene expression have to be expected to be significant for network reconstruction. Jan Kim (2005) proposed to use "transsys" simulations (Kim 2001) to explore the impact of morphogenesis and of other parameters on network reconstruction using the algorithm by Rung et al. (2002). This approach simulates reconstruction of a target network that does not organize morphogenesis, but may be informed by it. In this approach, it was chosen to enable attribution of differences to individual morphological structures, rather than to collections of mutant structures with complex and unfavorable statistical properties. This algorithm assumes that significant changes in expression levels resulting from a gene knockout indicate a direct target gene. Generation of knockout mutants and collection of gene expression measurements was implemented in Python (http://www.python.org/), based on the transsys framework (Kim 2001). The "R" language (Ihaka and Gentleman 1996) was used for programming data analysis and visualization. The code underlying the results presented is available on the transsys website (http://www2.cmp.uea.ac.uk/~jtk/transsys/).
Another approach that has been used to model gene regulation and interaction has been provided by fuzzy logic (Du et al. 2005). This work models interactions (also referred to as edges or links) in the network as fuzzy functions depending on the detail known about the network.
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