Many different approaches have been developed to model and simulate gene regulatory networks. We proposed the following categories for gene regulatory network models: network parts lists, network topology models, network control logic models, and dynamic models. Here we will describe some examples for each of these categories. We will study the topology of gene regulatory networks in yeast in more detail, comparing a direct network derived from transcription factor binding data and an indirect network derived from genome-wide expression data in mutants. Regarding the network dynamics we briefly describe discrete and continuous approaches to network modelling, then describe a hybrid model called Finite State Linear Model and demonstrate that some simple network dynamics can be simulated in this model.
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Gene networks are concerned with the control of transcription, i.e. how genes are up and down regulated in response to signals. In the 1960's genetic and biochemical experiments demonstrated the presence of regulatory sequences in the proximity of genes and the existence of proteins that are able to bind to those elements and to control the activity of genes by either activation or repression of transcription. These regulatory proteins are themselves encoded by genes (Figure 1). This allows the formation of complex regulatory networks, including positive and negative feedback loops. These principles of gene regulation apply to prokaryotes (e.g. bacteria) as well as to eukaryotes (e.g. higher organisms). The control of gene activity is much more complex than Figure 1 suggests. It involves many kinds of proteins thus allowing additional levels of control particularly in eukaryotes. Transcription factors, the proteins that recognize the regulatory elements in the DNA (the binding sites) need to interact with other proteins in order to activate gene expression. In addition to control of gene expression there are regulatory controls to determine the maturation, transport and degradation of the mRNA, as well as its translation. Just to illustrate the complexity of gene regulation: Gene Ontology (GO), a controlled vocabulary used to describe protein functions contains currently over 7500 different terms describing biological process 'transcription', including over 6500 terms under process 'regulation of transcription' [3].
Example for network logics. Genes A, B and C control the activity of gene D; D is active if A and B are bound, but not C; right: shows the FSLM representation for such a promoter. Reproduced from [2].
This approach assumes that the influence of one gene on another gene is linear. Note that the network topology will determine which of the weights w ij are equal to 0 (i.e., if there is no arc from gene i to gene j in the network topology, then w ij = 0). Like Boolean functions, linear functions are only approximations. For instance, it is not possible by linear functions to model a situation where the same transcription factor can play a role of an activator or repressor for the same gene, depending on the presence or absence of other transcription factors.
Although few promoters have been studied in great detail, there are excellent examples, such as the description of the promoter action logics of sea urchin developmental gene Endo16 [44]. The Endo16 promoter consists of almost 30 regulatory elements stretched over a region of 2.3 kb. Based on experimental data collected using modified promoter constructs Davidson and co-workers constructed a model expressed as an algorithm combining Boolean and linear functions. This algorithm takes as an input the occupancy information from 12 binding sites and outputs a value, that 'can be thought of as the factor by which, at any point of time, the endogenous transcription activity (...) is multiplied as a result of the interactions mediated by the cis-regulatory control system' [44]. Predictions of promoter manipulations based on this model have largely been confirmed in subsequent experiments. Extending their earlier work the group of Davidson compiled a regulatory network containing over 40 genes by the construction of a model that integrates extensive experimental evidence on early development of sea urchin embryos [45].
Recently Klamt et al. published an example for control logic networks [46], based on hypergraphs, which are an extension to the graphs described above. Several hyper-edges pointing to the same node represent OR relationships, but edges are allowed to combine to represent AND relationships. Weights on the edges distinguish positive and negative relationships. The authors provide a set of methods to analyse these networks, just to list a few examples: computation of all positive and negative signalling paths, computation of all positive and negative feedback loops and computation of minimal cut sets. These minimal cut sets report the smallest number of interventions necessary to force the network into a particular behaviour, for example, a minimal number of deletions necessary to block the activation of a particular downstream protein in a signalling cascade. These methods are implemented in the software tool CellNetAnalyzer and the example presented, a model of a signalling network for T-cell activation shows that these analyses are non-trivial for signalling networks of a typical size.
There are situations where neither Boolean rules nor linear functions are powerful enough to express the control logics: transcription factors might bind competitively, if one factor is bound, the other one is excluded, as is the case for example in the phage λ switch between lysis and lysogeny [54]. In some cases, transcription factors have to form homodimers or heterodimers to be fully functional. The transcription factors might have to bind sequentially or might act synergistically. In these situations it might be necessary to use more complex functions (here this would be solved by Boolean circuits with memory or delay). It remains an open question what is the minimum repertoire of functions to describe regulatory logics.
The knowledge of the parts list of a network, its topology and the control logics are necessary requirements in order to expand the model to capture dynamic changes during time. Compared to the approaches above, the dynamic models can be described as 'classical' approaches to gene network modelling, as many of them have been developed and studied long before the current genome era. Typically, they are relatively small, involving only a few genes. They aim at describing and often simulating the dynamic changes in the state of the system and predicting the network's response to various environmental changes and stimuli.
Various dynamic models have been proposed. Greller and Somogyi subclassified them [55] as follows: "Dichotomies for framing our thinking on how to best represent a particular biological network problem include the following distinguishing attributes: quantitative versus qualitative measurements; logical versus ordinal variables (e.g. Boolean versus abundances); deterministic versus probabilistic state transitions (e.g. differential equations versus hidden Markov); deterministic versus statistical overall system description (e.g. vector field versus Bayesian belief network probability distributions); continuous versus discrete state (e.g. continuous intensities or concentrations versus low, medium and high); nonlinear versus linear elementary interactions and state update rules (e.g. multiplicatives, sigmoids or non-monitonics versus linear ramps); high-dimensional versus low-dimensional (e.g. >> 100s of variables versus
As an example we will describe the finite state linear model (FSLM), more detailed descriptions of FSLM can be found in [2, 90, 91]. It combines the advantages of Boolean networks such as simplicity and low computational cost, with the advantages of continuous models, such as continuous representation of concentrations and time. The activity of genes is described by discrete states (e.g., gene is 'on' or 'off'), but the gene product concentrations are expressed as real numbers. Time is continuous in FSLM and the state of the network determines directly the concentration change rates, while the state is in turn affected by the concentrations themselves.
In FSLM there is only one class of molecules, represented by substances. There are three types of network elements: binding sites, control functions and substance generators (Figure 11A). The binding sites in the FLSM are comparable to DNA binding sites for transcription factors in the promoter regions of genes. A combination of binding site(s), control function(s) and a substance generator in the FSLM corresponds to a biological gene (Figure 11A). A gene network consists of one or more such genes, which influence each other via the substances they produce (Figure 12).
The building blocks of the finite state linear model. A Binding sites are represented by triangles, control functions by boxes and substance generators by diamonds. Dotted lines represent cases where the discrete output of one element is the input for another element. B Switching behaviour of the binding sites. The curve (left) is typical for processes with hysteresis characteristics of a system that does not instantly follow the forces applied to it, but reacts slowly, or does not return completely to their original state: that is, systems whose states depend on their immediate history. The threshold for switching the states of the binding sites in FSLM is state dependent and results in a similar curve (right). [c] concentration of substance binding to binding site j; asso j , disso j association and dissociation constants for binding site j; u binding site not occupied, o binding site occupied. Reproduced from [2]. 2ff7e9595c
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