style: update M0/M/D nomenclature

monomer_dimer.ipynb

@@ -17,52 +17,52 @@
    "source": [
     "## Monomer - Dimer kinetics\n",
     "\n",
-    "Lets start with the simple example of dimer formation from two identical monomer subunits:\n",
+    "Lets start with the simple example of dimer formation from two identical protomers:\n",
     "(TODO: mhchem package)\n",
     "\n",
     "$$\n",
-    "M + M <-> D\n",
+    "P_1 + P_1 <-> P_2\n",
     "$$\n",
     "\n",
     "From the kinetic scheme above, we can write down the differential equations describing this system. These differential equations tell us how the concentrations of the reaction (in this case dimerization) change in time. \n",
     "\n",
     "$$\n",
-    "\\frac{\\partial [M]}{\\partial t} = - 2 k_{on} [M][M] + 2 k_{off} [D]\n",
+    "\\frac{\\partial [P_1]}{\\partial t} = - 2 k_{on} [P_1][P_1] + 2 k_{off} [P_2]\n",
     "$$\n",
     "\n",
     "$$\n",
-    "\\frac{\\partial [D]}{\\partial t} =  k_{on} [M][M] - k_{off} [D]\n",
+    "\\frac{\\partial [P_2]}{\\partial t} =  k_{on} [P_1][P_1] - k_{off} [P_2]\n",
     "$$\n",
     "\n",
-    "Where square brackets are used to indicate the concentration of the species, ie $[M]$ is the concentration of the monomer at a given time point $t$.\n",
+    "Where square brackets are used to indicate the concentration of the species, ie $[P_1]$ is the concentration of the monomer at a given time point $t$.\n",
     "\n",
     "To sanity check our results, we can multiply the second equation with -2, such that both right-hand sides become equal. We then find that:\n",
-    " \n",
+    "\n",
     "$$\n",
-    "\\frac{\\partial [M]}{\\partial t} = - 2 \\frac{\\partial [D]}{\\partial t}\n",
+    "\\frac{\\partial [P_1]}{\\partial t} = - 2 \\frac{\\partial [P_2]}{\\partial t}\n",
     "$$\n",
     "\n",
-    "When reading the equation above, for the equality to hold true, we find that the dimer $D$ must appear at _half_ the rate at which the monomer $M$ disappears. Cross-checking this with our chemical intuition this makes sense, since two monomers are required to make one dimer. \n",
+    "When reading the equation above, for the equality to hold true, we find that the dimer $P_2$ must appear at _half_ the rate at which the monomer $P_1$ disappears. Cross-checking this with our chemical intuition this makes sense, since two monomers are required to make one dimer. \n",
     "\n",
-    "In order to now find which concentrations of monomer and dimer are formed given some starting concentration of the monomer $[M_0]$, we need three more pieces of information. \n",
+    "In order to now find which concentrations of monomer and dimer are formed given how much protomer is in the tube, we need three more pieces of information. \n",
     "\n",
     "First, we are looking for the concentration of the monomer and dimer at _steady-state_: the system has reached equillibrium and no more changes in the concentration of either the monomer or the dimer occur. This means we can set both differential equations to zero. \n",
     "\n",
-    "Second, we realize that the total amount of monomer in the tube never goes up or down; its either monomer or as a protomer in a dimer. The starting amount of monomer is thus:\n",
+    "Second, we realize that the total amount of protomer in the tube never goes up or down; its either monomer or as one of the protomers in a dimer. The total amount of protomer is thus:\n",
     "\n",
     "$$\n",
-    "[M_0] = [M] + 2[D]\n",
+    "[P_T] = [P_1] + 2[P_2]\n",
     "$$\n",
     "\n",
-    "We can think of [M_0] also as 'total amount of protomer', either in monomer or dimer form. This makes it easy to measure the [M_0] of a tube with a monomer/dimer mixture; we can simply calculate the extinction coefficient of the protomer and since extinction coefficients are generally additive we can use this to measure [M_0] directly. \n",
+    "Since extinction coefficients are additive ($\\epsilon_{P_2} = 2\\epsilon_{P_1}$) the total protomer concentration can be measured directly by UV-VIS spectroscopy. \n",
     "\n",
-    "Third, we need to have some value for the forward $k_f$ and backward rates $k_r$. These rates describe how fast the reaction takes place, but at the moment we are only interested in the steady-state concentrations. We can define a single new quantity which depends on the ratio of the forward and backward rates:\n",
+    "Third, we need to have some value for the forward $k_{on}$ and backward rates $k_{off}$. These rates describe how fast the reaction takes place, but at the moment we are only interested in the steady-state concentrations. We can define a single new quantity which depends on the ratio of the forward and backward rates:\n",
     "\n",
     "$$\n",
     "K_d = \\frac{k_{off}}{k_{on}}\n",
     "$$\n",
     "\n",
-    "If we look at our differential equations, the left-hand side describes the change of concentration of the monomer over time. The units are therefore molar per second ($\\underline{M} s^{-1}$; we use $\\underline{M}$ to discriminate from monomer $M$). To make the units match on the left and right-hand side, we can deduce that the units of $k_{on}$ must be $\\underline{M}^{-1} s^{-1}$. Similarly, the units of $k_{off}$ are $s^{-1}$. From this unit analysis, we can see that the lifetime or off rate from a complex is independent of concentratation and we can recognize $K_d$ as the dissociation constant with units $\\underline{M}$. "
+    "If we look at our differential equations, the left-hand side describes the change of concentration of the monomer over time. The units are therefore molar per second ($M s^{-1}$). To make the units match on the left and right-hand side, we can deduce that the units of $k_{on}$ must be $M^{-1} s^{-1}$. Similarly, the units of $k_{off}$ are $s^{-1}$. From this unit analysis, we can see that the lifetime or off rate from a complex is independent of concentratation and we can recognize $K_d$ as the dissociation constant with units $M$."
    ]
   },
   {
@@ -74,13 +74,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
     "import sympy as sp\n",
     "\n",
-    "M, D, M0, k_on, k_off, kD = sp.symbols(\"M D M0 k_on k_off k_D\", positive=True)"
+    "P1, P2, PT, k_on, k_off, kD = sp.symbols(\"P_1 P_2 P_T k_on k_off k_D\", positive=True)"
    ]
   },
   {
@@ -92,17 +92,30 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 3,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[{P_1: sqrt(k_D)*sqrt(8*P_T + k_D)/4 - k_D/4,\n",
+       "  P_2: P_T/2 - sqrt(k_D)*sqrt(8*P_T + k_D)/8 + k_D/8,\n",
+       "  k_off: k_D*k_on}]"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "sol = sp.solve(\n",
     "    [\n",
-    "        -2 * k_on * M * M + 2 * k_off * D,\n",
-    "        M + 2 * D - M0,\n",
+    "        -2 * k_on * P1 * P1 + 2 * k_off * P2,\n",
+    "        P1 + 2 * P2 - PT,\n",
     "        (k_off / k_on) - kD,\n",
     "    ],\n",
-    "    [M, D, k_on, k_off],\n",
+    "    [P1, P2, k_on, k_off],\n",
     "    dict=True,\n",
     ")\n",
     "sol"
@@ -117,37 +130,51 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 4,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\frac{P_{T}}{2} - \\frac{\\sqrt{k_{D}} \\sqrt{8 P_{T} + k_{D}}}{8} + \\frac{k_{D}}{8}$"
+      ],
+      "text/plain": [
+       "P_T/2 - sqrt(k_D)*sqrt(8*P_T + k_D)/8 + k_D/8"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "sol[0][D]"
+    "sol[0][P2]"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We can see that indeed the solution only depends on the dissociation constant $K_D$, and not the individual rates and indeed the 'mass balance' equation still holds ($M + 2D = M_0$), excercise for the reader."
+    "We can see that indeed the solution only depends on the dissociation constant $K_D$, and not the individual rates and indeed the 'mass balance' equation still holds ($P_1 + 2P_2 = P_T$), excercise for the reader."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Next, lets use [solara](https://solara.dev/) to quickly make an interactive component so that we can easily calculate the monomer and dimer concentrations, given a input concentration $M_0$ and the dissociation constant. \n",
+    "Next, lets use [solara](https://solara.dev/) to quickly make an interactive component so that we can easily calculate the monomer and dimer concentrations, given a total protomer concentration $P_T$ and the dissociation constant. \n",
     "\n",
     "First, we take the symbolic solutions from `sympy` and lambdify them so that we can input numbers and calculate the output:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [],
    "source": [
-    "solve_for = [M, D]\n",
-    "inputs = [M0, kD]\n",
+    "solve_for = [P1, P2]\n",
+    "inputs = [PT, kD]\n",
     "\n",
     "lambdas = {s: sp.lambdify(inputs, sol[0][s]) for s in solve_for}"
    ]
@@ -156,25 +183,43 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Next, we make a `solara` component, where a user can input the values for $M_0$ and $K_{D}$ and directly obtain $M$ and $D$ at steady-state:"
+    "Next, we make a `solara` component, where a user can input the values for $P_T$ and $K_{D}$ and directly obtain $P_1$ and $P_2$ at steady-state:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 6,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "7b7432b1958f4bdb91256f7c93718220",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/html": [
+       "Cannot show widget. You probably want to rerun the code cell above (<i>Click in the code cell, and press Shift+Enter <kbd>⇧</kbd>+<kbd>↩</kbd></i>)."
+      ],
+      "text/plain": [
+       "Cannot show ipywidgets in text"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "import solara\n",
     "\n",
     "@solara.component\n",
     "def Page():\n",
-    "    M0 = solara.use_reactive(10.)\n",
+    "    PT = solara.use_reactive(10.)\n",
     "    kD = solara.use_reactive(1.)\n",
     "\n",
-    "    ans = {k: ld(M0.value, kD.value) for k, ld in lambdas.items()}\n",
+    "    ans = {k: ld(PT.value, kD.value) for k, ld in lambdas.items()}\n",
     "\n",
-    "    solara.InputFloat('M0', value=M0)\n",
+    "    solara.InputFloat('PT', value=PT)\n",
     "    solara.InputFloat('kD', value=kD)\n",
     "\n",
     "    for k, v in ans.items():\n",
@@ -187,7 +232,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Note that the values are all unitless, so if we enter a dissociation constant of 1 $\\mu \\underline{M}$, the concentration $M_0$ should also be given in $\\mu \\underline{M}$, and the outputs are then $\\mu \\underline{M}$ as well. \n",
+    "Note that the values are all unitless, so if we enter a dissociation constant of 1 $\\mu M$, the concentration $P_T$ should also be given in $\\mu M$, and the outputs are then $\\mu M$ as well. \n",
     "\n",
     "This little widget now tells us that if the monomer concentration if 10 times over the value of $k_D$, in the steady-state equillibrium one out of three species is still a monomer!"
    ]
@@ -201,23 +246,98 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 9,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div>\n",
+       "<style scoped>\n",
+       "    .dataframe tbody tr th:only-of-type {\n",
+       "        vertical-align: middle;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe tbody tr th {\n",
+       "        vertical-align: top;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe thead th {\n",
+       "        text-align: right;\n",
+       "    }\n",
+       "</style>\n",
+       "<table border=\"1\" class=\"dataframe\">\n",
+       "  <thead>\n",
+       "    <tr style=\"text-align: right;\">\n",
+       "      <th></th>\n",
+       "      <th>PT</th>\n",
+       "      <th>P_1</th>\n",
+       "      <th>P_2</th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>0</th>\n",
+       "      <td>0.100000</td>\n",
+       "      <td>0.921311</td>\n",
+       "      <td>0.078689</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>1</th>\n",
+       "      <td>0.109750</td>\n",
+       "      <td>0.915248</td>\n",
+       "      <td>0.084752</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>2</th>\n",
+       "      <td>0.120450</td>\n",
+       "      <td>0.908825</td>\n",
+       "      <td>0.091175</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>3</th>\n",
+       "      <td>0.132194</td>\n",
+       "      <td>0.902035</td>\n",
+       "      <td>0.097965</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>4</th>\n",
+       "      <td>0.145083</td>\n",
+       "      <td>0.894871</td>\n",
+       "      <td>0.105129</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "         PT       P_1       P_2\n",
+       "0  0.100000  0.921311  0.078689\n",
+       "1  0.109750  0.915248  0.084752\n",
+       "2  0.120450  0.908825  0.091175\n",
+       "3  0.132194  0.902035  0.097965\n",
+       "4  0.145083  0.894871  0.105129"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "import numpy as np\n",
     "import pandas as pd\n",
     "\n",
     "\n",
     "# create a new function which calculates total molarity (monomer + dimer)\n",
-    "ld_total = sp.lambdify(inputs, sol[0][M] + sol[0][D])\n",
+    "ld_total = sp.lambdify(inputs, sol[0][P1] + sol[0][P2])\n",
     "\n",
-    "# create a vector of M0 values ranging from 0.1 times kD to 1000 times kD\n",
-    "M0_values = np.logspace(-1, 3, endpoint=True, num=100)\n",
-    "ans = {k: ld(M0_values, 1) / ld_total(M0_values, 1) for k, ld in lambdas.items()}\n",
+    "# create a vector of PT values ranging from 0.1 times kD to 1000 times kD\n",
+    "PT_values = np.logspace(-1, 3, endpoint=True, num=100)\n",
+    "ans = {k: ld(PT_values, 1) / ld_total(PT_values, 1) for k, ld in lambdas.items()}\n",
     "\n",
-    "# put the results in a dataframe, together with input M0 values\n",
-    "df = pd.DataFrame(dict(M0=M0_values) | {k.name: v for k, v in ans.items()})\n",
+    "# put the results in a dataframe, together with input PT values\n",
+    "df = pd.DataFrame(dict(PT=PT_values) | {k.name: v for k, v in ans.items()})\n",
     "\n",
     "df.head()"
    ]
@@ -231,21 +351,100 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 11,
    "metadata": {},
-   "outputs": [],
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0.12045035402587821, \"species\": \"P_1\", \"y\": 0.9088252563655043}, {\"PT\": 0.1321941148466029, \"species\": \"P_1\", \"y\": 0.9020350932162419}, {\"PT\": 0.14508287784959398, \"species\": \"P_1\", \"y\": 0.8948707231715299}, {\"PT\": 0.1592282793341092, \"species\": \"P_1\", \"y\": 0.887326964051884}, {\"PT\": 0.17475284000076838, \"species\": \"P_1\", \"y\": 0.8794001529111963}, {\"PT\": 0.19179102616724888, \"species\": \"P_1\", \"y\": 0.8710882296412226}, {\"PT\": 0.2104904144512021, \"species\": \"P_1\", \"y\": 0.8623908028192255}, {\"PT\": 0.23101297000831597, \"species\": \"P_1\", \"y\": 0.8533091966764864}, {\"PT\": 0.2535364493970112, \"species\": \"P_1\", \"y\": 0.8438464785616561}, {\"PT\": 0.2782559402207124, \"species\": \"P_1\", \"y\": 0.8340074667631514}, {\"PT\": 0.30538555088334157, \"species\": \"P_1\", \"y\": 0.8237987190163906}, {\"PT\": 0.33516026509388425, \"species\": \"P_1\", \"y\": 0.813228502436007}, {\"PT\": 0.36783797718286343, \"species\": \"P_1\", 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{\"PT\": 432.87612810830615, \"species\": \"P_1\", \"y\": 0.06466649684044665}, {\"PT\": 475.0810162102798, \"species\": \"P_1\", \"y\": 0.061864433157825666}, {\"PT\": 521.400828799969, \"species\": \"P_1\", \"y\": 0.05917805770877593}, {\"PT\": 572.236765935022, \"species\": \"P_1\", \"y\": 0.0566030849591937}, {\"PT\": 628.029144183426, \"species\": \"P_1\", \"y\": 0.05413534565517292}, {\"PT\": 689.2612104349702, \"species\": \"P_1\", \"y\": 0.05177078755281282}, {\"PT\": 756.463327554629, \"species\": \"P_1\", \"y\": 0.04950547569414219}, {\"PT\": 830.2175681319752, \"species\": \"P_1\", \"y\": 0.04733559227414581}, {\"PT\": 911.1627561154896, \"species\": \"P_1\", \"y\": 0.045257436141088785}, {\"PT\": 1000.0, \"species\": \"P_1\", \"y\": 0.04326742196959632}, {\"PT\": 0.1, \"species\": \"P_2\", \"y\": 0.07868932583326321}, {\"PT\": 0.10974987654930562, \"species\": \"P_2\", \"y\": 0.08475218572961124}, {\"PT\": 0.12045035402587821, \"species\": \"P_2\", \"y\": 0.09117474363449568}, {\"PT\": 0.1321941148466029, \"species\": \"P_2\", \"y\": 0.09796490678375804}, {\"PT\": 0.14508287784959398, \"species\": \"P_2\", \"y\": 0.10512927682846993}, {\"PT\": 0.1592282793341092, \"species\": \"P_2\", \"y\": 0.11267303594811592}, {\"PT\": 0.17475284000076838, \"species\": \"P_2\", \"y\": 0.12059984708880354}, {\"PT\": 0.19179102616724888, \"species\": \"P_2\", \"y\": 0.12891177035877757}, {\"PT\": 0.2104904144512021, \"species\": \"P_2\", \"y\": 0.1376091971807744}, {\"PT\": 0.23101297000831597, \"species\": \"P_2\", \"y\": 0.14669080332351347}, {\"PT\": 0.2535364493970112, \"species\": \"P_2\", \"y\": 0.1561535214383439}, {\"PT\": 0.2782559402207124, \"species\": \"P_2\", \"y\": 0.1659925332368486}, {\"PT\": 0.30538555088334157, \"species\": \"P_2\", \"y\": 0.17620128098360954}, {\"PT\": 0.33516026509388425, \"species\": \"P_2\", \"y\": 0.18677149756399297}, {\"PT\": 0.36783797718286343, \"species\": \"P_2\", \"y\": 0.19769325403333335}, {\"PT\": 0.40370172585965547, \"species\": \"P_2\", \"y\": 0.20895502326942395}, {\"PT\": 0.4430621457583881, \"species\": \"P_2\", \"y\": 0.22054375813767554}, {\"PT\": 0.4862601580065355, \"species\": \"P_2\", \"y\": 0.2324449824358711}, {\"PT\": 0.533669923120631, \"species\": \"P_2\", \"y\": 0.24464289280762855}, {\"PT\": 0.5857020818056667, \"species\": \"P_2\", \"y\": 0.257120469792299}, {\"PT\": 0.6428073117284322, \"species\": \"P_2\", \"y\": 0.26985959620441546}, {\"PT\": 0.7054802310718644, \"species\": \"P_2\", \"y\": 0.2828411810977828}, {\"PT\": 0.7742636826811272, \"species\": \"P_2\", \"y\": 0.2960452876580207}, {\"PT\": 0.8497534359086445, \"species\": \"P_2\", \"y\": 0.3094512634739916}, {\"PT\": 0.9326033468832199, \"species\": \"P_2\", \"y\": 0.3230378717555378}, {\"PT\": 1.0235310218990263, \"species\": \"P_2\", \"y\": 0.33678342218635515}, {\"PT\": 1.1233240329780276, \"species\": \"P_2\", \"y\": 0.3506659002222288}, {\"PT\": 1.2328467394420666, \"species\": \"P_2\", \"y\": 0.36466309376325495}, {\"PT\": 1.3530477745798075, \"species\": \"P_2\", \"y\": 0.37875271624230866}, {\"PT\": 1.484968262254465, \"species\": \"P_2\", \"y\": 0.39291252528008624}, {\"PT\": 1.6297508346206444, \"species\": \"P_2\", \"y\": 0.40712043615947846}, {\"PT\": 1.7886495290574351, \"species\": \"P_2\", \"y\": 0.42135462946919144}, {\"PT\": 1.9630406500402715, \"species\": \"P_2\", \"y\": 0.4355936523590508}, {\"PT\": 2.1544346900318843, \"species\": \"P_2\", \"y\": 0.4498165129380179}, {\"PT\": 2.3644894126454084, \"species\": \"P_2\", \"y\": 0.4640027674312564}, {\"PT\": 2.5950242113997373, \"species\": \"P_2\", \"y\": 0.4781325997951208}, {\"PT\": 2.848035868435802, \"species\": \"P_2\", \"y\": 0.49218689356900924}, {\"PT\": 3.1257158496882367, \"species\": \"P_2\", \"y\": 0.5061472958207289}, {\"PT\": 3.4304692863149193, \"species\": \"P_2\", \"y\": 0.5199962731172615}, {\"PT\": 3.764935806792469, \"species\": \"P_2\", \"y\": 0.533717159525297}, {\"PT\": 4.132012400115339, \"species\": \"P_2\", \"y\": 0.5472941967152098}, {\"PT\": 4.5348785081285845, \"species\": \"P_2\", \"y\": 0.5607125663077462}, {\"PT\": 4.977023564332112, \"species\": \"P_2\", \"y\": 0.5739584146640166}, {\"PT\": 5.462277217684343, \"species\": \"P_2\", \"y\": 0.5870188703758467}, {\"PT\": 5.994842503189412, \"species\": \"P_2\", \"y\": 0.5998820547646017}, {\"PT\": 6.5793322465756825, \"species\": \"P_2\", \"y\": 0.6125370857417747}, {\"PT\": 7.220809018385467, \"species\": \"P_2\", \"y\": 0.6249740754235427}, {\"PT\": 7.924828983539178, \"species\": \"P_2\", \"y\": 0.6371841219238735}, {\"PT\": 8.697490026177835, \"species\": \"P_2\", \"y\": 0.6491592957764619}, {\"PT\": 9.545484566618342, \"species\": \"P_2\", \"y\": 0.6608926214547796}, {\"PT\": 10.476157527896651, \"species\": \"P_2\", \"y\": 0.6723780544719333}, {\"PT\": 11.497569953977369, \"species\": \"P_2\", \"y\": 0.6836104545480887}, {\"PT\": 12.61856883066021, \"species\": \"P_2\", \"y\": 0.6945855553332363}, {\"PT\": 13.848863713938732, \"species\": \"P_2\", \"y\": 0.7052999311675234}, {\"PT\": 15.199110829529348, \"species\": \"P_2\", \"y\": 0.7157509613506929}, {\"PT\": 16.68100537200059, \"species\": \"P_2\", \"y\": 0.7259367923769716}, {\"PT\": 18.307382802953697, \"species\": \"P_2\", \"y\": 0.7358562985725914}, {\"PT\": 20.09233002565048, \"species\": \"P_2\", \"y\": 0.7455090415506367}, {\"PT\": 22.051307399030456, \"species\": \"P_2\", \"y\": 0.7548952288727201}, {\"PT\": 24.201282647943835, \"species\": \"P_2\", \"y\": 0.7640156722796789}, {\"PT\": 26.560877829466868, \"species\": \"P_2\", \"y\": 0.7728717458246792}, {\"PT\": 29.150530628251786, \"species\": \"P_2\", \"y\": 0.7814653442123345}, {\"PT\": 31.992671377973846, \"species\": \"P_2\", \"y\": 0.7897988416172199}, {\"PT\": 35.111917342151344, \"species\": \"P_2\", \"y\": 0.7978750512249496}, {\"PT\": 38.535285937105314, \"species\": \"P_2\", \"y\": 0.8056971857091976}, {\"PT\": 42.29242874389499, \"species\": \"P_2\", \"y\": 0.8132688188290305}, {\"PT\": 46.41588833612782, \"species\": \"P_2\", \"y\": 0.8205938483029968}, {\"PT\": 50.9413801481638, \"species\": \"P_2\", \"y\": 0.8276764600898192}, {\"PT\": 55.90810182512229, \"species\": \"P_2\", \"y\": 0.8345210941804637}, {\"PT\": 61.35907273413176, \"species\": \"P_2\", \"y\": 0.8411324119829602}, {\"PT\": 67.34150657750828, \"species\": \"P_2\", \"y\": 0.8475152653597271}, {\"PT\": 73.90722033525783, \"species\": \"P_2\", \"y\": 0.8536746673573522}, {\"PT\": 81.11308307896873, \"species\": \"P_2\", \"y\": 0.8596157646508609}, {\"PT\": 89.02150854450393, \"species\": \"P_2\", \"y\": 0.8653438117084111}, {\"PT\": 97.70099572992257, \"species\": \"P_2\", \"y\": 0.8708641466681002}, {\"PT\": 107.22672220103243, \"species\": \"P_2\", \"y\": 0.8761821689060604}, {\"PT\": 117.68119524349991, \"species\": \"P_2\", \"y\": 0.8813033182642038}, {\"PT\": 129.1549665014884, \"species\": \"P_2\", \"y\": 0.8862330558967654}, {\"PT\": 141.7474162926806, \"species\": \"P_2\", \"y\": 0.8909768466870598}, {\"PT\": 155.56761439304722, \"species\": \"P_2\", \"y\": 0.8955401431795483}, {\"PT\": 170.7352647470692, \"species\": \"P_2\", \"y\": 0.8999283709672571}, {\"PT\": 187.3817422860385, \"species\": \"P_2\", \"y\": 0.9041469154707042}, {\"PT\": 205.65123083486534, \"species\": \"P_2\", \"y\": 0.9082011100416706}, {\"PT\": 225.70197196339214, \"species\": \"P_2\", \"y\": 0.9120962253232613}, {\"PT\": 247.70763559917114, \"species\": \"P_2\", \"y\": 0.9158374597966633}, {\"PT\": 271.85882427329426, \"species\": \"P_2\", \"y\": 0.9194299314446935}, {\"PT\": 298.364724028334, \"species\": \"P_2\", \"y\": 0.9228786704625612}, {\"PT\": 327.4549162877732, \"species\": \"P_2\", \"y\": 0.9261886129471482}, {\"PT\": 359.3813663804629, \"species\": \"P_2\", \"y\": 0.9293645954974501}, {\"PT\": 394.420605943766, \"species\": \"P_2\", \"y\": 0.932411350660551}, {\"PT\": 432.87612810830615, \"species\": \"P_2\", \"y\": 0.9353335031595534}, {\"PT\": 475.0810162102798, \"species\": \"P_2\", \"y\": 0.9381355668421743}, {\"PT\": 521.400828799969, \"species\": \"P_2\", \"y\": 0.9408219422912241}, {\"PT\": 572.236765935022, \"species\": \"P_2\", \"y\": 0.9433969150408062}, {\"PT\": 628.029144183426, \"species\": \"P_2\", \"y\": 0.9458646543448271}, {\"PT\": 689.2612104349702, \"species\": \"P_2\", \"y\": 0.9482292124471873}, {\"PT\": 756.463327554629, \"species\": \"P_2\", \"y\": 0.9504945243058579}, {\"PT\": 830.2175681319752, \"species\": \"P_2\", \"y\": 0.9526644077258541}, {\"PT\": 911.1627561154896, \"species\": \"P_2\", \"y\": 0.9547425638589112}, {\"PT\": 1000.0, \"species\": \"P_2\", \"y\": 0.9567325780304037}]}}, {\"mode\": \"vega-lite\"});\n",
+       "</script>"
+      ],
+      "text/plain": [
+       "alt.LayerChart(...)"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "import altair as alt\n",
     "\n",
-    "source = df.melt(\"M0\", var_name=\"species\", value_name=\"y\")\n",
+    "source = df.melt(\"PT\", var_name=\"species\", value_name=\"y\")\n",
     "\n",
     "# Create a selection that chooses the nearest point & selects based on x-value\n",
     "nearest = alt.selection_point(nearest=True, on=\"pointerover\",\n",
-    "                              fields=[\"M0\"], empty=False)\n",
+    "                              fields=[\"PT\"], empty=False)\n",
     "\n",
     "# The basic line\n",
     "line = alt.Chart(source).mark_line(interpolate=\"basis\").encode(\n",
-    "    x=alt.X(\"M0:Q\", scale=alt.Scale(type=\"log\"), title='Ratio M0/kD'),\n",
+    "    x=alt.X(\"PT:Q\", scale=alt.Scale(type=\"log\"), title='Ratio PT/kD'),\n",
     "    y=alt.Y(\"y:Q\", title='Fraction of total'),\n",
     "    color=\"species:N\",\n",
     ")\n",
@@ -259,9 +458,9 @@
     "rules = alt.Chart(source).transform_pivot(\n",
     "    \"species\",\n",
     "    value=\"y\",\n",
-    "    groupby=[\"M0\"]\n",
+    "    groupby=[\"PT\"]\n",
     ").mark_rule(color=\"black\").encode(\n",
-    "    x=\"M0:Q\",\n",
+    "    x=\"PT:Q\",\n",
     "    opacity=alt.condition(nearest, alt.value(0.3), alt.value(0)),\n",
     "    tooltip=[alt.Tooltip(c, type=\"quantitative\", format=\".2f\") for c in df.columns],\n",
     ").add_params(nearest)\n",
@@ -290,8 +489,13 @@
    "source": [
     "Meanwhile, we can think about _why_ its so hard to reach full dimerization even if concentrations used are far higher than the dissociation constant. If we think about the reaction from an maximum entropy point of view, out intuition might tell us that the entropic equivalent of making all promomers into a dimer is the equivalent of putting all 'air molecules' into one corner of the room: it has a vanishingly low probability of happing because its so far away from the maximium entropy state of the system. \n",
     "\n",
-    "Second, a closer look at the differential equation describing the change of $[D]$ also tells us the answer. There are two terms in this equation: one is positive and depends on $[M]$, the second is negative and depends on $[D]$. Therefore, even when there is a lot of initial monomer $[M_0]$ compared to the $k_D$, as more and more dimer is formed $[M]$ will go down while $[D]$ goes up. Thus the positive term becomes smaller while the negative term becomes bigger, slowing down the formation of dimer, and equillibrium is reached at a point with still a large fraction of $M$ in solution.   "
+    "Second, a closer look at the differential equation describing the change of $[P_2]$ also tells us the answer. There are two terms in this equation: one is positive and depends on $[P_1]$, the second is negative and depends on $[P_2]$. Therefore, even when there is a lot of total protomer $[P_T]$ compared to the $k_D$, as more and more dimer is formed $[P_1]$ will go down while $[P_2]$ goes up. Thus the positive term becomes smaller while the negative term becomes bigger, slowing down the formation of dimer, and equillibrium is reached at a point with still a large fraction of $P_1$ in solution.   "
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
   }
  ],
  "metadata": {