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 # R in cal / mol K
  66 to.kcal <- function(k,temp=300) {
  67   gasconst <- 1.985
  68   return(-gasconst*temp*log(k)/1000)
  69 }
  70 @
  71
  72 \section{State Equation}
  73 % double check this with the bits in the paper
  74 \begin{equation}
  75   \frac{d C_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} -
  76   k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}}
  77   \label{eq:state}
  78 \end{equation}
  79
  80 For $k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]$,
  81 $k_{fi}$ has units of $\frac{\mathrm{m}}{\mathrm{s}}$,
  82 $k_{fi\mathrm{adj}}$ and $k_{bi\mathrm{adj}}$ are unitless,
  83 concentration is in units of $\frac{\mathrm{n}}{\mathrm{L}}$, surface
  84 area is in units of $\mathrm{m}^2$, $k_{bi}$ has units of
  85 $\frac{1}{\mathrm{s}}$ and $C_{i_\mathrm{ves}}$ has units of
  86 $\mathrm{n}$, Thus, we have
  87
  88 \begin{equation}
  89   \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} -
  90   \frac{1}{\mathrm{s}} \mathrm{n}
  91   =
  92   \frac{\mathrm{m^3}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{\mathrm{n}}{\mathrm{s}}=
  93   \frac{\mathrm{n}}{\mathrm{s}} = 1000 \frac{\mathrm{n}}{\mathrm{s}} - \frac{\mathrm{n}}{\mathrm{s}}
  94   \label{eq:state_units}
  95 \end{equation}
  96
  97 The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
  98
  99 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
 100
 101 \begin{equation}
 102   k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
 103   \label{eq:kf_adj}
 104 \end{equation}
 105
 106 \newpage
 107 \subsubsection{Unsaturation Forward}
 108 \begin{equation}
 109   un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
 110   \label{eq:unsaturation_forward}
 111 \end{equation}
 112
 113 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
 114 curve(2^x,from=0,to=sd(c(0,4)),
 115       main="Unsaturation forward",
 116       xlab="Standard Deviation of Unsaturation of Vesicle",
 117       ylab="Unsaturation Forward Adjustment")
 118 @
 119
 120 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
 121 curve(to.kcal(2^x),from=0,to=sd(c(0,4)),
 122       main="Unsaturation forward",
 123       xlab="Standard Deviation of Unsaturation of Vesicle",
 124       ylab="Unsaturation Forward (kcal/mol)")
 125 @
 126
 127
 128 \newpage
 129 \subsubsection{Charge Forward}
 130 \begin{equation}
 131   ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
 132   \label{eq:charge_forward}
 133 \end{equation}
 134
 135 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 136 x <- seq(-1,0,length.out=20)
 137 y <- seq(-1,0,length.out=20)
 138 grid <- expand.grid(x=x,y=y)
 139 grid$z <- as.vector(60^(-outer(x,y)))
 140 print(wireframe(z~x*y,grid,cuts=50,
 141           drape=TRUE,
 142           scales=list(arrows=FALSE),
 143           xlab=list("Average Vesicle Charge",rot=30),
 144           ylab=list("Component Charge",rot=-35),
 145           zlab=list("Charge Forward",rot=93)))
 146 rm(x,y,grid)
 147 @
 148
 149 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 150 x <- seq(-1,0,length.out=20)
 151 y <- seq(-1,0,length.out=20)
 152 grid <- expand.grid(x=x,y=y)
 153 grid$z <- as.vector(to.kcal(60^(-outer(x,y))))
 154 print(wireframe(z~x*y,grid,cuts=50,
 155           drape=TRUE,
 156           scales=list(arrows=FALSE),
 157           xlab=list("Average Vesicle Charge",rot=30),
 158           ylab=list("Component Charge",rot=-35),
 159           zlab=list("Charge Forward (kcal/mol)",rot=93)))
 160 rm(x,y,grid)
 161 @
 162
 163
 164 \newpage
 165 \subsubsection{Curvature Forward}
 166 \begin{equation}
 167   cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
 168   \label{eq:curvature_forward}
 169 \end{equation}
 170
 171 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 172 curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
 173                     sd(abs(log(c(1,1.33)))),
 174                     sd(abs(log(c(0.8,1)))))),
 175       main="Curvature forward",
 176       xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
 177       ylab="Curvature Forward Adjustment")
 178 @
 179
 180 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 181 curve(to.kcal(10^x),from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
 182                     sd(abs(log(c(1,1.33)))),
 183                     sd(abs(log(c(0.8,1)))))),
 184       main="Curvature forward",
 185       xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
 186       ylab="Curvature Forward Adjustment (kcal/mol)")
 187 @
 188
 189
 190 \newpage
 191 \subsubsection{Length Forward}
 192 \begin{equation}
 193   l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
 194   \label{eq:length_forward}
 195 \end{equation}
 196
 197 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 198 curve(3^x,from=0,to=sd(c(12,24)),
 199       main="Length forward",
 200       xlab="Standard Deviation of Length of Vesicle",
 201       ylab="Length Forward Adjustment")
 202 @
 203
 204 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 205 curve(to.kcal(3^x),from=0,to=sd(c(12,24)),
 206       main="Length forward",
 207       xlab="Standard Deviation of Length of Vesicle",
 208       ylab="Length Forward Adjustment (kcal/mol)")
 209 @
 210
 211
 212 \subsubsection{Complex Formation}
 213 \begin{equation}
 214   CF1_f=1
 215   \label{eq:complex_formation_forward}
 216 \end{equation}
 217
 218 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
 219
 220 \begin{equation}
 221   k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
 222   \label{eq:kf_adj}
 223 \end{equation}
 224
 225 \newpage
 226 \subsubsection{Unsaturation Backward}
 227 \begin{equation}
 228   un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
 229   \label{eq:unsaturation_backward}
 230 \end{equation}
 231
 232 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 233 grid <- expand.grid(x=seq(0,4,length.out=20),
 234                     y=seq(0,4,length.out=20))
 235 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
 236 print(wireframe(z~x*y,grid,cuts=50,
 237           drape=TRUE,
 238           scales=list(arrows=FALSE),
 239           xlab=list("Average Vesicle Unsaturation",rot=30),
 240           ylab=list("Monomer Unsaturation",rot=-35),
 241           zlab=list("Unsaturation Backward",rot=93)))
 242 rm(grid)
 243 @
 244
 245 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 246 grid <- expand.grid(x=seq(0,4,length.out=20),
 247                     y=seq(0,4,length.out=20))
 248 grid$z <- to.kcal(10^(abs(3.5^-grid$x-3.5^-grid$y)))
 249 print(wireframe(z~x*y,grid,cuts=50,
 250           drape=TRUE,
 251           scales=list(arrows=FALSE),
 252           xlab=list("Average Vesicle Unsaturation",rot=30),
 253           ylab=list("Monomer Unsaturation",rot=-35),
 254           zlab=list("Unsaturation Backward (kcal/mol)",rot=93)))
 255 rm(grid)
 256 @
 257
 258
 259 \newpage
 260 \subsubsection{Charge Backwards}
 261 \begin{equation}
 262   ch_b = 20^{\left<{ch}_v\right> {ch}_m}
 263   \label{eq:charge_backwards}
 264 \end{equation}
 265
 266 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 267 x <- seq(-1,0,length.out=20)
 268 y <- seq(-1,0,length.out=20)
 269 grid <- expand.grid(x=x,y=y)
 270 grid$z <- as.vector(20^(outer(x,y)))
 271 print(wireframe(z~x*y,grid,cuts=50,
 272           drape=TRUE,
 273           scales=list(arrows=FALSE),
 274           xlab=list("Average Vesicle Charge",rot=30),
 275           ylab=list("Component Charge",rot=-35),
 276           zlab=list("Charge Backwards",rot=93)))
 277 rm(x,y,grid)
 278 @
 279
 280 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 281 x <- seq(-1,0,length.out=20)
 282 y <- seq(-1,0,length.out=20)
 283 grid <- expand.grid(x=x,y=y)
 284 grid$z <- to.kcal(as.vector(20^(outer(x,y))))
 285 print(wireframe(z~x*y,grid,cuts=50,
 286           drape=TRUE,
 287           scales=list(arrows=FALSE),
 288           xlab=list("Average Vesicle Charge",rot=30),
 289           ylab=list("Component Charge",rot=-35),
 290           zlab=list("Charge Backwards (kcal/mol)",rot=93)))
 291 rm(x,y,grid)
 292 @
 293
 294 \newpage
 295 \subsubsection{Curvature Backwards}
 296 \begin{equation}
 297   cu_f = 7^{1-\left(20\left(\log_{e} cu_\mathrm{vesicle}-\log_{e} cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
 298   \label{eq:curvature_backwards}
 299 \end{equation}
 300
 301 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 302 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
 303                     y=seq(0.8,1.33,length.out=20))
 304 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
 305 print(wireframe(z~x*y,grid,cuts=50,
 306           drape=TRUE,
 307           scales=list(arrows=FALSE),
 308           xlab=list("Vesicle Curvature",rot=30),
 309           ylab=list("Monomer Curvature",rot=-35),
 310           zlab=list("Curvature Backward",rot=93)))
 311 rm(grid)
 312 @
 313
 314 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 315 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
 316                     y=seq(0.8,1.33,length.out=20))
 317 grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)))
 318 print(wireframe(z~x*y,grid,cuts=50,
 319           drape=TRUE,
 320           scales=list(arrows=FALSE),
 321           xlab=list("Vesicle Curvature",rot=30),
 322           ylab=list("Monomer Curvature",rot=-35),
 323           zlab=list("Curvature Backward (kcal/mol)",rot=93)))
 324 rm(grid)
 325 @
 326
 327
 328 \newpage
 329 \subsubsection{Length Backwards}
 330 \begin{equation}
 331   l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
 332   \label{eq:length_backward}
 333 \end{equation}
 334
 335 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 336 grid <- expand.grid(x=seq(12,24,length.out=20),
 337                     y=seq(12,24,length.out=20))
 338 grid$z <- 3.2^(abs(grid$x-grid$y))
 339 print(wireframe(z~x*y,grid,cuts=50,
 340           drape=TRUE,
 341           scales=list(arrows=FALSE),
 342           xlab=list("Average Vesicle Length",rot=30),
 343           ylab=list("Monomer Length",rot=-35),
 344           zlab=list("Length Backward",rot=93)))
 345 rm(grid)
 346 @
 347
 348 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 349 grid <- expand.grid(x=seq(12,24,length.out=20),
 350                     y=seq(12,24,length.out=20))
 351 grid$z <- to.kcal(3.2^(abs(grid$x-grid$y)))
 352 print(wireframe(z~x*y,grid,cuts=50,
 353           drape=TRUE,
 354           scales=list(arrows=FALSE),
 355           xlab=list("Average Vesicle Length",rot=30),
 356           ylab=list("Monomer Length",rot=-35),
 357           zlab=list("Length Backward (kcal/mol)",rot=93)))
 358 rm(grid)
 359 @
 360
 361
 362 \newpage
 363 \subsubsection{Complex Formation Backward}
 364 \begin{equation}
 365   CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
 366   \label{eq:complex_formation_backward}
 367 \end{equation}
 368
 369 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 370 grid <- expand.grid(x=seq(-1,3,length.out=20),
 371                     y=seq(-1,3,length.out=20))
 372 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
 373 print(wireframe(z~x*y,grid,cuts=50,
 374           drape=TRUE,
 375           scales=list(arrows=FALSE),
 376           xlab=list("Vesicle Complex Formation",rot=30),
 377           ylab=list("Monomer Complex Formation",rot=-35),
 378           zlab=list("Complex Formation Backward",rot=93)))
 379 rm(grid)
 380 @
 381
 382 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 383 grid <- expand.grid(x=seq(-1,3,length.out=20),
 384                     y=seq(-1,3,length.out=20))
 385 grid$z <- to.kcal(3.2^(grid$x*grid$y-abs(grid$x*grid$y)))
 386 print(wireframe(z~x*y,grid,cuts=50,
 387           drape=TRUE,
 388           scales=list(arrows=FALSE),
 389           xlab=list("Vesicle Complex Formation",rot=30),
 390           ylab=list("Monomer Complex Formation",rot=-35),
 391           zlab=list("Complex Formation Backward (kcal/mol)",rot=93)))
 392 rm(grid)
 393 @
 394
 395
 396
 397
 398
 399 % \bibliographystyle{plainnat}
 400 % \bibliography{references.bib}
 401
 402
 403 \end{document}