\usepackage{array}
\usepackage{dcolumn}
\usepackage{booktabs}
-\usepackage[noblocks]{authblk}
\usepackage[backend=biber,natbib=true,hyperref=true,citestyle=numeric-comp,style=nature,autocite=inline]{biblatex}
\addbibresource{references.bib}
\usepackage[hyperfigures,bookmarks,colorlinks,citecolor=black,filecolor=black,linkcolor=black,urlcolor=black]{hyperref}
+\usepackage[noblocks,auth-sc]{authblk}
\usepackage[capitalise]{cleveref}
+\usepackage[markifdraft,raisemark=0.01\paperheight,draft]{gitinfo2}
%\usepackage[sectionbib,sort&compress,numbers]{natbib}
% \usepackage[nomargin,inline,draft]{fixme}
%\usepackage[x11names,svgnames]{xcolor}
\clubpenalty = 10000
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-\author{}
-\title{Supplemental Material}
-\date{}
-% rubber: module bibtex
-% rubber: module natbib
+\title{Kinetic formalism for R-GARD Simulations}
+\author[1,2]{Don Armstrong}
+%\ead{don@donarmstrong.com}
+\author[2]{Raphael Zidovetzki}
+%\ead{raphael.zidovetzki@ucr.edu}
+\affil[1]{Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-Champaign, Urbana, IL, USA}
+\affil[2]{Cell Biology and Neuroscience, University of California at Riverside, Riverside, CA, USA}
+\affil[ ]{don@donarmstrong.com, raphael.zidovetzki@ucr.edu}
+%\date{}
\onehalfspacing
\begin{document}
\maketitle
<<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)
+options(scipen = -1, digits = 2)
library("lattice")
library("grid")
library("Hmisc")
@
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 $\left({Å}^2\right)$ & 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 \\
-% 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 $\left(\mathrm{Å}^2\right)$ & 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 \cref{tab:kinetic_parameters_lipid_types} with
$\frac{1}{\mathrm{M} \mathrm{s}}$. We assume that this value is close
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} found a value for $k_\mathrm{f}$ of $2.3 \times 10^{6}$~%
+\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
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 \ac{PS}. Thus, we can determine the $k_\mathrm{b}$ of \ac{PE} and
+\ac{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
$0.89$~$\mathrm{min}^{-1}$, \ac{PE} of $0.45$~$\mathrm{min}^{-1}$ and PG of
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
+measured the surface area of \ac{POPC} with a Langmuir film balance, and
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
+\ac{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 \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}
+\ac{DPPE}-d$_{62}$ to be 55.4 Å$^2$. \citet{Robinson1995:mds_chol_area}
found an area for \ac{CHOL} of 38~Å$^2$ using molecular dynamic
simulations.
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
$-0.013$, which leads to a range of $\Delta \Delta G^\ddagger$ from
$\Sexpr{format(digits=3,to.kcal(7^(1-1/(20*(-0.013-log(0.8))^2+1))))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature 0.8 to
+to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature
+near 1
$\Sexpr{format(digits=3,to.kcal(7^(1-1/(20*(-0.013-log(1.3))^2+1))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
-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
-\cref{fig:cub_graph}
+for monomers with curvature 1.3. The full range of values possible for
+$cu_\mathrm{b}$ are shown in \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 \cref{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
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 maximum size of the vesicle
+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.
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
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 \cref{eq:state}).
+$\left[C_{i\mathrm{monomer}}\right]$ (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)
% 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}
projects / ool / lipid_simulation_formalism.git / blobdiff