\usepackage{booktabs}
\usepackage[noblocks]{authblk}
\usepackage[hyperfigures,bookmarks,colorlinks,citecolor=black,filecolor=black,linkcolor=black,urlcolor=black]{hyperref}
+\usepackage[capitalise]{cleveref}
\usepackage[sectionbib,sort&compress,numbers]{natbib}
\usepackage[nomargin,inline,draft]{fixme}
\usepackage[x11names,svgnames]{xcolor}
\newcommand{\OM}[1]{\textcolor{red}{\fxnote{OM: #1}}}
-%\SweaveOpts{prefix.string=pub_318_phys_bio_submission_figures_suppl/pub_318_phys_bio_sub_suppl_fig,pdf=TRUE,eps=TRUE}
\oddsidemargin 0.0in
\textwidth 6.5in
\raggedbottom
\begin{document}
\maketitle
-<<results=hide,echo=FALSE>>=
-require(lattice)
-require(grid)
-require(Hmisc)
-require(gridBase)
+<<load.libraries,echo=FALSE,results="hide",warning=FALSE,message=FALSE,error=FALSE,cache=FALSE>>=
+opts_chunk$set(dev="CairoPDF",out.width="\\columnwidth",out.height="0.7\\textheight",out.extra="keepaspectratio")
+opts_chunk$set(cache=TRUE, autodep=TRUE)
+options(scipen = -2, digits = 1)
+library("lattice")
+library("grid")
+library("Hmisc")
+library("gridBase")
to.latex <- function(x){
gsub("\\\\","\\\\\\\\",latexSN(x))
}
\renewcommand{\thetable}{S\@arabic\c@table}
\makeatother
-\section{Competition Implementation}
-\subsection{Implementation changes}
-
-\begin{itemize}
-\item settable maximum number of vesicles to track (default $10^4$)
-\item start with 1~L ($10^{-3}$~m$^3$) cube
-\item if at any point the number of vesicles exceeds the maximum
- number, chop the volume and environment molecule number into tenths,
- randomly select one tenth of the vesicles, and continue tracking.
-\item generations will be counted per vesicle, and each progeny
- vesicle will have a generation number one greater than the parental
- vesicle.
-\item 100 generations can result in as many as $2^{100}$
- ($\Sexpr{to.latex(format(digits=3,2^100))}$) vesicles or as few as
- 101 vesicles.
-\item Environment will use a specific number of each component instead
- of a constant concentration; as the number may be larger than
- \texttt{long long} ($2^{64}$), we use libgmp to handle an arbitrary
- precision number of components
-\end{itemize}
-
-\subsection{Infrastructure changes}
-\begin{itemize}
-\item Rewrite core bits in C
-\item Use libgmp for handling large ints
-\item Use openmpi to split the calculations out over multiple
- machines/processors and allow deploying to larger
- clusters/supercomputers
-\end{itemize}
+% \section{Competition Implementation}
+% \subsection{Implementation changes}
+%
+% \begin{itemize}
+% \item settable maximum number of vesicles to track (default $10^4$)
+% \item start with 1~L ($10^{-3}$~m$^3$) cube
+% \item if at any point the number of vesicles exceeds the maximum
+% number, chop the volume and environment molecule number into tenths,
+% randomly select one tenth of the vesicles, and continue tracking.
+% \item generations will be counted per vesicle, and each progeny
+% vesicle will have a generation number one greater than the parental
+% vesicle.
+% \item 100 generations can result in as many as $2^{100}$
+% ($\Sexpr{2^100}$) vesicles or as few as
+% 101 vesicles.
+% \item Environment will use a specific number of each component instead
+% of a constant concentration; as the number may be larger than
+% \texttt{long long} ($2^{64}$), we use libgmp to handle an arbitrary
+% precision number of components
+% \end{itemize}
+%
+% \subsection{Infrastructure changes}
+% \begin{itemize}
+% \item Rewrite core bits in C
+% \item Use libgmp for handling large ints
+% \item Use openmpi to split the calculations out over multiple
+% machines/processors and allow deploying to larger
+% clusters/supercomputers
+% \end{itemize}
and is dependent on the particular lipid type (PC, PS, SM, etc.). The
forward adjustment parameter, $k_{\mathrm{f}i\mathrm{adj}}$, is based on the
properties of the vesicle and the specific component (type, length,
-unsaturation, etc.) (see Equation~\ref{eq:kf_adj}, and
-Section~\ref{sec:kinetic_adjustments}).
+unsaturation, etc.) (see \cref{eq:kf_adj,sec:kinetic_adjustments}).
$\left[C_{i_\mathrm{monomer}}\right]$ is the molar concentration of
monomer of the $i$th component. $\left[S_\mathrm{vesicle}\right]$ is the surface
area of the vesicle per volume. The base backwards kinetic parameter
for the $i$th component is $k_{\mathrm{b}i}$ and its adjustment parameter
-$k_{\mathrm{b}i\mathrm{adj}}$ (see Equation~\ref{eq:kb_adj}, and
-Section~\ref{sec:kinetic_adjustments}).
+$k_{\mathrm{b}i\mathrm{adj}}$ (see \cref{eq:kb_adj,sec:kinetic_adjustments}).
$\left[C_{i_\mathrm{vesicle}}\right]$ is the molar concentration of
the $i$th component in the vesicle.
\subsection{Per-Lipid Kinetic Parameters}
-<<echo=FALSE,results=hide>>=
+<<kf_prime,echo=FALSE,results='hide'>>=
kf.prime <- c(3.7e6,3.7e6,5.1e7,3.7e6,2.3e6)
kf <- (as.numeric(kf.prime)*10^-3)/(63e-20*6.022e23)
@
% \centering
% \begin{tabular}{c c c c c c c c}
% \toprule
-% Type & $k_\mathrm{f}$ $\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$ & $k'_\mathrm{f}$ $\left(\frac{1}{\mathrm{M} \mathrm{s}}\right)$ & $k_\mathrm{b}$ $\left(\mathrm{s}^{-1}\right)$ & Area $\left({\AA}^2\right)$ & Charge & $\mathrm{CF}1$ & Curvature \\
+% Type & $k_\mathrm{f}$ $\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$ & $k'_\mathrm{f}$ $\left(\frac{1}{\mathrm{M} \mathrm{s}}\right)$ & $k_\mathrm{b}$ $\left(\mathrm{s}^{-1}\right)$ & Area $\left({Å}^2\right)$ & Charge & $\mathrm{CF}1$ & Curvature \\
% \midrule
% PC & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[1]))}$ & $3.7 \times 10^6$ & $2 \times 10^{-5}$ & 63 & 0 & 2 & 0.8 \\
% PS & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[2]))}$ & $3.7 \times 10^6$ & $1.25\times 10^{-5}$ & 54 & -1 & 0 & 1 \\
% \end{table}
%%% \DLA{I think we may just reduce these three sections; area, $k_\mathrm{f}$
-%%% and $k_\mathrm{b}$ to Table~\ref{tab:kinetic_parameters_lipid_types} with
+%%% and $k_\mathrm{b}$ to \cref{tab:kinetic_parameters_lipid_types} with
%%% references.}
%%%
%%% \RZ{Yes, but we also have to have then as comments the numbers that
to be 9.6 hr. As this is a first order reaction, and the primary
limiting step in exchange is lipid desorption, $k_\mathrm{b}$ for DMPC is
$k_{\mathrm{b}_\mathrm{PC}}=\frac{\log 2}{9.6 \times 60 \times 60} \approx
-\Sexpr{to.latex(format(log(2)/(9.6*60*60),digits=2))}
+\Sexpr{log(2)/(9.6*60*60)}
\mathrm{s}^{-1}$. We assume that $k_\mathrm{b}$ for SM is the same as for PC.
To estimate the $k_\mathrm{b}$ of PE and PS, we used the data from
\citet{Nichols1982:ret_amphiphile_transfer} who measured the rate of
\frac{k_{\mathrm{b}_\mathrm{PC}}\times\mathrm{PE}}{\mathrm{PC}} \approx
\frac{2\times 10^{-5}\,\mathrm{s}^{-1} \times
0.45\,\mathrm{min}^{-1}}{0.89\,\mathrm{min}^{-1}} \approx
-\Sexpr{to.latex(format(log(2)/(9.6*60*60)*0.45/0.89,digits=2))}$~$\mathrm{s}^{-1}$
+\Sexpr{log(2)/(9.6*60*60)*0.45/0.89}$~$\mathrm{s}^{-1}$
and likewise, $k_{\mathrm{b}_\mathrm{PS}}\approx
-\Sexpr{to.latex(format(log(2)/(9.6*60*60)*0.55/0.89,digits=3))}$~$\mathrm{s}^{-1}$.
+\Sexpr{log(2)/(9.6*60*60)*0.55/0.89}$~$\mathrm{s}^{-1}$.
The $k_\mathrm{b}$ of SM was determined using the work of
\citet{Bai1997:lipid_movementbodipy}, who measured spontaneous
transfer of C$_5$-DMB-SM and C$_5$-DMB-PC from donor and acceptor
$2.2\times10^{-3}$~$\mathrm{s}^{-1}$ respectively; using the ratio of
$k_\mathrm{b}$ of C$_5$-DMB-SM to the $k_\mathrm{b}$ of C$_5$-DMB-PC times the $k_\mathrm{b}$ of
PC ($\frac{3.4 \times 10^{-2}\mathrm{s}^{-1}}{2.2 \times
- 10^{-3}\mathrm{s}^{-1}}
-\Sexpr{to.latex(format(log(2)/(9.6*60*60),digits=2))}\mathrm{s}^{-1}$,
+ 10^{-3}\mathrm{s}^{-1}}\approx
+\Sexpr{log(2)/(9.6*60*60)}\mathrm{s}^{-1}$),
we obtain $k_{\mathrm{b}_\mathrm{SM}} \approx
-\Sexpr{to.latex(format(log(2)/(9.6*60*60)*3.4e-2/2.2e-3,digits=2,scientific=TRUE))}$.
+\Sexpr{log(2)/(9.6*60*60)*3.4e-2/2.2e-3}$.
In the case of CHOL, \citet{Jones1990:lipid_transfer} measured the
$t_{1/2}$ of [$^3$H] transfer from POPC vesicles and found it to be 41
minutes, leading to a $k_{\mathrm{b}_\mathrm{CHOL}} = \frac{\log 2}{41\times
60} \approx
-\Sexpr{to.latex(format(log(2)/(41*60),digits=2,scientific=TRUE))}$~$\mathrm{s}^{-1}$.
+\Sexpr{log(2)/(41*60)}$~$\mathrm{s}^{-1}$.
\subsubsection{Headgroup Surface Area for lipid types}
Different lipids have different headgroup surface areas, which contributes to
$\left[S_\mathrm{vesicle}\right]$. \citet{Smaby1997:pc_area_with_chol}
measured the surface area of POPC with a Langmuir film balance, and
-found it to be 63~$\AA^2$ at $30$~$\frac{\mathrm{mN}}{\mathrm{m}}$.
-Molecular dynamic simulations found an area of 54 $\AA^2$ for
+found it to be 63~Å$^2$ at $30$~$\frac{\mathrm{mN}}{\mathrm{m}}$.
+Molecular dynamic simulations found an area of 54 Å$^2$ for
DPPS\citep{Cascales1996:mds_dpps_area,Pandit2002:mds_dpps}, which is
-in agreement with the experimental value of 56~$\AA^2$ found using a
+in agreement with the experimental value of 56~Å$^2$ found using a
Langmuir balance by \citet{Demel1987:ps_area}.
\citet{Shaikh2002:pe_phase_sm_area} measured the area of SM using a
-Langmuir film balance, and found it to be 61~$\AA^2$. Using $^2$H NMR,
+Langmuir film balance, and found it to be 61~Å$^2$. Using $^2$H NMR,
\citet{Thurmond1991:area_of_pc_pe_2hnmr} found the area of
-DPPE-d$_{62}$ to be 55.4 $\AA^2$. \citet{Robinson1995:mds_chol_area}
-found an area for CHOL of 38~$\AA^2$ using molecular dynamic
+DPPE-d$_{62}$ to be 55.4 Å$^2$. \citet{Robinson1995:mds_chol_area}
+found an area for CHOL of 38~Å$^2$ using molecular dynamic
simulations.
% robinson's chol area is kind of crappy; they did it using MDS, but
simulations is around $1.5$, which leads to a $\Delta \Delta
G^\ddagger$ of $\Sexpr{to.latex(format(digits=3,to.kcal(2^1.5)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$, and a total range of possible
-values depicted in Figure~\ref{fig:unf_graph}.
+values depicted in \cref{fig:unf_graph}.
% \RZ{Explain here, or even earlier that the formulas were ad hoc
% adjusted to correspond to ``reasonable'' changes in the Eyring
% not include a term for it in this formalism.
-\setkeys{Gin}{width=6.4in}
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=10,height=5>>=
+<<unf_graph,echo=FALSE,results='hide',fig.width=10,fig.height=5,cache=FALSE>>=
layout(matrix(1:2,nrow=1,ncol=2))
curve(2^x,from=0,to=sd(c(0,4)),
xlab=expression(paste(stdev,group("(",italic(un[vesicle]),")"))),
$\Sexpr{format(digits=3,to.kcal(60^(-.165*-1)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$ to
$0\frac{\mathrm{kcal}}{\mathrm{mol}}$, and the total range of possible
-values seen in Figure~\ref{fig:chf_graph}.
+values seen in \cref{fig:chf_graph}.
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=7>>=
+<<chf_graph,echo=FALSE,results='hide',fig.width=14,fig.height=7>>=
trellis.device(new=F,color=TRUE)
trellis.par.set(list(axis.line =list(col="transparent")))
pushViewport(viewport(layout=grid.layout(nrow=1,ncol=2),clip="off"))
of $\Sexpr{format(digits=3,to.kcal(10^(0.13*0.213)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$. This is a consequence of the
relatively matched curvatures in our environment. The full range of
-$cu_\mathrm{f}$ values possible are shown in Figure~\ref{fig:cuf_graph}.
+$cu_\mathrm{f}$ values possible are shown in \cref{fig:cuf_graph}.
% 1.5 to 0.75 3 to 0.33
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=7>>=
+<<cuf_graph,echo=FALSE,results='hide',fig.width=14,fig.height=7>>=
trellis.device(new=F,color=TRUE)
trellis.par.set(list(axis.line =list(col="transparent")))
pushViewport(viewport(layout=grid.layout(nrow=1,ncol=2),clip="off"))
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=5>>=
+<<lf_graph,echo=FALSE,results='hide',fig.width=14,fig.height=5>>=
layout(matrix(1:2,nrow=1,ncol=2))
curve(2^x,from=0,to=sd(c(12,24)),
xlab=expression(paste(stdev,group("(",italic(l[vesicle]),")"))),
\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with 0 unsaturation
to
$\Sexpr{format(digits=3,to.kcal(7^(1-1/(5*(2^-1.7-2^-4)^2+1))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
-for monomers with 4 unsaturations. See Figure~\ref{fig:unb_graph} for
+for monomers with 4 unsaturations. See \cref{fig:unb_graph} for
the full range of possible values.
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=7>>=
+<<unb_graph,echo=FALSE,results='hide',fig.width=14,fig.height=7>>=
trellis.device(new=F,color=TRUE)
trellis.par.set(list(axis.line =list(col="transparent")))
pushViewport(viewport(layout=grid.layout(nrow=1,ncol=2),clip="off"))
$\Sexpr{format(digits=3,to.kcal(20^(-.164*-1)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with charge $-1$ to
$0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with charge $0$.
-See Figure~\ref{fig:chb_graph} for the full range of possible values
+See \cref{fig:chb_graph} for the full range of possible values
of $ch_\mathrm{b}$.
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=7>>=
+<<chb_graph,echo=FALSE,results='hide',fig.width=14,fig.height=7>>=
trellis.device(new=F,color=TRUE)
trellis.par.set(list(axis.line =list(col="transparent")))
pushViewport(viewport(layout=grid.layout(nrow=1,ncol=2),clip="off"))
for monomers with curvature 1.3 to
$0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature near
1. The full range of values possible for $cu_\mathrm{b}$ are shown in
-Figure~\ref{fig:cub_graph}
+\cref{fig:cub_graph}
% \RZ{What about the opposite curvatures that actually do fit to each
% other?}
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=7>>=
+<<cub_graph,echo=FALSE,results='hide',fig.width=14,fig.height=7>>=
trellis.device(new=F,color=TRUE)
trellis.par.set(list(axis.line =list(col="transparent")))
pushViewport(viewport(layout=grid.layout(nrow=1,ncol=2),clip="off"))
$\Sexpr{format(digits=3,to.kcal(3.2^abs(24-17.75)))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
for monomers with length 24 to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$
for monomers with length near 18. The full range of possible values of
-$l_\mathrm{b}$ are shown in Figure~\ref{fig:lb_graph}
+$l_\mathrm{b}$ are shown in \cref{fig:lb_graph}
% (for methods? From McLean84LIB: The activation free energies and free
% energies of transfer from self-micelles to water increase by 2.2 and
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=7>>=
+<<lb_graph,echo=FALSE,results='hide',fig.width=14,fig.height=7>>=
trellis.device(new=F,color=TRUE)
trellis.par.set(list(axis.line =list(col="transparent")))
pushViewport(viewport(layout=grid.layout(nrow=1,ncol=2),clip="off"))
for monomers with complex formation $2$ to
$0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex
formation $0$. The full range of possible values for $CF1_\mathrm{b}$ are
-depicted in Figure~\ref{fig:cf1b_graph}.
+depicted in \cref{fig:cf1b_graph}.
\begin{figure}
-<<fig=TRUE,echo=FALSE,results=hide,width=14,height=7>>=
+<<cf1b_graph,echo=FALSE,results='hide',fig.width=14,fig.height=7>>=
trellis.device(new=F,color=TRUE)
trellis.par.set(list(axis.line =list(col="transparent")))
pushViewport(viewport(layout=grid.layout(nrow=1,ncol=2),clip="off"))
can also be altered at the initialization of the experiment to
specific values for specific lipid types or components.
+The environment is a volume which is the maximum number of vesicles
+from a single simulation (4096) times the maximum size of the vesicle
+(usually 10000) divided by Avagadro's number divided by the total
+environmental lipid concentration, or usually
+\Sexpr{4096*10000/6.022E23/141E-10}~L.
+
\subsection{Initial Vesicle Creation}
Initial vesicles are seeded by selecting lipid molecules from the
Once the environment has been created and the initial vesicle has been
formed, molecules join and leave the vesicle based on the kinetic
parameters and state equation discussed above until the vesicle splits
-forming two progeny vesicles, one of which the program continues to
-follow.
+forming two progeny vesicles. The program then follows both vesicles
+and their progeny until the simulation reaches either the maximum
+number of vesicles (usually 4096), or the maximum simulation time
+(usually 100~s).
\subsubsection{Calculation of Vesicle Properties}
Determining the number of molecules to add to the lipid membrane
($n_i$) requires knowing $k_{\mathrm{f}i_\mathrm{adj}}$, the surface area of the
-vesicle $S_\mathrm{vesicle}$ (see Section \ref{sec:ves_prop}), the time interval
+vesicle $S_\mathrm{vesicle}$ (see \cref{sec:ves_prop}), the time interval
$dt$ during which lipids are added, the base $k_{\mathrm{f}i}$, and the
concentration of the monomer in the environment
-$\left[C_{i\mathrm{vesicle}}\right]$ (see Equation~\ref{eq:state}).
-$k_{\mathrm{f}i\mathrm{adj}}$ is calculated (see Equation~\ref{eq:kf_adj}) based on the
+$\left[C_{i\mathrm{vesicle}}\right]$ (see \cref{eq:state}).
+$k_{\mathrm{f}i\mathrm{adj}}$ is calculated (see \cref{eq:kf_adj}) based on the
vesicle properties and their hypothesized effect on the rate (in as
many cases as possible, experimentally based)
-(see Section~\ref{sec:kinetic_adjustments} for details). $dt$ can be varied
-(see Section~\ref{sec:step_duration}), but for a given step is constant. This
+(see \cref{sec:kinetic_adjustments} for details). $dt$ can be varied
+(see \cref{sec:step_duration}), but for a given step is constant. This
leads to the following:
$n_i = k_{\mathrm{f}i}k_{\mathrm{f}i_\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{vesicle}\mathrm{N_A}dt$
immediately before and immediately after splitting are stored
to produce later output.
-\section{Analyzing output}
-
-The analysis of output is handled by a separate perl program which
-shares many common modules with the simulation program. Current output
-includes simulation progress, summary tables, summary statistics, and
-various graphs.
-
-\subsection{PCA plots}
-
-Two major groups of axes that contribute to vesicle variation between
-subsequent generations are the components and properties of vesicles.
-Each component in a vesicle is an axis on its own; it can be measured
-either as an absolute number of molecules in each component, or the
-fraction of molecules of that component over the total number of
-molecules; the second approach is often more convenient, as it allows
-vesicles with different number of molecules to be directly compared
-(though it hides any effect of vesicle size).
-
-In order to visualize the variation between and transition of
-subsequent generations of vesicles from their initial state in the
-simulation, to their final state at the termination of the simulation,
-we plot the projection of the generations onto the two principle PCA
-axes (and alternatively, any pairing of the axes).
-
-\subsection{Carpet plots}
-
-Carpet plots show the distance/similarity between the composition or
-properties of all generations in a single run. The difference between
-each group of vesicle can be calculated using Euclidean distance
-(properties and compositions) or H similarity (composition only). We
-must use Euclidean distance for properties because the H distance
-metric is invalid when comparing negative vector elements to positive
-vector elements.
-
-In addition to showing distance or similarity, carpet plots also
-depict propersomes and composomes as square boxes on the diagonals and
-propertypes and compotypes as rectangles off the diagonals, each
-propertype or compotype with a distinct color.
-
-\subsection{Propersomes, propertypes, composomes and compotypes}
-
-A generation is considered to be a propersome if it is less than $0.6$
-units (by Euclidean distance of normalized properties) away from the
-generation immediately following and preceding. Likewise, a generation
-is in a composome if its H similarity is more than $0.9$ (by the
-normalized H metric) from the preceding and following generations.
-Propersomes and composomes are then assigned to propertypes and
-compotypes using Paritioning Around Medioids
-(PAM). All values of $k$ between 2 and 15
-(or the number of propersomes and composomes, if that is less than 15)
-are tried, and the clustering with the smallest
-silhouette~\citep{Rousseeuw1987:silhouettes} is chosen as the ideal
-clustering~\citep{Shenhav2005:pgard}.
+% \section{Analyzing output}
+%
+% The analysis of output is handled by a separate perl program which
+% shares many common modules with the simulation program. Current output
+% includes simulation progress, summary tables, summary statistics, and
+% various graphs.
+%
+% \subsection{PCA plots}
+%
+% Two major groups of axes that contribute to vesicle variation between
+% subsequent generations are the components and properties of vesicles.
+% Each component in a vesicle is an axis on its own; it can be measured
+% either as an absolute number of molecules in each component, or the
+% fraction of molecules of that component over the total number of
+% molecules; the second approach is often more convenient, as it allows
+% vesicles with different number of molecules to be directly compared
+% (though it hides any effect of vesicle size).
+%
+% In order to visualize the variation between and transition of
+% subsequent generations of vesicles from their initial state in the
+% simulation, to their final state at the termination of the simulation,
+% we plot the projection of the generations onto the two principle PCA
+% axes (and alternatively, any pairing of the axes).
+%
+% \subsection{Carpet plots}
+%
+% Carpet plots show the distance/similarity between the composition or
+% properties of all generations in a single run. The difference between
+% each group of vesicle can be calculated using Euclidean distance
+% (properties and compositions) or H similarity (composition only). We
+% must use Euclidean distance for properties because the H distance
+% metric is invalid when comparing negative vector elements to positive
+% vector elements.
+%
+% In addition to showing distance or similarity, carpet plots also
+% depict propersomes and composomes as square boxes on the diagonals and
+% propertypes and compotypes as rectangles off the diagonals, each
+% propertype or compotype with a distinct color.
+%
+% \subsection{Propersomes, propertypes, composomes and compotypes}
+%
+% A generation is considered to be a propersome if it is less than $0.6$
+% units (by Euclidean distance of normalized properties) away from the
+% generation immediately following and preceding. Likewise, a generation
+% is in a composome if its H similarity is more than $0.9$ (by the
+% normalized H metric) from the preceding and following generations.
+% Propersomes and composomes are then assigned to propertypes and
+% compotypes using Paritioning Around Medioids
+% (PAM). All values of $k$ between 2 and 15
+% (or the number of propersomes and composomes, if that is less than 15)
+% are tried, and the clustering with the smallest
+% silhouette~\citep{Rousseeuw1987:silhouettes} is chosen as the ideal
+% clustering~\citep{Shenhav2005:pgard}.
\bibliographystyle{unsrtnat}
projects / ool / lipid_simulation_formalism.git / blobdiff