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Towards more biologically realistic update functions in real-life regulatory network models
Mario Giacobini
Faculty of Veterinary Medicine
Molecular Biotechnology Center
University of Torino, Italy


It has been suggested that the cells of living organisms are functioning in a near-chaotic regime called critical, which offers a trade-off between stability and evolvability. Abstract models for genetic regulatory networks (GRNs) such as Kauffmans Random Boolean Networks (RBNs) [4] certainly point in that direction. Although very attractive for its simplicity, the original RBN model suffers a number of limitations that can be overcome by incorporating recent discoveries. We previously addressed some of these weaknesses [2], namely the synchronicity and the random gene interaction. Here, we tackle different aspects of the RBNs original shortcomings. Firstly, even if their exact values are unknown, it is clear that gene update functions should not be random. Gene expression rests on the combined effect of incoming proteins that can have a promoting or repressing action on their target genes. Stoll et al. [7] have proposed a simple additive dynamical rule that characterizes the temporal evolution of the genes state. They consider that both the activating and repressing factors have the same weight, and thus, the state of a target gene at the next time-step will be: active if it receives a majority of promoting components from already active genes, inactive it receives a majority of repressing components, or the state of the target gene will remain unchanged in case the number of promoting and repressing inputs are equal. Inspired by their work, we propose an update function shared by all genes that takes into account the fact that promoting and repressing components could have asymmetrical effects. A gene could require a majority by more than one active gene to switch states, and therefore, we introduced a threshold parameter which has to be met in order for a gene to activate. This update function is equivalent to Stolls in the case where the threshold value is 0.5. Additionally, all rules in this class can be proven to correspond to a subset of the original RBN update functions. This offers the advantage of making the new model state space exhaustively enumerable. Another questionable assumption of the original RBNs model is the totally random interaction among genes with a fixed connectivity. Thanks to the recent developments in high throughput genomic and biochemical techniques, small parts of real-life GRNs have been discovered with data sufficiently reliable to specify highly probable interaction, with different confidence rates. We have selected the core transcriptional network in embryonic cells published by Chen et al. [1] and a portion of the yeast cell-cycle by Li et al. [5] as a substrate for our RBN model. Original RBNs go through a phase transition by tuning parameters such as the degree of the nodes and probability of a gene to be expressed. Considering current knowledge about GRNs, some of Kauffmans properties of the model are now subject to criticism. In our case, we use real-life networks and not artificial ones, and thus we cannot tune any property of the network topology to obtain the desired critical regime. Instead, we can adjust the threshold value in the update function to set the system into a critical regime. Finally, to validate our model we use a third dynamic Boolean sub-network from plant biology presented by Li et al. [6]. In this model the actual Boolean function of each component in the network were established and our analysis clearly shows it operates in the critical regime. The results are excellent, and prove that in this particular case, our Boolean functions are much closer to biological ones than random ones with an overlap of approximately 92%. Taking into account recent years advances in the field of cellular biology, we have proposed to identify under what conditions Kauffmans hypothesis that living organism cells operate in a region bordering order and chaos holds. This property confers to living organisms both the stability to resist transcriptional errors and external disruptions, and, at the same time, the flexibility necessary to evolution.


gene regulation, random boolean network


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  2. Ch. Darabos, M. Tomassini, and M. Giacobini.. (2009) Dynamics of unperturbed and noisy generalized boolean networks., Journal of Theoretical Biology., 4(260):531 544.
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