\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

<<results=hide,echo=FALSE>>=
require(lattice)
require(grid)
@ 

\section{State Equation}
% double check this with the bits in the paper
\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 ($k_{fi\mathrm{adj}}$)}

\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}
\begin{equation}
  un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
  \label{eq:unsaturation_forward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
curve(2^x,from=0,to=sd(c(0,4)),
      main="Unsaturation forward",
      xlab="Standard Deviation of Unsaturation of Vesicle",
      ylab="Unsaturation Forward Adjustment")
@ 

\newpage
\subsubsection{Charge Forward}
\begin{equation}
  ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
  \label{eq:charge_forward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
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),
          xlab=list("Average Vesicle Charge",rot=30),
          ylab=list("Component Charge",rot=-35),
          zlab=list("Charge Forward",rot=93)))
rm(x,y,grid)
@ 

\newpage
\subsubsection{Curvature Forward}
\begin{equation}
  cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
  \label{eq:curvature_forward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
                    sd(abs(log(c(1,1.33)))),
                    sd(abs(log(c(0.8,1)))))),
      main="Curvature forward",
      xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
      ylab="Curvature Forward Adjustment")
@ 

\newpage
\subsubsection{Length Forward}
\begin{equation}
  l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
  \label{eq:length_forward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
curve(3^x,from=0,to=sd(c(12,24)),
      main="Length forward",
      xlab="Standard Deviation of Length of Vesicle",
      ylab="Length Forward Adjustment")
@ 

\subsubsection{Complex Formation}
\begin{equation}
  CF1_f=1
  \label{eq:complex_formation_forward}
\end{equation}

\subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}

\begin{equation}
  k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
  \label{eq:kf_adj}
\end{equation}

\newpage
\subsubsection{Unsaturation Backward}
\begin{equation}
  un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
  \label{eq:unsaturation_backward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
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)
@ 

\newpage
\subsubsection{Charge Backwards}
\begin{equation}
  ch_b = 20^{\left<{ch}_v\right> {ch}_m}
  \label{eq:charge_backwards}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
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)
@ 

\newpage
\subsubsection{Curvature Backwards}
\begin{equation}
  cu_f = 7^{1-\left(20\left(\log cu_\mathrm{vesicle}-\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
  \label{eq:curvature_backwards}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
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)
@ 

\newpage
\subsubsection{Length Backwards}
\begin{equation}
  l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
  \label{eq:length_backward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
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)
@ 

\newpage
\subsubsection{Complex Formation Backward}
\begin{equation}
  CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
  \label{eq:complex_formation_backward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
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)
@ 




% \bibliographystyle{plainnat}
% \bibliography{references.bib}


\end{document}