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            "source": [
                "# The lac operon\n",
                "\n",
                "The *E. coli* lac operon controls the production of enzymes needed to metabolize lactose and is regulated in large part by lactose availability.\n",
                "This example model includes\n",
                "- LacI (`I`) repression of the lac operon via blocking of the operator (`O`),\n",
                "- lactose (`L`) transport by LacY (`Y`) permease,\n",
                "- conversion of lactose to allolactose (`A`) by LacZ β-galactosidase (`Z`),\n",
                "- and allolactose-mediated derepression of the operon."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {
                "nbsphinx": "hidden"
            },
            "outputs": [],
            "source": [
                "import random\n",
                "random.seed(0)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": [
                "from pykappa.system import System\n",
                "\n",
                "system = System.from_ka(\n",
                "    \"\"\"\n",
                "    %init: 1 O(i[.])\n",
                "    %init: 1 I(o[.])\n",
                "    %init: 10 Y()\n",
                "    %init: 10 Z(l[.])\n",
                "\n",
                "    %obs: 'Extracellular lactose' |L(loc{out})| / 100\n",
                "    %obs: 'LacY' |Y()|\n",
                "    %obs: 'Free Lac operon' |O(i[.])|\n",
                "\n",
                "    I(o[.]), O(i[.]) <-> I(o[1]), O(i[1]) @ 1, 0.1      // Repress the lac operon\n",
                "    Y(), L(loc{out}) -> Y(), L(loc{in}) @ 0.01          // Transport lactose into the cell\n",
                "    Z(l[.]), L(loc{in}) -> Z(l[.]), A(z[.], i[.]) @ 1   // Convert lactose into allolactose\n",
                "    A(i[.]), I(o[.]) <-> A(i[1]), I(o[1]) @ 1, 0.1      // Deactivate the repressor by allolactose\n",
                "    O(i[.]), ., . -> O(i[.]), Z(l[.]), Y() @ 1          // Express the lac operon\n",
                "    L(loc{in}) -> E() @ 1                               // Metabolize lactose\n",
                "\n",
                "    // Degradation\n",
                "    A() -> . @ 0.1\n",
                "    Y() -> . @ 0.5\n",
                "    Z() -> . @ 0.5\n",
                "    \"\"\"\n",
                ")"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "We simulate the system first without lactose, then add extracellular lactose to observe the regulatory response."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": [
                "# Simulate a while with no extracellular lactose\n",
                "while system.time < 300:\n",
                "    system.update()\n",
                "\n",
                "# Add extracellular lactose and continue simulating\n",
                "system.mixture.instantiate(\"L(loc{out})\", 1000)\n",
                "while system.time < 800:\n",
                "   system.update()"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": [
                "system.monitor.plot();"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Without lactose, the operon remains mostly repressed.\n",
                "When extracellular lactose is added,\n",
                "- existing LacY transports some lactose into the cell,\n",
                "- intracellular lactose is converted to allolactose by LacZ,\n",
                "- allolactose inactivates the LacI repressor,\n",
                "- derepressing the operon and leading to increased LacY and LacZ production.\n",
                "\n",
                "As lactose is metabolized the operon is again repressed."
            ]
        }
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