From 34ab2dce170021ce3d2452dff699a9c95dbc35e5 Mon Sep 17 00:00:00 2001
From: don <don@25fa0111-c432-4dab-af88-9f31a2f6ac42>
Date: Fri, 14 May 2010 02:15:48 +0000
Subject: [PATCH] update formalism

git-svn-id: svn+ssh://hemlock.ucr.edu/srv/svn/misc/trunk/origins_of_life@497 25fa0111-c432-4dab-af88-9f31a2f6ac42
---
 kinetic_formalism.Rnw | 25 ++++++++++++++++++++++---
 1 file changed, 22 insertions(+), 3 deletions(-)

diff --git a/kinetic_formalism.Rnw b/kinetic_formalism.Rnw
index 018c44e..1ed5a8b 100644
--- a/kinetic_formalism.Rnw
+++ b/kinetic_formalism.Rnw
@@ -67,11 +67,30 @@ require(grid)
 \section{State Equation}
 % double check this with the bits in the paper
 \begin{equation}
-  \frac{d C^{j}_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C^j_{i_\mathrm{monomer}}\right] -
-  k_{bi}k_{bi\mathrm{adj}}C^j_{i_\mathrm{ves}}
+  \frac{d C_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} -
+  k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}}
   \label{eq:state}
 \end{equation}
 
+For $k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]$,
+$k_{fi}$ has units of $\frac{\mathrm{m}}{\mathrm{s}}$,
+$k_{fi\mathrm{adj}}$ and $k_{bi\mathrm{adj}}$ are unitless,
+concentration is in units of $\frac{\mathrm{n}}{\mathrm{L}}$, surface
+area is in units of $\mathrm{m}^2$, $k_{bi}$ has units of
+$\frac{1}{\mathrm{s}}$ and $C_{i_\mathrm{ves}}$ has units of
+$\mathrm{n}$, Thus, we have
+
+\begin{equation}
+  \frac{\mathrm{n}}{\mathrm{s}} = \frac{\mathrm{m}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \mathrm{m}^2 \frac{1000\mathrm{L}}{\mathrm{m}^3} - 
+  \frac{1}{\mathrm{s}} \mathrm{n}
+  =
+  \frac{\mathrm{m^3}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{\mathrm{n}}{\mathrm{s}}=
+  \frac{\mathrm{n}}{\mathrm{s}} = 1000 \frac{\mathrm{n}}{\mathrm{s}} - \frac{\mathrm{n}}{\mathrm{s}}
+  \label{eq:state_units}
+\end{equation}
+
+The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
+
 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
 
 \begin{equation}
@@ -160,7 +179,7 @@ curve(3^x,from=0,to=sd(c(12,24)),
 \newpage
 \subsubsection{Unsaturation Backward}
 \begin{equation}
-  un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-\left<un_\mathrm{monomer}\right>}\right|}
+  un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
   \label{eq:unsaturation_backward}
 \end{equation}
 
-- 
2.39.2