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ENH use consistent symbols for equations

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Nathan Dwarshuis 2021-09-07 18:40:37 -04:00
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tex/thesis.tex
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@ -4,7 +4,7 @@
\usepackage{siunitx}
\usepackage{multicol}
\setlength{\columnsep}{1cm}
\usepackage[acronym]{glossaries}
\usepackage[acronym,toc,style=index]{glossaries}
\usepackage[T1]{fontenc}
\usepackage{enumitem}
\usepackage{titlesec}
@ -123,7 +123,8 @@
cells}]{#1}{T\textsubscript{#2}#4}{#3 T cell}
}
\renewcommand{\glossarysection}[2][]{} % remove glossary title
\newglossary[slg]{symbolslist}{syi}{syg}{Symbolslist}
\makeglossaries
\tcellacronym{tn}{n}{naive}{}
@ -244,6 +245,35 @@
\newacronym{tocsy}{TOCSY}{total correlation spectroscopy}
\newacronym{hplc}{HPLC}{high-performance liquid chromatography}
% symbols to make me sound mathier than I really am
\newcommand{\evalat}[2]{#1\rvert_{#2}}
\newcommand{\newsymbol}[3]{
\newglossaryentry{sym:#1}{name=\ensuremath{#2},
description={#3},
type=symbolslist}
}
\newsymbol{diff}{D}{diffusion coefficient of ligand}
\newsymbol{appdiff}{D_{app}}{apparent diffusion coefficient of ligand}
\newsymbol{geodiff}{\beta}{geometric diffusivity, which is a fractional
parameter representing the tortuousity and void fraction of the microcarrier}
\newsymbol{mcligconc}{C_{L,m}}{concentration of ligand in microcarrier}
\newsymbol{bulkligconc}{C_{L,b}}{concentration of ligand outside microcarriers}
\newsymbol{mcrecconc}{C_{R,m}}{concentration of receptor inside microcarriers}
\newsymbol{flowrate}{Q}{molar flow rate of ligand}
\newsymbol{mcflux}{N_{m}}{flux of ligand in microcarrier}
\newsymbol{rad}{r}{radial position in the microcarrier}
\newsymbol{interrad}{r_i}{radius of unbound:bound receptor interface in
microcarriers}
\newsymbol{intervol}{V_i}{volume of unbound:bound receptor interface in
microcarriers}
\newsymbol{mcrad}{R}{average radius of microcarriers}
\newsymbol{mcnum}{n}{number of microcarriers in bulk}
\newsymbol{vol}{V}{volume of bulk liquid in which microcarriers are suspended}
\newsymbol{time}{t}{time}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SI units for uber nerds
@ -466,8 +496,8 @@ question for which you have no words.
\clearpage
\chapter*{acknowledgements}
\addcontentsline{toc}{chapter}{Acknowledgments}
\chapter*{ACKNOWLEDGEMENTS}
\addcontentsline{toc}{chapter}{ACKNOWLEDGMENTS}
There are many people without which this work would not have been possible.
Firstly, I would like to thank my advisor, Krish Roy, for his mentoring and
@ -510,10 +540,12 @@ for obvious reasons :)
\clearpage
\chapter*{LIST OF SYMBOLS AND ABBREVIATIONS}
\addcontentsline{toc}{chapter}{LIST OF SYMBOLS AND ABBREVIATIONS}
\printglossary[type=\acronymtype,title=LIST OF ABBREVIATIONS,toctitle=LIST OF
ABBREVIATIONS]
\printglossary[type=\acronymtype]
\clearpage
\printglossary[type=symbolslist,title=LIST OF SYMBOLS,toctitle=LIST OF SYMBOLS]
\clearpage
\pagenumbering{arabic}
@ -1669,45 +1701,37 @@ quantified for \gls{stp} protein using the \gls{bca} assay.
The geometric diffusivity of the microcarriers was determined using a
pseudo-steady-state model. Each microcarrier was assumed to be a porous sphere
with a fixed number of uniformly distributed ``\gls{stp} binding sites'' equal
to the number of \gls{stp} molecules experimentally determined to bind to the
microcarriers. Because the reaction rate between biotin and \gls{stp} is so fast
(it is the strongest non-covalent bond in known existence), we assumed that the
interface of free biotin shrunk as a function of \gls{stp} diffusing to the
unbound biotin interface until 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} and surfaces would be small, thus the
geometric diffusivity could be represented as a fraction of the diffusion
coefficient of \gls{stp} in water. This model was given by
\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2} (see \cref{sec:appendix_binding}
for derivations):
with a fixed number of uniformly distributed ``receptors'' equal to the number
of \gls{stp} molecules (here called ``ligands'') experimentally determined to
bind to the microcarriers. Because the reaction rate between biotin and
\gls{stp} is so fast (it is the strongest non-covalent bond in known existence),
we assumed that the interface of unbound receptors (free biotin) shrunk as a
function of \gls{stp} diffusing to the unbound biotin interface until 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} and
surfaces would be small, thus the geometric diffusivity could be represented as
a fraction of the diffusion coefficient of \gls{stp} in water. This model was
given by \cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2,eqn:stp_diffusion_3}:
\begin{equation}
\label{eqn:stp_diffusion_1}
\frac{dr}{dt} = \frac{-D_{app}C_b}{Br(1-r/R)}
\frac{d\gls{sym:rad}}{d\gls{sym:time}} =
\frac{- \gls{sym:appdiff} \gls{sym:bulkligconc}}
{\gls{sym:rad} (1 - \gls{sym:rad} / \gls{sym:mcrad})
\evalat{\gls{sym:mcrecconc}}{\gls{sym:time} = 0}}
\end{equation}
\begin{equation}
\label{eqn:stp_diffusion_2}
\frac{dC_b}{dt} = \frac{-4 \pi n D_{app} C_b}{V(1/r-1/R)}
\frac{d\gls{sym:bulkligconc}}{d\gls{sym:time}} =
\frac{-4 \pi \gls{sym:mcnum} \gls{sym:appdiff}\gls{sym:bulkligconc}}
{\gls{sym:vol} (1 / \gls{sym:rad} - 1 / \gls{sym:mcrad})}
\end{equation}
\noindent where
\begin{itemize}[label={}]
\item $D_{app}$ is the apparent diffusion rate of species $X$ which is equal to
$D\beta$
\item $D$ the diffusion rate of species $X$ in water at room temperature
(where $X$ is \gls{stp} in this example and \glspl{mab} later in this section)
\item $\beta$ a fractional parameter representing the tortuousity and void
fraction of the microcarriers (here called the `geometric diffusivity')
\item $r$ is the interfatial radius of the unbound binding sites for species $X$
within a microcarrier
\item $t$ is the reaction time
\item $C_b$ is the concentration of species $X$ in the bulk solution
\item $V$ is the volume of the bulk medium
\item $R$ is the average radius of the microcarriers
\item $n$ is the number of microcarriers in the reaction volume
\end{itemize}
\begin{equation}
\label{eqn:stp_diffusion_3}
\gls{sym:appdiff}=\gls{sym:diff} \gls{sym:geodiff}
\end{equation}
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
@ -1725,39 +1749,33 @@ 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)
\frac{\partial \gls{sym:mcligconc}}{\partial \gls{sym:time}} =
\frac{1}{\gls{sym:rad}^2}
\frac{\partial}{\partial \gls{sym:rad}}
\left(\gls{sym:rad}^2
\gls{sym:appdiff}
\frac{\partial \gls{sym:mcligconc}}{\partial \gls{sym:rad}}
\right)
\end{equation}
\begin{equation}
\label{eqn:stp_washing_time_bc}
\evalat{\gls{sym:mcligconc}}{\gls{sym:time}=0} =
\evalat{\gls{sym:bulkligconc}}{\gls{sym:time}=0}
\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
\gls{sym:mcflux}\rvert_{\gls{sym:rad}=0} = 0
\end{equation}
\begin{equation}
\label{eqn:stp_washing_right_bc}
C_i(R, t) = (C_{b,0}+C_{b,\infty}) / 2
\evalat{\gls{sym:mcligconc}}{\gls{sym:rad} = \gls{sym:mcrad}} =
(\evalat{\gls{sym:bulkligconc}}{\gls{sym:time} = 0} +
\evalat{\gls{sym:bulkligconc}}{\gls{sym:time} = \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 species $X$ inside the microcarriers
\item $C_i$ is the concentration of species $X$ inside the microcarriers
\item $C_{i,0}$ is the initial concentration of species $X$ 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 species $X$ outside the
microcarriers after the initial wash volume has been added
\item $C_{b,\infty}$ is the final bulk concentration of species $X$ 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
@ -1775,7 +1793,7 @@ water, which is a safe assumption given that our reaction medium was 1X
\gls{pbs}.
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 derivations, as well as output in the case of the washing steps.
\subsection{Luminex Analysis}\label{sec:luminex_analysis}
@ -4902,47 +4920,44 @@ The code is available here: \url{https://github.gatech.edu/ndwarshuis3/mdma}.
\chapter{BINDING KINETICS}\label{sec:appendix_binding}
\newcommand{\lig}{\textit{ligand}}
\newcommand{\rcp}{\textit{receptor}}
\newcommand{\ligs}{\textit{ligands}}
\newcommand{\rcps}{\textit{receptors}}
The binding kinetics of \gls{stp} or \glspl{mab} were simulated using a
receptor:ligand model, where the free-floating species in question was the
ligand which bound to receptors attached to the microcarriers. Each microcarrier
was assumed to be a porous sphere with a fixed number of receptors uniformly
distributed throughout its interior matrix. The receptor/ligand 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 receptors in the interior were unbound and all on the exterior were
bound. At $\gls{sym:time}=0$ this interface was assumed to start with a radius
equal to that of the microcarrier, and shrunk down to radius of zero as ligand
flowed into the porous microcarriers and bound. We assumed the concentration of
ligand 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
moved slowly relative to the diffusion of ligand 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).
% TODO make notation consistent
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
The concentration profile of ligand 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)
0 = \frac{1}{\gls{sym:rad}^2} \frac{d}{d\gls{sym:rad}} \left( \gls{sym:rad}^2
\frac{d\gls{sym:mcligconc}}{d\gls{sym:rad}} \right)
\end{equation}
\begin{equation}
\label{eqn:binding_bc_left}
C_L\rvert_{r_i} = 0
\evalat{\gls{sym:mcligconc}}{\gls{sym:rad}=\gls{sym:interrad}} = 0
\end{equation}
\begin{equation}
\label{eqn:binding_bc_right}
C_L\rvert_R = C_{L,b}
\evalat{\gls{sym:mcligconc}}{\gls{sym:rad}=\gls{sym:mcrad}} =
\gls{sym:bulkligconc}
\end{equation}
Solving \cref{eqn:binding_ficks} we find the a relation for the concentration
@ -4950,7 +4965,10 @@ 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)
\gls{sym:mcligconc} =
\frac{\gls{sym:bulkligconc}}
{(1 / \gls{sym:interrad} - 1 / \gls{sym:mcrad})}
\left( \frac{1}{\gls{sym:interrad}} - \frac{1}{\gls{sym:rad}} \right)
\end{equation}
Solving \cref{eqn:binding_conc} for flux, the molar flow rate into the
@ -4958,7 +4976,10 @@ 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}
\gls{sym:flowrate} = 4 \pi \gls{sym:mcrad}^2
\evalat{\gls{sym:mcflux}}{\gls{sym:rad} = \gls{sym:mcrad}} =
\frac{-4 \pi \gls{sym:appdiff} \gls{sym:bulkligconc}}
{1 / \gls{sym:interrad} - 1 / \gls{sym:mcrad}}
\end{equation}
Using the quasi-steady-state assumption, we can now find time-dependent
@ -4967,21 +4988,25 @@ 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
\evalat{\gls{sym:mcrecconc}}{\gls{sym:time} = 0}
\frac{d\gls{sym:intervol}}{d\gls{sym:time}} = -\gls{sym:flowrate}
\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}}
\frac{d\gls{sym:interrad}}{d\gls{sym:time}} =
\frac{-\gls{sym:flowrate}}
{4 \pi \gls{sym:interrad}^2 \evalat{\gls{sym:mcrecconc}}{\gls{sym:time} = 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}
\frac{d\gls{sym:bulkligconc}}{d\gls{sym:time}} =
\frac{-\gls{sym:mcnum}\gls{sym:flowrate}}{\gls{sym:vol}}
\end{equation}
Substituting \cref{eqn:binding_molar_flow} into \cref{eqn:radial_radial_change}