\documentclass[english,12pt]{article} \usepackage{fancyhdr} %\usepackage[pdftex]{graphicx} \usepackage{graphicx} \usepackage[bf]{caption2} \usepackage{rotating} \usepackage{multirow} \usepackage{textcomp} \usepackage{mathrsfs} \usepackage{amssymb} \usepackage{setspace} \usepackage{txfonts} \usepackage[light,all]{draftcopy} \usepackage{fancyref} \usepackage[hyperfigures,backref,bookmarks,colorlinks]{hyperref} \usepackage[sectionbib,sort&compress,square,numbers]{natbib} \usepackage[margin,inline,draft]{fixme} \usepackage[x11names,svgnames]{xcolor} \usepackage{texshade} \newenvironment{narrow}[2]{% \begin{list}{}{% \setlength{\topsep}{0pt}% \setlength{\leftmargin}{#1}% \setlength{\rightmargin}{#2}% \setlength{\listparindent}{\parindent}% \setlength{\itemindent}{\parindent}% \setlength{\parsep}{\parskip}}% \item[]}{\end{list}} \newenvironment{paperquote}{% \begin{quote}% \it }% {\end{quote}} \renewcommand{\textfraction}{0.15} \renewcommand{\topfraction}{0.85} \renewcommand{\bottomfraction}{0.65} \renewcommand{\floatpagefraction}{0.60} %\renewcommand{\baselinestretch}{1.8} \newenvironment{enumerate*}% {\begin{enumerate}% \setlength{\itemsep}{0pt}% \setlength{\parskip}{0pt}}% {\end{enumerate}} \newenvironment{itemize*}% {\begin{itemize}% \setlength{\itemsep}{0pt}% \setlength{\parskip}{0pt}}% {\end{itemize}} \oddsidemargin 0.0in \textwidth 6.5in \raggedbottom \clubpenalty = 10000 \widowpenalty = 10000 \pagestyle{fancy} \author{Don Armstrong} \title{OOL Kinetic Formalisms} %\date{} \onehalfspacing \begin{document} %\maketitle <>= require(lattice) require(grid) # R in cal / mol K to.kcal <- function(k,temp=300) { gasconst <- 1.985 return(-gasconst*temp*log(k)/1000) } @ \section{State Equation} % double check this with the bits in the paper Given a base forward kinetic parameter for the $i$th specie $k_{fi}$ (which is dependent on lipid type, that is PC, PE, PS, etc.), an adjustment parameter $k_{fi\mathrm{adj}}$ based on the vesicle and the specific specie (length, unsaturation, etc.) (see~\fref{eq:kf_adj}), the molar concentration of monomer of the $i$th specie $\left[C_{i_\mathrm{monomer}}\right]$, the surface area of the vesicle $S_\mathrm{ves}$, the base backwards kinetic parameter for the $i$th specie $k_{bi}$ which is also dependent on lipid type, its adjustment parameter $k_{bi\mathrm{adj}}$ (see~\fref{eq:kb_adj}), and the molar concentration of the $i$th specie in the vesicle $C_{i_\mathrm{ves}}$, the change in concentration of the $i$th specie in the vesicle per change in time $\frac{d C_{i_\mathrm{ves}}}{dt}$ can be calculated: \begin{equation} \frac{d C_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} - k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}} \label{eq:state} \end{equation} For $k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]$, $k_{fi}$ has units of $\frac{\mathrm{m}}{\mathrm{s}}$, $k_{fi\mathrm{adj}}$ and $k_{bi\mathrm{adj}}$ are unitless, concentration is in units of $\frac{\mathrm{n}}{\mathrm{L}}$, surface area is in units of $\mathrm{m}^2$, $k_{bi}$ has units of $\frac{1}{\mathrm{s}}$ and $C_{i_\mathrm{ves}}$ has units of $\mathrm{n}$, Thus, we have \begin{equation} \frac{\mathrm{n}}{\mathrm{s}} = \frac{\mathrm{m}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \mathrm{m}^2 \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{1}{\mathrm{s}} \mathrm{n} = \frac{\mathrm{m^3}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{\mathrm{n}}{\mathrm{s}}= \frac{\mathrm{n}}{\mathrm{s}} = 1000 \frac{\mathrm{n}}{\mathrm{s}} - \frac{\mathrm{n}}{\mathrm{s}} \label{eq:state_units} \end{equation} The 1000 isn't in \fref{eq:state} above, because it is unit-dependent. \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)} The forward rate constant adjustment, $k_{fi\mathrm{adj}}$ takes into account unsaturation ($un_f$), charge ($ch_f$), curvature ($cu_f$), length ($l_f$), and complex formation ($CF1_f$), each of which are modified depending on the specific specie and the vesicle into which the specie is entering. \begin{equation} k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f \label{eq:kf_adj} \end{equation} \newpage \subsubsection{Unsaturation Forward} In order for a lipid to be inserted into a membrane, a void has to be formed for it to fill. Voids can be generated by the combination of unsaturated and saturated lipids forming herterogeneous domains. Void 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. Assuming that an increase in width of the distribution linearly decreases the free energy of activation, the $un_f$ parameter must follow $a^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}$ where $a > 1$, so a convenient starting base for $a$ is $2$: \begin{equation} un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)} \label{eq:unsaturation_forward} \end{equation} The most common $\mathrm{stdev}\left(un_\mathrm{ves}\right)$ is around $1.5$, which leads to a $\Delta \Delta G^\ddagger$ of $\Sexpr{format(digits=3,to.kcal(2^1.5))} \frac{\mathrm{kcal}}{\mathrm{mol}}$. \setkeys{Gin}{width=3.2in} <>= curve(2^x,from=0,to=sd(c(0,4)), main="Unsaturation Forward", xlab="Standard Deviation of Unsaturation of Vesicle", ylab="Unsaturation Forward Adjustment") @ <>= curve(to.kcal(2^x),from=0,to=sd(c(0,4)), main="Unsaturation forward", xlab="Standard Deviation of Unsaturation of Vesicle", ylab="Unsaturation Forward (kcal/mol)") @ \newpage \subsubsection{Charge Forward} A charged lipid such as 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 vesicle. If either the vesicle or the monomer has no charge, there should be no effect of charge upon the rate. This leads us to the following equation, $a^{-\left ch_m}$, where $\left$ is the average charge of the vesicle, and $ch_m$ is the charge of the monomer. If either $\left$ or $ch_m$ is 0, the adjustment parameter is 1 (no change), whereas it decreases if both are positive or negative, as the product of two real numbers with the same sign is always positive. A convenient base for $a$ is 60, resulting in the following equation: \begin{equation} ch_f = 60^{-\left<{ch}_v\right> {ch}_m} \label{eq:charge_forward} \end{equation} The most common $\left<{ch}_v\right>$ is around $-0.165$, which leads to a range of $\Delta \Delta G^\ddagger$ from $\Sexpr{format(digits=3,to.kcal(60^(-.165*-1)))} \frac{\mathrm{kcal}}{\mathrm{mol}}$ to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$. <>= x <- seq(-1,0,length.out=20) y <- seq(-1,0,length.out=20) grid <- expand.grid(x=x,y=y) grid$z <- as.vector(60^(-outer(x,y))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), main="Charge Forward", xlab=list("Average Vesicle Charge",rot=30), ylab=list("Component Charge",rot=-35), zlab=list("Charge Forward",rot=93))) rm(x,y,grid) @ <>= x <- seq(-1,0,length.out=20) y <- seq(-1,0,length.out=20) grid <- expand.grid(x=x,y=y) grid$z <- as.vector(to.kcal(60^(-outer(x,y)))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), main="Charge Forward (kcal/mol)", xlab=list("Average Vesicle Charge",rot=30), ylab=list("Component Charge",rot=-35), zlab=list("Charge Forward (kcal/mol)",rot=93))) rm(x,y,grid) @ \newpage \subsubsection{Curvature Forward} Curvature is a measure of the intrinsic propensity of specific lipids to form micelles (positive curvature), inverted micelles (negative curvature), or planar sheets (zero curvature). In this formalism, curvature is measured as the ratio of the size of the head to that of the base, so negative curvature is bounded by $(0,1)$, zero curvature is 1, and positive curvature is bounded by $(1,\infty)$. The curvature can be transformed into the typical postive/negative mapping using $\log$, which has the additional property of making the range of positive and negative curvature equal, and distributed about 0. As in the case of unsaturation, void formation is increased by the presence of lipids with mismatched curvature. Thus, a larger distribution of curvature in the vesicle increases the rate of lipid insertion into the vesicle. However, a species with curvature $e^{-1}$ will cancel out a species with curvature $e$, so we have to log transform (turning these into -1 and 1), then take the absolute value (1 and 1), and finally measure the width of the distribution. Thus, by using the log transform to make the range of the lipid curvature equal between positive and negative, and taking the average to cancel out exactly mismatched curvatures, we come to an equation with the shape $a^{\left<\log cu_\mathrm{vesicle}\right>}$. A convenient base for $a$ is $10$, yielding: \begin{equation} % cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|} cu_f = 10^{\left|\left<\log cu_\mathrm{vesicle} \right>\right|\mathrm{stdev} \left|\log cu_\mathrm{vesicle}\right|} \label{eq:curvature_forward} \end{equation} The most common $\left|\left<\log {cu}_v\right>\right|$ is around $0.013$, which with the most common $\mathrm{stdev} \log cu_\mathrm{vesicle}$ of $0.213$ leads to a $\Delta \Delta G^\ddagger$ of $\Sexpr{format(digits=3,to.kcal(10^(0.13*0.213)))} \frac{\mathrm{kcal}}{\mathrm{mol}}$ % 1.5 to 0.75 3 to 0.33 <>= grid <- expand.grid(x=seq(0,max(c(sd(abs(log(c(1,3)))), sd(abs(log(c(1,0.33)))),sd(abs(log(c(0.33,3)))))),length.out=20), y=seq(0,max(c(mean(log(c(1,3)), mean(log(c(1,0.33))), mean(log(c(0.33,3)))))),length.out=20)) grid$z <- 10^(grid$x*grid$y) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Vesicle stdev log curvature",rot=30), ylab=list("Vesicle average log curvature",rot=-35), zlab=list("Vesicle Curvature Forward",rot=93))) rm(grid) @ <>= grid <- expand.grid(x=seq(0,max(c(sd(abs(log(c(1,3)))), sd(abs(log(c(1,0.33)))),sd(abs(log(c(0.33,3)))))),length.out=20), y=seq(0,max(c(mean(log(c(1,3)), mean(log(c(1,0.33))), mean(log(c(0.33,3)))))),length.out=20)) grid$z <- to.kcal(10^(grid$x*grid$y)) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Vesicle stdev log curvature",rot=30), ylab=list("Vesicle average log curvature",rot=-35), zlab=list("Vesicle Curvature Forward (kcal/mol)",rot=93))) rm(grid) @ \newpage \subsubsection{Length Forward} As in the case of unsaturation, void formation is easier when vesicles are made up of components of widely different lengths. Thus, when the width of the distribution of lengths is larger, the forward rate should be greater as well, leading us to an equation of the form $x^{\mathrm{stdev} l_\mathrm{ves}}$, where $\mathrm{stdev} l_\mathrm{ves}$ is the standard deviation of the length of the components of the vesicle, which has a maximum possible value of 8 and a minimum of 0 in this set of experiments. A convenient base for $x$ is 2, leading to: \begin{equation} l_f = 2^{\mathrm{stdev} l_\mathrm{ves}} \label{eq:length_forward} \end{equation} <>= curve(2^x,from=0,to=sd(c(12,24)), main="Length forward", xlab="Standard Deviation of Length of Vesicle", ylab="Length Forward Adjustment") @ <>= curve(to.kcal(2^x),from=0,to=sd(c(12,24)), main="Length forward", xlab="Standard Deviation of Length of Vesicle", ylab="Length Forward Adjustment (kcal/mol)") @ \subsubsection{Complex Formation} There is no contribution of complex formation to the forward reaction rate in the current formalism. \begin{equation} CF1_f=1 \label{eq:complex_formation_forward} \end{equation} \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)} Just as the forward rate constant adjustment $k_{fi\mathrm{adj}}$ does, the backwards rate constant adjustment $k_{bi\mathrm{adj}}$ takes into account unsaturation ($un_b$), charge ($ch_b$), curvature ($cu_b$), length ($l_b$), and complex formation ($CF1_b$), each of which are modified depending on the specific specie and the vesicle into which the specie is entering: \begin{equation} k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b \label{eq:kb_adj} \end{equation} \subsubsection{Unsaturation Backward} Unsaturation also influences the ability of a lipid molecule to leave a membrane. If a molecule has an unsaturation level which is different from the surrounding membrane, it will be more likely to leave the membrane. The more different the unsaturation level is, the greater the propensity for the lipid molecule to leave. However, a vesicle with some unsaturation is more favorable for lipids with more unsaturation than the equivalent amount of less unsatuturation, so the difference in energy between unsaturation is not linear. Therefore, an equation with the shape $x^{\left|y^{-\left}-y^{-un_\mathrm{monomer}}\right|}$ where $\left$ is the average unsaturation of the vesicle, and $un_\mathrm{monomer}$ is the average unsaturation. In this equation, as the average unsaturation of the vesicle is larger, \textcolor{red}{I don't like this equation; the explanation above seems really contrived. Need to discuss.} \begin{equation} un_b = 10^{\left|3.5^{-\left}-3.5^{-un_\mathrm{monomer}}\right|} \label{eq:unsaturation_backward} \end{equation} <>= grid <- expand.grid(x=seq(0,4,length.out=20), y=seq(0,4,length.out=20)) grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y)) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Average Vesicle Unsaturation",rot=30), ylab=list("Monomer Unsaturation",rot=-35), zlab=list("Unsaturation Backward",rot=93))) rm(grid) @ <>= grid <- expand.grid(x=seq(0,4,length.out=20), y=seq(0,4,length.out=20)) grid$z <- to.kcal(10^(abs(3.5^-grid$x-3.5^-grid$y))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Average Vesicle Unsaturation",rot=30), ylab=list("Monomer Unsaturation",rot=-35), zlab=list("Unsaturation Backward (kcal/mol)",rot=93))) rm(grid) @ \newpage \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 $ch_b = x^{\left ch_m}$ is then appropriate, and using a base of 20 for $x$ yields: \begin{equation} ch_b = 20^{\left<{ch}_v\right> {ch}_m} \label{eq:charge_backwards} \end{equation} <>= x <- seq(-1,0,length.out=20) y <- seq(-1,0,length.out=20) grid <- expand.grid(x=x,y=y) grid$z <- as.vector(20^(outer(x,y))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Average Vesicle Charge",rot=30), ylab=list("Component Charge",rot=-35), zlab=list("Charge Backwards",rot=93))) rm(x,y,grid) @ <>= x <- seq(-1,0,length.out=20) y <- seq(-1,0,length.out=20) grid <- expand.grid(x=x,y=y) grid$z <- to.kcal(as.vector(20^(outer(x,y)))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Average Vesicle Charge",rot=30), ylab=list("Component Charge",rot=-35), zlab=list("Charge Backwards (kcal/mol)",rot=93))) rm(x,y,grid) @ \newpage \subsubsection{Curvature Backwards} 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, $cu_f$ needs to be one. To map negative and positive curvature to the same range, we also need take the logarithm. Increasing mismatches in curvature increase the rate of efflux, but asymptotically. \textcolor{red}{It is this property which the unsaturation backwards equation does \emph{not} satisfy, which I think it should.} An equation which satisfies this critera has the form $cu_f = a^{1-\left(b\left(\left<\log cu_\mathrm{vesicle} \right> -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}$. An alternative form would use the aboslute value of the difference, however, this yields a cusp and sharp increase about the curvature equilibrium, which is decidedly non-elegant. We have chosen bases of $a=7$ and $b=20$. \begin{equation} cu_f = 7^{1-\left(20\left(\left<\log cu_\mathrm{vesicle} \right> -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}} \label{eq:curvature_backwards} \end{equation} <>= grid <- expand.grid(x=seq(0.8,1.33,length.out=20), y=seq(0.8,1.33,length.out=20)) grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Vesicle Curvature",rot=30), ylab=list("Monomer Curvature",rot=-35), zlab=list("Curvature Backward",rot=93))) rm(grid) @ <>= grid <- expand.grid(x=seq(0.8,1.33,length.out=20), y=seq(0.8,1.33,length.out=20)) grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Vesicle Curvature",rot=30), ylab=list("Monomer Curvature",rot=-35), zlab=list("Curvature Backward (kcal/mol)",rot=93))) rm(grid) @ \newpage \subsubsection{Length Backwards} In a model membrane, the dissociation constant decreases by a factor of approximately 3.2 per carbon increase in acyl chain length (Nichols 1985). Unfortunatly, the known experimental data only measures chain length less than or equal to the bulk lipid, and does not exceed it, and is only known for one bulk lipid species (DOPC). The dissociation constant decreases by approximately 3.2 per carbon increase in acyl chain length (Nichols 1985). We assume that this decrease is in relationship to the average vesicle length. \begin{equation} l_b = 3.2^{\left|\left-l_\mathrm{monomer}\right|} \label{eq:length_backward} \end{equation} <>= grid <- expand.grid(x=seq(12,24,length.out=20), y=seq(12,24,length.out=20)) grid$z <- 3.2^(abs(grid$x-grid$y)) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Average Vesicle Length",rot=30), ylab=list("Monomer Length",rot=-35), zlab=list("Length Backward",rot=93))) rm(grid) @ <>= grid <- expand.grid(x=seq(12,24,length.out=20), y=seq(12,24,length.out=20)) grid$z <- to.kcal(3.2^(abs(grid$x-grid$y))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Average Vesicle Length",rot=30), ylab=list("Monomer Length",rot=-35), zlab=list("Length Backward (kcal/mol)",rot=93))) rm(grid) @ \newpage \subsubsection{Complex Formation Backward} \begin{equation} CF1_b=1.5^{\left CF1_\mathrm{monomer}-\left|\left CF1_\mathrm{monomer}\right|} \label{eq:complex_formation_backward} \end{equation} <>= grid <- expand.grid(x=seq(-1,3,length.out=20), y=seq(-1,3,length.out=20)) grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y)) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Vesicle Complex Formation",rot=30), ylab=list("Monomer Complex Formation",rot=-35), zlab=list("Complex Formation Backward",rot=93))) rm(grid) @ <>= grid <- expand.grid(x=seq(-1,3,length.out=20), y=seq(-1,3,length.out=20)) grid$z <- to.kcal(3.2^(grid$x*grid$y-abs(grid$x*grid$y))) print(wireframe(z~x*y,grid,cuts=50, drape=TRUE, scales=list(arrows=FALSE), xlab=list("Vesicle Complex Formation",rot=30), ylab=list("Monomer Complex Formation",rot=-35), zlab=list("Complex Formation Backward (kcal/mol)",rot=93))) rm(grid) @ % \bibliographystyle{plainnat} % \bibliography{references.bib} \end{document}