tag:blogger.com,1999:blog-77799167980889640112017-07-23T07:31:58.552-04:00Coexistence Boundaryfield notes from the interface between physics, chemistry, and biologyZou Ling-Nanhttp://www.blogger.com/profile/00676789182316516345noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-7779916798088964011.post-28700593324554096212015-09-04T01:55:00.002-04:002015-09-17T14:35:32.806-04:00A mean field model of the cell<p>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: <ol><li>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). </li><li>Transitions are directed by a small set of molecular signals (morphogens). However, different cell types can respond to the same signal in distinct ways. </li></ol>I don't know how widely true these characteristics are in very "simple" animals such as sponges or the hydra, or in plants (I know nothing about plant biology). <p>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 <i>S. cerevisiae</i> (yeast), ~1000 TFs and $\mathcal{O}(10)$ cell types in <i>C. elegans</i> (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. <p>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. <p>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. <p>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 <i>S</i>-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). <div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-FhlHDCRxPcA/VenP6Ur_IDI/AAAAAAAAAfc/EoEd7Uwx-78/s1600/r.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-FhlHDCRxPcA/VenP6Ur_IDI/AAAAAAAAAfc/EoEd7Uwx-78/s1600/r.png" width="40%"/></a></br><b>Figure 1.</b> The assumed functional form of $r(\phi_i)$.</div> <p>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 $i$. <p>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 <i>stable</i> 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. <p>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. <p><b>Updated September 6, 2015.</b> 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 $\sigma_J \to \sigma_J/\sqrt{N/2}$.Zou Ling-Nanhttp://www.blogger.com/profile/00676789182316516345noreply@blogger.com0