ENH use consistent symbols for equations
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tex/thesis.tex
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tex/thesis.tex
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@ -4,7 +4,7 @@
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\usepackage{siunitx}
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\usepackage{siunitx}
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\usepackage{multicol}
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\usepackage{multicol}
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\setlength{\columnsep}{1cm}
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\setlength{\columnsep}{1cm}
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\usepackage[acronym]{glossaries}
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\usepackage[acronym,toc,style=index]{glossaries}
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\usepackage[T1]{fontenc}
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\usepackage[T1]{fontenc}
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\usepackage{enumitem}
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\usepackage{enumitem}
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\usepackage{titlesec}
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\usepackage{titlesec}
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@ -123,7 +123,8 @@
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cells}]{#1}{T\textsubscript{#2}#4}{#3 T cell}
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cells}]{#1}{T\textsubscript{#2}#4}{#3 T cell}
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}
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}
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\renewcommand{\glossarysection}[2][]{} % remove glossary title
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\newglossary[slg]{symbolslist}{syi}{syg}{Symbolslist}
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\makeglossaries
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\makeglossaries
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\tcellacronym{tn}{n}{naive}{}
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\tcellacronym{tn}{n}{naive}{}
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@ -244,6 +245,35 @@
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\newacronym{tocsy}{TOCSY}{total correlation spectroscopy}
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\newacronym{tocsy}{TOCSY}{total correlation spectroscopy}
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\newacronym{hplc}{HPLC}{high-performance liquid chromatography}
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\newacronym{hplc}{HPLC}{high-performance liquid chromatography}
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% symbols to make me sound mathier than I really am
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\newcommand{\evalat}[2]{#1\rvert_{#2}}
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\newcommand{\newsymbol}[3]{
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\newglossaryentry{sym:#1}{name=\ensuremath{#2},
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description={#3},
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type=symbolslist}
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}
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\newsymbol{diff}{D}{diffusion coefficient of ligand}
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\newsymbol{appdiff}{D_{app}}{apparent diffusion coefficient of ligand}
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\newsymbol{geodiff}{\beta}{geometric diffusivity, which is a fractional
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parameter representing the tortuousity and void fraction of the microcarrier}
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\newsymbol{mcligconc}{C_{L,m}}{concentration of ligand in microcarrier}
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\newsymbol{bulkligconc}{C_{L,b}}{concentration of ligand outside microcarriers}
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\newsymbol{mcrecconc}{C_{R,m}}{concentration of receptor inside microcarriers}
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\newsymbol{flowrate}{Q}{molar flow rate of ligand}
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\newsymbol{mcflux}{N_{m}}{flux of ligand in microcarrier}
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\newsymbol{rad}{r}{radial position in the microcarrier}
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\newsymbol{interrad}{r_i}{radius of unbound:bound receptor interface in
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microcarriers}
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\newsymbol{intervol}{V_i}{volume of unbound:bound receptor interface in
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microcarriers}
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\newsymbol{mcrad}{R}{average radius of microcarriers}
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\newsymbol{mcnum}{n}{number of microcarriers in bulk}
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\newsymbol{vol}{V}{volume of bulk liquid in which microcarriers are suspended}
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\newsymbol{time}{t}{time}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% SI units for uber nerds
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% SI units for uber nerds
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@ -466,8 +496,8 @@ question for which you have no words.
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\clearpage
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\clearpage
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\chapter*{acknowledgements}
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\chapter*{ACKNOWLEDGEMENTS}
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\addcontentsline{toc}{chapter}{Acknowledgments}
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\addcontentsline{toc}{chapter}{ACKNOWLEDGMENTS}
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There are many people without which this work would not have been possible.
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There are many people without which this work would not have been possible.
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Firstly, I would like to thank my advisor, Krish Roy, for his mentoring and
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Firstly, I would like to thank my advisor, Krish Roy, for his mentoring and
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@ -510,10 +540,12 @@ for obvious reasons :)
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\clearpage
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\clearpage
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\chapter*{LIST OF SYMBOLS AND ABBREVIATIONS}
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\printglossary[type=\acronymtype,title=LIST OF ABBREVIATIONS,toctitle=LIST OF
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\addcontentsline{toc}{chapter}{LIST OF SYMBOLS AND ABBREVIATIONS}
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ABBREVIATIONS]
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\printglossary[type=\acronymtype]
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\clearpage
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\printglossary[type=symbolslist,title=LIST OF SYMBOLS,toctitle=LIST OF SYMBOLS]
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\clearpage
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\clearpage
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\pagenumbering{arabic}
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\pagenumbering{arabic}
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@ -1669,45 +1701,37 @@ quantified for \gls{stp} protein using the \gls{bca} assay.
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The geometric diffusivity of the microcarriers was determined using a
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The geometric diffusivity of the microcarriers was determined using a
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pseudo-steady-state model. Each microcarrier was assumed to be a porous sphere
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pseudo-steady-state model. Each microcarrier was assumed to be a porous sphere
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with a fixed number of uniformly distributed ``\gls{stp} binding sites'' equal
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with a fixed number of uniformly distributed ``receptors'' equal to the number
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to the number of \gls{stp} molecules experimentally determined to bind to the
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of \gls{stp} molecules (here called ``ligands'') experimentally determined to
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microcarriers. Because the reaction rate between biotin and \gls{stp} is so fast
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bind to the microcarriers. Because the reaction rate between biotin and
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(it is the strongest non-covalent bond in known existence), we assumed that the
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\gls{stp} is so fast (it is the strongest non-covalent bond in known existence),
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interface of free biotin shrunk as a function of \gls{stp} diffusing to the
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we assumed that the interface of unbound receptors (free biotin) shrunk as a
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unbound biotin interface until the center of the microcarriers was reached. We
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function of \gls{stp} diffusing to the unbound biotin interface until the center
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also assumed that the pores in the microcarriers were large enough that the
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of the microcarriers was reached. We also assumed that the pores in the
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interactions between the \gls{stp} and surfaces would be small, thus the
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microcarriers were large enough that the interactions between the \gls{stp} and
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geometric diffusivity could be represented as a fraction of the diffusion
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surfaces would be small, thus the geometric diffusivity could be represented as
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coefficient of \gls{stp} in water. This model was given by
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a fraction of the diffusion coefficient of \gls{stp} in water. This model was
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\cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2} (see \cref{sec:appendix_binding}
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given by \cref{eqn:stp_diffusion_1,eqn:stp_diffusion_2,eqn:stp_diffusion_3}:
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for derivations):
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\begin{equation}
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\begin{equation}
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\label{eqn:stp_diffusion_1}
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\label{eqn:stp_diffusion_1}
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\frac{dr}{dt} = \frac{-D_{app}C_b}{Br(1-r/R)}
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\frac{d\gls{sym:rad}}{d\gls{sym:time}} =
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\frac{- \gls{sym:appdiff} \gls{sym:bulkligconc}}
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{\gls{sym:rad} (1 - \gls{sym:rad} / \gls{sym:mcrad})
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\evalat{\gls{sym:mcrecconc}}{\gls{sym:time} = 0}}
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\end{equation}
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\end{equation}
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\begin{equation}
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\begin{equation}
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\label{eqn:stp_diffusion_2}
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\label{eqn:stp_diffusion_2}
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\frac{dC_b}{dt} = \frac{-4 \pi n D_{app} C_b}{V(1/r-1/R)}
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\frac{d\gls{sym:bulkligconc}}{d\gls{sym:time}} =
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\frac{-4 \pi \gls{sym:mcnum} \gls{sym:appdiff}\gls{sym:bulkligconc}}
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{\gls{sym:vol} (1 / \gls{sym:rad} - 1 / \gls{sym:mcrad})}
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\end{equation}
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\end{equation}
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\noindent where
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\begin{equation}
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\begin{itemize}[label={}]
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\label{eqn:stp_diffusion_3}
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\item $D_{app}$ is the apparent diffusion rate of species $X$ which is equal to
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\gls{sym:appdiff}=\gls{sym:diff} \gls{sym:geodiff}
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$D\beta$
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\end{equation}
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\item $D$ the diffusion rate of species $X$ in water at room temperature
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(where $X$ is \gls{stp} in this example and \glspl{mab} later in this section)
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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 binding sites for species $X$
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within a microcarrier
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\item $t$ is the reaction time
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\item $C_b$ is the concentration of species $X$ in the bulk solution
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\item $V$ is the volume of the bulk medium
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\item $R$ is the average radius of the microcarriers
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\item $n$ is the number of microcarriers in the reaction volume
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\end{itemize}
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The diffusion rate of \gls{stp} was assumed to be
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The diffusion rate of \gls{stp} was assumed to be
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\SI{6.2e-7}{\cm\squared\per\second}\cite{Kamholz2001}. Since all but $\beta$ was
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\SI{6.2e-7}{\cm\squared\per\second}\cite{Kamholz2001}. Since all but $\beta$ was
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@ -1725,39 +1749,33 @@ partial differential equation and boundary conditions:
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\begin{equation}
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\begin{equation}
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\label{eqn:stp_washing}
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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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\frac{\partial \gls{sym:mcligconc}}{\partial \gls{sym:time}} =
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r}\left(r^2 D_{app} \frac{\partial C_i}{\partial r}\right)
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\frac{1}{\gls{sym:rad}^2}
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\frac{\partial}{\partial \gls{sym:rad}}
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\left(\gls{sym:rad}^2
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\gls{sym:appdiff}
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\frac{\partial \gls{sym:mcligconc}}{\partial \gls{sym:rad}}
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\right)
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\end{equation}
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\begin{equation}
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\label{eqn:stp_washing_time_bc}
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\evalat{\gls{sym:mcligconc}}{\gls{sym:time}=0} =
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\evalat{\gls{sym:bulkligconc}}{\gls{sym:time}=0}
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\end{equation}
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\end{equation}
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\begin{equation}
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\begin{equation}
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\label{eqn:stp_washing_left_bc}
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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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\gls{sym:mcflux}\rvert_{\gls{sym:rad}=0} = 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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\end{equation}
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\begin{equation}
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\begin{equation}
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\label{eqn:stp_washing_right_bc}
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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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\evalat{\gls{sym:mcligconc}}{\gls{sym:rad} = \gls{sym:mcrad}} =
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(\evalat{\gls{sym:bulkligconc}}{\gls{sym:time} = 0} +
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\evalat{\gls{sym:bulkligconc}}{\gls{sym:time} = \infty}) / 2
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\end{equation}
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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 species $X$ inside the microcarriers
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\item $C_i$ is the concentration of species $X$ inside the microcarriers
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\item $C_{i,0}$ is the initial concentration of species $X$ 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 species $X$ outside the
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microcarriers after the initial wash volume has been added
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\item $C_{b,\infty}$ is the final bulk concentration of species $X$ outside the
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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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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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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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the final and initial concentration expected to be observed in bulk. This should
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@ -1775,7 +1793,7 @@ water, which is a safe assumption given that our reaction medium was 1X
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\gls{pbs}.
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\gls{pbs}.
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See \cref{sec:appendix_binding} and \cref{sec:appendix_washing} for the MATLAB
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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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code and derivations, as well as output in the case of the washing steps.
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\subsection{Luminex Analysis}\label{sec:luminex_analysis}
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\subsection{Luminex Analysis}\label{sec:luminex_analysis}
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@ -4902,47 +4920,44 @@ The code is available here: \url{https://github.gatech.edu/ndwarshuis3/mdma}.
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\chapter{BINDING KINETICS}\label{sec:appendix_binding}
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\chapter{BINDING KINETICS}\label{sec:appendix_binding}
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\newcommand{\lig}{\textit{ligand}}
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The binding kinetics of \gls{stp} or \glspl{mab} were simulated using a
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\newcommand{\rcp}{\textit{receptor}}
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receptor:ligand model, where the free-floating species in question was the
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\newcommand{\ligs}{\textit{ligands}}
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ligand which bound to receptors attached to the microcarriers. Each microcarrier
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\newcommand{\rcps}{\textit{receptors}}
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was assumed to be a porous sphere with a fixed number of receptors uniformly
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distributed throughout its interior matrix. The receptor/ligand reaction was
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assumed to be instantaneous (which is reasonable given that these are reactions
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between biotin and \gls{stp} which are extremely strong). From this, we further
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assumed a spherical interface within each microcarrier and aligned at the center
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wherein all receptors in the interior were unbound and all on the exterior were
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bound. At $\gls{sym:time}=0$ this interface was assumed to start with a radius
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equal to that of the microcarrier, and shrunk down to radius of zero as ligand
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flowed into the porous microcarriers and bound. We assumed the concentration of
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ligand to be zero at the interface and equal to the bulk concentration at the
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exterior surface of the microcarrier. Furthermore, we assumed that the interface
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moved slowly relative to the diffusion of ligand into the microcarriers, and
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thus we used a quasi-steady-state model to avoid solving a boundary value
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problem with two movable boundaries (the interface radius and the concentration
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in bulk).
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% TODO make notation consistent
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The concentration profile of ligand in the microcarriers is given by Fick's
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To model binding kinetics of either \gls{stp} or \glspl{mab} (here called
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\ligs{}), each microcarrier was assumed to be a porous sphere with a given
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number of binding sites for the \ligs{} (here called \rcps{}). The \rcp{}/\lig{}
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reaction was assumed to be instantaneous (which is reasonable given that these
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are reactions between biotin and \gls{stp} which are extremely strong). From
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this, we further assumed a spherical interface within each microcarrier and
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aligned at the center wherein all \rcps{} in the interior were unbound and all
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on the exterior were bound. At $t=0$ this interface was assumed to start with a
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radius equal to that of the microcarrier, and shrunk down to radius of zero as
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\ligs{} flowed into the porous microcarriers and bound. We assumed the
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concentration of \lig{} to be zero at the interface and equal to the bulk
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concentration at the exterior surface of the microcarrier. Furthermore, we
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assumed that the interface move slowly relative to the diffusion of \lig{} into
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the microcarriers, and thus we used a quasi-steady-state model to avoid solving
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a boundary value problem with two movable boundaries (the interface radius and
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the concentration in bulk).
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The concentration profile of \lig{} in the microcarriers is given by Fick's
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Second Law in spherical coordinates assuming only radial flux and steady state.
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Second Law in spherical coordinates assuming only radial flux and steady state.
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This with the boundary conditions as stated is:
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This with the boundary conditions as stated is:
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\begin{equation}
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\begin{equation}
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\label{eqn:binding_ficks}
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\label{eqn:binding_ficks}
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0 = \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dC_L}{dr}\right)
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0 = \frac{1}{\gls{sym:rad}^2} \frac{d}{d\gls{sym:rad}} \left( \gls{sym:rad}^2
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\frac{d\gls{sym:mcligconc}}{d\gls{sym:rad}} \right)
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\end{equation}
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\end{equation}
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\begin{equation}
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\begin{equation}
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\label{eqn:binding_bc_left}
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\label{eqn:binding_bc_left}
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C_L\rvert_{r_i} = 0
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\evalat{\gls{sym:mcligconc}}{\gls{sym:rad}=\gls{sym:interrad}} = 0
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\end{equation}
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\end{equation}
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\begin{equation}
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\begin{equation}
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\label{eqn:binding_bc_right}
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\label{eqn:binding_bc_right}
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C_L\rvert_R = C_{L,b}
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\evalat{\gls{sym:mcligconc}}{\gls{sym:rad}=\gls{sym:mcrad}} =
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\gls{sym:bulkligconc}
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\end{equation}
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\end{equation}
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Solving \cref{eqn:binding_ficks} we find the a relation for the concentration
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Solving \cref{eqn:binding_ficks} we find the a relation for the concentration
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@ -4950,7 +4965,10 @@ profile in terms of the interfacial radius:
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\begin{equation}
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\begin{equation}
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\label{eqn:binding_conc}
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\label{eqn:binding_conc}
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C_L = \frac{C_{L,b}}{r(1/r_i - 1/R)}\left(\frac{1}{r_i} - \frac{1}{r}\right)
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\gls{sym:mcligconc} =
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\frac{\gls{sym:bulkligconc}}
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{(1 / \gls{sym:interrad} - 1 / \gls{sym:mcrad})}
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\left( \frac{1}{\gls{sym:interrad}} - \frac{1}{\gls{sym:rad}} \right)
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\end{equation}
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\end{equation}
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Solving \cref{eqn:binding_conc} for flux, the molar flow rate into the
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Solving \cref{eqn:binding_conc} for flux, the molar flow rate into the
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@ -4958,7 +4976,10 @@ microcarriers is given by:
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\begin{equation}
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\begin{equation}
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\label{eqn:binding_molar_flow}
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\label{eqn:binding_molar_flow}
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Q = 4\pi R^2N\rvert_R = \frac{-4\pi DC}{1/r_i - 1/R}
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\gls{sym:flowrate} = 4 \pi \gls{sym:mcrad}^2
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\evalat{\gls{sym:mcflux}}{\gls{sym:rad} = \gls{sym:mcrad}} =
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\frac{-4 \pi \gls{sym:appdiff} \gls{sym:bulkligconc}}
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{1 / \gls{sym:interrad} - 1 / \gls{sym:mcrad}}
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\end{equation}
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\end{equation}
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Using the quasi-steady-state assumption, we can now find time-dependent
|
Using the quasi-steady-state assumption, we can now find time-dependent
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@ -4967,21 +4988,25 @@ volume in terms of molar flow rate is given by:
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\begin{equation}
|
\begin{equation}
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\label{eqn:binding_volume_change}
|
\label{eqn:binding_volume_change}
|
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C_{R,0}\frac{dV_i}{dt} = -Q
|
\evalat{\gls{sym:mcrecconc}}{\gls{sym:time} = 0}
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\frac{d\gls{sym:intervol}}{d\gls{sym:time}} = -\gls{sym:flowrate}
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\end{equation}
|
\end{equation}
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||||||
|
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||||||
Substituting volume of a sphere and applying the chain rule:
|
Substituting volume of a sphere and applying the chain rule:
|
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|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\label{eqn:radial_radial_change}
|
\label{eqn:radial_radial_change}
|
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\frac{dr_i}{dt} = \frac{-Q}{4\pi r_i^2C_{R,0}}
|
\frac{d\gls{sym:interrad}}{d\gls{sym:time}} =
|
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|
\frac{-\gls{sym:flowrate}}
|
||||||
|
{4 \pi \gls{sym:interrad}^2 \evalat{\gls{sym:mcrecconc}}{\gls{sym:time} = 0}}
|
||||||
\end{equation}
|
\end{equation}
|
||||||
|
|
||||||
The change in bulk concentration is simply given by:
|
The change in bulk concentration is simply given by:
|
||||||
|
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
\label{eqn:radial_conc_change}
|
\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}
|
\end{equation}
|
||||||
|
|
||||||
Substituting \cref{eqn:binding_molar_flow} into \cref{eqn:radial_radial_change}
|
Substituting \cref{eqn:binding_molar_flow} into \cref{eqn:radial_radial_change}
|
||||||
|
|
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Reference in New Issue