ADD equations for washing steps
This commit is contained in:
parent
75b1b9e588
commit
06431bd255
|
@ -124,11 +124,11 @@ diff = 1000000; % init to some huge number
|
||||||
while diff > tolerance
|
while diff > tolerance
|
||||||
t = linspace(t_0, t_f, 50);
|
t = linspace(t_0, t_f, 50);
|
||||||
|
|
||||||
Y = pdepe(m,@pde,@init,@bound,r,t);
|
Y = pdepe(m, @pde, @init, @bound, r, t);
|
||||||
C = Y(:,:,1);
|
C = Y(:, :, 1);
|
||||||
|
|
||||||
% final concentration in center of carrier at final time
|
% final concentration in center of carrier at final time
|
||||||
C_f_center = C(end,1);
|
C_f_center = C(end, 1);
|
||||||
|
|
||||||
% test to see how far off the center is from bulk
|
% test to see how far off the center is from bulk
|
||||||
diff = C_f_center / C_Lbf - 1;
|
diff = C_f_center / C_Lbf - 1;
|
||||||
|
@ -166,13 +166,13 @@ function u0 = init(r)
|
||||||
u0 = C_Lc_pw;
|
u0 = C_Lc_pw;
|
||||||
end
|
end
|
||||||
|
|
||||||
function [pl,ql,pr,qr] = bound(rl,cl,rr,cr,t)
|
function [pl, ql, pr, qr] = bound(rl, cl, rr, cr, t)
|
||||||
pl = 0;
|
pl = 0;
|
||||||
ql = 1;
|
ql = 1;
|
||||||
% assume that the concentration boundary
|
% assume that the concentration boundary
|
||||||
% is the average of the initial and
|
% is the average of the initial and
|
||||||
% theoretical final concentration in bulk
|
% theoretical final concentration in bulk
|
||||||
pr = cr - (C_Lbf + C_Lbf)/2;
|
pr = cr - (C_Lb0 + C_Lbf) / 2;
|
||||||
qr = 0;
|
qr = 0;
|
||||||
end
|
end
|
||||||
|
|
||||||
|
|
|
@ -1566,24 +1566,25 @@ 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}
|
the microcarriers were large enough that the interactions between the \gls{stp}
|
||||||
and surfaces would be small, thus the apparent diffusivity could be represented
|
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
|
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}:
|
given by \cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2}:
|
||||||
|
|
||||||
% TODO actually derive these equations, eg state the initial conditions and
|
% TODO actually derive these equations, eg state the initial conditions and
|
||||||
% governing equation
|
% governing equation
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\label{eqn:stp_diffision_1}
|
\label{eqn:stp_diffusion_1}
|
||||||
\frac{dr}{dt} = \frac{-D_{app}C}{Br(1-r/R)}
|
\frac{dr}{dt} = \frac{-D_{app}C}{Br(1-r/R)}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\label{eqn:stp_diffision_2}
|
\label{eqn:stp_diffusion_2}
|
||||||
\frac{dC}{dt} = \frac{-4 \pi n D_{app} C}{V(1/r-1/R)}
|
\frac{dC}{dt} = \frac{-4 \pi n D_{app} C}{V(1/r-1/R)}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
\noindent where
|
\noindent where
|
||||||
\begin{itemize}[label={}]
|
\begin{itemize}[label={}]
|
||||||
\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} (or \glspl{mab} for later
|
||||||
|
calculations) 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 (here called the `geometric diffusivity')
|
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
|
||||||
|
@ -1602,11 +1603,60 @@ 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
|
To model the washing of the microcarriers, they once again were assumed to be
|
||||||
were used in determining the kinetics of the washing steps, and
|
porous spheres filled with whatever amount of reagent was left unbound from the
|
||||||
\SI{5.0e-6}{\cm\squared\per\second}\cite{Niether2020} was used as the diffusion
|
previous step (which was assumed to be equal to concentration in the
|
||||||
coefficient for free biotin (which should be the only species left in solution
|
supernatent). The diffusion out of the microcarriers is given by the following
|
||||||
after all the \gls{snb} has hydrolyzed).
|
partial differential equation and boundary conditions:
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
\label{eqn:stp_washing}
|
||||||
|
\frac{\partial C_i}{\partial t} = \frac{1}{r^2}\frac{\partial}{\partial
|
||||||
|
r}\left(r^2 D_{app} \frac{\partial C_i}{\partial r}\right)
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
\label{eqn:stp_washing_left_bc}
|
||||||
|
C_i(r, 0) = C_{i,0}
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
\label{eqn:stp_washing_left_bc}
|
||||||
|
N_i(0, t) = 0
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\begin{equation}
|
||||||
|
\label{eqn:stp_washing_right_bc}
|
||||||
|
C_i(R, t) = (C_{b,0}+C_{b,\infty}) / 2
|
||||||
|
\end{equation}
|
||||||
|
|
||||||
|
\noindent where (in addition to the variables given already for
|
||||||
|
\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2})
|
||||||
|
\begin{itemize}[label={}]
|
||||||
|
\item $N_i$ is the radial flux of the species in question inside the
|
||||||
|
microcarriers
|
||||||
|
\item $C_i$ is the concentration of the species in question inside the
|
||||||
|
microcarriers
|
||||||
|
\item $C_{i,0}$ is the initial concentration of the species in question inside
|
||||||
|
the microcarriers (which is assumed to be the concentration in the bulk before
|
||||||
|
the wash volume is added)
|
||||||
|
\item $C_{b,0}$ is the initial bulk concentration of the species in question
|
||||||
|
outside the microcarriers after the initial wash volume has been added
|
||||||
|
\item $C_{b,\infty}$ is the final bulk concentration of the species in
|
||||||
|
question outside the microcarriers
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
Note that in order to avoid solving a moving boundary value problem, the
|
||||||
|
concentration at the boundary of the microcarriers was fixed at the average of
|
||||||
|
the final and initial concentration expected to be observed in bulk. This should
|
||||||
|
be a reasonable assumption given that the volume inside the microcarriers is
|
||||||
|
tiny compared to the amount of volume added in the wash, thus the boundary
|
||||||
|
concentration should change little.
|
||||||
|
|
||||||
|
The same diffusion 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}
|
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
|
water, which is a safe assumption given that our reaction medium was 1X
|
||||||
|
@ -1615,8 +1665,6 @@ water, which is a safe assumption given that our reaction medium was 1X
|
||||||
See \cref{sec:appendix_binding} and \cref{sec:appendix_washing} for the MATLAB
|
See \cref{sec:appendix_binding} and \cref{sec:appendix_washing} for the MATLAB
|
||||||
code (and output in the case of the washing steps) used.
|
code (and output in the case of the washing steps) used.
|
||||||
|
|
||||||
% METHOD add the equation governing the washing steps
|
|
||||||
|
|
||||||
\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
|
||||||
|
@ -1910,7 +1958,7 @@ this experimental binding data to fit a continuous model for the \gls{stp}
|
||||||
binding reaction. Using the diffusion rate of the \gls{stp}
|
binding reaction. Using the diffusion rate of the \gls{stp}
|
||||||
(\SI{6.2e-7}{\cm\squared\per\second}), we then calculated the geometric
|
(\SI{6.2e-7}{\cm\squared\per\second}), we then calculated the geometric
|
||||||
diffusivity of the microcarriers to be 0.190 (see
|
diffusivity of the microcarriers to be 0.190 (see
|
||||||
\cref{eqn:stp_diffision_1,eqn:stp_diffision_2}).
|
\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2}).
|
||||||
|
|
||||||
% RESULT state how I calculated the number of mab/surface area
|
% 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
|
||||||
|
|
Loading…
Reference in New Issue