improve wording

explanations/3_calculating_binding_energy.ipynb

@@ -10,7 +10,7 @@
     "## Justification\n",
     "\n",
     "\n",
-    "Here is a more detailed explanation of why we decided to parameterise the equilibrium constant rather than use the Gibbs free energy equation. The purpose of this reparameterisation is to find an equilibrium constant $K_{eq}$ that matches the reality of RNA binding for a given binding energy $G$, which is calculated by physics simulators. The Gibbs free energy equation does relate energy to $K_{eq}$, but does not match experimental data well and highly underestimates the $K_{eq}$. For two interacting RNAs, their binding energy and thus likelihood of interacting may be found through a variety of RNA simulators. \n",
+    "Here is a more detailed explanation of why we decided to parameterise the equilibrium constant rather than use the Gibbs free energy equation. The purpose of this reparameterisation is to find an equilibrium constant $K_{eq}$ that matches the reality of RNA binding for a given binding energy $G$, which is calculated by physics simulators. The Gibbs free energy equation does relate energy to $K_{eq}$, but does not match experimental data well and highly underestimates the $K_{eq}$ in practice. For two interacting RNAs, their binding energy and thus likelihood of interacting may be found through a variety of RNA simulators. \n",
     "\n",
     "In practice, this binding energy $\\Delta G$ may translate into actual binding and subsequent repression in a genetic circuit through the forward and reverse equilibrium rates of the reaction. Both of these will vary based on the RNA, so we have chosen to parameterise the RNA binding from existing data using the research paper [\"Metabolic engineering of Escherichia coli using synthetic small regulatory RNAs\" (2013)](https://www.nature.com/articles/nbt.2461#Fig1). The authors tested a variety of RNAs that could bind to a GFP mRNA with varying binding energies $G$, which they calculated through the RNA simulator NUPACK (rated very similarly to the RNA simulator used here, IntaRNA). They measured the subsequent relative fluorescence as a proxy for repression efficiency and thereby mRNA abundance, or the proportion of unbound to bound RNA. The more mRNA was bound, the less mRNA was free to be read into GFP. \n",
     "\n",
@@ -48,6 +48,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
+    "### Estimating the energy function from literature\n",
+    "\n",
+    "The paper mentioned above graphs the predicted $\\Delta G$ against the fluorescence of a reporter, which is silenced by an RNA that they designed with varying binding energies. The higher the binding energy, the higher the RNA binding efficiency, the lower the fluorescent signal. Therefore, the intensity of the fluorescence can roughly tell us the ratio of the bound to unbound mRNA available for transcription.\n",
+    "\n",
     "Estimating the function to best fit the data as a sigmoid function.\n",
     "\n",
     "Using the following points as guidelines:\n",
@@ -137,7 +141,17 @@
    "source": [
     "## Derivation of $K_{eq}$ from steady states\n",
     "\n",
-    "Because the fluorescence is actually relative fluorescence, it is convenient to use it as a measure for the relative amount of unbound RNA, so the fraction of all RNA that is unbound. \n",
+    "Here, we are deriving an expression that relates the equilibrium constant $K$ to the total number of RNA when the association and dissociation rates are equivalent for all RNAs in the system (eg simplest case of 2 RNAs). The Gibbs free energy is:\n",
+    "\n",
+    "$$ G = -RT ln(K) $$\n",
+    "$$ K = e^{-G/RT} $$\n",
+    "\n",
+    "where G is actually $\\Delta G$ and is the minimum free binding energy for 2 RNAs interacting, while $K$ is:\n",
+    "\n",
+    "$$ K_{eq} = \\frac{k_a}{k_d} $$\n",
+    "\n",
+    "\n",
+    "Because the fluorescence is actually relative fluorescence, it is convenient to use it as a measure for the relative amount of unbound RNA, so the fraction of all mRNA that is unbound. \n",
     "\n",
     "For this, we need to derive an equation for the amount of unbound RNA and for the total amount of RNA.\n",
     "\n",