ADD derivations to binding kinetics stuff

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Nathan Dwarshuis 2021-08-05 18:14:22 -04:00
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@ -4659,7 +4659,91 @@ hosted using \gls{aws} using their proprietary Aurora implementation.
The code is available here: \url{https://github.gatech.edu/ndwarshuis3/mdma}. The code is available here: \url{https://github.gatech.edu/ndwarshuis3/mdma}.
\chapter{BINDING KINETICS CODE}\label{sec:appendix_binding} \chapter{BINDING KINETICS}\label{sec:appendix_binding}
\newcommand{\lig}{\textit{ligand}}
\newcommand{\rcp}{\textit{receptor}}
\newcommand{\ligs}{\textit{ligands}}
\newcommand{\rcps}{\textit{receptors}}
To model binding kinetics of either \gls{stp} or \glspl{mab} (here called
\ligs{}), each microcarrier was assumed to be a porous sphere with a given
number of binding sites for the \ligs{} (here called \rcps{}). The \rcp{}/\lig{}
reaction was assumed to be instantaneous (which is reasonable given that these
are reactions between biotin and \gls{stp} which are extremely strong). From
this, we further assumed a spherical interface within each microcarrier and
aligned at the center wherein all \rcps{} in the interior were unbound and all
on the exterior were bound. At $t=0$ this interface was assumed to start with a
radius equal to that of the microcarrier, and shrunk down to radius of zero as
\ligs{} flowed into the porous microcarriers and bound. We assumed the
concentration of \lig{} to be zero at the interface and equal to the bulk
concentration at the exterior surface of the microcarrier. Furthermore, we
assumed that the interface move slowly relative to the diffusion of \lig{} into
the microcarriers, and thus we used a quasi-steady-state model to avoid solving
a boundary value problem with two movable boundaries (the interface radius and
the concentration in bulk).
The concentration profile of \lig{} in the microcarriers is given by Fick's
Second Law in spherical coordinates assuming only radial flux and steady state.
This with the boundary conditions as stated is:
\begin{equation}
\label{eqn:binding_ficks}
0 = \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dC_L}{dr}\right)
\end{equation}
\begin{equation}
\label{eqn:binding_bc_left}
C_L\rvert_{r_i} = 0
\end{equation}
\begin{equation}
\label{eqn:binding_bc_right}
C_L\rvert_R = C_{L,b}
\end{equation}
Solving \cref{eqn:binding_ficks} we find the a relation for the concentration
profile in terms of the interfacial radius:
\begin{equation}
\label{eqn:binding_conc}
C_L = \frac{C_{L,b}}{r(1/r_i - 1/R)}\left(\frac{1}{r_i} - \frac{1}{r}\right)
\end{equation}
Solving \cref{eqn:binding_conc} for flux, the molar flow rate into the
microcarriers is given by:
\begin{equation}
\label{eqn:binding_molar_flow}
Q = 4\pi R^2N\rvert_R = \frac{-4\pi DC}{1/r_i - 1/R}
\end{equation}
Using the quasi-steady-state assumption, we can now find time-dependent
equations for the interfatial radius and the bulk concentration. The interfacial
volume in terms of molar flow rate is given by:
\begin{equation}
\label{eqn:binding_volume_change}
C_{R,0}\frac{dV_i}{dt} = -Q
\end{equation}
Substituting volume of a sphere and applying the chain rule:
\begin{equation}
\label{eqn:radial_radial_change}
\frac{dr_i}{dt} = \frac{-Q}{4\pi r_i^2C_{R,0}}
\end{equation}
The change in bulk concentration is simply given by:
\begin{equation}
\label{eqn:radial_conc_change}
\frac{dC_{L,b}}{dt} = \frac{-nQ}{V}
\end{equation}
Substituting \cref{eqn:binding_molar_flow} into \cref{eqn:radial_radial_change}
and \cref{eqn:radial_conc_change} yields \cref{eqn:stp_diffusion_1} and
\cref{eqn:stp_diffusion_2}.
The \gls{stp} binding kinetic profile was fit and calculated using the following The \gls{stp} binding kinetic profile was fit and calculated using the following
MATLAB code. Note that the \inlinecode{geometry} parameter was varied so as to MATLAB code. Note that the \inlinecode{geometry} parameter was varied so as to