GeneticAlgorithm.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from simtk.openmm.app import *\n",
    "from simtk.openmm import *\n",
    "from simtk.unit import *\n",
    "from sys import stdout\n",
    "\n",
    "import numpy as np\n",
    "from joblib import Parallel, delayed\n",
    "import pickle\n",
    "\n",
    "%matplotlib notebook\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<Topology; 1 chains, 19 residues, 335 atoms, 338 bonds>\n"
     ]
    }
   ],
   "source": [
    "# load the PDB file containing the peptide information\n",
    "pdb = PDBFile('proteins/test.pdb')\n",
    "# initialize the forcefield and modeller\n",
    "forcefield = ForceField('amber14-all.xml', 'amber14/tip3pfb.xml')\n",
    "modeller = Modeller(pdb.topology, pdb.positions)\n",
    "print(modeller.topology)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "def get_index(name, atoms):\n",
    "    return [atom for atom in atoms if atom.name == name][0].index\n",
    "\n",
    "indices = []\n",
    "for res in modeller.topology.residues():\n",
    "    # get the chain indices of different atoms\n",
    "    indices.append(dict(N=get_index('N', res.atoms()),\n",
    "                        CA=get_index('CA', res.atoms()),\n",
    "                        CB=get_index('CB', res.atoms()),\n",
    "                        C=get_index('C', res.atoms())))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define hyperparameters\n",
    "k_start = 10\n",
    "population_size = 7\n",
    "n_survivors = 3\n",
    "n_random = 1\n",
    "n_residues = len(list(modeller.topology.residues()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def evaluate(parameters, return_simulation=False):\n",
    "    # initialize the system and integrator\n",
    "    system = forcefield.createSystem(modeller.topology, nonbondedMethod=CutoffNonPeriodic, nonbondedCutoff=1*nanometer, constraints=HBonds)\n",
    "    integrator = LangevinIntegrator(300*kelvin, 1/picosecond, 0.001*picoseconds)\n",
    "    integrator.setRandomNumberSeed(534456)\n",
    "    \n",
    "    # construct the angle forces from the parameter set\n",
    "    force = CustomAngleForce('k*(theta - theta0)^2')\n",
    "    force.addPerAngleParameter('k')\n",
    "    force.addPerAngleParameter('theta0')\n",
    "    for prev in range(len(indices) - 2):\n",
    "        curr = prev + 1\n",
    "        next_ = curr + 1\n",
    "        force.addAngle(indices[prev]['CA'],\n",
    "                       indices[curr]['CA'],\n",
    "                       indices[next_]['CA'],\n",
    "                       (parameters[prev][0], parameters[prev][2]))\n",
    "    angle_force_index = system.addForce(force)\n",
    "    \n",
    "    # construct the torsion forces from the parameter set\n",
    "    force = CustomTorsionForce('k*(theta - theta0)^2')\n",
    "    force.addPerTorsionParameter('k')\n",
    "    force.addPerTorsionParameter('theta0')\n",
    "    for prev in range(len(indices) - 1):\n",
    "        curr = prev + 1\n",
    "        force.addTorsion(indices[prev]['CA'],\n",
    "                         indices[prev]['CB'],\n",
    "                         indices[curr]['CA'],\n",
    "                         indices[curr]['CB'],\n",
    "                         (parameters[prev][1], parameters[prev][3]))\n",
    "    torsion_force_index = system.addForce(force)\n",
    "\n",
    "    # create the simulation\n",
    "    simulation = Simulation(modeller.topology, system, integrator)\n",
    "    simulation.context.setPositions(modeller.positions)\n",
    "    simulation.minimizeEnergy()\n",
    "    try:\n",
    "        simulation.step(20000)\n",
    "    except Exception as e:\n",
    "        print(f'Simulation step failed: {e}')\n",
    "        energy = np.inf * kilojoule/mole\n",
    "        if return_simulation:\n",
    "            # return the simulation object\n",
    "            return simulation, energy\n",
    "        return energy\n",
    "\n",
    "    # remove the forces to get a realstic energy level\n",
    "    simulation.system.removeForce(torsion_force_index)\n",
    "    simulation.system.removeForce(angle_force_index)\n",
    "    \n",
    "    # compute the current potential energy\n",
    "    energy = simulation.context.getState(getEnergy=True).getPotentialEnergy()\n",
    "    \n",
    "    if return_simulation:\n",
    "        # return the simulation object\n",
    "        return simulation, energy\n",
    "    return energy\n",
    "\n",
    "def random_params(count):\n",
    "    ks = np.ones((count, n_residues - 1, 2)) * k_start\n",
    "    thetas = np.random.uniform(0, 2 * np.pi, size=(count, n_residues - 1, 2))\n",
    "    return np.concatenate((ks, thetas), axis=2)\n",
    "\n",
    "def zero_k_params(count):\n",
    "    ks = np.zeros((count, n_residues - 1, 2))\n",
    "    thetas = np.random.uniform(0, 2 * np.pi, size=(count, n_residues - 1, 2))\n",
    "    return np.concatenate((ks, thetas), axis=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# initialize the population\n",
    "population = random_params(population_size)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "with open('population.pickle', 'rb') as file:\n",
    "    population = pickle.load(file)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1) Min: 7400.689161786126 kJ/mol, Max: inf kJ/mol\n",
      "(2) Min: 7083.909361583924 kJ/mol, Max: inf kJ/mol\n",
      "(3) Min: 7880.044556311283 kJ/mol, Max: inf kJ/mol\n",
      "(4) Min: 8125.537773226513 kJ/mol, Max: inf kJ/mol\n",
      "(5) Min: 7238.884051142578 kJ/mol, Max: inf kJ/mol\n",
      "(6) Min: 6970.0028388446635 kJ/mol, Max: inf kJ/mol\n",
      "(7) Min: 7773.6256591433785 kJ/mol, Max: inf kJ/mol\n",
      "(8) Min: 7458.487180953885 kJ/mol, Max: inf kJ/mol\n",
      "(9) Min: 7708.781573977129 kJ/mol, Max: inf kJ/mol\n",
      "(10) Min: 7438.925166045621 kJ/mol, Max: inf kJ/mol\n"
     ]
    }
   ],
   "source": [
    "def genetic_algorithm(survivor_params, count, mutation_rate=0.2):\n",
    "    if count == 1:\n",
    "        # perform crossover\n",
    "        parents = np.random.permutation(len(survivor_params))[:2]\n",
    "        params = np.array(survivor_params[parents[0]])\n",
    "        crossover = np.random.uniform(size=params.shape) < 0.5\n",
    "        params[crossover] = np.array(survivor_params[parents[1]][crossover])\n",
    "        \n",
    "        # get mutation for k parameter and apply mask\n",
    "        k_mutation = np.random.normal(0, 5, size=(len(params), 2))\n",
    "        k_mutation[np.random.uniform(size=k_mutation.shape)>mutation_rate] = 0\n",
    "        \n",
    "        # get mutation for theta parameter and apply mask\n",
    "        theta_mutation = np.random.normal(0, 2 * np.pi, size=(len(params), 2))\n",
    "        theta_mutation[np.random.uniform(size=theta_mutation.shape)>mutation_rate] = 0\n",
    "        \n",
    "        # add mutation to parameters\n",
    "        params[:,:2] += k_mutation\n",
    "        params[:,2:] += theta_mutation\n",
    "        \n",
    "        # apply constraint for k parameter\n",
    "        params[:,:2][params[:,:2]<0] = 0\n",
    "        return params\n",
    "    else:\n",
    "        #recursive call to generate single parameter sets\n",
    "        return [genetic_algorithm(survivor_params, 1) for _ in range(count)]\n",
    "\n",
    "# optimize the population for mutliple epochs\n",
    "for i in range(10):\n",
    "    # evaluate the population\n",
    "    scores = Parallel(n_jobs=-1)(delayed(evaluate)(params) for params in population)\n",
    "    population = population[np.argsort(scores)]\n",
    "    # generate new random parameter sets\n",
    "    population[n_survivors:n_survivors+n_random] = random_params(n_random)\n",
    "    # run the genetic algorithm\n",
    "    n_genetic_algorithm = population_size - n_survivors - n_random\n",
    "    population[n_survivors+n_random:] = genetic_algorithm(population[:n_survivors], n_genetic_algorithm)\n",
    "    print(f'({i + 1}) Min: {np.min(scores)}, Max: {np.max(scores)}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best energy: 9027.40798033378 kJ/mol\n"
     ]
    }
   ],
   "source": [
    "# get the best parameter set and create a simulation from it\n",
    "best_params = population[0]\n",
    "best_sim, best_energy = evaluate(best_params, return_simulation=True)\n",
    "print(f'Best energy: {best_energy}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "with open('population.pickle', 'wb') as file:\n",
    "    pickle.dump(population, file)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "<img 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width=\"900\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def draw_chain_atoms(residue_chain, positions, side_chain_viz='surface', aa_orientation=False):\n",
    "    residue_chain = list(residue_chain)\n",
    "    atoms = [np.array([positions[atom.index]._value for atom in res.atoms() if atom.name not in ['CA', 'N', 'C']]) for res in residue_chain]\n",
    "    ca_atoms = np.array(sum([[positions[atom.index]._value for atom in res.atoms() if atom.name == 'CA'] for res in residue_chain]))\n",
    "    c_atoms = np.array(sum([[positions[atom.index]._value for atom in res.atoms() if atom.name == 'C'] for res in residue_chain]))\n",
    "    n_atoms = np.array(sum([[positions[atom.index]._value for atom in res.atoms() if atom.name == 'N'] for res in residue_chain]))\n",
    "    aa_dir = c_atoms - n_atoms\n",
    "    \n",
    "    fig = plt.figure(figsize=(9, 5))\n",
    "    ax = fig.add_subplot(111, projection='3d')\n",
    "    cols = []\n",
    "    for i, res_atoms in enumerate(atoms):\n",
    "        col = ax.scatter(res_atoms[:,0], res_atoms[:,1], res_atoms[:,2], alpha=0).get_facecolor()\n",
    "        cols.append(col[0])\n",
    "        avg_side_pos = res_atoms.mean(axis=0)\n",
    "        \n",
    "        if side_chain_viz == 'arrow':\n",
    "            side_dir = avg_side_pos - ca_atoms[i]\n",
    "            ax.quiver(ca_atoms[i,0], ca_atoms[i,1], ca_atoms[i,2], side_dir[0], side_dir[1], side_dir[2], colors=col, alpha=0.6)\n",
    "        elif side_chain_viz == 'surface':\n",
    "            ax.plot_trisurf([n_atoms[i,0], avg_side_pos[0], c_atoms[i,0]], [n_atoms[i,1], avg_side_pos[1], c_atoms[i,1]], [n_atoms[i,2], avg_side_pos[2], c_atoms[i,2]])\n",
    "        elif side_chain_viz is not None:\n",
    "            print(f'Side chain visualization \"{side_chain_viz}\" unknown')\n",
    "    \n",
    "    ax.plot(ca_atoms[:,0], ca_atoms[:,1], ca_atoms[:,2], c='red')\n",
    "    if aa_orientation:\n",
    "        ax.quiver(n_atoms[:,0], n_atoms[:,1], n_atoms[:,2], aa_dir[:,0], aa_dir[:,1], aa_dir[:,2], colors='black', linewidth=2)\n",
    "\n",
    "    ax.axis('off')\n",
    "    plt.tight_layout()\n",
    "    plt.show()\n",
    "    \n",
    "# plot the protein structure\n",
    "positions = best_sim.context.getState(getPositions=True).getPositions(asNumpy=True)\n",
    "draw_chain_atoms(modeller.topology.residues(), positions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# save the protein structure to a file\n",
    "with open('proteins/ga_folding.pdb', 'w') as file:\n",
    "    PDBFile.writeModel(best_sim.topology, best_sim.context.getState(getPositions=True).getPositions(), file=file)"
   ]
  }
 ],
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  "language_info": {
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    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
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 },
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