ENH add diffusion method section

tex/references.bib

@@ -1167,6 +1167,19 @@ CONCLUSIONS: We developed a simplified, semi-closed system for the initial selec
   publisher = {Public Library of Science ({PLoS})},
 }
 
+@Article{Sherwood1992,
+  author    = {Jill K. Sherwood and Richard B. Dause and W. Mark Saltzman},
+  journal   = {Nature Biotechnology},
+  title     = {Controlled Antibody Delivery Systems},
+  year      = {1992},
+  month     = {nov},
+  number    = {11},
+  pages     = {1446--1449},
+  volume    = {10},
+  doi       = {10.1038/nbt1192-1446},
+  publisher = {Springer Science and Business Media {LLC}},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}
 
 @Comment{jabref-meta: grouping:

tex/thesis.tex

@@ -975,7 +975,63 @@ at \SI{260}{\nm} were taken every minute for \SI{2}{\hour}.
 
 \subsection{reaction kinetics quantification}
 
-% METHOD add reaction kinetics diffusion mathy stuff
+The diffusion of \gls{stp} into biotin-coated microcarriers was determined
+experimentally. \SI{40}{\ug\per\ml} \gls{stp} was added to multiple batches of
+biotin-coated microcarriers, and supernatents were taken at fixed intervals and
+quantified for \gls{stp} protein using the \gls{bca} assay.
+
+% TODO defend why the microcarriers were saturated with stp
+The effective diffusivity of the microcarriers was determined using a
+pseudo-steady-state model. Each microcarrier was assumed to be a porous sphere
+with a fixed number of `\gls{stp} binding sites' equal to the maximum number of
+\gls{stp} molecules that could binding to the surface per area (eg, we assumed
+the surface was fully covered by \gls{stp}). Because the reaction rate between
+biotin and \gls{stp} was so fast, we assumed that the interface of free biotin
+shrunk as a function of \gls{stp} bound until the center of the microcarriers
+was reached. We also assumed that the pores in the microcarriers were large
+enough that the interactions between the \gls{stp} and surfaces would be small,
+thus the apparent diffusivity could be represented as a fraction of the
+diffusion coefficient of \gls{stp} in water. This model was given by
+\cref{eqn:stp_diffision_1,eqn:stp_diffision_2}:
+
+% TODO actually derive these equations, eg state the initial conditions and
+% governing equation
+\begin{equation}
+  \label{eqn:stp_diffision_1}
+  \frac{dr}{dt} = \frac{-D_{app}C}{Br(1-r/R)}
+\end{equation}
+
+\begin{equation}
+  \label{eqn:stp_diffision_2}
+  \frac{dC}{dt} = \frac{-4 \pi n D_{app} C}{V(1/r-1/R)}
+\end{equation}
+
+\noindent where
+\begin{itemize}[label={}]
+\item $D_{app}$ is the apparent diffusion rate which is equal to $D_{STP}\beta$
+\item $D_{STP}$ the diffusion rate of \gls{stp} in water
+\item $\beta$ a fractional parameter representing the tortuousity and void
+  fraction of the microcarriers.
+\item $r$ is the interfatial radius of the unbound biotin within a microcarrier
+\item $t$ is the reaction time
+\item $C$ is the concentration of \gls{stp} in the bulk solution
+\item $V$ is the volume of the bulk medium
+\item $R$ is the average radius of the microcarriers
+\item $n$ is the number of microcarriers in the reaction volume
+\end{itemize}
+
+% TODO cite the diffusion rate of stp
+The diffusion rate of \gls{stp} was assumed to be
+\SI{3.89e-7}{\cm\squared\per\second} {\#}{diffusion rate citation}. Since all
+but $\beta$ was known, the experimental data was fit using these equations using
+\inlinecode{ode45} in MATLAB and least squares as the fitting error.
+
+% TODO this diffusion rate isn't actually reflected in the code
+These equations were then used analogously to describe the reaction profile of
+\glspl{mab} assuming a diffusion rate of
+\SI{4.8e-7}{\cm\squared\per\second}\cite{Sherwood1992}.
+
+% METHOD add the equation governing the washing steps
 
 \subsection{Luminex Analysis}