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    "# 2. Design principles governing the rate of gene expression\n",
    "\n",
    "<hr>"
   ]
  },
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    "**Concept**\n",
    "\n",
    "- Protein degradation rates determine response times for simple gene expression systems.\n",
    "\n",
    "\n",
    "**Design principle**\n",
    "\n",
    "- Negative autoregulation accelerates turn-on times, but not turn-off times.\n",
    "\n",
    "\n",
    "**Techniques**\n",
    "\n",
    "- Steady state normalization allows analysis of response times. \n",
    "- Numerical solution of ODEs using Python.\n",
    "- Interactive plotting using Bokeh.\n",
    "\n",
    "<hr>\n"
   ]
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Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    const el = document.getElementById(\"p1001\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error(url) {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (let i = 0; i < css_urls.length; i++) {\n      const url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    for (let i = 0; i < js_urls.length; i++) {\n      const url = js_urls[i];\n      const element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.async = false;\n      element.src = url;\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.1.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.1.0.min.js\"];\n  const css_urls = [];\n\n  const inline_js = [    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\nfunction(Bokeh) {\n    }\n  ];\n\n  function run_inline_js() {\n    if (root.Bokeh !== undefined || force === true) {\n          for (let i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\nif (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      const cell = $(document.getElementById(\"p1001\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Colab setup ------------------\n",
    "import os, sys, subprocess\n",
    "if \"google.colab\" in sys.modules:\n",
    "    cmd = \"pip install --upgrade biocircuits watermark\"\n",
    "    process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n",
    "    stdout, stderr = process.communicate()\n",
    "    data_path = \"https://biocircuits.github.io/chapters/data/\"\n",
    "else:\n",
    "    data_path = \"data/\"\n",
    "# ------------------------------\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import scipy.integrate\n",
    "\n",
    "import colorcet\n",
    "\n",
    "import biocircuits.jsplots\n",
    "\n",
    "import bokeh.io\n",
    "import bokeh.plotting\n",
    "\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "## Protein stability determines the response time to a change in gene expression\n",
    "\n",
    "So far, we have focused on what happens to a system at steady state. However, biological circuits change dynamically, even in constant environmental conditions. \n",
    "\n",
    "Imagine a cell suddenly finds itself in a new environment, requiring it to turn up or down the expression of specific genes. How rapidly can it switch proteins to their new desired levels? \n",
    "\n",
    "Starting with our basic equation for gene expression, and neglecting the mRNA level for the moment, we write:\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}x}{\\mathrm{d}t} = \\beta - \\gamma x\n",
    "\\end{align}\n",
    "\n",
    "Let's assume the gene has been off for a long time, so that $x(0)=0$ and suddenly turns on at $t=0$. Solving the above differential equation yields the general solution, \n",
    "\n",
    "\\begin{align}\n",
    "x(t) = \\frac{\\beta }{\\gamma} (1-e^{-\\gamma t}).\n",
    "\\end{align}\n",
    "\n",
    "Thus, the parameter that determines the response time is $\\gamma$, the degradation + dilution rate of the protein. In the above expression, we see that the characteristic response time scale is $\\gamma^{-1}$. A plot of the concentration of gene product over time is shown below for $\\beta = 100$ and $\\gamma = 1$. Note that $\\beta$ has dimension of concentration per time and $\\gamma$ has dimension of inverse time. We take their respective units as arbitrary in the plots that follow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-04-05T17:20:07.356315Z",
     "start_time": "2019-04-05T17:20:07.055723Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
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       "  const render_items = [{\"docid\":\"954f3b12-6e4f-4e92-8aa7-a2927c5471f2\",\"roots\":{\"p1002\":\"769bf1dd-8f19-41a3-81d0-eec4c7b29b7b\"},\"root_ids\":[\"p1002\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1002"
      }
     },
     "output_type": "display_data"
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   ],
   "source": [
    "# Parameters\n",
    "beta = 100\n",
    "gamma = 1\n",
    "\n",
    "# Dynamics\n",
    "t = np.linspace(0, 6, 400)\n",
    "x = beta / gamma * (1 - np.exp(-gamma * t))\n",
    "\n",
    "# Plot response\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_height=275,\n",
    "    frame_width=375,\n",
    "    x_axis_label=\"time\",\n",
    "    y_axis_label=\"x(t)\",\n",
    "    x_range=[0, 6],\n",
    ")\n",
    "p.line(t, x, line_width=2)\n",
    "\n",
    "# Mark the response time (when we get to level 1-1/e)\n",
    "t0 = 1 / gamma\n",
    "x0 = beta / gamma * (1 - np.exp(-1))\n",
    "\n",
    "# Add the glyph with label\n",
    "source = bokeh.models.ColumnDataSource(\n",
    "    data=dict(t0=[t0], x0=[x0], text=[\"response time = 1/γ\"])\n",
    ")\n",
    "p.circle(x=\"t0\", y=\"x0\", source=source, size=10)\n",
    "p.add_layout(\n",
    "    bokeh.models.LabelSet(\n",
    "        x=\"t0\", y=\"x0\", text=\"text\", source=source, x_offset=10, y_offset=-10\n",
    "    )\n",
    ")\n",
    "p.add_layout(\n",
    "    bokeh.models.Span(\n",
    "        location=t0,\n",
    "        level=\"underlay\",\n",
    "        dimension=\"height\",\n",
    "        line_color=\"black\",\n",
    "        line_dash=\"dashed\",\n",
    "    )\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "## How can we speed up responses?\n",
    "\n",
    "So far, it seems as if the cell's ability to modulate the concentration of a stable protein is quite limited, apparently requiring timescales on the order of the cell cycle for both increases and decreases. This seems rather pathetic for such successful, billion year old creatures. You might think you could beat this timescale and up-regulate the protein level faster by cranking up the promoter strength (increasing $\\beta$). Indeed, this would allow the cell to hit a specific threshold earlier. However, it would also increase the final steady state concentration ($\\beta/\\gamma$), and therefore leave the timescale over which the system reaches its new steady-state unaffected.\n",
    "\n",
    "One simple way to speed up the response time of the protein is to destabilize it, by increasing  $\\gamma$, while also increasing $\\beta$ by the same factor, and thereby preserving the steady state protein concentration. This strategy pays the cost of a **futile cycle** of protein synthesis and degradation to provide a benefit in terms of the speed with which the regulatory system can reach a new steady state. \n",
    "\n",
    "To see this, it is useful to normalize these plots by their steady states, as in the bottom plot below, which disentangles amplitude and timescale. Comparing normalized response curves in this way is reasonable, because mutations can alter the expression level of a gene allowing evolution to optimize expression level independently of the regulatory system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
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       "  const render_items = [{\"docid\":\"4a921173-3574-46dc-a838-b5e75ea9c3be\",\"roots\":{\"p1421\":\"34adc9b6-f7c7-4443-9803-3369c77c7a42\"},\"root_ids\":[\"p1421\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1421"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Parameters\n",
    "beta_1 = 100\n",
    "gamma = np.array([1, 2, 3])\n",
    "\n",
    "# Compute dynamics\n",
    "t = np.linspace(0, 6, 400)\n",
    "x = [beta_1 / g * (1 - np.exp(-g * t)) for g in gamma]\n",
    "\n",
    "# Set up plots\n",
    "colors = bokeh.palettes.Blues5\n",
    "p1 = bokeh.plotting.figure(\n",
    "    frame_height=175,\n",
    "    frame_width=300,\n",
    "    x_axis_label=\"time\",\n",
    "    y_axis_label=\"x(t)\",\n",
    "    x_range=[0, 6],\n",
    ")\n",
    "p2 = bokeh.plotting.figure(\n",
    "    frame_height=175,\n",
    "    frame_width=300,\n",
    "    x_axis_label=\"time\",\n",
    "    y_axis_label=\"x(t)/xₛₛ\",\n",
    "    x_range=[0, 6],\n",
    ")\n",
    "p2.x_range = p1.x_range\n",
    "\n",
    "# Populate graphs\n",
    "for x_vals, g, color in zip(x, gamma, colors):\n",
    "    p1.line(t, x_vals, color=color, line_width=2)\n",
    "    p1.circle(1 / g, beta_1 / g * (1 - np.exp(-1)), color=color, size=10)\n",
    "    p2.line(t, x_vals / x_vals.max(), color=color, line_width=2)\n",
    "    p2.circle(1 / g, 1 - np.exp(-1), color=color, size=10)\n",
    "    \n",
    "\n",
    "# Label lines\n",
    "p1.text(\n",
    "    x=[4],\n",
    "    y=[95],\n",
    "    text=[f\"γ = {gamma[0]}\"],\n",
    "    text_color=colors[0],\n",
    "    text_font_size=\"10pt\",\n",
    "    text_align=\"left\",\n",
    "    text_baseline=\"top\",\n",
    ")\n",
    "p1.text(\n",
    "    x=[4],\n",
    "    y=[53],\n",
    "    text=[f\"γ = {gamma[1]}\"],\n",
    "    text_color=colors[1],\n",
    "    text_font_size=\"10pt\",\n",
    "    text_align=\"left\",\n",
    "    text_baseline=\"bottom\",\n",
    ")\n",
    "p1.text(\n",
    "    x=[4],\n",
    "    y=[30],\n",
    "    text=[f\"γ = {gamma[2]}\"],\n",
    "    text_color=colors[2],\n",
    "    text_font_size=\"10pt\",\n",
    "    text_align=\"left\",\n",
    "    text_baseline=\"top\",\n",
    ")\n",
    "\n",
    "bokeh.io.show(bokeh.layouts.column([p1, bokeh.models.Spacer(height=20), p2]))"
   ]
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   "cell_type": "markdown",
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    "<!-- The plots above show the same y axis label -- we need to indicate normalization -->\n",
    "\n",
    "This analysis reveals an important **design principle**: _Increased protein degradation can speed up the response time of a gene expression system, at the cost of additional protein production, to reach the same steady state concentration._"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "## Network motifs identify functionally important circuits\n",
    "\n",
    "We have just seen that destabilizing a protein can speed its response time. However, most bacterial proteins, transcription factors in particular, are stable. Do they use other mechanisms to accelerate response times? \n",
    "\n",
    "The answer to this question will turn out to be yes, there is a circuit that can  accelerate responses. But before we get there, let's step back for a moment to ask how one can go about discovering such functionally important circuits in the first place. Do you have to guess them? It would be nice if there were some kind of catalog of important circuits and their functions that we could browse.\n",
    "\n",
    "The concept of **network motifs** is one way to obtain such a catalog. We define a network motif as a regulatory pattern, or sub-circuit, that is statistically over-represented in natural networks (circuits), compared to what one might expect from random networks with similar numbers of genes and regulatory interactions. In 2002, Alon and colleagues showed that network motif analysis could reveal recurring circuit modules with specific functions ([Shen-Orr et al, Nat. Gen. 2002](https://doi.org/10.1038/ng881)). \n",
    "\n",
    "More specifically, imagine the transcriptional regulatory network of an organism as a **graph** consisting of **nodes** and **directed edges** (arrows). In bacteria, each node represents an operon, while each arrow represents regulation of the target operon (tip of the arrow) by a transcription factor in the originating operon (base of the arrow), as shown schematically here.\n",
    "\n",
    "<div style=\"width: 300px; margin: auto;\">\n",
    "\n",
    "![simple graph](figs/simple_graph_2.png)\n",
    "\n",
    "</div>\n",
    "\n",
    "The transcriptional regulatory network of _E. coli_ has been mapped (see [RegulonDB](http://regulondb.ccg.unam.mx)). It contains ≈424 operons (nodes), ≈519 transcriptional regulatory interactions (arrows), involving ≈116 transcription factors. If the target of each arrow was chosen randomly, the probability of any given arrow being autoregulatory is low (≈1/424). One might expect only about one such event in the entire network. However, ≈40 such autoregulatory arrows are observed. (If we further consider whether the arrows are activating (+) or repressing (-), then we find 32 negative autoregulatory operons and 8 positive autoregulatory ones. Later on, we will discuss both types.) Autoregulation thus appears to be statistically over-represented compared to the null hypothesis of random regulatory interactions. It is a motif.\n",
    "\n",
    "The **motif principle** states that statistically over-represented patterns in networks have been selected repeatedly because they provide key cellular functions. A similar concept is useful in other aspects of biology. For example, sequence motifs are statistically over-represented sequences within the genome that are enriched for functionally important features, such as protein binding sites. Motifs of various kinds represent a cross-cutting concept in bioinformatics. "
   ]
  },
  {
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   "source": [
    "## Negative autoregulation accelerates response times\n",
    "\n",
    "Identifying motifs in regulatory networks is one way to discover functionally important circuits, and one of the strongest motifs of them all is autoregulation. Here, we will analyze autorepression (we will explore autoactivation in the next chapter). You can guess that since we are discussing speeding up circuit response that autorepression does just that. So, how does it work?\n",
    "\n",
    "We start by writing down a differential equation describing the dynamics of the concentration $x$ of the repressor. We represent the regulation of production using a repressive Hill function.\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}x}{\\mathrm{d}t} = \\frac{\\beta}{1+(x/k)^n} - \\gamma x.\n",
    "\\end{align}\n",
    "\n",
    "For arbitrary $n$, this differential equation has no analytical solution. As is often the case when reasoning about circuits, it is worthwhile to consider extreme cases. We can obtain approximate expressions for the dynamics in the regime of, $\\beta/\\gamma k \\gg 1$. This is a mathematical statement of strong repression; the concentration of repressor necessary to invoke repression, $k$, is less than the steady state level for unregulated expression, $\\beta/\\gamma$. Here, we summarize the results derived in [Technical Appendix 2a](02a_approximate_autorepression_dynamics.ipynb)."
   ]
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   "source": [
    "### Highly ultrasensitive, strong repression\n",
    "\n",
    "For the case where repression is highly ultrasensitive ($\\beta/\\gamma k \\gg 1$ and $n \\gg 1$), we have\n",
    "\n",
    "\\begin{align}\n",
    "x(t) \\approx \\left\\{\\begin{array}{lll}\n",
    "\\beta t & & t < k/\\beta,\\\\[1em]\n",
    "k & & t \\ge k/\\beta\n",
    "\\end{array}\n",
    "\\right.\n",
    "\\end{align}\n",
    "\n",
    "A plot of the this approximate $x(t)$ is shown below."
   ]
  },
  {
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    "# Curve\n",
    "t = [0, 1, 2]\n",
    "x = [0, 1, 1]\n",
    "\n",
    "# Set up plot\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_height=175,\n",
    "    frame_width=300,\n",
    "    x_axis_label=\"time\",\n",
    "    y_axis_label=\"x(t)\",\n",
    "    x_range=[0, 2],\n",
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    "\n",
    "# Custom axis labels\n",
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    "p.yaxis.ticker = bokeh.models.tickers.FixedTicker(ticks=[0, 1])\n",
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    "\n",
    "# Populate plot\n",
    "p.line(t, x, line_width=2)\n",
    "\n",
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    "\n",
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    "Initially, $x$ builds up approximately linearly, at rate $\\beta$, since the repression is step-like and no repression is occurring for small $x$. Eventually, its concentration is high enough to shut its own production off when $x \\approx k$, after which the steady state is maintained at $x \\approx k$.\n",
    "\n",
    "The half-time to the steady state of $k$ is $t_{1/2} \\approx k/2\\beta$. \n",
    "\n",
    "In the unregulated case,\n",
    "\n",
    "\\begin{align}\n",
    "x(t) = \\frac{\\beta}{\\gamma}\\left(1 - \\mathrm{e}^{-\\gamma t}\\right),\n",
    "\\end{align}\n",
    "\n",
    "giving a half time of $t_\\mathrm{1/2} = \\ln 2 / \\gamma$. Comparing the half times,\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\text{autorepressed }t_{1/2}}{\\text{unregulated }t_{1/2}} \\approx \\frac{k/2\\beta}{\\ln 2/\\gamma} = \\frac{1}{2\\ln 2}\\,\\frac{\\gamma k}{\\beta} \\ll 1,\n",
    "\\end{align}\n",
    "\n",
    "since $\\beta/\\gamma k \\gg 1$ for strong repression. Thus, the time to reach steady state is much shorter for the autorepressed circuit in the highly ultrasensitive, strong repression regime."
   ]
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   "source": [
    "### Less ultrasensitive strong autorepression\n",
    "\n",
    "If we relax the condition that $n \\gg 1$, we can derive an approximate expression for the dynamics for approach to steady state, assuming that $x_\\mathrm{ss} \\gg k$, which is typically only valid for $n$ not too large. The result, again derived in [Technical Appendix 2a](02a_approximate_autorepression_dynamics.ipynb), is\n",
    "\n",
    "\\begin{align}\n",
    "x(t) \\approx k\\left(\\frac{\\beta}{\\gamma k}\\right)^{\\frac{1}{n+1}}\\left(1 - \\mathrm{e}^{-(n+1)\\gamma t}\\right)^{\\frac{1}{n+1}},\n",
    "\\end{align}\n",
    "\n",
    "with the approximate steady state being\n",
    "\n",
    "\\begin{align}\n",
    "x_\\mathrm{ss}(t) \\approx k\\left(\\frac{\\beta}{\\gamma k}\\right)^{\\frac{1}{n+1}}.\n",
    "\\end{align}\n",
    "\n",
    "We then have\n",
    "\n",
    "\\begin{align}\n",
    "t_{1/2} \\approx \\frac{1}{(n+1)\\gamma}\\,\\ln\\left(\\frac{2^{n+1}}{2^{n+1}-1}\\right).\n",
    "\\end{align}\n",
    "\n",
    "Comparing to the unregulated case,\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\text{autorepressed }t_{1/2}}{\\text{unregulated }t_{1/2}} \\approx \\frac{\\ln\\left(\\frac{2^{n+1}}{2^{n+1}-1}\\right)}{(n+1)\\ln 2} < 1,\n",
    "\\end{align}\n",
    "\n",
    "meaning that the autorepressed circuit still have faster response. If we consider the non-ultrasensitive case with $n = 1$, the ratio of half times is\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\text{autorepressed }t_{1/2}}{\\text{unregulated }t_{1/2}} \\approx \\frac{\\ln (4/3)}{2 \\ln 2} \\approx 0.2,\n",
    "\\end{align}\n",
    "\n",
    "with the autorepressed circuit being five times faster.\n",
    "\n",
    "These approximate solutions for the dynamics are quite useful for reasoning about the speed boosts given by autorepression, but we need to resort to numerical solutions of the ODE to get the complete dynamics. We do this in the following sections, pausing first to consider how we might explicitly model inducers in the dynamics."
   ]
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    "### Incorporating explicit input dependence into a model of regulation\n",
    "\n",
    "Throughout this chapter we have been invoking a situation where the gene has been off for some time before it is suddenly turned on. We previously saw that it is possible for an inducer molecule to inhibit the action of a repressor, but it is also possible for there to be molecules that can activate the action of a repressor, or inhibit the action of an activator, or many other types of interactions. For now, we will ignore specific biomolecular details and simply write out a generic dependence of the autorepressive gene's expression on the presence of some external signal, $s$.\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}x}{\\mathrm{d}t} = \\frac{\\beta(s)}{1 + (x/k)^n} - \\gamma x.\n",
    "\\end{align}\n",
    "\n",
    "What is the functional form of $\\beta(s)$? We expect $\\beta$ should increase with $s$ and have a value close to $0$ when $s$ is close to $0$. Furthermore, there should be a limit to how rapidly the gene product can be produced, so $\\beta(s)$ must be finite, even for large $s$. As we saw in the previous chapter, the Hill function satisfies all of these properties, and therefore provides a simple phenomenological model of activation by $s$. Additionally, the two parameters of the Hill function, $k$ and $n$, allow flexibility in the $s$ concentration required for activation as well as the degree of ultrasensitivity. We will therefore represent $\\beta(s)$ as an activating Hill function of $s$:\n",
    "\n",
    "\\begin{align}\n",
    "\\beta (s) = \\beta_0\\, \\frac{(s/k_s)^{n_s}}{1 + (s/k_s)^{n_s}}.\n",
    "\\end{align}\n",
    "\n",
    "Choosing $n_s = 1$ leads to a linear relationship between $\\beta$ and $s$ in the regime where $s \\ll k_s$. Alternatively, the same versatile function can produce a step-like response in $\\beta$ at $s = k_s$ if we choose $n_s \\gg 1$. It is important to keep in mind that the $\\beta(s)$ Hill function does not represent a process of transcriptional regulation like the Hill term in our original $\\mathrm{d}x/\\mathrm{d}t$ expression, but rather  phenomenologically represents a complex process of enzymatic inactivation of the repressor $x$ by the external signal $s$.\n",
    "\n",
    "Having chosen to represent $\\beta(s)$ as an activating Hill function, our ODE model for our autoregulated gene now contains a product of activating and repressing Hill functions:\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}x}{\\mathrm{d}t} = \\beta_0\\,\\frac{(s/k_s)^{n_s}}{1 + (s/k_s)^{n_s}}\\, \\frac{1}{(1 + (x/k)^n)} - \\gamma x.\n",
    "\\end{align}"
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    "### Numerically solving for the dynamics\n",
    "\n",
    "As our dynamical equations become more complex, as they are starting to here, numerical solutions become essential. Indeed, numerical solution of ordinary differential equations is a ubiquitous technique in the study and design of biological circuits. We introduce this technique in [Technical Appendix 2b](../technical_appendices/02b_numerical_odes.ipynb) of this chapter.\n",
    "\n",
    "In our following analysis, we will consider the non-ultrasensitive case of $n = 1$."
   ]
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    "### Solving for a constant input\n",
    "\n",
    "We will first consider the case where we initially have no $s$ present, but a step increase in $s$ at time $t=0$ causes a corresponding step-like increase in the value of $\\beta$, which we represent by having $n_s \\gg 1$ and $s \\gg k_s$. After solving for $x(t)$ numerically, we will plot it along with the unregulated case,\n",
    "\n",
    "\\begin{align}\n",
    "x(t) = \\frac{\\beta_0}{\\gamma}\\,\\frac{(s/k_s)^{n_s}}{1 + (s/k_s)^{n_s}}\\left(1 - \\mathrm{e}^{-\\gamma t}\\right)\n",
    "\\end{align}\n",
    "\n",
    "for comparison. The result, detailed in [Technical Appendix 2b](../technical_appendices/02a_numerical_odes.ipynb), is shown below."
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       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
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      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def neg_auto_rhs(x, t, beta0, gamma, k, n, ks, ns, s):\n",
    "    \"\"\"\n",
    "    Right hand side for negative autoregulation motif with s dependence.\n",
    "    Return dx/dt.\n",
    "    \"\"\"\n",
    "    # Compute dx/dt\n",
    "    return (\n",
    "        beta0 * (s / ks) ** ns / (1 + (s / ks) ** ns) / (1 + (x / k) ** n) - gamma * x\n",
    "    )\n",
    "\n",
    "\n",
    "# Time points we want for the solution\n",
    "t = np.linspace(0, 10, 200)\n",
    "\n",
    "# Initial condition\n",
    "x0 = 0.0\n",
    "\n",
    "# Parameters\n",
    "beta0 = 100\n",
    "gamma = 1.0\n",
    "k = 1.0\n",
    "n = 1.0\n",
    "s = 100.0\n",
    "ns = 10.0\n",
    "ks = 0.1\n",
    "\n",
    "# Package parameters into a tuple\n",
    "args = (beta0, gamma, k, n, ks, ns, s)\n",
    "\n",
    "# Integrate ODES\n",
    "x = scipy.integrate.odeint(neg_auto_rhs, x0, t, args=args)\n",
    "\n",
    "# Extract time course for first (in this case only) species\n",
    "x = x.transpose()[0]\n",
    "\n",
    "# Unregulated solution\n",
    "x_unreg = (\n",
    "    beta0 * (s / ks) ** ns / (1 + (s / ks) ** ns) / gamma * (1 - np.exp(-gamma * t))\n",
    ")\n",
    "\n",
    "# Set up color palette for this notebook\n",
    "colors = colorcet.b_glasbey_category10\n",
    "\n",
    "# Set up figure\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_width=325,\n",
    "    frame_height=250,\n",
    "    x_axis_label=\"time\",\n",
    "    y_axis_label=\"x\",\n",
    "    x_range=[np.min(t), np.max(t)],\n",
    ")\n",
    "\n",
    "cds = bokeh.models.ColumnDataSource(\n",
    "    dict(t=t, x=x, x_unreg=x_unreg)\n",
    ")\n",
    "\n",
    "# Populate glyphs\n",
    "p.line(\n",
    "    source=cds,\n",
    "    x=\"t\",\n",
    "    y=\"x\",\n",
    "    line_width=2,\n",
    "    color=colors[0],\n",
    "    legend_label=\"autorepressive\",\n",
    ")\n",
    "p.line(\n",
    "    source=cds,\n",
    "    x=\"t\",\n",
    "    y=\"x_unreg\",\n",
    "    line_width=2,\n",
    "    color=colors[1],\n",
    "    legend_label=\"unregulated\",\n",
    ")\n",
    "\n",
    "# Place the legend\n",
    "p.legend.location = \"center_right\"\n",
    "\n",
    "# Allow hiding some plots\n",
    "p.legend.click_policy = \"hide\"\n",
    "\n",
    "# Write the title\n",
    "p.title.text = \"Constant-input dynamics\"\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we want to compare the speed with which the system approaches steady state, we should instead plot the normalized expression level."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
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       "  const render_items = [{\"docid\":\"1bb37e52-0669-41b1-a7a3-50df2cd8f9d5\",\"roots\":{\"p2356\":\"3957ab16-c4c7-4d03-9bc4-20fc695fc644\"},\"root_ids\":[\"p2356\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p2356"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "cds.data[\"x\"] /= cds.data[\"x\"].max()\n",
    "cds.data[\"x_unreg\"] /= cds.data[\"x_unreg\"].max()\n",
    "\n",
    "# Update axis label\n",
    "p.yaxis.axis_label = \"x(t)/xₛₛ\"\n",
    "\n",
    "# Show plot\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use these numerical solutions to compare the ratios of the half times of the autorepressed and unregulated case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "half time ratio = 0.23\n"
     ]
    }
   ],
   "source": [
    "# Find half time by finding time where reaches half maximum value\n",
    "half_time_ratio = t[x <= x[-1] / 2][-1] / t[x_unreg <= x_unreg[-1] / 2][-1]\n",
    "\n",
    "print(\"half time ratio = %.2f\" % half_time_ratio)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is close to the ratio of 0.2 that we derived from the approximate analytical solution above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "### Design principle: Negative autoregulation of a transcription factor accelerates its response to a change in input\n",
    "\n",
    "Comparing the unnormalized and normalized plots exposes two related effects of autoregulation. First, adding negative autoregulation reduces the steady-state expression level. Second, as a consequence, the approach to steady state is accelerated. This can be seen in the second plot where all the concentrations are normalized to their steady state. All told, in this example, negative autoregulation accelerated the dynamics by about 5-fold compared to the unregulated system. We have thus arrived at another **design principle**: _Negative autoregulation of a transcription factor accelerates its response to a change in input_."
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    "### Experimental demonstration of negative autoregulation speeding response time\n",
    "\n",
    "Can this acceleration be observed experimentally? To find out, Rosenfeld and coworkers engineered a simple synthetic system based on a bacterial repressor called TetR, fused to a fluorescent protein for readout, and studied its turn-on dynamics in bacterial populations. The plot below show the response to sudden induction of gene expression of GFP-fused TetR, using data digitized from Fig. 3 of the [Rosenfeld, et al. paper](https://doi.org/10.1016/S0022-2836(02)00994-4)."
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    "The curves in blue are three trials with the autorepressive circuit. The gray line shows the response in the presence of aTc, a small molecule that inactivates TetR. The black line shows the response of an \"open loop\" circuit (simple repression) that lacks feedback.\n",
    "\n",
    "Interestingly, these dynamics show the expected acceleration, as well as some oscillations around steady-state, which may be explained by time delays in the regulatory system (which we will discuss more in an upcoming chapter)."
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    "## Solving for a time-varying input\n",
    "\n",
    "This acceleration in response occurs when we turn the gene on. What happens when we turn the gene off, for example by suddenly stopping the gene's expression at some later time point? We can address this question by including $s$ as a time-varying signal.\n",
    "\n",
    "Let's imagine that the input signal $s$, rather than turning on instantaneously at $t=0$ and remaining constant for the rest of the simulation, instead occurs as a pulse of duration $\\tau$ that peaks at time $t_0$. Then $s$ would be present roughly within the time window $(t_0-\\tau/2, t_0+\\tau/2)$, but absent outside of this time window. We will model the peak as a Gaussian function of unit height,\n",
    "\n",
    "\\begin{align}\n",
    "s(t) = \\exp\\left[-\\frac{4(t-t_0)^2}{\\tau^2}\\right].\n",
    "\\end{align}\n",
    "\n",
    "Let's take a look at what the pulse looks like for visualization purposes."
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    "def s_pulse(t, t_0, tau):\n",
    "    \"\"\"\n",
    "    Returns s value for a pulse centered at t_0 with duration tau.\n",
    "    \"\"\"\n",
    "    # Return 0 is tau is zero, otherwise Gaussian\n",
    "    return 0 if tau == 0 else np.exp(-4 * (t - t_0) ** 2 / tau ** 2)\n",
    "\n",
    "\n",
    "# Plot the pulse\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_width=350,\n",
    "    frame_height=150,\n",
    "    x_axis_label=\"time\",\n",
    "    y_axis_label=\"s\",\n",
    "    x_range=[0, 10],\n",
    ")\n",
    "\n",
    "# Populate glyph\n",
    "p.line(t, s_pulse(t, 4.0, 2.0), line_width=2)\n",
    "\n",
    "# Show plot\n",
    "bokeh.io.show(p)"
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    "The input time course has the desired pulsatile shape. We proceed to numerically solve for the dynamics,"
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   "source": [
    "def neg_auto_rhs_s_fun(x, t, beta0, gamma, k, n, ks, ns, s_fun, s_args):\n",
    "    \"\"\"\n",
    "    Right hand side for negative autoregulation function, with s variable.\n",
    "    Returns dx/dt.\n",
    "    \n",
    "    s_fun is a function of the form s_fun(t, *s_args), so s_args is a tuple\n",
    "    containing the arguments to pass to s_fun.\n",
    "    \"\"\"\n",
    "    # Compute s\n",
    "    s = s_fun(t, *s_args)\n",
    "    \n",
    "    # Correct for x possibly being numerically negative as odeint() adjusts step size\n",
    "    x = np.maximum(0, x)\n",
    "    \n",
    "    # Plug in this value of s to the RHS of the negative autoregulation model\n",
    "    return neg_auto_rhs(x, t, beta0, gamma, k, n, ks, ns, s)\n",
    "\n",
    "# Set up parameters for the pulse\n",
    "s_args = (4.0, 2.0)\n",
    "\n",
    "# Package parameters into a tuple\n",
    "args = (beta0, gamma, k, n, ks, ns, s_pulse, s_args)\n",
    "\n",
    "# Integrate ODEs\n",
    "x = scipy.integrate.odeint(neg_auto_rhs_s_fun, x0, t, args=args).transpose()[0]\n",
    "\n",
    "# Plot the normalized values\n",
    "x /= x.max()\n",
    "\n",
    "# Also calculate the pulse for plotting purposes\n",
    "s = s_pulse(t, *s_args)\n",
    "\n",
    "# Plot the results\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_width=400,\n",
    "    frame_height=150,\n",
    "    x_axis_label=\"time\",\n",
    "    y_axis_label=\"normalized concentration\",\n",
    "    x_range=[0, 10],\n",
    ")\n",
    "\n",
    "# Populate glyphs\n",
    "p.line(t, s, line_width=2, color=colors[0], legend_label=\"s\")\n",
    "p.line(t, x, line_width=2, color=colors[1], legend_label=\"x\")\n",
    "\n",
    "# Place the legend\n",
    "p.legend.location = \"top_right\"\n",
    "\n",
    "# Allow hiding curves\n",
    "p.legend.click_policy = \"hide\"\n",
    "\n",
    "# Show plot\n",
    "bokeh.io.show(p)"
   ]
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   "source": [
    "As expected, we see the qualitative result that $x$ does not turn on until the pulse of $s$ begins, and once $s$ stops, $x$ begins an exponential decay back down to zero. The specific quantitative details of the relationship between the $x$ trajectory and the $s$ pulse, such as the threshold value of $s$ where $x$ begins to turn on, are dependent on the specific parametrization of the system, such as the value of $k_s$.\n",
    "\n",
    "We now want to ask how the dynamics of $x$'s rise and fall compare to the unregulated case. In order to do this, we simply need to write another function that plots out the unregulated time course as an explicit function of $s$.\n",
    "\n",
    "The attentive reader might, at this point, raise an objection: How can we claim that the *unregulated* gene $x$ is nonetheless still governed by the presence or absence of this input signal $s$? Indeed, although an unregulated gene does not have any transcriptional activators or repressors that modulate its expression, all genes in an organism can always have their expression level be affected by higher-level factors such as the concentration of RNA polymerase or ribosomes, the global metabolic rate, or in the case of eukaryotic organisms, chromatin accessibility. So in this case we could imagine that our $s$ pulse represents a very simplified model of a scenario where a bacterium containing our gene is persisting in a nutrient-starved environment, where most of its genes have globally been turned off, and suddenly comes across a pulse of nutrients that allow it to transiently ramp up its metabolism and gene expression before returning to a starvation state when the nutrients disappear.\n",
    "\n",
    "Let's now put together a function that takes our unregulated ODE and converts it to a form that includes an explicit Hill-like $s$-dependence for the activation term, as in\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}x}{\\mathrm{d}t} = \\beta(s) - \\gamma x = \\beta_0\\, \\frac{(s/k_s)^{n_s}}{1 + (s/k_s)^{n_s}} - \\gamma x.\n",
    "\\end{align}\n",
    "\n",
    "We can again numerically solve and add the unregulated case to our plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "tags": [
     "hide-input"
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   "source": [
    "def unreg_rhs(x, t, beta0, gamma, ks, ns, s):\n",
    "    \"\"\"\n",
    "    Right hand side for constitutive gene expression\n",
    "    modulated to only be active in the presence of s.\n",
    "    Returns dx/dt.\n",
    "    \"\"\"\n",
    "    return beta0 * (s / ks) ** ns / (1 + (s / ks) ** ns) - gamma * x\n",
    "\n",
    "\n",
    "def unreg_rhs_s_fun(x, t, beta0, gamma, ks, ns, s_fun, s_args):\n",
    "    \"\"\"\n",
    "    Right hand side for unregulated function, with s variable.\n",
    "    Returns dx/dt.\n",
    "\n",
    "    s_fun is a function of the form s_fun(t, *s_args), so s_args is a tuple\n",
    "    containing the arguments to pass to s_fun.\n",
    "    \"\"\"\n",
    "    # Compute s\n",
    "    s = s_fun(t, *s_args)\n",
    "\n",
    "    # Plug in this value of s to the RHS of the negative autoregulation model\n",
    "    return unreg_rhs(x, t, beta0, gamma, ks, ns, s)\n",
    "\n",
    "\n",
    "# Package parameters into a tuple\n",
    "args_unreg = (beta0, gamma, ks, ns, s_pulse, s_args)\n",
    "\n",
    "# Integrate ODEs\n",
    "x_unreg = scipy.integrate.odeint(unreg_rhs_s_fun, x0, t, args=args_unreg).transpose()[0]\n",
    "\n",
    "# Normalize\n",
    "x_unreg /= x_unreg.max()\n",
    "\n",
    "# Add to the plot\n",
    "p.line(t, x_unreg, line_width=2, color=colors[2], legend_label=\"unregulated x\")\n",
    "\n",
    "# Show plot\n",
    "bokeh.io.show(p)"
   ]
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    "We can see from the above plot that although negative autoregulation speeds up the rise time of the gene's expression in response to the appearance of stimulus, it has no impact on the speed of the fall time in response to the disappearance of the stimulus. We actually could have known this result would occur just from looking at the equations governing the gene's dynamics: recall that in the regulated case, $x$ is governed by\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}x}{\\mathrm{d}t}=\\frac{\\beta(s)}{(1+x/k)^n} - \\gamma x,\n",
    "\\end{align}\n",
    "\n",
    "while in the unregulated case, $x$ is governed by\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}x}{\\mathrm{d}t}=\\beta(s) - \\gamma x.\n",
    "\\end{align}\n",
    "\n",
    "If we set $\\beta(s)=0$, then both ODEs become the same expression,\n",
    "\n",
    "\\begin{align}\n",
    "\\left.\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right\\vert_{\\beta(s)=0}=- \\gamma x,\n",
    "\\end{align}\n",
    "\n",
    "meaning that the dynamics of the fall in gene expression will become identical between the two systems as $\\beta(s)$ approaches $0$.\n"
   ]
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   "source": [
    "## Interactive plotting and varying parameters\n",
    "\n",
    "Although we have now convincingly demonstrated that it is possible for negative autoregulation to speed the response time of a gene's expression, we do not yet have a clear understanding of how the values of the parameters themselves, particularly $k$ and $n$, affect the magnitude of this speed-up. To gain insight into these effects, we turn to interactive plotting.\n",
    "\n",
    "Plotting with Bokeh in Jupyter notebooks allows for interactivity with plots that can help to rapidly gain insights about how parameter values might affect the dynamics. We have found that this is a useful tool to rapidly explore parameter dependence on circuit behavior. We detail how to build an interactive plot in [Technical Appendix 2c](../technical_appendices/02c_interactive_plotting.ipynb). The result is shown below."
   ]
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     "hide-input"
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transpose(A) {\\n    var m = A.length;\\n    var n = A[0].length;\\n    var AT = [];\\n\\n    for (var j = 0; j < n; j++) {\\n        var ATj = [];\\n        for (var i = 0; i < m; i++) {\\n            ATj.push(A[i][j]);\\n        }\\n        AT.push(ATj);\\n    }\\n\\n    return AT;\\n}\\n\\n\\nfunction dot(v1, v2) {\\n    /*\\n     * Compute dot product v1 . v2.\\n     */\\n\\n    var n = v1.length;\\n    var result = 0.0;\\n    for (var i = 0; i < n; i++) result += v1[i] * v2[i];\\n\\n    return result;\\n}\\n\\n\\nfunction norm(v) {\\n    /*\\n     * 2-norm of a vector\\n     */\\n\\n    return Math.sqrt(dot(v, v));\\n}\\n\\n\\nfunction mvMult(A, v, diagonalA) {\\n    /*\\n     * Compute dot product A . v, where A is a matrix.\\n     * If diagonalA is true, then A must be a 1-D array.\\n     */\\n\\n    if (diagonalA) return elementwiseVectorMult(A, v);\\n    else {\\n        return A.map(function (Arow) {\\n            return dot(Arow, v);\\n        });\\n    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var result = [];\\n    n = v1.length;\\n\\n    for (var i = 0; i < n; i++) {\\n        result.push(v1[i] / v2[i]);\\n    }\\n\\n    return result;\\n}\\n\\n\\nfunction elementwiseVectorMult(v1, v2) {\\n    /*\\n     * Compute v1 * v2 elementwise.\\n     */\\n\\n    var result = [];\\n    n = v1.length;\\n\\n    for (var i = 0; i < n; i++) {\\n        result.push(v1[i] * v2[i]);\\n    }\\n\\n    return result;\\n}\\n\\n\\nfunction svMultAdd(scalars, vectors) {\\n    /*\\n     * Add a set of vectors together, each multiplied by a scalar.\\n     */\\n\\n    var m = vectors[0].length;\\n    var n = scalars.length;\\n\\n    if (vectors.length != n) {\\n        console.warn(\\\"svMultAdd: Difference number of scalars and vectors.\\\");\\n        return null;\\n    }\\n\\n    var result = [];\\n    for (var i = 0; i < m; i++) {\\n        var element = 0.0;\\n        for (var j = 0; j < n; j++) {\\n            element += scalars[j] * vectors[j][i];\\n        }\\n        result.push(element);\\n    }\\n\\n    return result;\\n}\\n\\n\\nfunction absVector(v) {\\n    var result = [];\\n    for (var i = 0; i < v.length; i++) {\\n        result[i] = Math.abs(v[i]);\\n    }\\n\\n    return result;\\n}\\n\\n\\nfunction LUPDecompose(A, eps) {\\n    /*\\n     * LUP decomposition.\\n     */\\n\\n    var i, j, k, imax;\\n    var maxA, absA;\\n    var Arow;\\n    var p = [];\\n    var n = A.length;\\n    var LU = shallowCopyMatrix(A);\\n\\n    // Permutation matrix\\n    for (i = 0; i <= n; i++) p.push(i);\\n\\n    for (i = 0; i < n; i++) {\\n        maxA = 0.0;\\n        imax = i;\\n\\n        for (k = i; k < n; k++) {\\n            absA = Math.abs(LU[k][i]);\\n            if (absA > maxA) {\\n                maxA = absA;\\n                imax = k;\\n            }\\n        }\\n\\n        // Failure; singular matrix\\n        if (maxA < eps) return [null, null];\\n\\n        if (imax != i) {\\n            // Pivot\\n            j = p[i];\\n            p[i] = p[imax];\\n            p[imax] = j;\\n\\n            // Pivot rows of A\\n            Arow = LU[i];\\n            LU[i] = LU[imax];\\n            LU[imax] = Arow;\\n\\n            // Count pivots\\n            p[n]++;\\n        }\\n\\n        for (j = i + 1; j < n; j++) {\\n            LU[j][i] /= LU[i][i];\\n\\n            for (k = i + 1; k < n; k++) LU[j][k] -= LU[j][i] * LU[i][k];\\n        }\\n    }\\n\\n    return [LU, p];\\n}\\n\\nfunction LUPSolve(LU, p, b) {\\n    /*\\n     * Solve a linear system where LU and p are stored as the\\n     * output of LUPDecompose().\\n     */\\n\\n    var n = b.length;\\n    var x = [];\\n\\n    for (var i = 0; i < n; i++) {\\n        x.push(b[p[i]]);\\n        for (var k = 0; k < i; k++) x[i] -= LU[i][k] * x[k];\\n    }\\n\\n    for (i = n - 1; i >= 0; i--) {\\n        for (k = i + 1; k < n; k++) x[i] -= LU[i][k] * x[k];\\n\\n        x[i] /= LU[i][i];\\n    }\\n\\n    return x;\\n}\\n\\nfunction solve(A, b) {\\n    /*\\n     * Solve a linear system using LUP decomposition.\\n     *\\n     * Returns null if singular.\\n     */\\n\\n    var eps = 1.0e-14;\\n    var LU, p;\\n\\n    [LU, p] = LUPDecompose(A, eps);\\n\\n    // Return null if singular\\n    if (LU === null) return null;\\n\\n    return LUPSolve(LU, p, b);\\n}\\n\\n\\nfunction rkf45(\\n    f,\\n    initialCondition,\\n    timePoints,\\n    args,\\n    dt,\\n    tol,\\n    relStepTol,\\n    maxDeadSteps,\\n    sBounds,\\n    hMin,\\n    enforceNonnegative,\\n    debugMode,\\n) {\\n    // Set up return variables\\n    let tSol = [timePoints[0]];\\n    let t = timePoints[0];\\n    let iMax = timePoints.length;\\n    let y = [initialCondition];\\n    let y0 = initialCondition;\\n    let i = 1;\\n    let nDeadSteps = 0;\\n    let deadStep = false;\\n\\n    // DEBUG\\n    let nSteps = 0;\\n    // END EDEBUG\\n\\n    // Default parameters\\n    let h;\\n    if (dt === undefined) h = timePoints[1] - timePoints[0];\\n    else h = dt;\\n\\n    if (tol === undefined) tol = 1e-7;\\n    if (relStepTol === undefined) relStepTol = 0.0;\\n    if (sBounds === undefined) sBounds = [0.1, 10.0];\\n    if (hMin === undefined) hMin = 0.0;\\n    if (enforceNonnegative === undefined) enforceNonnegative = true;\\n    if (maxDeadSteps === undefined) maxDeadSteps = 10;\\n    if (debugMode === undefined) debugMode = false;\\n\\n    while (i < iMax && nDeadSteps < maxDeadSteps) {\\n        nDeadSteps = 0;\\n        while (t < timePoints[i] && nDeadSteps < maxDeadSteps) {\\n            [y0, t, h, deadStep] = rkf45Step(\\n                f,\\n                y0,\\n                t,\\n                args,\\n                h,\\n                tol,\\n                relStepTol,\\n                sBounds,\\n                hMin\\n            );\\n            nDeadSteps = deadStep ? nDeadSteps + 1 : 0;\\n            if (enforceNonnegative) {\\n                y0 = y0.map(function (x) {\\n                    if (x < 0.0) return 0.0;\\n                    else return x;\\n                });\\n            }\\n            // DEBUG\\n            nSteps += 1;\\n            // END DEBUG\\n        }\\n        if (t > tSol[tSol.length - 1]) {\\n            y.push(y0);\\n            tSol.push(t);\\n        }\\n        i += 1;\\n    }\\n\\n    // DEBUG\\n    if (debugMode) console.log(nSteps);\\n    // END DEBUG\\n\\n    let yInterp;\\n    if (nDeadSteps == maxDeadSteps) {\\n    \\tyInterp = nanArray(initialCondition.length, iMax);\\n    }\\n    else yInterp = interpolateSolution(timePoints, tSol, transpose(y));\\n\\n    return yInterp;\\n}\\n\\n\\nfunction rkf45Step(f, y, t, args, h, tol, relStepTol, sBounds, hMin) {\\n    let k1 = svMult(h, f(y, t, ...args));\\n\\n    let y2 = svMultAdd([0.25, 1.0], [k1, y]);\\n    let k2 = svMult(h, f(y2, t + 0.25 * h, ...args));\\n\\n    let y3 = svMultAdd([0.09375, 0.28125, 1.0], [k1, k2, y]);\\n    let k3 = svMult(h, f(y3, t + 0.375 * h, ...args));\\n\\n    let y4 = svMultAdd(\\n        [1932.0 / 2197.0, -7200.0 / 2197.0, 7296.0 / 2197.0, 1.0],\\n        [k1, k2, k3, y]\\n    );\\n    let k4 = svMult(h, f(y4, t + (12.0 * h) / 13.0, ...args));\\n\\n    let y5 = svMultAdd(\\n        [\\n            8341.0 / 4104.0,\\n            -32832.0 / 4104.0,\\n            29440.0 / 4104.0,\\n            -845.0 / 4104.0,\\n            1.0,\\n        ],\\n        [k1, k2, k3, k4, y]\\n    );\\n    let k5 = svMult(h, f(y5, t + h, ...args));\\n\\n    let y6 = svMultAdd(\\n        [\\n            -6080.0 / 20520.0,\\n            41040.0 / 20520.0,\\n            -28352.0 / 20520.0,\\n            9295.0 / 20520.0,\\n            -5643.0 / 20520.0,\\n            1.0,\\n        ],\\n        [k1, k2, k3, k4, k5, y]\\n    );\\n    let k6 = svMult(h, f(y6, t + h / 2.0, ...args));\\n\\n    // Calculate new step\\n    let yNew = svMultAdd(\\n        [\\n            2375.0 / 20520.0,\\n            11264.0 / 20520.0,\\n            10985.0 / 20520.0,\\n            -4104.0 / 20520.0,\\n            1.0,\\n        ],\\n        [k1, k3, k4, k5, y]\\n    );\\n\\n    // Relative difference between steps\\n\\tlet relChangeStep = norm(vectorAdd(yNew, svMult(-1.0, y))) / norm(yNew);\\n\\n    // Calculate error (note that k2's contribution to the error is zero)\\n    let errorVector = svMultAdd(\\n        [\\n            209.0 / 75240.0,\\n            -2252.8 / 75240.0,\\n            -2197.0 / 75240.0,\\n            1504.8 / 75240.0,\\n            2736.0 / 75240.0,\\n        ],\\n        [k1, k3, k4, k5, k6]\\n    );\\n    let error = Math.max(...absVector(errorVector));\\n\\n    // Either don't take a step or use the RK4 step\\n    let deadStep;\\n    if (error < tol || relChangeStep < relStepTol || h <= hMin) {\\n        t += h;\\n        deadStep = false;\\n    } else {\\n        yNew = y;\\n        deadStep = true;\\n    }\\n\\n    // Compute scaling for new step size\\n    let s;\\n    if (error === 0.0) {\\n        s = sBounds[1];\\n    } else {\\n        s = Math.pow((tol * h) / 2.0 / error, 0.25);\\n    }\\n    if (s < sBounds[0]) {\\n        s = sBounds[0];\\n    } else if (s > sBounds[1]) {\\n        s = sBounds[1];\\n    }\\n\\n    // Return new y-values, new time, and updated step size h\\n    return [yNew, t, Math.max(s * h, hMin), deadStep];\\n}\\n\\n\\nfunction dydtIMEX(y, t, f, cfun, Afun, fArgs, cfunArgs, AfunArgs, diagonalA) {\\n    /*\\n     * Right hand side of ODEs for initializing IMEX method with RKF.\\n     */\\n\\n    n = y.length;\\n    let rhs = zeros(n);\\n\\n    let A = Afun(t, ...AfunArgs);\\n    let c = cfun(t, ...cfunArgs);\\n\\n    // Linear part\\n    let nonConstantLinear = diagonalA\\n        ? elementwiseVectorMult(A, y)\\n        : mvMult(A, y, diagonalA);\\n    let linearPart = vectorAdd(nonConstantLinear, c);\\n\\n    // Nonlinear part\\n    let nonlinearPart = f(y, t, ...fArgs);\\n\\n    return vectorAdd(nonlinearPart, linearPart);\\n}\\n\\n\\nfunction cnab2Step(u, c, A, f1, f0, g1, omega, k, diagonalA) {\\n    /*\\n     * Take a CNAB2 step.\\n     *\\n     * - u is the current value of the solution.\\n     * - c is the constant term.\\n     * - A is the matrix for the linear function.\\n     * - f1 is the nonlinear function evaluated at the current value of y.\\n     * - f0 is the nonlinear function evaluated at the previous value of y.\\n     * - g1 is the linear function evaluated at the current value of y.\\n     * - omega is the ratio of the most recent step size to the one before that.\\n     * - k is the current step size.\\n     * - diagonalA is true if A is diagonal. This leads to a *much* faster time step.\\n     *   If diagonalA is true, then A is provided only as the diagonal.\\n     */\\n\\n    let invk = 1.0 / k;\\n    let b = vectorAdd(\\n        svMult(0.5, c),\\n        svMult(invk, u),\\n        svMult(1.0 + omega / 2.0, f1),\\n        svMult(-omega / 2.0, f0),\\n        svMult(0.5, g1)\\n    );\\n\\n    if (diagonalA) {\\n        let Aaug = svAdd(invk, svMult(-0.5, A));\\n        let result = elementwiseVectorDivide(b, Aaug);\\n    } else {\\n        let n = A.length;\\n        let Aaug = smMult(-0.5, A);\\n        for (i = 0; i < n; i++) {\\n            Aaug[i][i] += invk;\\n        }\\n        let result = solve(Aaug, b);\\n    }\\n\\n    return result;\\n}\\n\\n\\nfunction vsimexAdjustStepSizePID(\\n    k,\\n    relChange,\\n    relChangeStep,\\n    tol,\\n    kP,\\n    kI,\\n    kD,\\n    kBounds,\\n    sBounds\\n) {\\n    /*\\n     * Adjust step size using a PID controller.\\n     */\\n    let mult =\\n        Math.pow(relChange[1] / relChangeStep, kP) *\\n        Math.pow(tol / relChangeStep, kI) *\\n        Math.pow(Math.pow(relChange[0], 2) / relChange[1] / relChangeStep, kD);\\n    if (mult > sBounds[1]) mult = sBounds[1];\\n    else if (mult < sBounds[0]) mult = sBounds[0];\\n\\n    let newk = mult * k;\\n\\n    if (newk > kBounds[1]) newk = kBounds[1];\\n    else if (newk < kBounds[0]) newk = kBounds[0];\\n\\n    return newk;\\n}\\n\\n\\nfunction vsimexAdjustStepSizeRejectedStep(\\n    k,\\n    relChangeStep,\\n    tol,\\n    kBounds,\\n    sBounds\\n) {\\n    /*\\n     * Adjust step for rejected step\\n     */\\n\\n    let mult = tol / relChangeStep;\\n    if (mult < sBounds[0]) mult = sBounds[0];\\n\\n    let newk = mult * k;\\n    if (newk < kBounds[0]) newk = kBounds[0];\\n\\n    return newk;\\n}\\n\\n\\nfunction vsimexAdjustStepSizeFailedSolve(k, failedSolveS) {\\n    /*\\n     * Adjust step for failed solve. Bringing step size down will\\n     * eventually make matrix for linear solve positive definite.\\n     */\\n\\n    return k * failedSolveS;\\n}\\n\\n\\nfunction vsimex(\\n    f,\\n    cfun,\\n    Afun,\\n    initialCondition,\\n    timePoints,\\n    fArgs,\\n    cfunArgs,\\n    AfunArgs,\\n    diagonalA,\\n    k0,\\n    kBounds,\\n    tol,\\n    tolBuffer,\\n    kP,\\n    kI,\\n    kD,\\n    sBounds,\\n    failedSolveS,\\n    enforceNonnegative,\\n    maxDeadSteps\\n) {\\n    /*\\n     *\\n     */\\n\\n    // Defaults\\n    if (k0 === undefined) k0 = 1.0e-5;\\n    if (kBounds === undefined) kBounds = [1.0e-6, 100.0];\\n    if (tol === undefined) tol = 0.001;\\n    if (tolBuffer === undefined) tolBuffer = 0.01;\\n    if (kP === undefined) kP = 0.075;\\n    if (kI === undefined) kI = 0.175;\\n    if (kD === undefined) kD = 0.01;\\n    if (sBounds === undefined) sBounds = [0.1, 10.0];\\n    if (failedSolveS === undefined) failedSolveS = 0.1;\\n    if (enforceNonnegative == undefined) enforceNonnegative = true;\\n    if (maxDeadSteps === undefined) maxDeadSteps = 10;\\n\\n    // Do RKF to get the first few time points\\n    let rkf45TimePoints = [\\n        timePoints[0],\\n        timePoints[0] + k0,\\n        timePoints[0] + 2.0 * k0,\\n    ];\\n\\n    let args = [f, cfun, Afun, fArgs, cfunArgs, AfunArgs, diagonalA];\\n    let yRKF = rkf45(\\n        dydtIMEX,\\n        initialCondition,\\n        rkf45TimePoints,\\n        args,\\n        k0 / 10.0,\\n        tol,\\n        sBounds,\\n        0.0,\\n        enforceNonnegative,\\n        maxDeadSteps\\n    );\\n\\n    yRKF = transpose(yRKF);\\n\\n    // Set up variables for running CNAB2 VSIMEX\\n    let tSol = [timePoints[0]];\\n    let iMax = timePoints.length;\\n    let y = [initialCondition];\\n    let k = 2.0 * k0;\\n    let newk;\\n    let t = rkf45TimePoints[2];\\n    let y0 = yRKF[2];\\n    let i = 1;\\n    let nDeadSteps = 0;\\n    let deadStep = false;\\n    let c = cfun(t, ...cfunArgs);\\n    let A = Afun(t, ...AfunArgs);\\n    let f0 = f(initialCondition, timePoints[0], ...fArgs);\\n    let f1 = f(y0, t, ...fArgs);\\n    let g1 = vectorAdd(c, mvMult(A, y0, diagonalA));\\n    let omega = 1.0;\\n    let yStep;\\n    let relChangeStep;\\n    let relTol = tol * (1.0 + tolBuffer);\\n    let relChange = [\\n        norm(vectorAdd(y0, svMult(-1.0, yRKF[1]))) / norm(y0),\\n        norm(vectorAdd(yRKF[1], svMult(-1.0, initialCondition))) /\\n            norm(yRKF[1]),\\n    ];\\n\\n    // DEBUG\\n    let nSteps = 3;\\n    // END EDEBUG\\n\\n    while (i < iMax && nDeadSteps < maxDeadSteps) {\\n        nDeadSteps = 0;\\n        while (t < timePoints[i] && nDeadSteps < maxDeadSteps) {\\n            // Take CNAB2 step\\n            yStep = cnab2Step(y0, c, A, f1, f0, g1, omega, k, diagonalA);\\n\\n            // Reject the step if failed to solve\\n            if (yStep === null) {\\n                newk = vsimexAdjustStepSizeFailedSolve(k, failedSolveS);\\n                omega *= newk / k;\\n                k = newk;\\n                nDeadSteps += 1;\\n                console.log(\\\"null yStep\\\");\\n            } else {\\n                // Relative change\\n                relChangeStep =\\n                    norm(vectorAdd(yStep, svMult(-1.0, y0))) / norm(yStep);\\n\\n                // Take step if below tolerance\\n                if (relChangeStep <= relTol) {\\n                    f0 = f(y0, t, ...fArgs);\\n                    t += k;\\n                    y0 = yStep;\\n                    f1 = f(y0, t, ...fArgs);\\n                    c = cfun(t, ...cfunArgs);\\n                    A = Afun(t, ...AfunArgs);\\n                    g1 = vectorAdd(c, mvMult(A, y0, diagonalA));\\n                    newk = vsimexAdjustStepSizePID(\\n                        k,\\n                        relChange,\\n                        relChangeStep,\\n                        tol,\\n                        kP,\\n                        kI,\\n                        kD,\\n                        kBounds,\\n                        sBounds\\n                    );\\n                    relChange = [relChange[1], relChangeStep];\\n                    omega = newk / k;\\n                    k = newk;\\n                    nDeadSteps = 0;\\n                }\\n                // Reject the step is not within tolerance\\n                else {\\n                    newk = vsimexAdjustStepSizeRejectedStep(\\n                        k,\\n                        relChangeStep,\\n                        tol,\\n                        kBounds,\\n                        sBounds\\n                    );\\n                    omega *= newk / k;\\n                    k = newk;\\n                    nDeadSteps += 1;\\n                }\\n            }\\n            if (enforceNonnegative) {\\n                y0 = y0.map(function (x) {\\n                    if (x < 0.0) return 0.0;\\n                    else return x;\\n                });\\n            }\\n\\n            // DEBUG\\n\\t\\t    nSteps += 1;\\n\\t\\t    // END EDEBUG\\n        }\\n        if (t > tSol[tSol.length - 1]) {\\n            y.push(y0);\\n            tSol.push(t);\\n        }\\n        i += 1;\\n    }\\n\\n    // DEBUG\\n    console.log(nSteps);\\n    // END DEBUG\\n\\n    if (nDeadSteps == maxDeadSteps) {\\n        return nanArray(initialCondition, iMax);\\n    }\\n    let yInterp = interpolateSolution(timePoints, tSol, transpose(y));\\n\\n    return yInterp;\\n}\\n\\n\\nfunction interpolate1d(x, xs, ys) {\\n    let y2s = naturalSplineSecondDerivs(xs, ys);\\n\\n    let yInterp = x.map(function (xVal) {\\n        return splineEvaluate(xVal, xs, ys, y2s);\\n    });\\n\\n    return yInterp;\\n}\\n\\n\\nfunction interpolateSolution(timePoints, t, y) {\\n    // Interpolate each row of y\\n    let yInterp = y.map(function (yi) {\\n        return interpolate1d(timePoints, t, yi);\\n    });\\n\\n    return yInterp;\\n}\\n\\n\\nfunction naturalSplineSecondDerivs(xs, ys) {\\n    /*\\n     * Compute the second derivatives for a cubic spline data\\n     * measured at positions xs, ys.\\n     *\\n     * The second derivatives are then used to evaluate the spline.\\n     */\\n\\n    let n = xs.length;\\n\\n    // Storage used in tridiagonal solve\\n    let u = zeros(n);\\n\\n    // Return value\\n    let y2s = zeros(n);\\n\\n    // Solve trigiadonal matrix by decomposition\\n    for (let i = 1; i < n - 1; i++) {\\n        let fracInterval = (xs[i] - xs[i - 1]) / (xs[i + 1] - xs[i - 1]);\\n        let p = fracInterval * y2s[i - 1] + 2.0;\\n        y2s[i] = (fracInterval - 1.0) / p;\\n        u[i] =\\n            (ys[i + 1] - ys[i]) / (xs[i + 1] - xs[i]) -\\n            (ys[i] - ys[i - 1]) / (xs[i] - xs[i - 1]);\\n        u[i] =\\n            ((6.0 * u[i]) / (xs[i + 1] - xs[i - 1]) - fracInterval * u[i - 1]) /\\n            p;\\n    }\\n\\n    // Tridiagonal solve back substitution\\n    for (let k = n - 2; k >= 0; k--) {\\n        y2s[k] = y2s[k] * y2s[k + 1] + u[k];\\n    }\\n\\n    return y2s;\\n}\\n\\n\\nfunction splineEvaluate(x, xs, ys, y2s) {\\n    /*\\n     * Evaluate a spline computed from points xs, ys, with second derivatives\\n     * y2s, as compute by naturalSplineSecondDerivs().\\n     *\\n     * Assumes that x and xs are sorted.\\n     */\\n    let n = xs.length;\\n\\n    // Indices bracketing where x is\\n    let lowInd = 0;\\n    let highInd = n - 1;\\n\\n    // Perform bisection search to find index of x\\n    while (highInd - lowInd > 1) {\\n        let i = (highInd + lowInd) >> 1;\\n        if (xs[i] > x) {\\n            highInd = i;\\n        } else {\\n            lowInd = i;\\n        }\\n    }\\n    let h = xs[highInd] - xs[lowInd];\\n    let a = (xs[highInd] - x) / h;\\n    let b = (x - xs[lowInd]) / h;\\n\\n    let y = a * ys[lowInd] + b * ys[highInd];\\n    y +=\\n        (((Math.pow(a, 3) - a) * y2s[lowInd] + (Math.pow(b, 3) - b) * y2s[highInd]) * Math.pow(h, 2)) /\\n        6.0;\\n\\n    return y;\\n}\\n\\n\\n// module.exports = {\\n//   vsimex,\\n//   rkf45,\\n//   zeros, \\n//   linspace\\n// };\\n\\n// vsimex(lotkaVolterra, [1.0, 3.0], linspace(0.0, 20.0, 200), [1.0, 2.0, 3.0, 4.0], 0.01, 1e-7, [0.1, 10.0], 0.0)\\n// let lv = lotkaVolterraIMEX(1.0, 2.0, 3.0, 4.0);\\n// let sol = vsimex(lv.f, lv.cfun, lv.Afun, [1.0, 3.0], linspace(0.0, 20.0, 200), [], [], [], lv.diagonalA)\\n\\n\\nfunction ij(i, j, n) {\\n    /*\\n     * Lexicographic indexing of 2D array represented as 1D.\\n     */\\n\\n    return i * n + j;\\n}\\n\\n\\nfunction twoDto1D(A) {\\n    /*\\n     * Convert a 2D matrix to a 1D representation with row-based (C)\\n     * lexicographic ordering.\\n     */\\n\\n    var m = A.length;\\n    var n = A[0].length;\\n\\n    var A1d = [];\\n    for (var i = 0; i < m; i++) {\\n        for (var j = 0; j < n; j++) {\\n            A1d.push(A[i][j]);\\n        }\\n    }\\n\\n    return A1d;\\n}\\n\\n\\nfunction linspace(start, stop, n) {\\n    var x = [];\\n    var currValue = start;\\n    var step = (stop - start) / (n - 1);\\n    for (var i = 0; i < n; i++) {\\n        x.push(currValue);\\n        currValue += step;\\n    }\\n    return x;\\n}\\n\\n\\nfunction zeros(n) {\\n    var x = [];\\n    for (var i = 0; i < n; i++) x.push(0.0);\\n    return x;\\n}\\n\\n\\nfunction shallowCopyMatrix(A) {\\n    /*\\n     * Make a shallow copy of a matrix.\\n     */\\n\\n    var Ac = [];\\n    var n = A.length;\\n    for (i = 0; i < n; i++) {\\n        Ac.push([...A[i]]);\\n    }\\n\\n    return Ac;\\n}\\n\\n\\nfunction nanArray() {\\n    /*\\n     * Return a NaN array of shape given by arguments.\\n     */\\n    if (arguments.length == 1) {\\n        var x = [];\\n        for (var i = 0; i < arguments[0]; i++) x.push(NaN);\\n    }\\n    else if (arguments.length == 2) {\\n        var x = [];\\n        for (var i = 0; i < arguments[0]; i++) {\\n            var xRow = [];\\n            for (var j = 0; j < arguments[1]; j++) xRow.push(NaN);\\n            x.push(xRow);\\n        }\\n    }\\n    else {\\n        throw 'Must only have one or two arguments to nanArray().'\\n    }\\n\\n    return x;\\n}\\n\\nfunction negAutoRHS(x, t, beta0, gamma, k, n, ks, ns, sFun, sArgs=[]) {\\n\\tlet xScalar = x[0]\\n\\n\\tlet s = sFun(t, ...sArgs);\\n\\n\\t// Correct for x being numerically negative\\n\\tlet xCorr = (xScalar > 0.0) ? xScalar : 0.0;\\n\\n\\tlet beta = beta0 * Math.pow(s / ks, ns) / (1 + Math.pow(s / ks, ns));\\n\\n\\treturn [beta / (1 + Math.pow(xCorr / k, n)) - gamma * xCorr];\\n} \\n\\n\\nfunction unregRHS(x, t, beta0, gamma, ks, ns, sFun, sArgs=[]) {\\n\\tlet xScalar = x[0]\\n\\tlet s = sFun(t, ...sArgs);\\n\\n\\tlet beta = beta0 * Math.pow(s / ks, ns) / (1 + Math.pow(s / ks, ns));\\n\\n\\treturn [beta - gamma * xScalar];\\n}\\n\\n\\nfunction sPulse(t, t0, tau) {\\n\\treturn Math.exp(-4.0 * Math.pow((t - t0) / tau, 2));\\n}\\n\\n\\nfunction callback() {\\n\\tlet xRangeMax = xRange.end;\\n\\tlet dt = 0.01;\\n\\tlet x0 = [0.0];\\n\\tlet t0 = t0Slider.value;\\n\\tlet tau = tauSlider.value;\\n\\tlet s = cds.data['s'];\\n\\n\\tlet t = linspace(0.0, xRangeMax, cds.data['t'].length);\\n\\tlet sArgs = [t0, tau];\\n\\tlet args = [\\n\\t\\tMath.pow(10, logBeta0Slider.value),\\n\\t\\tMath.pow(10, logGammaSlider.value),\\n\\t\\tMath.pow(10, logkSlider.value),\\n\\t\\tnSlider.value,\\n\\t\\tMath.pow(10, logksSlider.value),\\n\\t\\tnsSlider.value,\\n\\t\\tsPulse,\\n\\t\\tsArgs\\n\\t];\\n\\tlet argsUnreg = [args[0], args[1], args[4], args[5], args[6], args[7]];\\n\\n\\t// Integrate ODES\\n\\tlet xSolve = rkf45(negAutoRHS, x0, t, args, dt)[0];\\n\\tlet xUnregSolve = rkf45(unregRHS, x0, t, argsUnreg)[0];\\n\\n\\t// Pulse for plotting\\n\\tfor (let i = 0; i < t.length; i++) {\\n\\t\\ts[i] = sPulse(t[i], t0, tau);\\n\\t}\\n\\n\\t// Normalize if necessary\\n\\tif (normalizeToggle.active) {\\n\\t\\tlet xMax = Math.max(...xSolve);\\n\\t\\tif (xMax > 0.0) xSolve = svMult(1.0 / xMax, xSolve);\\n\\n\\t\\tlet xUnregMax = Math.max(...xUnregSolve);\\n\\t\\tif (xUnregMax > 0.0) xUnregSolve = svMult(1.0 / xUnregMax, xUnregSolve);\\n\\n\\t\\tlet sMax = Math.max(...s);\\n\\t\\tif (sMax > 0.0) s = svMult(1.0 / sMax, s);\\n\\n\\t\\tyaxis.axis_label = 'normalized concentration';\\n\\t}\\n\\telse yaxis.axis_label = 'concentration';\\n\\n\\t// Toggle legend visibility\\n\\tlegend.visible = (legendToggle.active) ? true : false;\\n\\n\\tcds.data['t'] = t;\\n\\tcds.data['x'] = xSolve;\\n\\tcds.data['x_unreg'] = xUnregSolve;\\n\\n\\tcds.change.emit();\\n}\\n\\ncallback()\"}}]]]},\"end\":10.0}},\"y_range\":{\"type\":\"object\",\"name\":\"DataRange1d\",\"id\":\"p4552\"},\"x_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p4563\"},\"y_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p4565\"},\"title\":{\"type\":\"object\",\"name\":\"Title\",\"id\":\"p4554\"},\"renderers\":[{\"id\":\"p4602\"},{\"id\":\"p4613\"},{\"id\":\"p4623\"}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p4556\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p4581\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p4582\"},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p4583\",\"attributes\":{\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p4584\",\"attributes\":{\"syncable\":false,\"level\":\"overlay\",\"visible\":false,\"left_units\":\"canvas\",\"right_units\":\"canvas\",\"bottom_units\":\"canvas\",\"top_units\":\"canvas\",\"line_color\":\"black\",\"line_alpha\":1.0,\"line_width\":2,\"line_dash\":[4,4],\"fill_color\":\"lightgrey\",\"fill_alpha\":0.5}}}},{\"type\":\"object\",\"name\":\"SaveTool\",\"id\":\"p4585\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p4586\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p4587\"}]}},\"left\":[{\"id\":\"p4574\"}],\"below\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p4567\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p4569\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p4570\"},\"axis_label\":\"time\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p4568\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p4573\",\"attributes\":{\"axis\":{\"id\":\"p4567\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p4580\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p4574\"}}},{\"id\":\"p4605\"}],\"frame_width\":375,\"frame_height\":250}},{\"type\":\"object\",\"name\":\"Spacer\",\"id\":\"p4638\",\"attributes\":{\"width\":30}},{\"type\":\"object\",\"name\":\"Column\",\"id\":\"p4639\",\"attributes\":{\"children\":[{\"id\":\"p4627\"},{\"id\":\"p4628\"},{\"id\":\"p4629\"},{\"id\":\"p4630\"},{\"id\":\"p4636\"}]}},{\"type\":\"object\",\"name\":\"Column\",\"id\":\"p4640\",\"attributes\":{\"children\":[{\"id\":\"p4631\"},{\"id\":\"p4632\"},{\"id\":\"p4633\"},{\"id\":\"p4634\"},{\"id\":\"p4635\"}]}}]}}],\"callbacks\":{\"type\":\"map\"}}};\n",
       "  const render_items = [{\"docid\":\"22b76506-e9ce-410d-b95f-e8c38e741590\",\"roots\":{\"p4641\":\"2d7642f3-780d-4e2c-aaab-216c4e226b3d\"},\"root_ids\":[\"p4641\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p4641"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "bokeh.io.show(biocircuits.jsplots.autorepressor_response_to_pulse())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By moving around the sliders, we can see that increasing the strength of the repression (by decreasing $k$) accentuates the speed-up provided by negative autoregulation. Furthermore, we see that increasing the cooperativitity of the repressor (by increasing $n$) makes the initial rise in $x$ \"sharper\". What other properties can you find through interacting with this plot?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "## References\n",
    "\n",
    "- Rosenfeld, et al., Negative autoregulation speeds the response times of transcription networks, _J. Molec. Biol._, 323, 785–793, 2002. ([link](https://doi.org/10.1016/S0022-2836(02)00994-4))\n",
    "- Shen-Orr S. S., et al., Network motifs in the transcriptional regulation network of _Escherichia coli_, _Nature Genetics_, 31, 64–68, 2002. ([link](https://doi.org/10.1038/ng881))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "## Technical appendices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx-toctree": {
     "maxdepth": 1
    },
    "tags": []
   },
   "source": [
    "- [2a. Approximate solution of autorepressor dynamics](../technical_appendices/02a_approximate_autorepression_dynamics.ipynb)\n",
    "- [2b. Numerical solutions to ODEs](../technical_appendices/02b_numerical_odes.ipynb)\n",
    "- [2c. Interactive plotting with Bokeh](../technical_appendices/02c_interactive_plotting.ipynb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "<hr>\n",
    "\n",
    "## Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx-toctree": {
     "maxdepth": 1
    },
    "tags": []
   },
   "source": [
    "- [2.1: The cost of a steady state](../problems/02/problem_2.1.ipynb)\n",
    "- [2.2: Event handling for discontinuous derivatives](../problems/02/problem_2.2.ipynb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.10.10\n",
      "IPython version      : 8.10.0\n",
      "\n",
      "numpy      : 1.23.5\n",
      "pandas     : 1.5.3\n",
      "scipy      : 1.10.0\n",
      "bokeh      : 3.1.0\n",
      "colorcet   : 3.0.1\n",
      "biocircuits: 0.1.9\n",
      "jupyterlab : 3.5.3\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -p numpy,pandas,scipy,bokeh,colorcet,biocircuits,jupyterlab"
   ]
  }
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