ADD a bunch of stuff about how I calculated diffusion and such

tex/thesis.tex

@@ -1561,7 +1561,7 @@ diffusion coefficient of \gls{stp} in water. This model was given by
 \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.
+  fraction of the microcarriers (here called the `geometric diffusivity')
 \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
@@ -1573,18 +1573,18 @@ diffusion coefficient of \gls{stp} in water. This model was given by
 The diffusion rate of \gls{stp} was assumed to be
 \SI{6.2e-7}{\cm\squared\per\second}\cite{Kamholz2001}. 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
+\inlinecode{ode45} in MATLAB and least squares as the fitting error. 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}.
+\SI{4.8e-7}{\cm\squared\per\second}\cite{Sherwood1992}. These same coefficients
+were used in determining the kinetics of the washing steps, and
+\SI{5.0e-6}{\cm\squared\per\second}\cite{Niether2020} was used as the diffusion
+coefficient for free biotin (which should be the only species left in solution
+after all the \gls{snb} has hydrolyzed). All diffusion coefficients were taken
+to be valid at \gls{rt} and in \gls{di} water, which is a safe assumption given
+that our reaction medium was 1X \gls{pbs}.
 
 % METHOD add the equation governing the washing steps
 
-The diffusion coefficient used for biotin was
-\SI{5e-6}{\cm\squared\per\second}\cite{Niether2020}
-
 \subsection{Luminex Analysis}\label{sec:luminex_analysis}
 
 Luminex was performed using a \product{ProcartaPlex kit}{\thermo}{custom} for
@@ -1863,10 +1863,11 @@ We observed that for either concentration, the reaction was over in
 \SIrange{20}{30}{\minute} (\cref{fig:dms_biotin_rxn_mass}). Furthermore, when
 put in terms of fraction of input \gls{snb}, we observed that the curves are
 almost identical (\cref{fig:dms_biotin_rxn_frac}). Given this, the reaction step
-for biotin attached was set to \SI{30}{\minute}.
+for biotin attached was set to \SI{30}{\minute}\footnote{we actually used
+  \SI{60}{\minute} for most of the runs as outlined in methods, which shouldn't
+  make any difference except save for being excessive according to this result}.
 
-% TODO these numbers might be totally incorrect
-% TODO state what the effective diffusivity is
+% RESULT state how we calculated the number of stp/site
 Next, we quantified the amount of \gls{stp} reacted with the surface of the
 biotin-coated microcarriers. Different batches of biotin-coated \glspl{dms} were
 coated with \SI{40}{\ug\per\ml} \gls{stp} and sampled at intermediate timepoints
@@ -1874,9 +1875,12 @@ using the \gls{bca} assay to indirectly quantify the amount of attached
 \gls{stp} mass. We found this reaction took approximately \SI{30}{\minute}
 (\cref{fig:dms_stp_per_time}). Assuming a quasi-steady-state paradigm, we used
 this experimental binding data to fit a continuous model for the \gls{stp}
-binding reaction. Using the diffusion rate of the \gls{stp}, we then calculated
-the effective diffusivity of the microcarriers to be 0.2.
+binding reaction. Using the diffusion rate of the \gls{stp}
+(\SI{6.2e-7}{\cm\squared\per\second}), we then calculated the geometric
+diffusivity of the microcarriers to be 0.190 (see
+\cref{eqn:stp_diffision_1,eqn:stp_diffision_2}).
 
+% RESULT state how I calculated the number of mab/surface area
 Using this effective diffusivity and the known diffusion coefficient of a
 \gls{mab} protein in water, we calculated predict the binding of \glspl{mab} per
 time onto the microcarriers (this obviously assumes that the effectively
@@ -1884,32 +1888,48 @@ diffusivity is independent of the protein used, which should be reasonable given
 that the pores of the microcarriers are huge compared to the proteins, and we
 don't expect any significant reaction between the protein and the microcarrier
 surface save for the \gls{stp}-biotin binding reaction). According to this
-model, the \gls{mab} binding reaction should be complete within \SI{15}{\minute}
-under the conditions used for our protocol (\cref{fig:dms_mab_per_time}). Note
-that our unoptimized coated steps were done in \SI{45}{\minute}, which seemed
-reasonable given the slightly larger hydrodynamic radius of \glspl{mab} compared
-to \gls{stp} which was shown to react in \SI{30}{\minute} experimentally. The
-results of this model should be experimentally verified.
+model, the \gls{mab} binding reaction should be complete within \SI{75}{\minute}
+under the conditions used for our protocol
+(\cref{fig:dms_mab_per_time})\footnote{We actually used \SI{60}{\minute} as
+  describe in the method section as this model was not updated with new
+  parameters until recently; however, we should point out that even at
+  \SI{60}{\minute} the reaction appears to be >\SI{95}{\percent} complete}.
 
-% TODO find the actual numbers for this
-Finally, we used the effective diffusivity of the microcarriers to predict the
-time needed for wash steps. This is important, as failing to wash out residual
-free \gls{snb} (for example) could occupy binding sites on the \gls{stp}
-molecules, lowering the effective binding capacity of the \gls{mab} downstream.
-Once again, we assumed the microcarriers to be porous spheres, this time with an
-initial concentration of \gls{snb}, \gls{stp}, or \glspl{mab} equal to the final
-concentration of the bulk concentration of the previous binding step, and
-calculated the amount of time it would take for the concentration profile inside
-the microcarriers to equilibrate to the bulk in the wash step. Using this model,
-we found that the wash times for \gls{snb}, \gls{stp}, and \glspl{mab} was
-\SI{10}{\minute}, {\#} minutes, and {\#} minutes respectively. Note that
-\gls{snb}, \gls{stp}, and \glspl{mab} each required 3, 2, and 2 washes to reduce
-the concentration down to a level that was 1/1000 of the starting concentration
-(which was deemed to be acceptable for preventing downstream inhibition). Using
-this in our protocol, we verified that the \gls{snb} was totally undetectable
-after washing (\cref{fig:dms_biotin_washed}). The other two species need to be
-verified, but note that the consequences of residual \gls{stp} or \gls{mab} are
-far less severe than that of \gls{snb}.
+Finally, we calculated the number of wash steps needed to remove the reagents
+between each step, including the time for each wash which required the geometric
+diffusivity of the microcarriers as calculated above. This is important, as
+failing to wash out residual free \gls{snb} (for example) could occupy binding
+sites on the \gls{stp} molecules, lowering the effective binding capacity of the
+\gls{mab} downstream. Each wash was a 1:15 dilution (\SI{1}{\ml} reaction volume
+in a \SI{15}{\ml} conical tube), and in the case of \gls{snb} we wished to wash
+out enough biotin such that less than \SI{1}{\percent} of the binding sites in
+\gls{stp} would be occupied. Given this dilution factor, a maximum of
+\SI{20}{\nmol} of biotin remaining \cref{fig:biotin_coating} \SI{2.9}{\nmol}
+biotin binding sites on \SI{40}{\ug} \gls{stp} (assuming 4 binding sites per
+\gls{stp} protein), this turned out to be 3 washes. By similar logic, using 2
+washes after the \gls{stp} binding step will ensure that the number of free
+\gls{stp} binding sites is less than 20X the number of \gls{mab} molecules
+added\footnote{This step may benefit from an additional wash, as the number of
+  washes used here was develop when \SI{40}{\ug} rather than \SI{4}{\ug}
+  \gls{mab} was used to coat the \gls{dms}, yielding a much wider margin.
+  However, it is also not clear to what extent this matters, as the \gls{mab}
+  have multiple biotin molecules per \gls{mab} protein, and thus one \gls{mab}
+  would require binding to several \gls{stp} molecules to be prevented from
+  binding at all.}
+
+To determine the length of time required for each wash, we again assumed the
+microcarriers to be porous spheres, this time with an initial concentration of
+\gls{snb}, \gls{stp}, or \glspl{mab} equal to the final concentration of the
+bulk concentration of the previous binding step, and calculated the amount of
+time it would take for the concentration profile inside the microcarriers to
+equilibrate to the bulk in the wash step. Using this model, we found that the
+wash times for \gls{snb}, \gls{stp}, and \glspl{mab} was \SI{3}{\minute},
+\SI{15}{\minute}, and \SI{17}{\minute} respectively. We verified that the
+\gls{snb} was totally undetectable after washing (\cref{fig:dms_biotin_washed}).
+The other two species need to be verified in a similar manner; however, we
+should not that the washing time for both the \gls{stp} and \gls{mab} coating
+steps were \SI{30}{\minute}, which is a significant margin of safety (albeit
+one that could be optimized).
 
 \subsection{DMSs can efficiently expand T cells compared to beads}