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\author{Don Armstrong}
\title{OOL Kinetic Formalisms}
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\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^{j}_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C^j_{i_\mathrm{monomer}}\right] -
  k_{bi}k_{bi\mathrm{adj}}C^j_{i_\mathrm{ves}}
  \label{eq:state}
\end{equation}

\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=5>>=
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="Average Vesicle Charge",
          ylab="Component Charge",
          zlab="Charge Forward"))
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=sd(c(0.8,1.33)),
      main="Curvature forward",
      xlab="Standard Deviation of Curvature of Vesicle",
      ylab="Curvature Forward Adjustment")
@ 

\newpage
\subsubsection{Length Forward}
\begin{equation}
  l_f = 3^{\mathrm{stdev}\left|\log l_\mathrm{ves}\right|}
  \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^{-\left<un_\mathrm{monomer}\right>}\right|}
  \label{eq:unsaturation_backward}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
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="Average Vesicle Unsaturation",
          ylab="Monomer Unsaturation",
          zlab="Unsaturation Backward"))
rm(grid)
@ 

\newpage
\subsubsection{Charge Backwards}
\begin{equation}
  ch_b = 60^{-\left<{ch}_v\right> {ch}_m}
  \label{eq:charge_backwards}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
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="Average Vesicle Charge",
          ylab="Component Charge",
          zlab="Charge Backwards"))
rm(x,y,grid)
@ 

\newpage
\subsubsection{Curvature Backwards}
\begin{equation}
  cu_f = 4^{\left|\left|\log cu_\mathrm{vesicle}\right|-\left|\log cu_\mathrm{monomer}\right|\right|}
  \label{eq:curvature_backwards}
\end{equation}

<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
                    y=seq(0.8,1.33,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="Vesicle Curvature",
          ylab="Monomer Curvature",
          zlab="Curvature Backward"))
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=5>>=
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="Average Vesicle Length",
          ylab="Monomer Length",
          zlab="Length Backward"))
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=5>>=
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="Vesicle Complex Formation",
          ylab="Monomer Complex Formation",
          zlab="Complex Formation Backward"))
rm(grid)
@ 




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


\end{document}