![](\"caltech_rosen.png\")
\n",
"\n",
"*This tutorial was generated from a Jupyter notebook. You can download the notebook [here](l05_ffls.ipynb).*\n",
"\n",
"\\n\"+\n", " \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n", " \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n", " \"
\\n\"+\n", " \"\\n\"+\n",
" \"from bokeh.resources import INLINE\\n\"+\n",
" \"output_notebook(resources=INLINE)\\n\"+\n",
" \"\\n\"+\n",
" \"\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"
\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"![](\"ffls_alon.png\")
\n",
"\n",
"Half of the FFL architectures are **coherent** and half are **incoherent**. For those that are coherent, X's direct regulation of Z and its indirect regulation of Z are the same, either both activating or both repressing. For incoherent FFLs, X's direct and indirect regulation are not the same."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## The most-encountered FFLs\n",
"\n",
"While FFLs in general are motifs, some FFLS are more often encountered than others. In the figure below, also taken from [the Alon review](https://doi.org/10.1038/nrg2102), we see relative abundance of the eight different FFLs in *E. coli* and *S. cerevisiae*. Two FFLs, C1-FFL and I1-FFL stand out as having much higher abundance than the other six. Let's take a look at these two FFLs in turn to see what features they might provide.\n",
"\n",
"![](\"ffl_abundance.png\")
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Logic of regulation by two transcription factors\n",
"\n",
"Because X and Y both regulate Z in an FFL, we need to specify how they do that. \n",
"\n",
"For the sake of illustration, let us assume we are discussing C1-FFL, where X activates Z and Y also activates Z. One can imagine a scenario where both X and Y need to be present to turn on Z. For example, they could be binding partners that together serve to recruit polymerase to the promoter. We call this **AND logic**. In other words, to get expression of Z, we must have `X AND Y`. Conversely, if either X or Y may alone activate Z, we have **OR** logic. That is, to get expression of Z, we must have `X OR Y`.\n",
"\n",
"So, to fully specify an FFL, we need to also specify the logic of how Z is regulated. So, we might have, for example, a C1-FFL with OR logic. This gives a total of 16 possible FFLs.\n",
"\n",
"We are now left with the task of figuring out how to encode AND and OR logic. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Logic with two activators\n",
"\n",
"Let us start first with Y and Z both activating with AND logic. We can construct a **truth table** for whether or not Z is on, given the on/off status of X and Y. The truth table is shown below, with a zero entry meaning that the gene is not on and a one entry meaning it is on.\n",
"\n",
"|X | Y | Z|\n",
"|:---:|:---:|:----------:|\n",
"|0 | 0 | 0|\n",
"|0 | 1 | 0|\n",
"|1 | 0 | 0|\n",
"|1 | 1 | 1|\n",
"\n",
"We can also construct a truth table for OR logic with X and Y both activating.\n",
"\n",
"X | Y | Z\n",
":---:|:---:|:----------:\n",
"0 | 0 | 0\n",
"0 | 1 | 1\n",
"1 | 0 | 1\n",
"1 | 1 | 1\n",
"\n",
"As a reminder, the dynamics of the concentration of Z may be written as\n",
"\n",
"\\begin{align}\n",
"\\frac{\\mathrm{d}z}{\\mathrm{d}t} = \\beta_0\\,f(x, y) - \\gamma z.\n",
"\\end{align}\n",
"\n",
"Our goal is to specify the dimensionless function $f(x, y)$. How do we encode it with Hill functions? We could derive the functional form from the molecular details of the promoter region and the activators. Instead, we will again use a phenomenological Hill function. (Again, we could show that these Hill functions match certain promoter architectures.) The following functions work.\n",
"\n",
"\\begin{align}\n",
" &\\text{AND logic: } f(x,y) = \\frac{(x/k_x)^{n_x} (y/k_y)^{n_y}}{1 + (x/k_x)^{n_x} (y/k_y)^{n_y}},\\\\[1em]\n",
" &\\text{OR logic: } f(x,y) = \\frac{(x/k_x)^{n_x} + (y/k_y)^{n_y}}{1 + (x/k_x)^{n_x} + (y/k_y)^{n_y}}.\n",
"\\end{align}\n",
"\n",
"In what follows, we will take $x/k_x \\to x$ and $y/k_y \\to y$ such that $x$ and $y$ are not dimensionless, nondimensionalized by their Hill $k$. In this notation, we have\n",
"\n",
"\\begin{align}\n",
" &\\text{AND logic: } f(x,y) = \\frac{x^{n_x} y^{n_y}}{1 + x^{n_x} y^{n_y}},\\\\[1em]\n",
" &\\text{OR logic: } f(x,y) = \\frac{x^{n_x} + y^{n_y}}{1 + x^{n_x} + y^{n_y}}.\n",
"\\end{align}\n",
"\n",
"We can make plots of these functions to demonstrate how they resprensent the respective logic."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"![](\"i1-ffl.png\")
\n",
"\n",
"X activates Y and Z, but Y represses Z. We can use the expressions for production rate under AND and OR logic for one activator/one repressor that we showed above in writing down our dynamical equations. Here, we will consider AND logic. Nondimensionalization proceeds as in the C1-FFL example. The dimensionless dynamical equations are\n",
"\n",
"\\begin{align}\n",
"\\frac{\\mathrm{d}y}{\\mathrm{d}t} &= \\beta_y\\,\\frac{(\\kappa x)^{n_{xy}}}{1+(\\kappa x)^{n_{xy}}} - y,\\\\[1em]\n",
"\\gamma^{-1}\\,\\frac{\\mathrm{d}z}{\\mathrm{d}t} &= \\frac{x^{n_{xz}}}{(1 + x^{n_{xz}} + y^{n_{yz}})} - z.\n",
"\\end{align}\n",
"\n",
"We can code this up."
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [],
"source": [
"def i1ffl(yz, t, beta_y, kappa, gamma, n_xy, n_xz, n_yz, logic, x_fun, x_args):\n",
" \"\"\"Right hand side of dynamics of C1-FFL\"\"\"\n",
" # Unpack\n",
" y, z = yz\n",
" \n",
" # Evaluate x\n",
" x = x_fun(t, *x_args)\n",
" \n",
" # y-derivative\n",
" dy_dt = beta_y * (kappa*x)**n_xy / (1 + (kappa*x)**n_xy) - y\n",
"\n",
" # x-derivative\n",
" if logic == 'and':\n",
" dz_dt = gamma * (ar_and(x, y, n_xz, n_yz) - z)\n",
" elif logic == 'or':\n",
" dz_dt = gamma * (ar_or(x, y, n_xz, n_yz) - z)\n",
" \n",
" return np.array([dy_dt, dz_dt])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now, let's use it to observe the dynamics of an I1-FFL. We will investigate the response in Z to a sudden, sustained step in stimulus X."
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
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