loopless_m_models.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving M Models without Loops"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A type III loop is a cycle with no exchange of metabolites. Therefore, type III loops do not have flux through exchange, demand and biomass reactions, which exchange mass with the system. If flux flows through a type III loop, it has no thermodynamic driving force, which is not realistic. As long as the reaction bounds do not force flux through any reactions in the loop, a solution with flux through a loop can be collapsed into a solution without it which still allows flux through the biomass.\n",
    "\n",
    "Proof: Take the following model, where all fluxes are positive or zero (conversion of a model with negative fluxes to this form is trivial), with a flux solution $v$. All exchange, demand, and biomass, reactions (which by definition can not be a part of a loop) are denoted with $_i$ subscripts, while all others are denoted with $_j$.\n",
    "$$ \\mathbf S \\cdot v = 0$$\n",
    "$$ ub \\ge v \\ge 0 $$\n",
    "$$ ub_i \\ge v_i \\ge lb_i $$\n",
    "\n",
    "Let $v_L$ be any flux loop within $v$. Because exchange, demand, and biomass can not be in the loop, we have the following:\n",
    "$$v_{L,i} = 0 $$\n",
    "\n",
    "By definition, no metabolites are exchanged, so\n",
    "$$ \\mathbf S \\cdot v_L = 0 $$\n",
    "$$ \\mathbf S \\cdot (v - v_L) = 0 $$\n",
    "\n",
    "Because $v_L$ is a loop within $v$, the following are true:\n",
    "$$ ub \\ge v \\ge v_L \\ge 0 $$\n",
    "$$ ub \\ge v - v_L \\ \\ge 0 $$\n",
    "\n",
    "$\\therefore$ $\\forall v$ and $\\forall v_L$ within it, $v - v_L$ is also a valid solution which will produce biomass flux."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This proof does not apply to models where flux is forced through a reaction (i.e. where the lower bound is greater than 0). To optain a flux state with no flux through thermodynamically infeasible loops for all models, loopless FBA was developed by [Schellenberger et al.](http://dx.doi.org/10.1016/j.bpj.2010.12.3707) To compute this for all of these correctly parsed-models, we use the loopless FBA [implementation](http://cobrapy.readthedocs.org/en/latest/loopless.html) in [cobrapy](http://opencobra.github.io/cobrapy/) (version 0.4.0b1 or later) and the [gurobi](http://www.gurobi.com) solver (version 6.0). Because loopless FBA is an MILP, it is not guarenteed to find the optimum, but for every model we are able to find a feasible loopless solution with the algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import os\n",
    "\n",
    "import pandas\n",
    "\n",
    "import cobra\n",
    "\n",
    "pandas.set_option(\"display.max_rows\", 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load in the models, as parsed by this [script](load_models.ipynb). We will also break up all reversible reactions into the model into two irreversible reactions in opposite directions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "models = cobra.DictList()\n",
    "for filename in sorted(os.listdir(\"sbml3\")):\n",
    "    if not filename.endswith(\".xml\"):\n",
    "        continue\n",
    "    models.append(cobra.io.read_sbml_model(os.path.join(\"sbml3\", filename)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct the loopless models, which impose additional constraints which prevent flux from going around one of these loops."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "loopless_models = cobra.DictList((cobra.flux_analysis.construct_loopless_model(m) for m in models))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve the models using gurobi, allowing the branch-and-bound solver to proceed for two minutes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "loopless_solutions = {}\n",
    "for loopless_model in loopless_models:\n",
    "    solution = loopless_model.optimize(solver=\"gurobi\", time_limit=120)\n",
    "    loopless_solutions[loopless_model.id] = solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Constrain the models based the loopless result. Only reactions which carry flux in the loopless solution will be allowed to carry flux."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "constrained_loopless_models = cobra.DictList()\n",
    "for model, loopless_model in zip(models, loopless_models):\n",
    "    constrained = model.copy()\n",
    "    constrained_loopless_models.append(constrained)\n",
    "    for reaction in constrained.reactions:\n",
    "        loopless_flux = loopless_model.reactions.get_by_id(reaction.id).x\n",
    "        reaction.upper_bound = min(reaction.upper_bound, loopless_flux + 1e-6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve these problems which are constrained to be loopless exactly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "loopless_exact_solutions = {\n",
    "    model.id: model.optimize(solver=\"esolver\", rational_solution=True)\n",
    "    for model in constrained_loopless_models}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Format the results as a table. This demonstrates these models have a feasible flux state which produce biomass without using these loops, and this loopless flux state can also be solved using exact arithmetic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>FBA</th>\n",
       "      <th>loopless</th>\n",
       "      <th>constrained looples exact</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
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       "      <td>119.233012</td>\n",
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       "      <th>iMA871</th>\n",
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       "    <tr>\n",
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       "      <td>0.030942</td>\n",
       "      <td>0.030942</td>\n",
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       "    <tr>\n",
       "      <th>iMH551</th>\n",
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       "    </tr>\n",
       "    <tr>\n",
       "      <th>iMM1415</th>\n",
       "      <td>1.363428</td>\n",
       "      <td>1.333707</td>\n",
       "      <td>10000001/7497900</td>\n",
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       "    <tr>\n",
       "      <th>iMM904</th>\n",
       "      <td>0.287866</td>\n",
       "      <td>0.287866</td>\n",
       "      <td>145400000/505096641</td>\n",
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       "    <tr>\n",
       "      <th>iMO1056</th>\n",
       "      <td>1.047929</td>\n",
       "      <td>1.047929</td>\n",
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       "    <tr>\n",
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       "      <td>0.804432</td>\n",
       "      <td>0.804432</td>\n",
       "      <td>7063000000/8780103713</td>\n",
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       "    <tr>\n",
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       "      <td>0.052199</td>\n",
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       "    <tr>\n",
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       "      <td>250</td>\n",
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       "    <tr>\n",
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       "      <td>90000000/40197209</td>\n",
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       "    <tr>\n",
       "      <th>iRC1080</th>\n",
       "      <td>6.156851</td>\n",
       "      <td>6.156851</td>\n",
       "      <td>1000000000000000000000/162420679450988000257</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iRM588</th>\n",
       "      <td>1000.000000</td>\n",
       "      <td>1000.000000</td>\n",
       "      <td>1000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iRR1083</th>\n",
       "      <td>77.947363</td>\n",
       "      <td>77.947363</td>\n",
       "      <td>1250000000/16036463</td>\n",
       "    </tr>\n",
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       "      <th>iRS1563</th>\n",
       "      <td>0.079962</td>\n",
       "      <td>0.079959</td>\n",
       "      <td>309291382289604/3868108826410625</td>\n",
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       "      <td>8.969346</td>\n",
       "      <td>8.969346</td>\n",
       "      <td>37500000/4180907</td>\n",
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       "    <tr>\n",
       "      <th>iRS605_fixed</th>\n",
       "      <td>2.116780</td>\n",
       "      <td>0.495108</td>\n",
       "      <td>460211973096629/929518769000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iRsp1095</th>\n",
       "      <td>9.686899</td>\n",
       "      <td>9.686899</td>\n",
       "      <td>131447171192069/13570000000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iSB619</th>\n",
       "      <td>0.158050</td>\n",
       "      <td>0.158050</td>\n",
       "      <td>10000/63271</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iSH335</th>\n",
       "      <td>1.159211</td>\n",
       "      <td>1.159211</td>\n",
       "      <td>87700000000/75654904569</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iSO783</th>\n",
       "      <td>17.920308</td>\n",
       "      <td>17.918524</td>\n",
       "      <td>900000000100/50227352087</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iSR432</th>\n",
       "      <td>11.482570</td>\n",
       "      <td>11.482570</td>\n",
       "      <td>72500000000000000/6313917669853967</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iSS724</th>\n",
       "      <td>22.421525</td>\n",
       "      <td>22.421525</td>\n",
       "      <td>5000/223</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iSS884</th>\n",
       "      <td>70.359286</td>\n",
       "      <td>70.359286</td>\n",
       "      <td>7513000000000/106780503717</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iSyn669</th>\n",
       "      <td>0.179090</td>\n",
       "      <td>0.179090</td>\n",
       "      <td>6802000000/37980929237</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iTH366</th>\n",
       "      <td>27890.888638</td>\n",
       "      <td>27890.888638</td>\n",
       "      <td>10741489258500000/385125386207</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iTL885</th>\n",
       "      <td>7.098836</td>\n",
       "      <td>7.098836</td>\n",
       "      <td>1868614144114307/263228246898000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iTY425_fixed</th>\n",
       "      <td>0.886833</td>\n",
       "      <td>0.886833</td>\n",
       "      <td>5000000/5638037</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iVM679</th>\n",
       "      <td>2.826570</td>\n",
       "      <td>2.826570</td>\n",
       "      <td>26520000000/9382396471</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iVS941_fixed</th>\n",
       "      <td>0.523385</td>\n",
       "      <td>0.523385</td>\n",
       "      <td>10000000/19106407</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iWV1314</th>\n",
       "      <td>25.682402</td>\n",
       "      <td>25.682402</td>\n",
       "      <td>10504392200845731410/409011287584573141</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iWZ663</th>\n",
       "      <td>34.075881</td>\n",
       "      <td>34.075881</td>\n",
       "      <td>350000000000/10271194511</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iYL1228</th>\n",
       "      <td>1.042637</td>\n",
       "      <td>1.042637</td>\n",
       "      <td>1590492000/1525450603</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iYO844</th>\n",
       "      <td>1.582798</td>\n",
       "      <td>1.582798</td>\n",
       "      <td>256250000000/161896806353</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iZM363</th>\n",
       "      <td>2.532606</td>\n",
       "      <td>2.532606</td>\n",
       "      <td>181250000/71566597</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>iZmobMBEL601</th>\n",
       "      <td>0.350080</td>\n",
       "      <td>0.350080</td>\n",
       "      <td>61718750/176298791</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>mus_musculus</th>\n",
       "      <td>129.100556</td>\n",
       "      <td>129.100556</td>\n",
       "      <td>10000000/77459</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>textbook</th>\n",
       "      <td>0.873922</td>\n",
       "      <td>0.873922</td>\n",
       "      <td>686440/785471</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                             FBA      loopless  \\\n",
       "AbyMBEL891            119.233012    115.359718   \n",
       "AraGEM                 10.000000     10.000000   \n",
       "GSMN_TB                 0.131262      0.119309   \n",
       "PpaMBEL1254            78.700594     78.694551   \n",
       "PpuMBEL1071           132.365353    132.365353   \n",
       "STM_v1_0                0.477834      0.477475   \n",
       "S_coilicolor_fixed    860.088835    860.088835   \n",
       "SpoMBEL1693            63.785658     63.785658   \n",
       "T_Maritima              0.359469      0.359469   \n",
       "VvuMBEL943             96.402322     96.402322   \n",
       "iAC560                  4.042612      4.042543   \n",
       "iAF1260                 0.736701      0.728690   \n",
       "iAF692                  0.027250      0.027250   \n",
       "iAI549                 84.174193     84.174100   \n",
       "iAN840m                 0.340136      0.340136   \n",
       "iAO358                255.180548    255.180548   \n",
       "iAbaylyiV4             12.306545     12.306545   \n",
       "iBT721_fixed            0.500000      0.500000   \n",
       "iBsu1103             1229.657244   1229.657244   \n",
       "iCA1273                 0.740242      0.740240   \n",
       "iCB925                  0.119428      0.118840   \n",
       "iCR744                 41.476742     41.476742   \n",
       "iCS291                  0.140549      0.140549   \n",
       "iCS400                  0.215639      0.215639   \n",
       "iCac802                 0.209597      0.209597   \n",
       "iFF708                 14.917106     14.917106   \n",
       "iGB555_fixed          127.789084    127.789084   \n",
       "iHD666_fixed            0.255672      0.255653   \n",
       "iIB711                 84.688415     84.688415   \n",
       "iIT341                  0.692813      0.692813   \n",
       "iJL432                  0.328741      0.328741   \n",
       "iJN678                  0.063150      0.063150   \n",
       "iJN746                  3.679412      3.679412   \n",
       "iJO1366                 0.982372      0.982298   \n",
       "iJP815                  0.861512      0.861512   \n",
       "iJR904                  0.921948      0.898181   \n",
       "iJS747                 34.172662     34.172662   \n",
       "iKF1028                 0.517807      0.517807   \n",
       "iLC915                 79.021232     79.021232   \n",
       "iLL672                  3.475283      3.475228   \n",
       "iMA871                 27.792976     27.792976   \n",
       "iMA945                 79.589584     79.589584   \n",
       "iMB745                  0.030942      0.030942   \n",
       "iMH551                  0.775081      0.775081   \n",
       "iMM1415                 1.363428      1.333707   \n",
       "iMM904                  0.287866      0.287866   \n",
       "iMO1056                 1.047929      1.047929   \n",
       "iMP429_fixed            0.804432      0.804432   \n",
       "iND750                  0.097323      0.097314   \n",
       "iNJ661                  0.052199      0.052199   \n",
       "iNJ661m                 0.052199      0.052199   \n",
       "iNV213                 11.538859     11.538859   \n",
       "iOG654                  0.355447      0.355447   \n",
       "iOR363                250.000000    250.000000   \n",
       "iPP668                  3.891474      3.891454   \n",
       "iPS189_fixed            2.238961      2.238961   \n",
       "iRC1080                 6.156851      6.156851   \n",
       "iRM588               1000.000000   1000.000000   \n",
       "iRR1083                77.947363     77.947363   \n",
       "iRS1563                 0.079962      0.079959   \n",
       "iRS1597                 8.969346      8.969346   \n",
       "iRS605_fixed            2.116780      0.495108   \n",
       "iRsp1095                9.686899      9.686899   \n",
       "iSB619                  0.158050      0.158050   \n",
       "iSH335                  1.159211      1.159211   \n",
       "iSO783                 17.920308     17.918524   \n",
       "iSR432                 11.482570     11.482570   \n",
       "iSS724                 22.421525     22.421525   \n",
       "iSS884                 70.359286     70.359286   \n",
       "iSyn669                 0.179090      0.179090   \n",
       "iTH366              27890.888638  27890.888638   \n",
       "iTL885                  7.098836      7.098836   \n",
       "iTY425_fixed            0.886833      0.886833   \n",
       "iVM679                  2.826570      2.826570   \n",
       "iVS941_fixed            0.523385      0.523385   \n",
       "iWV1314                25.682402     25.682402   \n",
       "iWZ663                 34.075881     34.075881   \n",
       "iYL1228                 1.042637      1.042637   \n",
       "iYO844                  1.582798      1.582798   \n",
       "iZM363                  2.532606      2.532606   \n",
       "iZmobMBEL601            0.350080      0.350080   \n",
       "mus_musculus          129.100556    129.100556   \n",
       "textbook                0.873922      0.873922   \n",
       "\n",
       "                                            constrained looples exact  \n",
       "AbyMBEL891                  155786642126569502217/1350442295638400000  \n",
       "AraGEM                                                             10  \n",
       "GSMN_TB             6166288634265443783207711/51683396091939360000...  \n",
       "PpaMBEL1254                       189843852710751543/2412414190360000  \n",
       "PpuMBEL1071                                   644062500000/4865793703  \n",
       "STM_v1_0                        432019017800787961/904799542500000000  \n",
       "S_coilicolor_fixed                 408741530000000000/475231759036371  \n",
       "SpoMBEL1693                                 1237500000000/19400913101  \n",
       "T_Maritima                                        180000004/500738635  \n",
       "VvuMBEL943                                      14000000000/145224717  \n",
       "iAC560                               2333293447244409/577184600000000  \n",
       "iAF1260                     41715149428753474983/57246783910000000000  \n",
       "iAF692                                           107424000/3942157643  \n",
       "iAI549                                        114000000015/1354335833  \n",
       "iAN840m                                                        50/147  \n",
       "iAO358                                        586000000000/2296413281  \n",
       "iAbaylyiV4          34263125839313509122373268880000000/2784138406...  \n",
       "iBT721_fixed                                                      1/2  \n",
       "iBsu1103                               1687500000000000/1372333638281  \n",
       "iCA1273                                           293776004/396865803  \n",
       "iCB925                                             42473700/357401351  \n",
       "iCR744                                        179687500000/4332247261  \n",
       "iCS291                                                   38125/271258  \n",
       "iCS400                                                   63750/295633  \n",
       "iCac802                                     199660004750/952590217137  \n",
       "iFF708                                        113942000000/7638344959  \n",
       "iGB555_fixed                            142500000000000/1115118722693  \n",
       "iHD666_fixed                    209383236634838473/819014644990680000  \n",
       "iIB711                                        100000000000/1180799053  \n",
       "iIT341                                     814668000000/1175884921963  \n",
       "iJL432                                         3750000000/11407166737  \n",
       "iJN678                                    869550000000/13769631712373  \n",
       "iJN746                                              40000000/10871301  \n",
       "iJO1366                                   6628393837351/6750000000000  \n",
       "iJP815                                            179500000/208354621  \n",
       "iJR904                          119572964779828413/133127939375000000  \n",
       "iJS747                                    15000000000000/438947370703  \n",
       "iKF1028                                               1900000/3669321  \n",
       "iLC915                                  786562500000000/9953812160053  \n",
       "iLL672                                416082085743217/119728000000000  \n",
       "iMA871                                        100000000000/3598031369  \n",
       "iMA945                                            5000000000/62822291  \n",
       "iMB745                                           161000000/5203208413  \n",
       "iMH551                                        27300000000/35222122883  \n",
       "iMM1415                                              10000001/7497900  \n",
       "iMM904                                            145400000/505096641  \n",
       "iMO1056                                            100000000/95426353  \n",
       "iMP429_fixed                                    7063000000/8780103713  \n",
       "iND750                               544231132811567/5592500000000000  \n",
       "iNJ661                                                894000/17126611  \n",
       "iNJ661m                              487433149672039/9338050000000000  \n",
       "iNV213                             12500000000000000/1083295987243349  \n",
       "iOG654                                                    12500/35167  \n",
       "iOR363                                                            250  \n",
       "iPP668                         1437362919286850321/369363939375000000  \n",
       "iPS189_fixed                                        90000000/40197209  \n",
       "iRC1080                  1000000000000000000000/162420679450988000257  \n",
       "iRM588                                                           1000  \n",
       "iRR1083                                           1250000000/16036463  \n",
       "iRS1563                              309291382289604/3868108826410625  \n",
       "iRS1597                                              37500000/4180907  \n",
       "iRS605_fixed                          460211973096629/929518769000000  \n",
       "iRsp1095                               131447171192069/13570000000000  \n",
       "iSB619                                                    10000/63271  \n",
       "iSH335                                        87700000000/75654904569  \n",
       "iSO783                                       900000000100/50227352087  \n",
       "iSR432                             72500000000000000/6313917669853967  \n",
       "iSS724                                                       5000/223  \n",
       "iSS884                                     7513000000000/106780503717  \n",
       "iSyn669                                        6802000000/37980929237  \n",
       "iTH366                                 10741489258500000/385125386207  \n",
       "iTL885                               1868614144114307/263228246898000  \n",
       "iTY425_fixed                                          5000000/5638037  \n",
       "iVM679                                         26520000000/9382396471  \n",
       "iVS941_fixed                                        10000000/19106407  \n",
       "iWV1314                       10504392200845731410/409011287584573141  \n",
       "iWZ663                                       350000000000/10271194511  \n",
       "iYL1228                                         1590492000/1525450603  \n",
       "iYO844                                      256250000000/161896806353  \n",
       "iZM363                                             181250000/71566597  \n",
       "iZmobMBEL601                                       61718750/176298791  \n",
       "mus_musculus                                           10000000/77459  \n",
       "textbook                                                686440/785471  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "results = pandas.DataFrame.from_dict(\n",
    "    {model.id: {\"loopless\": loopless_solutions[model.id].f,\n",
    "                \"FBA\": model.optimize().f,\n",
    "                \"constrained looples exact\": str(loopless_exact_solutions[model.id].f)}\n",
    "     for model in models},\n",
    "    orient=\"index\")\n",
    "results"
   ]
  }
 ],
 "metadata": {
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    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.6"
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 },
 "nbformat": 4,
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