Mathematical models

The models studied here have an arbitrary number of components. Each variable represents the intracellular concentration of a differentiation factor (called "switch element" in the following), which enhances its own expression and represses that of all others (the system is symmetrical, in that any element has the same relationship with all others, and in that all elements share a common set of parameters). The models can represent different forms of biological interactions. The terminology used below is that of transcriptional control: each factor is supposed to be a protein, which enhances the transcription of its own mRNA, and represses the transcription of the mRNAs for other switch elements, with or without physical interaction with other factors; as a simplification, the translation step is not taken into account in the model, and proteins are thus supposed to act directly on each other's concentrations. There is evidence that translational regulation can play a major role in some cases (Okano et al., 2002, Wickens et al., 2000). In the following models, different forms of post-transcriptional control (by means of regulation of mRNA stability, or translation of the proteins), can be represented in the same way as transcriptional control. Downregulation of cytokine receptors has been observed prior to commitment (Kondo et al., 2000), and downregulation of receptors promoting expression of competing factors could also be accounted for by the following models.

3 kinds of models are studied below:

• Mutual inhibition with autocatalysis: all switch elements repress one another, and enhance their own expression. This is one of the simplest models one can think of that is able to achieve dominant expression of each of its elements, depending on the initial conditions.
• Mutual inhibition with autocatalysis, and leak: the same as the previous, with a supplementary term that represents an identical, basal level of expression, which is independent of any element of the network. This could correspond for example to a gene upstream in the differentiation hierarchy, which "primes" the lower level of the differentiation network, as has been proposed within the hematopoietic differentiation network (Ye et al., 2003, reviewed by Orkin, 2003).
• bHLH dimerisation: based on the class A/class B bHLH dimerisation discussed above.

The first two models can be viewed as a generic representation of the interactions between switch elements, while the third is based on an explicit assumption. All are formulated according to standard kinetics.

These models are cell-autonomous, and do not take into account "differentiation cues" that cells receive. The models could be extended to take into account either different initial conditions, leading to various differentiated states, or different biases of the network (for example by providing a higher basal expression level of one of the factors).