In multicellular organisms, the same genetic material is shared by many distinct cell types. During embryonic development, tissue repair & growth, etc., cells can transition from one type to another. At least in "complex" animals (read "vertebrates") under normal conditions, cell type transitions across different species broadly share the following characteristics:

- Transitions are targeted (a cell can't just transit from any type to any other type) and directional (transitions are one way, i.e., bone marrow $\to$ blood, but not vice versa).
- Transitions are directed by a small set of molecular signals (morphogens). However, different cell types can respond to the same signal in distinct ways.

It also seems approximately true that organisms with smaller gene-regulatory networks (GRNs) have fewer cell types. There are ~250 DNA-binding transcription factors (TFs) and $\mathcal{O}(1)$ cell type in *S. cerevisiae* (yeast), ~1000 TFs and $\mathcal{O}(10)$ cell types in *C. elegans* (nematode), ~2500 TFs and $\mathcal{O}(100)$ cell types in humans. However, the scaling of cell type number with GRN size is actually kind of odd, since generically the number of states of a $N$-component network grows as $e^{N}$. The problem is not that we humans have more cell types than yeast, but that we only have ~100 times more.

I think these facts are probably related on a deep, physical level, by which I mean they are probably independent of the detailed interaction between specific genes and specific proteins in specific organisms. I suspect any sufficiently complex GRN will exhibit these characteristics.

Let's consider a minimalist $N$-component GRN governed by \begin{equation}\label{model} \frac{ds_{i}}{dt} = -\frac{s_i}{\tau_i} + r_{i}(\vec{s}), \quad i=1, \dots, N. \end{equation} Here $s_i$ is the expression level of gene $i$. I make no distinction between mRNA and protein (more on this in a later post), and by "genes" I mean genes encoding TFs. The term $-s_i/\tau_i$ describes the degradation of gene products over time (for simplicity I will set $\tau_i=1$ for all $i$), and $r_i(\vec{s})$ is the rate of transcription/translation from gene $i$. The rate is $\vec{s}$-dependent due to possible gene-regulatory activity by $j, k, \dots$, as well as $i$ itself.

Now I assume the net gene-regulatory activity on $i$ from $i, j, k, \dots$ is described by a mean field $\phi_i(\vec{s})$, and that $r_i(\vec{s})=r[\phi_i(\vec{s})]$ is an universal function of $\phi_i$ only. In particular, I take $r(\phi_i) = r(\phi_i - \phi_0)$ to be a *S*-shaped function centered on a threshold $\phi_0 > 0$ and bounded by $0 \leq r(\phi_i) \leq 1$. The strength scale of TF-gene interactions is given by $\phi_0$: gene-regulatory interaction where $\vert J_{ij} \vert < \phi_0$ can be regarded as weak, while interaction where $\vert J_{ij} \vert > \phi_0$ can be regarded as strong. We will set $\phi_0 = 1$ without loss of generality (see update below).

I don't know what $\phi_i(\vec{s})$ looks like, but I can expand it as a series, \begin{equation}\label{phi_expansion} \phi_i(\vec{s}) = \sum_j J_{ij} s_{j} + \frac{1}{2}\sum_{j, k} K_{ijk}s_{j}s_{k} + \dots. \end{equation} The terms of the series can be interpreted as binding of $i$'s regulatory elements by one, two, or more TFs. I will truncate the expansion to retain only one-body TF-gene interactions parametrized by $J_{ij}$. The sign ($\pm$) of $J_{ij}$ describes whether $j$ promotes or inhibits the expression of $i$.

As defined, (\ref{model}) describes a large, non-linear dynamical system, and in general we expect such a system will be multi-stable. I will assume experimentally observable cell types correspond to *stable* fixed points of the model. The state $\vec{s}=0$, i.e. death, where nothing is being expressed, is a stable fixed point of the model. GRN states that are not fixed points will evolve on a time scale set by the lifetime of gene products (mRNA and protein), which is ~1 hour. Cells corresponding to a non-stationary state of the GRN will not be around long enough for us to capture and study it.

Does this model do anything interesting? Of course it does; otherwise I wouldn't be writing about it. This is an on-going research project, and these posts are essentially a set of notes-to-self to collect my thoughts in a semi-organized fashion.

**Updated September 6, 2015.** If $J_{ij}$s are i.i.d. with zero mean and variance $\sigma_J^2$, then $\phi_i = \sum_j J_{ij}s_j$ will be i.i.d. with zero mean and variance $\sigma_\phi^2 = N\sigma_J^2/2$. The typical size of $\vert\phi_i\vert \approx \sigma_\phi$ will grow as $\sqrt{N}$; this is dynamically equivalent letting $\phi_0 \to 0$ as $1/\sqrt{N}$. Maintaining dynamical similarity between GRNs of different sizes requires either rescaling $\phi_0 \to \sqrt{N/2}\phi_0$, or rescaling $\sigma_J \to \sigma_J/\sqrt{N/2}$.

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