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\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}
% double check this with the bits in the paper
For a system with monomers $(_\mathrm{monomer})$ and a vesicle
-$(_\mathrm{vesicle})$, the change in composition of the $i$th component of
+$(_\mathrm{vesicle})$, the change in concentration of the $i$th component of
a lipid vesicle per change in time ($d \left[C_{i_\mathrm{vesicle}}\right]/dt$)
can be described by a modification of the basic mass action law:
\end{equation}
The base forward kinetic parameter for the $i$th component is $k_{\mathrm{f}i}$
-and is dependent on the particular lipid type (PC, PS, SM, etc.). The
+and is dependent on the particular lipid type (\ac{PC}, \ac{PS}, \ac{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)
@
Each of the 5 lipid types has different kinetic parameters; where
-available, these were taken from literature.
-
-% \begin{table}
-% \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 \\
-% \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 \\
-% CHOL & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[3]))}$ & $5.1 \times 10^7$ & $2.8 \times 10^{-4}$ & 38 & 0 & -1 & 1.21 \\
-% SM & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[4]))}$ & $3.7 \times 10^6$ & $3.1 \times 10^{-3}$ & 61 & 0 & 3 & 0.8 \\
-% PE & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[5]))}$ & $2.3 \times 10^6$ & $1 \times 10^{-5}$ & 55 & 0 & 0 & 1.33 \\
-% \bottomrule
-% \end{tabular}
-% \caption{Kinetic parameters and molecular properties of lipid types}
-% \label{tab:kinetic_parameters_lipid_types}
-% \end{table}
+available, these were taken from literature (\cref{tab:kinetic_parameters_lipid_types}).
+
+\begin{table}
+ \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 $\left({Å}^2\right)$ & Charge & $\mathrm{CF}1$ & Curvature \\
+ \midrule
+ PC & $\Sexpr{kf[1]}$ & $3.7 \times 10^6$ & $2 \times 10^{-5}$ & 63 & 0 & 2 & 0.8 \\
+ PS & $\Sexpr{kf[2]}$ & $3.7 \times 10^6$ & $1.25\times 10^{-5}$ & 54 & -1 & 0 & 1 \\
+ CHOL & $\Sexpr{kf[3]}$ & $5.1 \times 10^7$ & $2.8 \times 10^{-4}$ & 38 & 0 & -1 & 1.21 \\
+ SM & $\Sexpr{kf[4]}$ & $3.7 \times 10^6$ & $3.1 \times 10^{-3}$ & 61 & 0 & 3 & 0.8 \\
+ PE & $\Sexpr{kf[5]}$ & $2.3 \times 10^6$ & $1 \times 10^{-5}$ & 55 & 0 & 0 & 1.33 \\
+ \bottomrule
+ \end{tabular}
+ \caption{Kinetic parameters and molecular properties of lipid types}
+ \label{tab:kinetic_parameters_lipid_types}
+\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
$k_{\mathrm{f}_\mathrm{PC}}$ was measured by
\citet{Nichols1985:phospholipid_monomer_vesicle_thermodynamics} and
was found to be $3.7\times 10^6$~$\frac{1}{\mathrm{M}\mathrm{s}}$ by the
-partitioning of P-C$_6$-NBD-PC between DOPC vesicles and water. As
-similar references are not available for SM or PS, we assume that they have
-the same $k_\mathrm{f}$. For CHOL, no direct measurement of $k_\mathrm{f}$ is available,
+partitioning of \ac{NBDPC} between \ac{DOPC} vesicles and water. As
+similar references are not available for \ac{SM} or \ac{PS}, we assume that they have
+the same $k_\mathrm{f}$. For \ac{CHOL}, no direct measurement of $k_\mathrm{f}$ is available,
however, \citet{Estronca2007:dhe_kinetics} measured the transfer of
-DHE from BSA to LUV, and found a $k_\mathrm{f}$ of $5.1\times 10^7$~%
+\ac{DHE} from \ac{BSA} to \acp{LUV}, and found a $k_\mathrm{f}$ of $5.1\times 10^7$~%
$\frac{1}{\mathrm{M} \mathrm{s}}$. We assume that this value is close
-to that of CHOL, and use it for $k_{\mathrm{f}_\mathrm{CHOL}}$. In the case of
-PE, \citet{Abreu2004:kinetics_ld_lo} measured the association of
-NBD-DMPE with POPC LUV found a value for $k_\mathrm{f}$ of $2.3 \times 10^{6}$~%
+to that of \ac{CHOL}, and use it for $k_{\mathrm{f}_\mathrm{\ac{CHOL}}}$. In the case of
+\ac{PE}, \citet{Abreu2004:kinetics_ld_lo} measured the association of
+\ac{NBDDMPE} with \ac{POPC} \acp{LUV} and found a value for $k_\mathrm{f}$ of $2.3 \times 10^{6}$~%
$\frac{1}{\mathrm{M} \mathrm{s}}$. These three authors used a slightly
different kinetic formalism ($\frac{d\left[A_v\right]}{dt} =
k'_\mathrm{f}[A_m][L_v] - k_\mathrm{b}[A_v]$), so we correct their values of $k_\mathrm{f}$ by
\subsubsection{$k_\mathrm{b}$ for lipid types}
\citet{Wimley1990:dmpc_exchange} measured the half time for the
-exchange of [$^3$H]DMPC between LUV at 30\textdegree C, and found it
+exchange of [$^3$H]\ac{DMPC} between \acp{LUV} at 30\textdegree C, and found it
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
+limiting step in exchange is lipid desorption, $k_\mathrm{b}$ for \ac{DMPC} is
$k_{\mathrm{b}_\mathrm{PC}}=\frac{\log 2}{9.6 \times 60 \times 60} \approx
\Sexpr{log(2)/(9.6*60*60)}
-\mathrm{s}^{-1}$. 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
+\mathrm{s}^{-1}$. We assume that $k_\mathrm{b}$ for \ac{SM} is the same as for \ac{PC}.
+To estimate the $k_\mathrm{b}$ of \ac{PE} and \ac{PS}, we used the data from
\citet{Nichols1982:ret_amphiphile_transfer} who measured the rate of
-exchange of the fluorescent label C$_6$-NBD attached to different
-lipid species. Although the values of kb are different for the labeled
+exchange of the fluorescent label \ac{C6NBD} attached to different
+lipid species. Although the values of $k_\mathrm{b}$ are different for the labeled
and unlabeled lipids, we assume that the ratios of the kinetics
constants for the lipid types are the same. Furthermore we assume that
-PG behaves similarly to PS. Thus, we can determine the $k_\mathrm{b}$ of PE and
-PS from the already known $k_\mathrm{b}$ of PC. For C$_6$-NBD labeled PC,
+PG behaves similarly to \ac{PS}. Thus, we can determine the $k_\mathrm{b}$ of \ac{PE} and
+\ac{PS} from the already known $k_\mathrm{b}$ of \ac{PC}. For \ac{C6NBD} labeled \ac{PC},
\citet{Nichols1982:ret_amphiphile_transfer} obtained a $k_\mathrm{b}$ of
-$0.89$~$\mathrm{min}^{-1}$, PE of $0.45$~$\mathrm{min}^{-1}$ and PG of
+$0.89$~$\mathrm{min}^{-1}$, \ac{PE} of $0.45$~$\mathrm{min}^{-1}$ and PG of
$0.55$~$\mathrm{min}^{-1}$. Assuming the ratios of the rate of
exchange are the same for unlabeled lipids and labeled lipids, we can
-determine the $k_\mathrm{b}$ of PE and PS from the already known $k_\mathrm{b}$ of
-PC~\citep{Wimley1990:dmpc_exchange}. Calculating the ratio, this leads
+determine the $k_\mathrm{b}$ of \ac{PE} and \ac{PS} from the already known $k_\mathrm{b}$ of
+\ac{PC}~\citep{Wimley1990:dmpc_exchange}. Calculating the ratio, this leads
us to $k_{\mathrm{b}_\mathrm{PE}} =
\frac{k_{\mathrm{b}_\mathrm{PC}}\times\mathrm{PE}}{\mathrm{PC}} \approx
\frac{2\times 10^{-5}\,\mathrm{s}^{-1} \times
\Sexpr{log(2)/(9.6*60*60)*0.45/0.89}$~$\mathrm{s}^{-1}$
and likewise, $k_{\mathrm{b}_\mathrm{PS}}\approx
\Sexpr{log(2)/(9.6*60*60)*0.55/0.89}$~$\mathrm{s}^{-1}$.
-The $k_\mathrm{b}$ of SM was determined using the work of
+The $k_\mathrm{b}$ of \ac{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
vesicles, finding $3.4\times10^{-2}$~$\mathrm{s}^{-1}$ and
$2.2\times10^{-3}$~$\mathrm{s}^{-1}$ 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 ($\frac{3.4 \times 10^{-2}\mathrm{s}^{-1}}{2.2 \times
+\ac{PC} ($\frac{3.4 \times 10^{-2}\mathrm{s}^{-1}}{2.2 \times
10^{-3}\mathrm{s}^{-1}}\approx
\Sexpr{log(2)/(9.6*60*60)}\mathrm{s}^{-1}$),
we obtain $k_{\mathrm{b}_\mathrm{SM}} \approx
\Sexpr{log(2)/(9.6*60*60)*3.4e-2/2.2e-3}$.
-In the case of CHOL, \citet{Jones1990:lipid_transfer} measured the
+In the case of \ac{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 $k_{\mathrm{b}_\mathrm{CHOL}} = \frac{\log 2}{41\times
60} \approx
DPPS\citep{Cascales1996:mds_dpps_area,Pandit2002:mds_dpps}, which is
in agreement with the experimental value of 56~Å$^2$ found using a
Langmuir balance by \citet{Demel1987:ps_area}.
-\citet{Shaikh2002:pe_phase_sm_area} measured the area of SM using a
+\citet{Shaikh2002:pe_phase_sm_area} measured the area of \ac{SM} using a
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 Å$^2$. \citet{Robinson1995:mds_chol_area}
-found an area for CHOL of 38~Å$^2$ using molecular dynamic
+found an area for \ac{CHOL} of 38~Å$^2$ using molecular dynamic
simulations.
% robinson's chol area is kind of crappy; they did it using MDS, but
% vaguely; they don't indicate what it was at the end, and they only
-% had 2 molecules of CHOL.
+% had 2 molecules of \ac{CHOL}.
-% PC: 63 A (Smaby97rd) (Pandit (?)
-% PS: 54 A (Pandit02LIB) (Cascales 1996; J. Chem. Phys 104:2713, Mukhopadhyay2004)
-% CHOL: 38 A (Robinson 1995) (Lundberg 1982)
-% SM: 51 A (Shaikh2002, but 61(?))
+% \ac{PC}: 63 A (Smaby97rd) (Pandit (?)
+% \ac{PS}: 54 A (Pandit02LIB) (Cascales 1996; J. Chem. Phys 104:2713, Mukhopadhyay2004)
+% \ac{CHOL}: 38 A (Robinson 1995) (Lundberg 1982)
+% \ac{SM}: 51 A (Shaikh2002, but 61(?))
% PE: 55 A (Thurmond, 1991) (Pandit2002)
%
% Compare to results by Petrache2004 with DOPC of 72.5, DOPS of 65.3.
\subsubsection{Complex Formation 1}
-\citet{Thomas1988:chol_transfer} found that SM significantly decreases
-the rate of cholesterol transfer from PC, while PE and PS have no
+\citet{Thomas1988:chol_transfer} found that \ac{SM} significantly decreases
+the rate of cholesterol transfer from \ac{PC}, while \ac{PE} and \ac{PS} have no
effect at physiologically significant levels. We attribute this to the
-formation of complexes between SM and CHOL, which retards the rate of
-efflux of CHOL from the membrane. Similar complexes exist between PC
-and CHOL, where it was shown that two CHOL molecules complex with one
-PC~\citep{Huang1999:chol_solubility_model,
+formation of complexes between \ac{SM} and \ac{CHOL}, which retards the rate of
+efflux of \ac{CHOL} from the membrane. Similar complexes exist between \ac{PC}
+and \ac{CHOL}, where it was shown that two \ac{CHOL} molecules complex with one
+\ac{PC}~\citep{Huang1999:chol_solubility_model,
Huang1999:cholesterol_solubility,McConnell2006,McConnell2003}. It
-has also been shown that SM binds more CHOL molecules than
-PC~\citep{Katz1988:pl_packing_chol}. We assume that three CHOL complex
-with one SM, leading to $\mathrm{CF}1$ values of $3$, $2$, and $-1$
-for SM, PC, and CHOL, respectively.
+has also been shown that \ac{SM} binds more \ac{CHOL} molecules than
+\ac{PC}~\citep{Katz1988:pl_packing_chol}. We assume that three \ac{CHOL} complex
+with one \ac{SM}, leading to $\mathrm{CF}1$ values of $3$, $2$, and $-1$
+for \ac{SM}, \ac{PC}, and \ac{CHOL}, respectively.
\subsubsection{Curvature}
$l_c$ is the critical length of the acyl chain, $v$ is the volume of
the lipid, and $a$ is the area of the head group.
\citet{Kumar1991:lipid_packing} found a value for $S$ of 1.33 for PE,
-1.21 for CHOL, and 0.8 for diC$_{16}$ PC. We assume that PS has neutral
-curvature (value of 1), and that SM has the same curvature as PC
+1.21 for \ac{CHOL}, and 0.8 for diC$_{16}$ \ac{PC}. We assume that \ac{PS} has neutral
+curvature (value of 1), and that \ac{SM} has the same curvature as \ac{PC}
(0.8). It should also be noted that in reality the curvature parameter
changes with length, but at longer chain lengths, is effectively
constant; in the current model, curvature is held constant for each
formation is increased when the unsaturation of lipids in the vesicle
is widely distributed; in other words, the insertion of lipids into
the membrane is greater when the standard deviation of the
-unsaturation is larger %
+unsaturation is larger. %
%%% \RZ{May need to site (at least for us) works showing
%%% mismatch-dependent ``defects''}. %
Assuming that an increase in width of the distribution linearly
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
\subsubsection{Charge Forward}
-A charged lipid such as PS approaching a vesicle with an average
+A charged lipid such as \ac{PS} approaching a vesicle with an average
charge of the same sign will experience repulsion, whereas those with
different signs will experience attraction, the degree of which is
dependent upon the charge of the monomer and the average charge of the
$\Sexpr{format(digits=3,to.kcal(60^(-.165*-1)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$ 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}
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}
\end{equation}
The most common $\mathrm{stdev} l_\mathrm{vesicle}$ is around $3.4$, which leads to
-a range of $\Delta \Delta G^\ddagger$ of
+a $\Delta \Delta G^\ddagger$ of
$\Sexpr{format(digits=3,to.kcal(2^(3.4)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$.
does, the backwards rate constant adjustment $k_{\mathrm{b}i\mathrm{adj}}$
takes into account unsaturation ($un_\mathrm{b}$), charge ($ch_\mathrm{b}$), curvature
($cu_\mathrm{b}$), length ($l_\mathrm{b}$), and complex formation ($CF1_\mathrm{b}$), each of
-which are modified depending on the specific component and the vesicle
+which is modified depending on the specific component and the vesicle
from which the component is exiting:
\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.
\end{figure}
\subsubsection{Charge Backwards}
-As in the case of monomers entering a vesicle, monomers leaving a
-vesicle leave faster if their charge has the same sign as the average
-charge vesicle. An equation of the form $ch_\mathrm{b} = a^{\left<ch_v\right>
- ch_m}$ is then appropriate, and using a base of $a=20$ yields:
+As in the case of monomers entering a vesicle, opposites attract.
+Monomers leaving a vesicle leave faster if their charge has the same
+sign as the average charge vesicle. An equation of the form
+$ch_\mathrm{b} = a^{\left<ch_v\right> ch_m}$ is then appropriate, and
+using a base of $a=20$ yields:
\begin{equation}
ch_\mathrm{b} = 20^{\left<{ch}_v\right> {ch}_m}
$\Sexpr{format(digits=3,to.kcal(20^(-.164*-1)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$ 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 $ch_\mathrm{b}$.
The less a monomer's intrinsic curvature matches the average curvature
of the vesicle in which it is in, the greater its rate of efflux. If
-the difference is 0, $cu_\mathrm{f}$ needs to be one. To map negative and
+the curvatures match exactly, $cu_\mathrm{f}$ needs to be one. To map negative and
positive curvature to the same range, we also need take the logarithm.
-Positive and negative curvature lipids cancel each other out,
-resulting in an average curvature of 0; the average of the log also
-has this property. Increasing mismatches in curvature increase the
+Positive ($cu > 1$) and negative ($0 < cu < 1$) curvature lipids cancel each other out,
+resulting in an average curvature of 1; the average of the log also
+has this property (average curvature of 0). Increasing mismatches in curvature increase the
rate of efflux, but asymptotically. An equation which satisfies these
criteria has the form $cu_\mathrm{f} = a^{1-\left(b\left( \left< \log
cu_\mathrm{vesicle} \right> -\log
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?}
The most common $\left<l_\mathrm{vesicle}\right>$ is around $17.75$,
which leads to a range of $\Delta \Delta G^\ddagger$ from
$\Sexpr{format(digits=3,to.kcal(3.2^abs(12-17.75)))}
-\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with length 12 to
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with length 12
+to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$
+for monomers with length near 18 to
$\Sexpr{format(digits=3,to.kcal(3.2^abs(24-17.75)))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
-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}
+for monomers with length 24. The full range of possible values of
+$l_\mathrm{b}$ 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
\subsubsection{Complex Formation Backward}
-Complex formation ($CF1$) describes the interaction between CHOL and
-PC or SM, where PC or SM protects the hydroxyl group of CHOL from
+Complex formation ($CF1$) describes the interaction between \ac{CHOL} and
+\ac{PC} or \ac{SM}, where \ac{PC} or \ac{SM} protects the hydroxyl group of \ac{CHOL} from
interactions with water %
\citep{Huang1999:chol_solubility_model,
Huang1999:cholesterol_solubility, McConnell2006, McConnell2003}%
-. PC ($CF1=2$) can interact with two CHOL, and SM ($CF1=3$) with three
-CHOL ($CF1=-1$). If the average of $CF1$ is positive (excess of SM and
-PC with regards to complex formation), components with negative $CF1$
-(CHOL) will be retained. If average $CF1$ is negative, components with
+. \ac{PC} ($CF1=2$) can interact with two \ac{CHOL}, and \ac{SM} ($CF1=3$) with three
+\ac{CHOL} ($CF1=-1$). If the average of $CF1$ is positive (excess of \ac{SM} and
+\ac{PC} with regards to complex formation), components with negative $CF1$
+(\ac{CHOL}) will be retained. If average $CF1$ is negative, components with
positive $CF1$ are retained. An equation which has this property is
$CF1_\mathrm{b}=a^{\left<CF1_\mathrm{vesicle}\right>
CF1_\mathrm{monomer}-\left|\left<CF1_\mathrm{vesicle}\right>
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}.
\subsubsection{Components}
The environment contains concentrations of components. In the current
set of simulations, there are \Sexpr{1+4*5*7} different possible
-components, consisting of PC, PE, PS, SM, and CHOL, with all lipids
-except for CHOL having 5 possible unsaturations ranging from 0 to 4,
+components, consisting of \ac{PC}, PE, \ac{PS}, \ac{SM}, and \ac{CHOL}, with all lipids
+except for \ac{CHOL} having 5 possible unsaturations ranging from 0 to 4,
and 7 lengths from $12,14,...,22$ ($7\cdot
5\cdot4+1=\Sexpr{1+4*5*7}$). In cases where the environment has less
than the maximum number of components, the components are selected in
order without replacement from a randomly shuffled deck of components
-(with the exception of CHOL) represented once until the desired number
-of unique components is obtained. CHOL is overrepresented
+(with the exception of \ac{CHOL}) represented once until the desired number
+of unique components is obtained. \ac{CHOL} is overrepresented
$\Sexpr{5*7}$ times to be at the level of other lipid types, ensuring
-that the probability of CHOL being absent in the environment is the
-same as the probability of any one of the other lipid types (PS, PE,
+that the probability of \ac{CHOL} being absent in the environment is the
+same as the probability of any one of the other lipid types (\ac{PS}, PE,
etc.) being absent. This reduces the number of simulations with a
-small number of components which are devoid of CHOL.
+small number of components which are devoid of \ac{CHOL}.
\subsubsection{Concentrations}
Once the components of the environment have been selected, their
concentrations are determined. In experiments where the environmental
concentration is the same across all lipid components, the
-concentration is set to $10^{-10}\mathrm{M}$. In other cases, the
+concentration is set to $10^{-10}$~M. In other cases, the
environmental concentration is set to a random number from a gamma
distribution with shape parameter 2 and an average of
-$10^{-10}\mathrm{M}$. The base concentration ($10^{-10}\mathrm{M}$)
+$10^{-10}$~M. The base concentration ($10^{-10}$~M)
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 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
environment until the vesicle reaches a specific starting size. The
vesicle starting size has gamma distribution with shape parameter 2
and a mean of the per-simulation specified starting size, with a
-minimum of 5 lipid molecules. Lipid molecules are then selected to be
-added to the lipid membrane according to four different methods. In
-the constant method, molecules are added in direct proportion to their
+minimum of 5 lipid molecules, or can be specified to have a precise
+number of molecules. Lipid molecules are then selected to be added to
+the lipid membrane according to four different methods. In the
+constant method, molecules are added in direct proportion to their
concentration in the environment. The uniform method adds molecules in
proportion to their concentration in the environment scaled by a
uniform random value, whereas the random method adds molecules in
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
+$\left[C_{i\mathrm{monomer}}\right]$ (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}.
-
-
-\bibliographystyle{unsrtnat}
-\bibliography{references.bib}
+% \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*{Formalism}
+
+The most current revision of this formalism is available at
+\url{https://git.donarmstrong.com/ool/lipid_simulation_formalism.git}.
+This document is \gitMarkPref • \gitMark.
+
+%\bibliographystyle{unsrtnat}
+%\bibliography{references.bib}
+
+\printbibliography
\end{document}
projects / ool / lipid_simulation_formalism.git / blobdiff