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

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Nathan Dwarshuis 2021-08-03 18:21:08 -04:00
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@ -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 $D_{app}$ is the apparent diffusion rate which is equal to $D_{STP}\beta$
\item $D_{STP}$ the diffusion rate of \gls{stp} in water \item $D_{STP}$ the diffusion rate of \gls{stp} in water
\item $\beta$ a fractional parameter representing the tortuousity and void \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 $r$ is the interfatial radius of the unbound biotin within a microcarrier
\item $t$ is the reaction time \item $t$ is the reaction time
\item $C$ is the concentration of \gls{stp} in the bulk solution \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 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 \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 known, the experimental data was fit using these equations using
\inlinecode{ode45} in MATLAB and least squares as the fitting error. \inlinecode{ode45} in MATLAB and least squares as the fitting error. These equations were then used analogously to describe the reaction profile of
% 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 \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 % 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} \subsection{Luminex Analysis}\label{sec:luminex_analysis}
Luminex was performed using a \product{ProcartaPlex kit}{\thermo}{custom} for 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 \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 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 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 % RESULT state how we calculated the number of stp/site
% TODO state what the effective diffusivity is
Next, we quantified the amount of \gls{stp} reacted with the surface of the 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 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 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} \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 (\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} 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 binding reaction. Using the diffusion rate of the \gls{stp}
the effective diffusivity of the microcarriers to be 0.2. (\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 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 \gls{mab} protein in water, we calculated predict the binding of \glspl{mab} per
time onto the microcarriers (this obviously assumes that the effectively 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 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 don't expect any significant reaction between the protein and the microcarrier
surface save for the \gls{stp}-biotin binding reaction). According to this 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} 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}). Note under the conditions used for our protocol
that our unoptimized coated steps were done in \SI{45}{\minute}, which seemed (\cref{fig:dms_mab_per_time})\footnote{We actually used \SI{60}{\minute} as
reasonable given the slightly larger hydrodynamic radius of \glspl{mab} compared describe in the method section as this model was not updated with new
to \gls{stp} which was shown to react in \SI{30}{\minute} experimentally. The parameters until recently; however, we should point out that even at
results of this model should be experimentally verified. \SI{60}{\minute} the reaction appears to be >\SI{95}{\percent} complete}.
% TODO find the actual numbers for this Finally, we calculated the number of wash steps needed to remove the reagents
Finally, we used the effective diffusivity of the microcarriers to predict the between each step, including the time for each wash which required the geometric
time needed for wash steps. This is important, as failing to wash out residual diffusivity of the microcarriers as calculated above. This is important, as
free \gls{snb} (for example) could occupy binding sites on the \gls{stp} failing to wash out residual free \gls{snb} (for example) could occupy binding
molecules, lowering the effective binding capacity of the \gls{mab} downstream. sites on the \gls{stp} molecules, lowering the effective binding capacity of the
Once again, we assumed the microcarriers to be porous spheres, this time with an \gls{mab} downstream. Each wash was a 1:15 dilution (\SI{1}{\ml} reaction volume
initial concentration of \gls{snb}, \gls{stp}, or \glspl{mab} equal to the final in a \SI{15}{\ml} conical tube), and in the case of \gls{snb} we wished to wash
concentration of the bulk concentration of the previous binding step, and out enough biotin such that less than \SI{1}{\percent} of the binding sites in
calculated the amount of time it would take for the concentration profile inside \gls{stp} would be occupied. Given this dilution factor, a maximum of
the microcarriers to equilibrate to the bulk in the wash step. Using this model, \SI{20}{\nmol} of biotin remaining \cref{fig:biotin_coating} \SI{2.9}{\nmol}
we found that the wash times for \gls{snb}, \gls{stp}, and \glspl{mab} was biotin binding sites on \SI{40}{\ug} \gls{stp} (assuming 4 binding sites per
\SI{10}{\minute}, {\#} minutes, and {\#} minutes respectively. Note that \gls{stp} protein), this turned out to be 3 washes. By similar logic, using 2
\gls{snb}, \gls{stp}, and \glspl{mab} each required 3, 2, and 2 washes to reduce washes after the \gls{stp} binding step will ensure that the number of free
the concentration down to a level that was 1/1000 of the starting concentration \gls{stp} binding sites is less than 20X the number of \gls{mab} molecules
(which was deemed to be acceptable for preventing downstream inhibition). Using added\footnote{This step may benefit from an additional wash, as the number of
this in our protocol, we verified that the \gls{snb} was totally undetectable washes used here was develop when \SI{40}{\ug} rather than \SI{4}{\ug}
after washing (\cref{fig:dms_biotin_washed}). The other two species need to be \gls{mab} was used to coat the \gls{dms}, yielding a much wider margin.
verified, but note that the consequences of residual \gls{stp} or \gls{mab} are However, it is also not clear to what extent this matters, as the \gls{mab}
far less severe than that of \gls{snb}. 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} \subsection{DMSs can efficiently expand T cells compared to beads}