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$z <- as.vector(60^(-outer(x,y)))
 108 print(wireframe(z~x*y,grid,cuts=50,
 109           drape=TRUE,
 110           scales=list(arrows=FALSE),
 111           xlab="Average Vesicle Charge",
 112           ylab="Component Charge",
 113           zlab="Charge Forward"))
 114 rm(x,y,grid)
 115 @
 116
 117 \newpage
 118 \subsubsection{Curvature Forward}
 119 \begin{equation}
 120   cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
 121   \label{eq:curvature_forward}
 122 \end{equation}
 123
 124 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 125 curve(10^x,from=0,to=sd(c(0.8,1.33)),
 126       main="Curvature forward",
 127       xlab="Standard Deviation of Curvature of Vesicle",
 128       ylab="Curvature Forward Adjustment")
 129 @
 130
 131 \newpage
 132 \subsubsection{Length Forward}
 133 \begin{equation}
 134   l_f = 3^{\mathrm{stdev}\left|\log l_\mathrm{ves}\right|}
 135   \label{eq:length_forward}
 136 \end{equation}
 137
 138 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 139 curve(3^x,from=0,to=sd(c(12,24)),
 140       main="Length forward",
 141       xlab="Standard Deviation of Length of Vesicle",
 142       ylab="Length Forward Adjustment")
 143 @
 144
 145 \subsubsection{Complex Formation}
 146 \begin{equation}
 147   CF1_f=1
 148   \label{eq:complex_formation_forward}
 149 \end{equation}
 150
 151 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
 152
 153 \begin{equation}
 154   k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
 155   \label{eq:kf_adj}
 156 \end{equation}
 157
 158 \newpage
 159 \subsubsection{Unsaturation Backward}
 160 \begin{equation}
 161   un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-\left<un_\mathrm{monomer}\right>}\right|}
 162   \label{eq:unsaturation_backward}
 163 \end{equation}
 164
 165 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
 166 grid <- expand.grid(x=seq(0,4,length.out=20),
 167                     y=seq(0,4,length.out=20))
 168 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
 169 print(wireframe(z~x*y,grid,cuts=50,
 170           drape=TRUE,
 171           scales=list(arrows=FALSE),
 172           xlab="Average Vesicle Unsaturation",
 173           ylab="Monomer Unsaturation",
 174           zlab="Unsaturation Backward"))
 175 rm(grid)
 176 @
 177
 178 \newpage
 179 \subsubsection{Charge Backwards}
 180 \begin{equation}
 181   ch_b = 60^{-\left<{ch}_v\right> {ch}_m}
 182   \label{eq:charge_backwards}
 183 \end{equation}
 184
 185 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 186 x <- seq(-1,0,length.out=20)
 187 y <- seq(-1,0,length.out=20)
 188 grid <- expand.grid(x=x,y=y)
 189 grid$z <- as.vector(60^(-outer(x,y)))
 190 print(wireframe(z~x*y,grid,cuts=50,
 191           drape=TRUE,
 192           scales=list(arrows=FALSE),
 193           xlab="Average Vesicle Charge",
 194           ylab="Component Charge",
 195           zlab="Charge Backwards"))
 196 rm(x,y,grid)
 197 @
 198
 199 \newpage
 200 \subsubsection{Curvature Backwards}
 201 \begin{equation}
 202   cu_f = 4^{\left|\left|\log cu_\mathrm{vesicle}\right|-\left|\log cu_\mathrm{monomer}\right|\right|}
 203   \label{eq:curvature_backwards}
 204 \end{equation}
 205
 206 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 207 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
 208                     y=seq(0.8,1.33,length.out=20))
 209 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
 210 print(wireframe(z~x*y,grid,cuts=50,
 211           drape=TRUE,
 212           scales=list(arrows=FALSE),
 213           xlab="Vesicle Curvature",
 214           ylab="Monomer Curvature",
 215           zlab="Curvature Backward"))
 216 rm(grid)
 217 @
 218
 219 \newpage
 220 \subsubsection{Length Backwards}
 221 \begin{equation}
 222   l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
 223   \label{eq:length_backward}
 224 \end{equation}
 225
 226 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 227 grid <- expand.grid(x=seq(12,24,length.out=20),
 228                     y=seq(12,24,length.out=20))
 229 grid$z <- 3.2^(abs(grid$x-grid$y))
 230 print(wireframe(z~x*y,grid,cuts=50,
 231           drape=TRUE,
 232           scales=list(arrows=FALSE),
 233           xlab="Average Vesicle Length",
 234           ylab="Monomer Length",
 235           zlab="Length Backward"))
 236 rm(grid)
 237 @
 238
 239 \newpage
 240 \subsubsection{Complex Formation Backward}
 241 \begin{equation}
 242   CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
 243   \label{eq:complex_formation_backward}
 244 \end{equation}
 245
 246 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 247 grid <- expand.grid(x=seq(-1,3,length.out=20),
 248                     y=seq(-1,3,length.out=20))
 249 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
 250 print(wireframe(z~x*y,grid,cuts=50,
 251           drape=TRUE,
 252           scales=list(arrows=FALSE),
 253           xlab="Vesicle Complex Formation",
 254           ylab="Monomer Complex Formation",
 255           zlab="Complex Formation Backward"))
 256 rm(grid)
 257 @
 258
 259
 260
 261
 262 % \bibliographystyle{plainnat}
 263 % \bibliography{references.bib}
 264
 265
 266 \end{document}