kinetic_formalism.Rnw

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  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
  75 Given a base forward kinetic parameter for the $i$th specie $k_{fi}$
  76 (which is dependent on lipid type, that is PC, PE, PS, etc.), an
  77 adjustment parameter $k_{fi\mathrm{adj}}$ based on the vesicle and the
  78 specific specie (length, unsaturation, etc.) (see~\fref{eq:kf_adj}),
  79 the molar concentration of monomer of the $i$th specie
  80 $\left[C_{i_\mathrm{monomer}}\right]$, the surface area of the vesicle
  81 $S_\mathrm{ves}$, the base backwards kinetic parameter for the $i$th
  82 specie $k_{bi}$ which is also dependent on lipid type, its adjustment
  83 parameter $k_{bi\mathrm{adj}}$ (see~\fref{eq:kb_adj}), and the molar
  84 concentration of the $i$th specie in the vesicle $C_{i_\mathrm{ves}}$,
  85 the change in concentration of the $i$th specie in the vesicle per
  86 change in time $\frac{d C_{i_\mathrm{ves}}}{dt}$ can be calculated:
  87
  88 \begin{equation}
  89   \frac{d C_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} -
  90   k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}}
  91   \label{eq:state}
  92 \end{equation}
  93
  94 For $k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]$,
  95 $k_{fi}$ has units of $\frac{\mathrm{m}}{\mathrm{s}}$,
  96 $k_{fi\mathrm{adj}}$ and $k_{bi\mathrm{adj}}$ are unitless,
  97 concentration is in units of $\frac{\mathrm{n}}{\mathrm{L}}$, surface
  98 area is in units of $\mathrm{m}^2$, $k_{bi}$ has units of
  99 $\frac{1}{\mathrm{s}}$ and $C_{i_\mathrm{ves}}$ has units of
 100 $\mathrm{n}$, Thus, we have
 101
 102 \begin{equation}
 103   \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} -
 104   \frac{1}{\mathrm{s}} \mathrm{n}
 105   =
 106   \frac{\mathrm{m^3}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{\mathrm{n}}{\mathrm{s}}=
 107   \frac{\mathrm{n}}{\mathrm{s}} = 1000 \frac{\mathrm{n}}{\mathrm{s}} - \frac{\mathrm{n}}{\mathrm{s}}
 108   \label{eq:state_units}
 109 \end{equation}
 110
 111 The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
 112
 113 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
 114
 115 The forward rate constant adjustment, $k_{fi\mathrm{adj}}$ takes into
 116 account unsaturation ($un_f$), charge ($ch_f$), curvature ($cu_f$),
 117 length ($l_f$), and complex formation ($CF1_f$), each of which are
 118 modified depending on the specific specie and the vesicle into which
 119 the specie is entering.
 120
 121 \begin{equation}
 122   k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
 123   \label{eq:kf_adj}
 124 \end{equation}
 125
 126 \newpage
 127 \subsubsection{Unsaturation Forward}
 128
 129 In order for a lipid to be inserted into a membrane, a void has to be
 130 formed for it to fill. Voids can be generated by the combination of
 131 unsaturated and saturated lipids forming herterogeneous domains. Void
 132 formation is increased when the unsaturation of lipids in the vesicle
 133 is widely distributed; in other words, the insertion of lipids into
 134 the membrane is greater when the standard deviation of the
 135 unsaturation is larger. Assuming that an increase in width of the
 136 distribution linearly decreases the free energy of activation, the
 137 $un_f$ parameter must follow
 138 $x^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}$ where $x > 1$, so a
 139 convenient starting base for $x$ is 2:
 140
 141 \begin{equation}
 142   un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
 143   \label{eq:unsaturation_forward}
 144 \end{equation}
 145
 146 \setkeys{Gin}{width=3.2in}
 147 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
 148 curve(2^x,from=0,to=sd(c(0,4)),
 149       main="Unsaturation Forward",
 150       xlab="Standard Deviation of Unsaturation of Vesicle",
 151       ylab="Unsaturation Forward Adjustment")
 152 @
 153 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
 154 curve(to.kcal(2^x),from=0,to=sd(c(0,4)),
 155       main="Unsaturation forward",
 156       xlab="Standard Deviation of Unsaturation of Vesicle",
 157       ylab="Unsaturation Forward (kcal/mol)")
 158 @
 159
 160
 161 \newpage
 162 \subsubsection{Charge Forward}
 163
 164 A charged lipid such as PS approaching a vesicle with an average
 165 charge of the same sign will experience repulsion, whereas those with
 166 different signs will experience attraction, the degree of which is
 167 dependent upon the charge of the monomer and the average charge of the
 168 vesicle. If either the vesicle or the monomer has no charge, there
 169 should be no effect of charge upon the rate. This leads us to the
 170 following equation, $x^{-\left<ch_v\right> ch_m}$, where
 171 $\left<ch_v\right>$ is the average charge of the vesicle, and $ch_m$
 172 is the charge of the monomer. If either $\left<ch_v\right>$ or $ch_m$
 173 is 0, the adjustment parameter is 1 (no change), whereas it decreases
 174 if both are positive or negative, as the product of two real numbers
 175 with the same sign is always positive. A convenient base for $x$ is
 176 60, resulting in the following equation:
 177
 178
 179 \begin{equation}
 180   ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
 181   \label{eq:charge_forward}
 182 \end{equation}
 183
 184 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 185 x <- seq(-1,0,length.out=20)
 186 y <- seq(-1,0,length.out=20)
 187 grid <- expand.grid(x=x,y=y)
 188 grid$z <- as.vector(60^(-outer(x,y)))
 189 print(wireframe(z~x*y,grid,cuts=50,
 190                 drape=TRUE,
 191                 scales=list(arrows=FALSE),
 192                 main="Charge Forward",
 193                 xlab=list("Average Vesicle Charge",rot=30),
 194                 ylab=list("Component Charge",rot=-35),
 195                 zlab=list("Charge Forward",rot=93)))
 196 rm(x,y,grid)
 197 @
 198 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 199 x <- seq(-1,0,length.out=20)
 200 y <- seq(-1,0,length.out=20)
 201 grid <- expand.grid(x=x,y=y)
 202 grid$z <- as.vector(to.kcal(60^(-outer(x,y))))
 203 print(wireframe(z~x*y,grid,cuts=50,
 204                 drape=TRUE,
 205                 scales=list(arrows=FALSE),
 206                 main="Charge Forward (kcal/mol)",
 207                 xlab=list("Average Vesicle Charge",rot=30),
 208                 ylab=list("Component Charge",rot=-35),
 209                 zlab=list("Charge Forward (kcal/mol)",rot=93)))
 210 rm(x,y,grid)
 211 @
 212
 213
 214 \newpage
 215 \subsubsection{Curvature Forward}
 216
 217 Curvature is a measure of the intrinsic propensity of specific lipids
 218 to form micelles (positive curvature), inverted micelles (negative
 219 curvature), or planar sheets (zero curvature). In this formalism,
 220 curvature is measured as the ratio of the size of the head to that of
 221 the base, so negative curvature is bounded by $(0,1)$, zero curvature
 222 is 1, and positive curvature is bounded by $(1,\infty)$. The curvature
 223 can be transformed into the typical postive/negative mapping using
 224 $\log$, which has the additional property of making the range of
 225 positive and negative curvature equal, and distributed about 0.
 226
 227 As in the case of unsaturation, void formation is increased by the
 228 presence of lipids with mismatched curvature. Thus, a larger
 229 distribution of curvature in the vesicle increases the rate of lipid
 230 insertion into the vesicle. However, a species with curvature $e^-1$
 231 will cancel out a species with curvature $e$, so we have to log
 232 transform (turning these into -1 and 1), then take the absolute value
 233 (1 and 1), and finally measure the width of the distribution. Thus, by
 234 using the log transform to make the range of the lipid curvature equal
 235 between positive and negative, taking the absolute value to cancel out
 236 exactly mismatched curvatures, we come to an equation with the shape
 237 $x^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}$, and a
 238 convenient base for $x$ is 10, yielding:
 239
 240 {\color{red} Shouldn't a vesicle of -1,1,0 have the same activation
 241   energy as a vesicle of 0,0,0? It doesn't currently.}
 242
 243
 244 \begin{equation}
 245   cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
 246   \label{eq:curvature_forward}
 247 \end{equation}
 248
 249
 250 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 251 curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
 252                     sd(abs(log(c(1,1.33)))),
 253                     sd(abs(log(c(0.8,1)))))),
 254       main="Curvature forward",
 255       xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
 256       ylab="Curvature Forward Adjustment")
 257 @
 258 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 259 curve(to.kcal(10^x),from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
 260                     sd(abs(log(c(1,1.33)))),
 261                     sd(abs(log(c(0.8,1)))))),
 262       main="Curvature forward",
 263       xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
 264       ylab="Curvature Forward Adjustment (kcal/mol)")
 265 @
 266
 267
 268 \newpage
 269 \subsubsection{Length Forward}
 270
 271 As in the case of unsaturation, void formation is easier when vesicles
 272 are made up of components of widely different lengths. Thus, when the
 273 width of the distribution of lengths is larger, the forward rate
 274 should be greater as well, leading us to an equation of the form
 275 $x^{\mathrm{stdev} l_\mathrm{ves}}$, where $\mathrm{stdev}
 276 l_\mathrm{ves}$ is the standard deviation of the length of the
 277 components of the vesicle, which has a maximum possible value of 8 and
 278 a minimum of 0 in this set of experiments. A convenient base for $x$
 279 is 2, leading to:
 280
 281 \begin{equation}
 282   l_f = 2^{\mathrm{stdev} l_\mathrm{ves}}
 283   \label{eq:length_forward}
 284 \end{equation}
 285
 286 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 287 curve(2^x,from=0,to=sd(c(12,24)),
 288       main="Length forward",
 289       xlab="Standard Deviation of Length of Vesicle",
 290       ylab="Length Forward Adjustment")
 291 @
 292 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
 293 curve(to.kcal(2^x),from=0,to=sd(c(12,24)),
 294       main="Length forward",
 295       xlab="Standard Deviation of Length of Vesicle",
 296       ylab="Length Forward Adjustment (kcal/mol)")
 297 @
 298
 299
 300 \subsubsection{Complex Formation}
 301 There is no contribution of complex formation to the forward reaction
 302 rate in the current formalism.
 303
 304 \begin{equation}
 305   CF1_f=1
 306   \label{eq:complex_formation_forward}
 307 \end{equation}
 308
 309 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
 310
 311 \begin{equation}
 312   k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
 313   \label{eq:kb_adj}
 314 \end{equation}
 315
 316 \newpage
 317 \subsubsection{Unsaturation Backward}
 318 \begin{equation}
 319   un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
 320   \label{eq:unsaturation_backward}
 321 \end{equation}
 322
 323 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 324 grid <- expand.grid(x=seq(0,4,length.out=20),
 325                     y=seq(0,4,length.out=20))
 326 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
 327 print(wireframe(z~x*y,grid,cuts=50,
 328           drape=TRUE,
 329           scales=list(arrows=FALSE),
 330           xlab=list("Average Vesicle Unsaturation",rot=30),
 331           ylab=list("Monomer Unsaturation",rot=-35),
 332           zlab=list("Unsaturation Backward",rot=93)))
 333 rm(grid)
 334 @
 335 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 336 grid <- expand.grid(x=seq(0,4,length.out=20),
 337                     y=seq(0,4,length.out=20))
 338 grid$z <- to.kcal(10^(abs(3.5^-grid$x-3.5^-grid$y)))
 339 print(wireframe(z~x*y,grid,cuts=50,
 340           drape=TRUE,
 341           scales=list(arrows=FALSE),
 342           xlab=list("Average Vesicle Unsaturation",rot=30),
 343           ylab=list("Monomer Unsaturation",rot=-35),
 344           zlab=list("Unsaturation Backward (kcal/mol)",rot=93)))
 345 rm(grid)
 346 @
 347
 348
 349 \newpage
 350 \subsubsection{Charge Backwards}
 351 \begin{equation}
 352   ch_b = 20^{\left<{ch}_v\right> {ch}_m}
 353   \label{eq:charge_backwards}
 354 \end{equation}
 355
 356 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 357 x <- seq(-1,0,length.out=20)
 358 y <- seq(-1,0,length.out=20)
 359 grid <- expand.grid(x=x,y=y)
 360 grid$z <- as.vector(20^(outer(x,y)))
 361 print(wireframe(z~x*y,grid,cuts=50,
 362           drape=TRUE,
 363           scales=list(arrows=FALSE),
 364           xlab=list("Average Vesicle Charge",rot=30),
 365           ylab=list("Component Charge",rot=-35),
 366           zlab=list("Charge Backwards",rot=93)))
 367 rm(x,y,grid)
 368 @
 369 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 370 x <- seq(-1,0,length.out=20)
 371 y <- seq(-1,0,length.out=20)
 372 grid <- expand.grid(x=x,y=y)
 373 grid$z <- to.kcal(as.vector(20^(outer(x,y))))
 374 print(wireframe(z~x*y,grid,cuts=50,
 375           drape=TRUE,
 376           scales=list(arrows=FALSE),
 377           xlab=list("Average Vesicle Charge",rot=30),
 378           ylab=list("Component Charge",rot=-35),
 379           zlab=list("Charge Backwards (kcal/mol)",rot=93)))
 380 rm(x,y,grid)
 381 @
 382
 383 \newpage
 384 \subsubsection{Curvature Backwards}
 385 \begin{equation}
 386   cu_f = 7^{1-\left(20\left(\log_{e} cu_\mathrm{vesicle}-\log_{e} cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
 387   \label{eq:curvature_backwards}
 388 \end{equation}
 389
 390 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 391 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
 392                     y=seq(0.8,1.33,length.out=20))
 393 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
 394 print(wireframe(z~x*y,grid,cuts=50,
 395           drape=TRUE,
 396           scales=list(arrows=FALSE),
 397           xlab=list("Vesicle Curvature",rot=30),
 398           ylab=list("Monomer Curvature",rot=-35),
 399           zlab=list("Curvature Backward",rot=93)))
 400 rm(grid)
 401 @
 402 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 403 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
 404                     y=seq(0.8,1.33,length.out=20))
 405 grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)))
 406 print(wireframe(z~x*y,grid,cuts=50,
 407           drape=TRUE,
 408           scales=list(arrows=FALSE),
 409           xlab=list("Vesicle Curvature",rot=30),
 410           ylab=list("Monomer Curvature",rot=-35),
 411           zlab=list("Curvature Backward (kcal/mol)",rot=93)))
 412 rm(grid)
 413 @
 414
 415
 416 \newpage
 417 \subsubsection{Length Backwards}
 418 \begin{equation}
 419   l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
 420   \label{eq:length_backward}
 421 \end{equation}
 422
 423 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 424 grid <- expand.grid(x=seq(12,24,length.out=20),
 425                     y=seq(12,24,length.out=20))
 426 grid$z <- 3.2^(abs(grid$x-grid$y))
 427 print(wireframe(z~x*y,grid,cuts=50,
 428           drape=TRUE,
 429           scales=list(arrows=FALSE),
 430           xlab=list("Average Vesicle Length",rot=30),
 431           ylab=list("Monomer Length",rot=-35),
 432           zlab=list("Length Backward",rot=93)))
 433 rm(grid)
 434 @
 435 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 436 grid <- expand.grid(x=seq(12,24,length.out=20),
 437                     y=seq(12,24,length.out=20))
 438 grid$z <- to.kcal(3.2^(abs(grid$x-grid$y)))
 439 print(wireframe(z~x*y,grid,cuts=50,
 440           drape=TRUE,
 441           scales=list(arrows=FALSE),
 442           xlab=list("Average Vesicle Length",rot=30),
 443           ylab=list("Monomer Length",rot=-35),
 444           zlab=list("Length Backward (kcal/mol)",rot=93)))
 445 rm(grid)
 446 @
 447
 448
 449 \newpage
 450 \subsubsection{Complex Formation Backward}
 451 \begin{equation}
 452   CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
 453   \label{eq:complex_formation_backward}
 454 \end{equation}
 455
 456 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 457 grid <- expand.grid(x=seq(-1,3,length.out=20),
 458                     y=seq(-1,3,length.out=20))
 459 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
 460 print(wireframe(z~x*y,grid,cuts=50,
 461           drape=TRUE,
 462           scales=list(arrows=FALSE),
 463           xlab=list("Vesicle Complex Formation",rot=30),
 464           ylab=list("Monomer Complex Formation",rot=-35),
 465           zlab=list("Complex Formation Backward",rot=93)))
 466 rm(grid)
 467 @
 468 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
 469 grid <- expand.grid(x=seq(-1,3,length.out=20),
 470                     y=seq(-1,3,length.out=20))
 471 grid$z <- to.kcal(3.2^(grid$x*grid$y-abs(grid$x*grid$y)))
 472 print(wireframe(z~x*y,grid,cuts=50,
 473           drape=TRUE,
 474           scales=list(arrows=FALSE),
 475           xlab=list("Vesicle Complex Formation",rot=30),
 476           ylab=list("Monomer Complex Formation",rot=-35),
 477           zlab=list("Complex Formation Backward (kcal/mol)",rot=93)))
 478 rm(grid)
 479 @
 480
 481
 482
 483
 484
 485 % \bibliographystyle{plainnat}
 486 % \bibliography{references.bib}
 487
 488
 489 \end{document}