toy.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d968e0f7",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "R  = 8.314\n",
    "T  = 298.15\n",
    "B  = 1/(R*T)\n",
    "ph = np.arange(-14,14,0.1)\n",
    "\n",
    "# reference pKa\n",
    "ref_res={\n",
    "    \"D\":3.90,\n",
    "    \"E\":4.36,\n",
    "    \"C\":8.55,\n",
    "    \"AA\":4,\n",
    "    \"BB\":3,\n",
    "    \"CC\":8}\n",
    "\n",
    "# for finding where dG_protein = 0\n",
    "def find_nearest(array, value):\n",
    "    array = np.asarray(array)\n",
    "    idx = (np.abs(array - value)).argmin()\n",
    "    return idx\n",
    "\n",
    "\n",
    "# for getting pka from ddG and reference\n",
    "def getpKa(ddG,res,a=1):\n",
    "    return ref_res[res] + a*ddG/(R*T*np.log(10))\n",
    "\n",
    "\n",
    "# for getting dG_protein from ddG (i.e., subtract off peptide protonation)\n",
    "def deltaG(ddG, res, T=T, ph=np.arange(-14,14,0.01),a=1.0):\n",
    "    return ddG + a*(R*T*np.log(10)*(ref_res[res]-ph))\n",
    "\n",
    "\n",
    "def coupled_ddG(ddG0, ddG1, ddG2, res, ph=np.arange(-14,14,0.01),a=1.0):    \n",
    "    dG1 = deltaG(ddG1, res, T, ph, a)\n",
    "    dG2 = deltaG(ddG2, res, T, ph, a)    \n",
    "    \n",
    "    return (ddG0 +\\\n",
    "            (1/B)*np.log(1+np.exp(-B*dG1)) -\\\n",
    "            (1/B)*np.log(1+np.exp(-B*dG2)))\n",
    "\n",
    "# for resolving protonation probabilities as a function of dG\n",
    "def partition(dG1,dG2,dG3,ph=np.arange(-14,14,0.01)):\n",
    "    \n",
    "    B = 1/(R*T)\n",
    "    mu = (B**(-1))*np.log(10)*ph\n",
    "    \n",
    "    Z = 1 + np.exp(-B*(dG1-mu))+\\\n",
    "            np.exp(-B*(dG2-mu))+\\\n",
    "            np.exp(-B*((dG1+dG3)-2*mu))\n",
    "    \n",
    "    return 1/Z, np.exp(-B*(dG1-mu))/Z, np.exp(-B*(dG2-mu))/Z, np.exp(-B*((dG1+dG3)-2*mu))/Z\n",
    "\n",
    "\n",
    "# relating microscopic and macroscopic pKa values\n",
    "def macro(pk1,pk2,w):\n",
    "    pka1 = pk1\n",
    "    pka2 = pk2\n",
    "    w    = w\n",
    "        \n",
    "    Ka1 = np.exp(-pka1)\n",
    "    Ka2 = np.exp(-pka2)\n",
    "    \n",
    "    Ka1a = np.exp(-(pka1-w))\n",
    "    Ka2a = np.exp(-(pka2-w))\n",
    "    \n",
    "    \n",
    "    return -np.log((Ka1*Ka2)/(Ka1+Ka2)), -np.log(Ka1a+Ka2a)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "22825b6c",
   "metadata": {},
   "outputs": [],
   "source": [
    "#  Toy example 4\n",
    "#\n",
    "#     (F11)   -18     (F01)\n",
    "#   CH + BH  -->  C- + BH\n",
    "#\n",
    "#  8  |               | 28\n",
    "#     V               V\n",
    "#             2\n",
    "#   CH + B-  -->  C- + B-\n",
    "#    (F10)         (F00)\n",
    "\n",
    "res1=\"C\"\n",
    "res2=\"E\"\n",
    "\n",
    "ddG1 =   -18 * 1000\n",
    "ddG2 =    8 * 1000\n",
    "ddG3 =   28 * 1000\n",
    "ddG4 =    2 * 1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "81aec1b7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "plain\n",
      "pKa1: 8.90, pKa2: 9.27\n",
      "\n",
      "adjust\n",
      "pKa1: 5.60, pKa2: 9.00\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 216x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(1,1,figsize=(3,2))\n",
    "\n",
    "# compute apparent ddG(pH) due to forward and backwards\n",
    "# add peptide deprot to get dG_protein\n",
    "# (i.e., ddG = dGprotein - dG_peptide -> ddG + dG_peptide = dG_protein)\n",
    "dG_protein1 = coupled_ddG(ddG1,ddG2,ddG3,res2,ph=ph,a=1.0) + R*T*np.log(10)*(ref_res[res1]-ph) \n",
    "dG_protein2 = coupled_ddG(ddG2,ddG1,ddG4,res1,ph=ph,a=1.0) + R*T*np.log(10)*(ref_res[res2]-ph)\n",
    "\n",
    "plt.plot(ph,dG_protein1/1000,color='tab:orange')\n",
    "plt.plot(ph,dG_protein2/1000,color='tab:blue')\n",
    "\n",
    "# find when dG_protein = 0\n",
    "idx1 = find_nearest(dG_protein1,0)\n",
    "idx2 = find_nearest(dG_protein2,0)\n",
    "\n",
    "# get corresponding pH, which will be the pKa\n",
    "pka1 = ph[idx1]\n",
    "pka2 = ph[idx2]\n",
    "\n",
    "axs.plot(pka1,0,'o',ms=8,zorder=10,color='tab:orange')\n",
    "axs.plot(pka2,0,'o',ms=8,zorder=10,color='tab:blue')\n",
    "\n",
    "axs.axhline(0,color='k')\n",
    "axs.set_xlim(0,14)\n",
    "axs.set_ylim(-40,40)\n",
    "axs.set_xlabel(\"pH\")\n",
    "axs.set_ylabel(\"dG_protein (kJ/mol)\")\n",
    "\n",
    "print(\"plain\\npKa1: %.2f, pKa2: %.2f\\n\" % (getpKa(ddG4,res=res1),\n",
    "                                  getpKa(ddG3,res=res2)))\n",
    "\n",
    "print(\"adjust\\npKa1: %.2f, pKa2: %.2f\" % (ph[idx1],\n",
    "                                  ph[idx2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3bd38fac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 14.0)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 216x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axs = plt.subplots(1,1,figsize=(3,2))\n",
    "\n",
    "# get standard deprotonation free energy (i.e.dG_protein(pH=0))\n",
    "dG1 = deltaG(ddG1,res1,ph=0)\n",
    "dG2 = deltaG(ddG2,res2,ph=0)\n",
    "dG3 = deltaG(ddG3,res2,ph=0)\n",
    "dG4 = deltaG(ddG4,res1,ph=0)\n",
    "\n",
    "# utilize Hill formalism to get microstates\n",
    "a,b,c,d = partition(dG1,dG2,dG3,ph=ph)\n",
    "\n",
    "plt.plot(ph,a+c,color='tab:orange',label=\"<AH+BH> + <A-+BH>\")\n",
    "plt.plot(ph,c,color='tab:orange',ls='dashed')\n",
    "\n",
    "plt.plot(ph,a+b,color='tab:blue',label=\"<AH+BH> + <AH+B->\")\n",
    "plt.plot(ph,b,color='tab:blue',ls='dashed')\n",
    "\n",
    "plt.ylabel(\"<x>\")\n",
    "plt.xlabel(\"pH\")\n",
    "plt.legend(loc='upper center',bbox_to_anchor=(0.5,1.5))\n",
    "plt.xlim(0,14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e1a090e8",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "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": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}