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    "# Numerical simulatin of β-catenin destruction cycle\n",
    "\n",
    "(c) 2018 Justin Bois. This work is licensed under a [Creative Commons Attribution License CC-BY 4.0](https://creativecommons.org/licenses/by/4.0/). All code contained herein is licensed under an [MIT license](https://opensource.org/licenses/MIT).\n",
    "\n",
    "*This document was generated from a Jupyter notebook, which may be downloaded [here](l03_betacat_destruction_cycle.ipynb).*"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy.integrate\n",
    "\n",
    "import bokeh.models\n",
    "import bokeh.plotting\n",
    "import bokeh.io\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We first set up the parameters.  Most are measured and reported, at least approximately, in the Lee, et al. paper.  The parameters $k_8$, $k_{\\text{-}8}$ and $k_{12}$ are unknown.  However, $K_\\mathrm{d} = k_{\\text{-}8}/k_8$ is known, so we only have two parameters to set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Key for names\n",
    "names = ['Axin complex', 'Axin-βcat', 'Axin-βcat*', 'βcat*', 'βcat']\n",
    "\n",
    "# Define known parameters\n",
    "c_A = 50          # nM (given by fixed GSK-3 concentration)\n",
    "k9 = 206          # 1/min\n",
    "k10 = 206         # 1/min\n",
    "k11 = 0.417       # 1/min\n",
    "Kd8 = 120         # nM\n",
    "\n",
    "# Unknown parameters\n",
    "km8 = 1          # 1/min\n",
    "k12 = 100         # nM/min\n",
    "\n",
    "# k8 determined form Kd8 and km8\n",
    "k8 = km8 / Kd8    # 1/nM-min\n",
    "\n",
    "# Set up params for ODE\n",
    "params = np.array([k8, km8, k9, k10, k11, k12])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For our initial conditions, we will assume that we have no $\\beta$-catenin and that we have a fixed amount of APC/Axin/GKC-3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initial conditions\n",
    "c0 = np.array([c_A, 0, 0, 0, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we define the time derivative.  We use the functional form that is necessary to pass into `scipy.integrate.odeint()`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dcdt(c, t, k8, km8, k9, k10, k11, k12):\n",
    "    \"\"\"\n",
    "    Time derivative of concentrations.\n",
    "    c = (c3, c8, c9, c10, c11)\n",
    "    \"\"\"\n",
    "    # Unpack concentrations\n",
    "    c3, c8, c9, c10, c11 = c\n",
    "    \n",
    "    # Build derivatives\n",
    "    deriv = np.empty(5)\n",
    "    deriv[0] = -k8*c3*c11 + km8*c8 + k10*c9\n",
    "    deriv[1] = k8*c3*c11 - (km8 + k9)*c8\n",
    "    deriv[2] = k9*c8 - k10*c9\n",
    "    deriv[3] = k10*c9 - k11*c10\n",
    "    deriv[4] = -k8*c3*c11 + km8*c8 + k12\n",
    "    \n",
    "    return deriv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we just have to specify what time points we want and then run the solver!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set up time points and solve\n",
    "t = np.linspace(0, 20, 300)\n",
    "c = scipy.integrate.odeint(dcdt, c0, t, args=(k8, km8, k9, k10, k11, k12))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can plot the solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
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       "  var render_items = [{\"docid\":\"3afcf767-0d43-41a6-9811-558854fb820e\",\"elementid\":\"60f22f1b-e8f7-473b-b2d4-bac47ba13ec6\",\"modelid\":\"27178ee8-e028-4b7c-8a1a-a578d0f43217\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
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     "output_type": "display_data"
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   ],
   "source": [
    "# Set up canvas on which to paint plot\n",
    "p = bokeh.plotting.figure(plot_width=650,\n",
    "                          plot_height=400,\n",
    "                          x_axis_label='time (min)',\n",
    "                          y_axis_label='conc (nM)',\n",
    "                          border_fill_alpha=0,\n",
    "                          background_fill_alpha=0)\n",
    "\n",
    "# Specify colors\n",
    "colors = bokeh.palettes.d3['Category10'][10]\n",
    "\n",
    "# Add glyphs\n",
    "for i, name in enumerate(names):\n",
    "    p.line(t, c[:,i], line_width=3, color=colors[i], legend=name)\n",
    "\n",
    "# Place legend\n",
    "p.legend.location = 'center_right'\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Axin complexes almost immediately go to steady state, and then the β-catenin levels respond more slowly as they reach their steady state."
   ]
  }
 ],
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