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55 \author{Don Armstrong}
56 \title{OOL Kinetic Formalisms}
62 <<results=hide,echo=FALSE>>=
66 to.kcal <- function(k,temp=300) {
68 return(-gasconst*temp*log(k)/1000)
72 \section{State Equation}
73 % double check this with the bits in the paper
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:
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}}
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
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}
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}
111 The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
113 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
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.
122 k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
127 \subsubsection{Unsaturation Forward}
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 $a^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}$ where $a > 1$, so a
139 convenient starting base for $a$ is $2$:
142 un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
143 \label{eq:unsaturation_forward}
146 The most common $\mathrm{stdev}\left(un_\mathrm{ves}\right)$ is around
147 $1.5$, which leads to a $\Delta \Delta G^\ddagger$ of
148 $\Sexpr{format(digits=3,to.kcal(2^1.5))}
149 \frac{\mathrm{kcal}}{\mathrm{mol}}$.
151 \setkeys{Gin}{width=3.2in}
152 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
153 curve(2^x,from=0,to=sd(c(0,4)),
154 main="Unsaturation Forward",
155 xlab="Standard Deviation of Unsaturation of Vesicle",
156 ylab="Unsaturation Forward Adjustment")
158 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
159 curve(to.kcal(2^x),from=0,to=sd(c(0,4)),
160 main="Unsaturation forward",
161 xlab="Standard Deviation of Unsaturation of Vesicle",
162 ylab="Unsaturation Forward (kcal/mol)")
167 \subsubsection{Charge Forward}
169 A charged lipid such as PS approaching a vesicle with an average
170 charge of the same sign will experience repulsion, whereas those with
171 different signs will experience attraction, the degree of which is
172 dependent upon the charge of the monomer and the average charge of the
173 vesicle. If either the vesicle or the monomer has no charge, there
174 should be no effect of charge upon the rate. This leads us to the
175 following equation, $a^{-\left<ch_v\right> ch_m}$, where
176 $\left<ch_v\right>$ is the average charge of the vesicle, and $ch_m$
177 is the charge of the monomer. If either $\left<ch_v\right>$ or $ch_m$
178 is 0, the adjustment parameter is 1 (no change), whereas it decreases
179 if both are positive or negative, as the product of two real numbers
180 with the same sign is always positive. A convenient base for $a$ is
181 60, resulting in the following equation:
185 ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
186 \label{eq:charge_forward}
189 The most common $\left<{ch}_v\right>$ is around $-0.165$, which leads to
190 a range of $\Delta \Delta G^\ddagger$ from
191 $\Sexpr{format(digits=3,to.kcal(60^(-.165*-1)))}
192 \frac{\mathrm{kcal}}{\mathrm{mol}}$ to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$.
194 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
195 x <- seq(-1,0,length.out=20)
196 y <- seq(-1,0,length.out=20)
197 grid <- expand.grid(x=x,y=y)
198 grid$z <- as.vector(60^(-outer(x,y)))
199 print(wireframe(z~x*y,grid,cuts=50,
201 scales=list(arrows=FALSE),
202 main="Charge Forward",
203 xlab=list("Average Vesicle Charge",rot=30),
204 ylab=list("Component Charge",rot=-35),
205 zlab=list("Charge Forward",rot=93)))
208 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
209 x <- seq(-1,0,length.out=20)
210 y <- seq(-1,0,length.out=20)
211 grid <- expand.grid(x=x,y=y)
212 grid$z <- as.vector(to.kcal(60^(-outer(x,y))))
213 print(wireframe(z~x*y,grid,cuts=50,
215 scales=list(arrows=FALSE),
216 main="Charge Forward (kcal/mol)",
217 xlab=list("Average Vesicle Charge",rot=30),
218 ylab=list("Component Charge",rot=-35),
219 zlab=list("Charge Forward (kcal/mol)",rot=93)))
225 \subsubsection{Curvature Forward}
227 Curvature is a measure of the intrinsic propensity of specific lipids
228 to form micelles (positive curvature), inverted micelles (negative
229 curvature), or planar sheets (zero curvature). In this formalism,
230 curvature is measured as the ratio of the size of the head to that of
231 the base, so negative curvature is bounded by $(0,1)$, zero curvature
232 is 1, and positive curvature is bounded by $(1,\infty)$. The curvature
233 can be transformed into the typical postive/negative mapping using
234 $\log$, which has the additional property of making the range of
235 positive and negative curvature equal, and distributed about 0.
237 As in the case of unsaturation, void formation is increased by the
238 presence of lipids with mismatched curvature. Thus, a larger
239 distribution of curvature in the vesicle increases the rate of lipid
240 insertion into the vesicle. However, a species with curvature $e^{-1}$
241 will cancel out a species with curvature $e$, so we have to log
242 transform (turning these into -1 and 1), then take the absolute value
243 (1 and 1), and finally measure the width of the distribution. Thus, by
244 using the log transform to make the range of the lipid curvature equal
245 between positive and negative, and taking the average to cancel out
246 exactly mismatched curvatures, we come to an equation with the shape
247 $a^{\left<\log cu_\mathrm{vesicle}\right>}$. A convenient base for $a$
252 % cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
253 cu_f = 10^{\left<\log cu_\mathrm{vesicle} \right>}
254 \label{eq:curvature_forward}
257 The most common $\left<\log {cu}_v\right>$ is around $-0.165$, which leads to
258 a range of $\Delta \Delta G^\ddagger$ from
259 $\Sexpr{format(digits=3,to.kcal(60^(-.165*-1)))}
260 \frac{\mathrm{kcal}}{\mathrm{mol}}$ to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$.
262 % 1.5 to 0.75 3 to 0.33
263 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
264 curve(10^x,from=0,to=max(abs(c(mean(log(c(0.8,1.33))),
265 mean(log(c(1,1.33))),
266 mean(log(c(0.8,1)))))),
267 main="Curvature forward",
268 xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
269 ylab="Curvature Forward Adjustment")
271 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
272 curve(to.kcal(10^(x^2)),from=0,to=max(abs(c(mean(log(c(0.8,1.33))),
273 mean(log(c(1,1.33))),
274 mean(log(c(0.8,1)))))),
275 main="Curvature forward",
276 xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
277 ylab="Curvature Forward Adjustment (kcal/mol)")
282 \subsubsection{Length Forward}
284 As in the case of unsaturation, void formation is easier when vesicles
285 are made up of components of widely different lengths. Thus, when the
286 width of the distribution of lengths is larger, the forward rate
287 should be greater as well, leading us to an equation of the form
288 $x^{\mathrm{stdev} l_\mathrm{ves}}$, where $\mathrm{stdev}
289 l_\mathrm{ves}$ is the standard deviation of the length of the
290 components of the vesicle, which has a maximum possible value of 8 and
291 a minimum of 0 in this set of experiments. A convenient base for $x$
295 l_f = 2^{\mathrm{stdev} l_\mathrm{ves}}
296 \label{eq:length_forward}
299 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
300 curve(2^x,from=0,to=sd(c(12,24)),
301 main="Length forward",
302 xlab="Standard Deviation of Length of Vesicle",
303 ylab="Length Forward Adjustment")
305 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
306 curve(to.kcal(2^x),from=0,to=sd(c(12,24)),
307 main="Length forward",
308 xlab="Standard Deviation of Length of Vesicle",
309 ylab="Length Forward Adjustment (kcal/mol)")
313 \subsubsection{Complex Formation}
314 There is no contribution of complex formation to the forward reaction
315 rate in the current formalism.
319 \label{eq:complex_formation_forward}
322 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
324 Just as the forward rate constant adjustment $k_{fi\mathrm{adj}}$
325 does, the backwards rate constant adjustment $k_{bi\mathrm{adj}}$
326 takes into account unsaturation ($un_b$), charge ($ch_b$), curvature
327 ($cu_b$), length ($l_b$), and complex formation ($CF1_b$), each of
328 which are modified depending on the specific specie and the vesicle
329 into which the specie is entering:
333 k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
337 \subsubsection{Unsaturation Backward}
339 Unsaturation also influences the ability of a lipid molecule to leave
340 a membrane. If a molecule has an unsaturation level which is different
341 from the surrounding membrane, it will be more likely to leave the
342 membrane. The more different the unsaturation level is, the greater
343 the propensity for the lipid molecule to leave. However, a vesicle
344 with some unsaturation is more favorable for lipids with more
345 unsaturation than the equivalent amount of less unsatuturation, so the
346 difference in energy between unsaturation is not linear. Therefore, an
347 equation with the shape
348 $x^{\left|y^{-\left<un_\mathrm{ves}\right>}-y^{-un_\mathrm{monomer}}\right|}$
349 where $\left<un_\mathrm{ves}\right>$ is the average unsaturation of
350 the vesicle, and $un_\mathrm{monomer}$ is the average unsaturation. In
351 this equation, as the average unsaturation of the vesicle is larger,
353 \textcolor{red}{I don't like this equation; the explanation above
354 seems really contrived. Need to discuss.}
357 un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
358 \label{eq:unsaturation_backward}
361 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
362 grid <- expand.grid(x=seq(0,4,length.out=20),
363 y=seq(0,4,length.out=20))
364 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
365 print(wireframe(z~x*y,grid,cuts=50,
367 scales=list(arrows=FALSE),
368 xlab=list("Average Vesicle Unsaturation",rot=30),
369 ylab=list("Monomer Unsaturation",rot=-35),
370 zlab=list("Unsaturation Backward",rot=93)))
373 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
374 grid <- expand.grid(x=seq(0,4,length.out=20),
375 y=seq(0,4,length.out=20))
376 grid$z <- to.kcal(10^(abs(3.5^-grid$x-3.5^-grid$y)))
377 print(wireframe(z~x*y,grid,cuts=50,
379 scales=list(arrows=FALSE),
380 xlab=list("Average Vesicle Unsaturation",rot=30),
381 ylab=list("Monomer Unsaturation",rot=-35),
382 zlab=list("Unsaturation Backward (kcal/mol)",rot=93)))
388 \subsubsection{Charge Backwards}
389 As in the case of monomers entering a vesicle, monomers leaving a
390 vesicle leave faster if their charge has the same sign as the average
391 charge vesicle. An equation of the form $ch_b = x^{\left<ch_v\right>
392 ch_m}$ is then appropriate, and using a base of 20 for $x$ yields:
395 ch_b = 20^{\left<{ch}_v\right> {ch}_m}
396 \label{eq:charge_backwards}
399 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
400 x <- seq(-1,0,length.out=20)
401 y <- seq(-1,0,length.out=20)
402 grid <- expand.grid(x=x,y=y)
403 grid$z <- as.vector(20^(outer(x,y)))
404 print(wireframe(z~x*y,grid,cuts=50,
406 scales=list(arrows=FALSE),
407 xlab=list("Average Vesicle Charge",rot=30),
408 ylab=list("Component Charge",rot=-35),
409 zlab=list("Charge Backwards",rot=93)))
412 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
413 x <- seq(-1,0,length.out=20)
414 y <- seq(-1,0,length.out=20)
415 grid <- expand.grid(x=x,y=y)
416 grid$z <- to.kcal(as.vector(20^(outer(x,y))))
417 print(wireframe(z~x*y,grid,cuts=50,
419 scales=list(arrows=FALSE),
420 xlab=list("Average Vesicle Charge",rot=30),
421 ylab=list("Component Charge",rot=-35),
422 zlab=list("Charge Backwards (kcal/mol)",rot=93)))
427 \subsubsection{Curvature Backwards}
429 The less a monomer's intrinsic curvature matches the average curvature
430 of the vesicle in which it is in, the greater its rate of efflux. If
431 the difference is 0, $cu_f$ needs to be one. To map negative and
432 positive curvature to the same range, we also need take the logarithm.
433 Increasing mismatches in curvature increase the rate of efflux, but
434 asymptotically. \textcolor{red}{It is this property which the
435 unsaturation backwards equation does \emph{not} satisfy, which I
436 think it should.} An equation which satisfies this critera has the
437 form $cu_f = a^{1-\left(b\left(\left<\log cu_\mathrm{vesicle} \right>
438 -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}$. An
439 alternative form would use the aboslute value of the difference,
440 however, this yields a cusp and sharp increase about the curvature
441 equilibrium, which is decidedly non-elegant. We have chosen bases of
445 cu_f = 7^{1-\left(20\left(\left<\log cu_\mathrm{vesicle} \right> -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
446 \label{eq:curvature_backwards}
449 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
450 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
451 y=seq(0.8,1.33,length.out=20))
452 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
453 print(wireframe(z~x*y,grid,cuts=50,
455 scales=list(arrows=FALSE),
456 xlab=list("Vesicle Curvature",rot=30),
457 ylab=list("Monomer Curvature",rot=-35),
458 zlab=list("Curvature Backward",rot=93)))
461 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
462 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
463 y=seq(0.8,1.33,length.out=20))
464 grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)))
465 print(wireframe(z~x*y,grid,cuts=50,
467 scales=list(arrows=FALSE),
468 xlab=list("Vesicle Curvature",rot=30),
469 ylab=list("Monomer Curvature",rot=-35),
470 zlab=list("Curvature Backward (kcal/mol)",rot=93)))
476 \subsubsection{Length Backwards}
478 l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
479 \label{eq:length_backward}
482 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
483 grid <- expand.grid(x=seq(12,24,length.out=20),
484 y=seq(12,24,length.out=20))
485 grid$z <- 3.2^(abs(grid$x-grid$y))
486 print(wireframe(z~x*y,grid,cuts=50,
488 scales=list(arrows=FALSE),
489 xlab=list("Average Vesicle Length",rot=30),
490 ylab=list("Monomer Length",rot=-35),
491 zlab=list("Length Backward",rot=93)))
494 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
495 grid <- expand.grid(x=seq(12,24,length.out=20),
496 y=seq(12,24,length.out=20))
497 grid$z <- to.kcal(3.2^(abs(grid$x-grid$y)))
498 print(wireframe(z~x*y,grid,cuts=50,
500 scales=list(arrows=FALSE),
501 xlab=list("Average Vesicle Length",rot=30),
502 ylab=list("Monomer Length",rot=-35),
503 zlab=list("Length Backward (kcal/mol)",rot=93)))
509 \subsubsection{Complex Formation Backward}
511 CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
512 \label{eq:complex_formation_backward}
515 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
516 grid <- expand.grid(x=seq(-1,3,length.out=20),
517 y=seq(-1,3,length.out=20))
518 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
519 print(wireframe(z~x*y,grid,cuts=50,
521 scales=list(arrows=FALSE),
522 xlab=list("Vesicle Complex Formation",rot=30),
523 ylab=list("Monomer Complex Formation",rot=-35),
524 zlab=list("Complex Formation Backward",rot=93)))
527 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
528 grid <- expand.grid(x=seq(-1,3,length.out=20),
529 y=seq(-1,3,length.out=20))
530 grid$z <- to.kcal(3.2^(grid$x*grid$y-abs(grid$x*grid$y)))
531 print(wireframe(z~x*y,grid,cuts=50,
533 scales=list(arrows=FALSE),
534 xlab=list("Vesicle Complex Formation",rot=30),
535 ylab=list("Monomer Complex Formation",rot=-35),
536 zlab=list("Complex Formation Backward (kcal/mol)",rot=93)))
544 % \bibliographystyle{plainnat}
545 % \bibliography{references.bib}