Mathematical model

A simple kind of model trying to account for switch-like behavior is where a set of class B proteins activate their own transcription. If class A proteins are considered to be expressed in a constitutive way, and not subjected to regulated degradation, they are present at a constant level. Only the time-evolution of each of the class B species is thus of interest. Calling $x_i, i=1..n$, the concentrations of $B_i$ (class B species), the equations are

$$\frac{\mathrm{d}x_i}{\mathrm{d}t}=-d_i x_i+\sigma_i \frac{x_i^2}{\alpha D^2 + x_i^2},$$ (1)

with $D=1+\Sigma_{i=1}^nx_i$, $\alpha=K_{2}^2/a_{t}^2 \in {\sf R\hspace*{-0.9ex}\rule{0.15ex}{1.5ex}\hspace*{0.9ex}}_{*}^{+}$, where $K_2$ is the concentration of $A-B_i$ complex at which $B_i$ transcription is half-maximal, $a_t$ is the total quantity of class A proteins, $\sigma_i$ and $d_i$ are respectively the maximal synthesis rate and the degradation rate of $B_i$, and where each $x_i$ is normalized with respect to the dissociation constant for the $A-B_i$ complex (this normalization leads to each maximal synthesis rate $\sigma_i$ being divided by the dissociation constant of the $A-B_i$ complex, see Appendix A). The equations assume that for all $i$ the quantity of $A-B_i$ complexes is negligible compared to the total quantity of $B_i$ (see Cinquin, 2006, for a relaxation of that assumption).

This set of equations is the same as derived by Cinquin (2005), without the restriction $\forall i, d_i=1, \sigma_i=\sigma$. We perform a steady state analysis of the system, assuming that it equilibrates over a time scale much shorter than that of cellular differentiation; this assumption is supported by the fact that transcription factors commonly have very short half-lives, which can be as low as a few minutes, while cellular differentiation often takes place over the course of hours or days.