ENH add diffusion method section
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@ -1167,6 +1167,19 @@ CONCLUSIONS: We developed a simplified, semi-closed system for the initial selec
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publisher = {Public Library of Science ({PLoS})},
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publisher = {Public Library of Science ({PLoS})},
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}
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}
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@Article{Sherwood1992,
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author = {Jill K. Sherwood and Richard B. Dause and W. Mark Saltzman},
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journal = {Nature Biotechnology},
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title = {Controlled Antibody Delivery Systems},
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year = {1992},
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month = {nov},
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number = {11},
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pages = {1446--1449},
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volume = {10},
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doi = {10.1038/nbt1192-1446},
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publisher = {Springer Science and Business Media {LLC}},
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}
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@Comment{jabref-meta: databaseType:bibtex;}
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@Comment{jabref-meta: databaseType:bibtex;}
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@Comment{jabref-meta: grouping:
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@Comment{jabref-meta: grouping:
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@ -975,7 +975,63 @@ at \SI{260}{\nm} were taken every minute for \SI{2}{\hour}.
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\subsection{reaction kinetics quantification}
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\subsection{reaction kinetics quantification}
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% METHOD add reaction kinetics diffusion mathy stuff
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The diffusion of \gls{stp} into biotin-coated microcarriers was determined
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experimentally. \SI{40}{\ug\per\ml} \gls{stp} was added to multiple batches of
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biotin-coated microcarriers, and supernatents were taken at fixed intervals and
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quantified for \gls{stp} protein using the \gls{bca} assay.
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% TODO defend why the microcarriers were saturated with stp
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The effective diffusivity of the microcarriers was determined using a
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pseudo-steady-state model. Each microcarrier was assumed to be a porous sphere
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with a fixed number of `\gls{stp} binding sites' equal to the maximum number of
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\gls{stp} molecules that could binding to the surface per area (eg, we assumed
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the surface was fully covered by \gls{stp}). Because the reaction rate between
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biotin and \gls{stp} was so fast, we assumed that the interface of free biotin
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shrunk as a function of \gls{stp} bound until the center of the microcarriers
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was reached. We also assumed that the pores in the microcarriers were large
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enough that the interactions between the \gls{stp} and surfaces would be small,
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thus the apparent diffusivity could be represented as a fraction of the
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diffusion coefficient of \gls{stp} in water. This model was given by
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\cref{eqn:stp_diffision_1,eqn:stp_diffision_2}:
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% TODO actually derive these equations, eg state the initial conditions and
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% governing equation
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\begin{equation}
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\label{eqn:stp_diffision_1}
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\frac{dr}{dt} = \frac{-D_{app}C}{Br(1-r/R)}
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\end{equation}
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\begin{equation}
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\label{eqn:stp_diffision_2}
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\frac{dC}{dt} = \frac{-4 \pi n D_{app} C}{V(1/r-1/R)}
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\end{equation}
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\noindent where
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\begin{itemize}[label={}]
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\item $D_{app}$ is the apparent diffusion rate which is equal to $D_{STP}\beta$
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\item $D_{STP}$ the diffusion rate of \gls{stp} in water
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\item $\beta$ a fractional parameter representing the tortuousity and void
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fraction of the microcarriers.
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\item $r$ is the interfatial radius of the unbound biotin within a microcarrier
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\item $t$ is the reaction time
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\item $C$ is the concentration of \gls{stp} in the bulk solution
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\item $V$ is the volume of the bulk medium
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\item $R$ is the average radius of the microcarriers
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\item $n$ is the number of microcarriers in the reaction volume
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\end{itemize}
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% TODO cite the diffusion rate of stp
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The diffusion rate of \gls{stp} was assumed to be
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\SI{3.89e-7}{\cm\squared\per\second} {\#}{diffusion rate citation}. Since all
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but $\beta$ was known, the experimental data was fit using these equations using
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\inlinecode{ode45} in MATLAB and least squares as the fitting error.
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% TODO this diffusion rate isn't actually reflected in the code
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These equations were then used analogously to describe the reaction profile of
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\glspl{mab} assuming a diffusion rate of
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\SI{4.8e-7}{\cm\squared\per\second}\cite{Sherwood1992}.
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% METHOD add the equation governing the washing steps
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\subsection{Luminex Analysis}
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\subsection{Luminex Analysis}
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