kinetic_formalism.Rnw

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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_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} -
  71   k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}}
  72   \label{eq:state}
  73 \end{equation}
  74
  75 For $k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]$,
  76 $k_{fi}$ has units of $\frac{\mathrm{m}}{\mathrm{s}}$,
  77 $k_{fi\mathrm{adj}}$ and $k_{bi\mathrm{adj}}$ are unitless,
  78 concentration is in units of $\frac{\mathrm{n}}{\mathrm{L}}$, surface
  79 area is in units of $\mathrm{m}^2$, $k_{bi}$ has units of
  80 $\frac{1}{\mathrm{s}}$ and $C_{i_\mathrm{ves}}$ has units of
  81 $\mathrm{n}$, Thus, we have
  82
  83 \begin{equation}
  84   \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} -
  85   \frac{1}{\mathrm{s}} \mathrm{n}
  86   =
  87   \frac{\mathrm{m^3}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{\mathrm{n}}{\mathrm{s}}=
  88   \frac{\mathrm{n}}{\mathrm{s}} = 1000 \frac{\mathrm{n}}{\mathrm{s}} - \frac{\mathrm{n}}{\mathrm{s}}
  89   \label{eq:state_units}
  90 \end{equation}
  91
  92 The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
  93
  94 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
  95
  96 \begin{equation}
  97   k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
  98   \label{eq:kf_adj}
  99 \end{equation}
 100
 101 \newpage
 102 \subsubsection{Unsaturation Forward}
 103 \begin{equation}
 104   un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
 105   \label{eq:unsaturation_forward}
 106 \end{equation}
 107
 108 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
 109 curve(2^x,from=0,to=sd(c(0,4)),
 110       main="Unsaturation forward",
 111       xlab="Standard Deviation of Unsaturation of Vesicle",
 112       ylab="Unsaturation Forward Adjustment")
 113 @
 114
 115 \newpage
 116 \subsubsection{Charge Forward}
 117 \begin{equation}
 118   ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
 119   \label{eq:charge_forward}
 120 \end{equation}
 121
 122 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 123 x <- seq(-1,0,length.out=20)
 124 y <- seq(-1,0,length.out=20)
 125 grid <- expand.grid(x=x,y=y)
 126 grid$z <- as.vector(60^(-outer(x,y)))
 127 print(wireframe(z~x*y,grid,cuts=50,
 128           drape=TRUE,
 129           scales=list(arrows=FALSE),
 130           xlab=list("Average Vesicle Charge",rot=30),
 131           ylab=list("Component Charge",rot=-35),
 132           zlab=list("Charge Forward",rot=93)))
 133 rm(x,y,grid)
 134 @
 135
 136 \newpage
 137 \subsubsection{Curvature Forward}
 138 \begin{equation}
 139   cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
 140   \label{eq:curvature_forward}
 141 \end{equation}
 142
 143 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 144 curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
 145                     sd(abs(log(c(1,1.33)))),
 146                     sd(abs(log(c(0.8,1)))))),
 147       main="Curvature forward",
 148       xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
 149       ylab="Curvature Forward Adjustment")
 150 @
 151
 152 \newpage
 153 \subsubsection{Length Forward}
 154 \begin{equation}
 155   l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
 156   \label{eq:length_forward}
 157 \end{equation}
 158
 159 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 160 curve(3^x,from=0,to=sd(c(12,24)),
 161       main="Length forward",
 162       xlab="Standard Deviation of Length of Vesicle",
 163       ylab="Length Forward Adjustment")
 164 @
 165
 166 \subsubsection{Complex Formation}
 167 \begin{equation}
 168   CF1_f=1
 169   \label{eq:complex_formation_forward}
 170 \end{equation}
 171
 172 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
 173
 174 \begin{equation}
 175   k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
 176   \label{eq:kf_adj}
 177 \end{equation}
 178
 179 \newpage
 180 \subsubsection{Unsaturation Backward}
 181 \begin{equation}
 182   un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
 183   \label{eq:unsaturation_backward}
 184 \end{equation}
 185
 186 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 187 grid <- expand.grid(x=seq(0,4,length.out=20),
 188                     y=seq(0,4,length.out=20))
 189 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
 190 print(wireframe(z~x*y,grid,cuts=50,
 191           drape=TRUE,
 192           scales=list(arrows=FALSE),
 193           xlab=list("Average Vesicle Unsaturation",rot=30),
 194           ylab=list("Monomer Unsaturation",rot=-35),
 195           zlab=list("Unsaturation Backward",rot=93)))
 196 rm(grid)
 197 @
 198
 199 \newpage
 200 \subsubsection{Charge Backwards}
 201 \begin{equation}
 202   ch_b = 20^{\left<{ch}_v\right> {ch}_m}
 203   \label{eq:charge_backwards}
 204 \end{equation}
 205
 206 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 207 x <- seq(-1,0,length.out=20)
 208 y <- seq(-1,0,length.out=20)
 209 grid <- expand.grid(x=x,y=y)
 210 grid$z <- as.vector(20^(outer(x,y)))
 211 print(wireframe(z~x*y,grid,cuts=50,
 212           drape=TRUE,
 213           scales=list(arrows=FALSE),
 214           xlab=list("Average Vesicle Charge",rot=30),
 215           ylab=list("Component Charge",rot=-35),
 216           zlab=list("Charge Backwards",rot=93)))
 217 rm(x,y,grid)
 218 @
 219
 220 \newpage
 221 \subsubsection{Curvature Backwards}
 222 \begin{equation}
 223   cu_f = 7^{1-\left(20\left(\log cu_\mathrm{vesicle}-\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
 224   \label{eq:curvature_backwards}
 225 \end{equation}
 226
 227 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 228 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
 229                     y=seq(0.8,1.33,length.out=20))
 230 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
 231 print(wireframe(z~x*y,grid,cuts=50,
 232           drape=TRUE,
 233           scales=list(arrows=FALSE),
 234           xlab=list("Vesicle Curvature",rot=30),
 235           ylab=list("Monomer Curvature",rot=-35),
 236           zlab=list("Curvature Backward",rot=93)))
 237 rm(grid)
 238 @
 239
 240 \newpage
 241 \subsubsection{Length Backwards}
 242 \begin{equation}
 243   l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
 244   \label{eq:length_backward}
 245 \end{equation}
 246
 247 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 248 grid <- expand.grid(x=seq(12,24,length.out=20),
 249                     y=seq(12,24,length.out=20))
 250 grid$z <- 3.2^(abs(grid$x-grid$y))
 251 print(wireframe(z~x*y,grid,cuts=50,
 252           drape=TRUE,
 253           scales=list(arrows=FALSE),
 254           xlab=list("Average Vesicle Length",rot=30),
 255           ylab=list("Monomer Length",rot=-35),
 256           zlab=list("Length Backward",rot=93)))
 257 rm(grid)
 258 @
 259
 260 \newpage
 261 \subsubsection{Complex Formation Backward}
 262 \begin{equation}
 263   CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
 264   \label{eq:complex_formation_backward}
 265 \end{equation}
 266
 267 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 268 grid <- expand.grid(x=seq(-1,3,length.out=20),
 269                     y=seq(-1,3,length.out=20))
 270 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
 271 print(wireframe(z~x*y,grid,cuts=50,
 272           drape=TRUE,
 273           scales=list(arrows=FALSE),
 274           xlab=list("Vesicle Complex Formation",rot=30),
 275           ylab=list("Monomer Complex Formation",rot=-35),
 276           zlab=list("Complex Formation Backward",rot=93)))
 277 rm(grid)
 278 @
 279
 280
 281
 282
 283 % \bibliographystyle{plainnat}
 284 % \bibliography{references.bib}
 285
 286
 287 \end{document}