{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Problem 10.3: Controlling p53 levels\n",
"\n",
"\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import collections\n",
"import numpy as np"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"[p53](https://en.wikipedia.org/wiki/P53) is an important protein for tumor suppression, DNA damage repair, cell cycle regulation, and more. In their [2012 paper](https://doi.org/10.1126/science.1218351), the researchers in Galit Lahav's group investigated how temporal dynamics of p53 controls cell fate in [MCF-7 cells](https://en.wikipedia.org/wiki/MCF-7), a breast cancer cell line. It had been previously observed that p53 levels oscillate upon exposure to stress due to γ radiation. To instead study how cells respond to sustained levels of p53, the authors controlled p53 levels using the drug Nutlin-3. To do this, they devised a program for adjusting Nutlin-3 levels in the cell culture that would keep the p53 levels at a constant high level. They based this program on a mathematical model of the p53 circuit (shown in the figure below) that they developed and parametrized in an [earlier paper](https://doi.org/10.1016/j.molcel.2008.03.016).\n",
"\n",
"\n",
" \n",
"\n",
"\n",
"
\n",
"\n",
"The p53 levels oscillate due to a delays in the autoinhibitory loops mediated by Mdm2 and Wip1. The dynamical equations for this circuit, as presented in the paper, are\n",
"\n",
"\\begin{align}\n",
"\\frac{\\mathrm{d}p_i}{\\mathrm{d}t} &= \\beta_\\mathrm{p} - \\frac{\\delta_\\mathrm{mp}^i \\, m\\,p_i}{1+\\left(n(t-\\tau_\\mathrm{n})/k\\right)^{n_\\mathrm{n}}} - \\beta_\\mathrm{sp}\\,p_i\\,\\frac{(s/k_\\mathrm{s})^{n_\\mathrm{s}}}{1 + (s/k_\\mathrm{s})^{n_\\mathrm{s}}} - \\gamma_\\mathrm{p}^i\\, p_i,\\\\[1em]\n",
"\\frac{\\mathrm{d}p_a}{\\mathrm{d}t} &= \\beta_\\mathrm{sp}\\,p_i\\,\\frac{(s/k_\\mathrm{s})^{n_\\mathrm{s}}}{1 + (s/k_\\mathrm{s})^{n_\\mathrm{s}}} - \\frac{\\delta_\\mathrm{mp}^a \\, m\\,p_a}{1+\\left(n(t-\\tau_\\mathrm{n})/k\\right)^{n_\\mathrm{n}}},\\\\[1em]\n",
"\\frac{\\mathrm{d}m}{\\mathrm{d}t} &= \\beta_{m_i} + \\beta_\\mathrm{m}\\,p_a(t-\\tau_\\mathrm{m}) - \\delta_\\mathrm{sm}\\,s\\,m - \\gamma_m\\,m,\\\\[1em]\n",
"\\frac{\\mathrm{d}w}{\\mathrm{d}t} &= \\beta_\\mathrm{w}\\,p_a(t-\\tau_\\mathrm{w}) - \\gamma_\\mathrm{w}\\,w,\\\\[1em]\n",
"\\frac{\\mathrm{d}s}{\\mathrm{d}t} &= \\beta_\\mathrm{s}\\,\\theta(t) - \\delta_\\mathrm{ws}\\,s\\,\\frac{(w/k_\\mathrm{w})^{n_\\mathrm{w}}}{1 + (w/k_\\mathrm{w})^{n_\\mathrm{w}}} - \\gamma_\\mathrm{s}\\,s.\n",
"\\end{align}\n",
"\n",
"Here, $p_i$ and $p_a$ denote the concentrations of inactive and active p53, respectively. The variables $m$ and $w$ respectively represent the concentrations of Mdm2 and Wip1. The concentration of the DNA damage signal is denoted as $s$. Finally, $n$ is the externally imposed concentration of Nutlin-3. The function $\\theta(t)$ is the Heaviside function,\n",
"\n",
"\\begin{align}\n",
"\\theta(t) = \\left\\{\\begin{array}[ll] 00 & t < 0 \\\\\n",
"1 & t \\ge 0.\n",
"\\end{array}\n",
"\\right.\n",
"\\end{align}\n",
"\n",
"The parameters for the model are given in the code cell below as a named tuple for convenience. The units of all parameters are such that time is in hours and concentrations in micromolar. Before proceeding, take a moment and make sure you understand the physical meaning of each term in the equations."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"Params = collections.namedtuple(\n",
" \"Params\",\n",
" [\n",
" \"delta_mpi\",\n",
" \"gamma_pi\",\n",
" \"delta_mpa\",\n",
" \"delta_sm\",\n",
" \"gamma_m\",\n",
" \"gamma_w\",\n",
" \"delta_ws\",\n",
" \"gamma_s\",\n",
" \"beta_p\",\n",
" \"beta_sp\",\n",
" \"beta_m\",\n",
" \"beta_mi\",\n",
" \"beta_w\",\n",
" \"beta_s\",\n",
" \"tau_n\",\n",
" \"tau_m\",\n",
" \"tau_w\",\n",
" \"k\",\n",
" \"n_n\",\n",
" \"n_s\",\n",
" \"n_w\",\n",
" \"k_w\",\n",
" \"k_s\",\n",
" ],\n",
")\n",
"\n",
"params = Params(\n",
" delta_mpi=5,\n",
" gamma_pi=2,\n",
" delta_mpa=1.4,\n",
" delta_sm=0.5,\n",
" gamma_m=1,\n",
" gamma_w=0.7,\n",
" delta_ws=50,\n",
" gamma_s=7.5,\n",
" beta_p=0.9,\n",
" beta_sp=10,\n",
" beta_m=0.9,\n",
" beta_mi=0.2,\n",
" beta_w=0.25,\n",
" beta_s=10,\n",
" tau_n=0.4,\n",
" tau_m=0.7,\n",
" tau_w=1.25,\n",
" k=2.3,\n",
" n_n=3.3,\n",
" n_s=4,\n",
" n_w=4,\n",
" k_w=0.2,\n",
" k_s=1,\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**a)** Solve the equations numerically with $n = 0$. Plot the solution along with the experimental measurements from the [Purvis, et al. paper](https://doi.org/10.1126/science.1218351), given in the Numpy arrays below. This helps verify that the model gives results similar to what you would observe experimentally."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"# Time points (in units of hours)\n",
"time = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 12])\n",
"\n",
"# p53 concentration\n",
"relative_total_p53_conc = np.array(\n",
" [0.13, 0.69, 1.00, 0.87, 0.60, 0.35, 0.34, 0.54, 0.61, 0.42]\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**b)** Come up with a program for controlling the Nutlin-3 concentration so as to bring the total p53 level to a sustained value similar to that of its maximum value during oscillations in the absence of Nutlin-3. I.e., choose a function $N(t)$ to give a sustained level of p53 that is approximately unity (in the units specified by the parameter values). You might want to consider experimental constraints; for example that you might want to have only a few steps where the Nutlin-3 concentrations are adjusted to ease experimentation. Show a plot verifying that your scheme works."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
"
]
}
],
"metadata": {
"anaconda-cloud": {},
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.12"
}
},
"nbformat": 4,
"nbformat_minor": 4
}