kinetic_formalism.Rnw

   1 \documentclass[english,12pt]{article}
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  49 \oddsidemargin 0.0in
  50 \textwidth 6.5in
  51 \raggedbottom
  52 \clubpenalty = 10000
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  54 \pagestyle{fancy}
  55 \author{Don Armstrong}
  56 \title{OOL Kinetic Formalisms}
  57 %\date{}
  58 \onehalfspacing
  59 \begin{document}
  60 %\maketitle
  61
  62 <<results=hide,echo=FALSE>>=
  63 require(lattice)
  64 require(grid)
  65 @
  66
  67 \section{State Equation}
  68 % double check this with the bits in the paper
  69 \begin{equation}
  70   \frac{d C^{j}_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C^j_{i_\mathrm{monomer}}\right] -
  71   k_{bi}k_{bi\mathrm{adj}}C^j_{i_\mathrm{ves}}
  72   \label{eq:state}
  73 \end{equation}
  74
  75 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
  76
  77 \begin{equation}
  78   k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
  79   \label{eq:kf_adj}
  80 \end{equation}
  81
  82 \newpage
  83 \subsubsection{Unsaturation Forward}
  84 \begin{equation}
  85   un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
  86   \label{eq:unsaturation_forward}
  87 \end{equation}
  88
  89 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
  90 curve(2^x,from=0,to=sd(c(0,4)),
  91       main="Unsaturation forward",
  92       xlab="Standard Deviation of Unsaturation of Vesicle",
  93       ylab="Unsaturation Forward Adjustment")
  94 @
  95
  96 \newpage
  97 \subsubsection{Charge Forward}
  98 \begin{equation}
  99   ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
 100   \label{eq:charge_forward}
 101 \end{equation}
 102
 103 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 104 x <- seq(-1,0,length.out=20)
 105 y <- seq(-1,0,length.out=20)
 106 grid <- expand.grid(x=x,y=y)
 107 grid$z <- as.vector(60^(-outer(x,y)))
 108 print(wireframe(z~x*y,grid,cuts=50,
 109           drape=TRUE,
 110           scales=list(arrows=FALSE),
 111           xlab="Average Vesicle Charge",
 112           ylab="Component Charge",
 113           zlab="Charge Forward"))
 114 rm(x,y,grid)
 115 @
 116
 117 \newpage
 118 \subsubsection{Curvature Forward}
 119 \begin{equation}
 120   cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
 121   \label{eq:curvature_forward}
 122 \end{equation}
 123
 124 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 125 curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
 126                     sd(abs(log(c(1,1.33)))),
 127                     sd(abs(log(c(0.8,1)))))),
 128       main="Curvature forward",
 129       xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
 130       ylab="Curvature Forward Adjustment")
 131 @
 132
 133 \newpage
 134 \subsubsection{Length Forward}
 135 \begin{equation}
 136   l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
 137   \label{eq:length_forward}
 138 \end{equation}
 139
 140 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 141 curve(3^x,from=0,to=sd(c(12,24)),
 142       main="Length forward",
 143       xlab="Standard Deviation of Length of Vesicle",
 144       ylab="Length Forward Adjustment")
 145 @
 146
 147 \subsubsection{Complex Formation}
 148 \begin{equation}
 149   CF1_f=1
 150   \label{eq:complex_formation_forward}
 151 \end{equation}
 152
 153 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
 154
 155 \begin{equation}
 156   k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
 157   \label{eq:kf_adj}
 158 \end{equation}
 159
 160 \newpage
 161 \subsubsection{Unsaturation Backward}
 162 \begin{equation}
 163   un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-\left<un_\mathrm{monomer}\right>}\right|}
 164   \label{eq:unsaturation_backward}
 165 \end{equation}
 166
 167 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
 168 grid <- expand.grid(x=seq(0,4,length.out=20),
 169                     y=seq(0,4,length.out=20))
 170 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
 171 print(wireframe(z~x*y,grid,cuts=50,
 172           drape=TRUE,
 173           scales=list(arrows=FALSE),
 174           xlab="Average Vesicle Unsaturation",
 175           ylab="Monomer Unsaturation",
 176           zlab="Unsaturation Backward"))
 177 rm(grid)
 178 @
 179
 180 \newpage
 181 \subsubsection{Charge Backwards}
 182 \begin{equation}
 183   ch_b = 20^{\left<{ch}_v\right> {ch}_m}
 184   \label{eq:charge_backwards}
 185 \end{equation}
 186
 187 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 188 x <- seq(-1,0,length.out=20)
 189 y <- seq(-1,0,length.out=20)
 190 grid <- expand.grid(x=x,y=y)
 191 grid$z <- as.vector(20^(outer(x,y)))
 192 print(wireframe(z~x*y,grid,cuts=50,
 193           drape=TRUE,
 194           scales=list(arrows=FALSE),
 195           xlab="Average Vesicle Charge",
 196           ylab="Component Charge",
 197           zlab="Charge Backwards"))
 198 rm(x,y,grid)
 199 @
 200
 201 \newpage
 202 \subsubsection{Curvature Backwards}
 203 \begin{equation}
 204   cu_f = 7^{1-\left(20\left(\log cu_\mathrm{vesicle}-\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
 205   \label{eq:curvature_backwards}
 206 \end{equation}
 207
 208 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 209 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
 210                     y=seq(0.8,1.33,length.out=20))
 211 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
 212 print(wireframe(z~x*y,grid,cuts=50,
 213           drape=TRUE,
 214           scales=list(arrows=FALSE),
 215           xlab="Vesicle Curvature",
 216           ylab="Monomer Curvature",
 217           zlab="Curvature Backward"))
 218 rm(grid)
 219 @
 220
 221 \newpage
 222 \subsubsection{Length Backwards}
 223 \begin{equation}
 224   l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
 225   \label{eq:length_backward}
 226 \end{equation}
 227
 228 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 229 grid <- expand.grid(x=seq(12,24,length.out=20),
 230                     y=seq(12,24,length.out=20))
 231 grid$z <- 3.2^(abs(grid$x-grid$y))
 232 print(wireframe(z~x*y,grid,cuts=50,
 233           drape=TRUE,
 234           scales=list(arrows=FALSE),
 235           xlab="Average Vesicle Length",
 236           ylab="Monomer Length",
 237           zlab="Length Backward"))
 238 rm(grid)
 239 @
 240
 241 \newpage
 242 \subsubsection{Complex Formation Backward}
 243 \begin{equation}
 244   CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
 245   \label{eq:complex_formation_backward}
 246 \end{equation}
 247
 248 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 249 grid <- expand.grid(x=seq(-1,3,length.out=20),
 250                     y=seq(-1,3,length.out=20))
 251 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
 252 print(wireframe(z~x*y,grid,cuts=50,
 253           drape=TRUE,
 254           scales=list(arrows=FALSE),
 255           xlab="Vesicle Complex Formation",
 256           ylab="Monomer Complex Formation",
 257           zlab="Complex Formation Backward"))
 258 rm(grid)
 259 @
 260
 261
 262
 263
 264 % \bibliographystyle{plainnat}
 265 % \bibliography{references.bib}
 266
 267
 268 \end{document}