ADD equations for washing steps
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@ -124,11 +124,11 @@ diff = 1000000; % init to some huge number
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while diff > tolerance
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t = linspace(t_0, t_f, 50);
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Y = pdepe(m,@pde,@init,@bound,r,t);
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C = Y(:,:,1);
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Y = pdepe(m, @pde, @init, @bound, r, t);
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C = Y(:, :, 1);
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% final concentration in center of carrier at final time
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C_f_center = C(end,1);
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C_f_center = C(end, 1);
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% test to see how far off the center is from bulk
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diff = C_f_center / C_Lbf - 1;
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@ -166,13 +166,13 @@ function u0 = init(r)
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u0 = C_Lc_pw;
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end
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function [pl,ql,pr,qr] = bound(rl,cl,rr,cr,t)
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function [pl, ql, pr, qr] = bound(rl, cl, rr, cr, t)
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pl = 0;
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ql = 1;
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% assume that the concentration boundary
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% is the average of the initial and
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% theoretical final concentration in bulk
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pr = cr - (C_Lbf + C_Lbf)/2;
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pr = cr - (C_Lb0 + C_Lbf) / 2;
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qr = 0;
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end
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@ -1566,24 +1566,25 @@ the center of the microcarriers was reached. We also assumed that the pores in
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the microcarriers were large enough that the interactions between the \gls{stp}
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and surfaces would be small, thus the apparent diffusivity could be represented
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as a fraction of the diffusion coefficient of \gls{stp} in water. This model was
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given by \cref{eqn:stp_diffision_1,eqn:stp_diffision_2}:
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given by \cref{eqn:stp_diffusion_1,eqn:stp_diffusion_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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\label{eqn:stp_diffusion_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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\label{eqn:stp_diffusion_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 $D_{STP}$ the diffusion rate of \gls{stp} (or \glspl{mab} for later
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calculations) in water
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\item $\beta$ a fractional parameter representing the tortuousity and void
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fraction of the microcarriers (here called the `geometric diffusivity')
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\item $r$ is the interfatial radius of the unbound biotin within a microcarrier
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@ -1602,11 +1603,60 @@ 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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These same coefficients
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were used in determining the kinetics of the washing steps, and
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\SI{5.0e-6}{\cm\squared\per\second}\cite{Niether2020} was used as the diffusion
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coefficient for free biotin (which should be the only species left in solution
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after all the \gls{snb} has hydrolyzed).
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To model the washing of the microcarriers, they once again were assumed to be
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porous spheres filled with whatever amount of reagent was left unbound from the
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previous step (which was assumed to be equal to concentration in the
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supernatent). The diffusion out of the microcarriers is given by the following
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partial differential equation and boundary conditions:
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\begin{equation}
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\label{eqn:stp_washing}
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\frac{\partial C_i}{\partial t} = \frac{1}{r^2}\frac{\partial}{\partial
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r}\left(r^2 D_{app} \frac{\partial C_i}{\partial r}\right)
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\end{equation}
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\begin{equation}
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\label{eqn:stp_washing_left_bc}
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C_i(r, 0) = C_{i,0}
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\end{equation}
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\begin{equation}
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\label{eqn:stp_washing_left_bc}
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N_i(0, t) = 0
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\end{equation}
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\begin{equation}
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\label{eqn:stp_washing_right_bc}
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C_i(R, t) = (C_{b,0}+C_{b,\infty}) / 2
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\end{equation}
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\noindent where (in addition to the variables given already for
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\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2})
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\begin{itemize}[label={}]
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\item $N_i$ is the radial flux of the species in question inside the
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microcarriers
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\item $C_i$ is the concentration of the species in question inside the
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microcarriers
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\item $C_{i,0}$ is the initial concentration of the species in question inside
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the microcarriers (which is assumed to be the concentration in the bulk before
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the wash volume is added)
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\item $C_{b,0}$ is the initial bulk concentration of the species in question
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outside the microcarriers after the initial wash volume has been added
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\item $C_{b,\infty}$ is the final bulk concentration of the species in
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question outside the microcarriers
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\end{itemize}
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Note that in order to avoid solving a moving boundary value problem, the
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concentration at the boundary of the microcarriers was fixed at the average of
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the final and initial concentration expected to be observed in bulk. This should
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be a reasonable assumption given that the volume inside the microcarriers is
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tiny compared to the amount of volume added in the wash, thus the boundary
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concentration should change little.
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The same diffusion coefficients were used in determining the kinetics of the
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washing steps, and \SI{5.0e-6}{\cm\squared\per\second}\cite{Niether2020} was
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used as the diffusion coefficient for free biotin (which should be the only
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species left in solution after all the \gls{snb} has hydrolyzed).
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All diffusion coefficients were taken to be valid at \gls{rt} and in \gls{di}
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water, which is a safe assumption given that our reaction medium was 1X
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@ -1615,8 +1665,6 @@ water, which is a safe assumption given that our reaction medium was 1X
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See \cref{sec:appendix_binding} and \cref{sec:appendix_washing} for the MATLAB
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code (and output in the case of the washing steps) used.
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% METHOD add the equation governing the washing steps
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\subsection{Luminex Analysis}\label{sec:luminex_analysis}
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Luminex was performed using a \product{ProcartaPlex kit}{\thermo}{custom} for
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@ -1910,7 +1958,7 @@ this experimental binding data to fit a continuous model for the \gls{stp}
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binding reaction. Using the diffusion rate of the \gls{stp}
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(\SI{6.2e-7}{\cm\squared\per\second}), we then calculated the geometric
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diffusivity of the microcarriers to be 0.190 (see
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\cref{eqn:stp_diffision_1,eqn:stp_diffision_2}).
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\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2}).
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% RESULT state how I calculated the number of mab/surface area
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Using this effective diffusivity and the known diffusion coefficient of a
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