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Subsections
Spatial models
Many of the models described above do not include any consideration of the physical space in which gene regulation is occurring. However, there are at least two possible situations in which spatial information may be important. All regulatory events involve a physical interaction between molecules, some of which are present only in very small numbers and all of which are several orders of magnitude smaller than the size of a cell. Therefore a molecule may take some time, and require the assistance of some additional mechanism, before it is located in a position to act. This importance of localisation is compounded in eukaryotic cells, which have a complicated internal structure. As an example, while mRNA molecules are transcribed from DNA in the cell nucleus, they must be transported through the nucleur membrane and into the cytoplasm before they can be transcribed. While including the location and momentum of every single molecule would quickly become computationally infeasible, some models incorporate time delays to allow for the diffusion and transportation of molecules [111].
The second situation when spatial information may be required arises in models that incorporate interactions between cells. One of the most apparent distinctions between prokaryotic and eukaryotic organisms is that, while prokaryotes all consist of a single cell, a large majority of eukaryotes are multi-cellular. A human, for example, consists of around a trillion cells. Specifying the morphogenetic processes that transform a single cell into a complete organism requires a substantial increase in regulatory complexity. It also introduces several new issues related both to intercellular communication and to the mechanical processes of development, such as migration and cell adhesion.
Intercellular communication
One of the simplest ways of implementing intercellular communication is to simply allow network connections to exist not only between elements within a cell, but also between elements in adjacent cells [138,83].
Mechanisms of development
Controlling the formation of spatial patterns during development presents a significant computational challenge. In addition to the the internal dynamics of the cell, external factors such as protein gradients and physical interactions between cells also play a role. One of the earliest mathematical attempts at modelling pattern formation was by Turing. His approach used a pair of coupled reaction-diffusion equations to describe a system consisting of two chemicals, known as morphogens (see [16] for a review). As the two morphogens diffuse across a spatial field and react with one another, a variety of patterns emerge, depeding on parameter values. One problem with this approach is the lack of any evidence for morphogens actually existing in a biological system. A gap therefore exists between the phenomenological description of the pattern formation process and the regulatory process that controls it at a genetic level.
The gene circuit approach of Mjolsness et. al. [87] mentioned above goes some way towards addressing this issue. The geometric aspect of the model uses a diffusion mechanism to describe communication between cells. Solé and Salazar-Ciudad also use a reaction-diffusion mechanism linked directly to a regulatory network to investigate developmental dynamics [104,118]. Their model is based on that of Mjolsness:
where xij represents the concentration of gene product j in cell i, the first term specifies the production of xij, the second term its degradation, and the final term specifies the diffusion component, at rate Dj. The networks are connected together in a random fashion, inspired by the ensemble approach of random Boolean networks [69], and the behaviour of the networks under different parameter settings is explored. One finding of this study was that networks capable of producing spatial patterns such as gradients, stripes, spots and noise (chaos) are relatively common once a connectivity threshold is crossed [118].
Next: Further reading Up: Continuous models Previous: Hybrid models Nic Geard 2004-05-06