1 \documentclass[english,12pt]{article}
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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|\left<\log cu_\mathrm{vesicle} \right>\right|\mathrm{stdev} \log cu_\mathrm{vesicle}}
254 \label{eq:curvature_forward}
257 The most common $\left|\left<\log {cu}_v\right>\right|$ is around $0.013$, which
258 with the most common $\mathrm{stdev} \log cu_\mathrm{vesicle}$ of
259 $0.213$ leads to a $\Delta \Delta G^\ddagger$ of
260 $\Sexpr{format(digits=3,to.kcal(10^(0.13*0.213)))}
261 \frac{\mathrm{kcal}}{\mathrm{mol}}$
263 % 1.5 to 0.75 3 to 0.33
264 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
265 grid <- expand.grid(x=seq(0,max(c(sd(log(c(1,3))),
266 sd(log(c(1,0.33))),sd(log(c(0.33,3))))),length.out=20),
267 y=seq(0,max(c(mean(log(c(1,3)),
268 mean(log(c(1,0.33))),
269 mean(log(c(0.33,3)))))),length.out=20))
270 grid$z <- 10^(grid$x*grid$y)
271 print(wireframe(z~x*y,grid,cuts=50,
273 scales=list(arrows=FALSE),
274 xlab=list("Vesicle stdev log curvature",rot=30),
275 ylab=list("Vesicle average log curvature",rot=-35),
276 zlab=list("Vesicle Curvature Forward",rot=93)))
279 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
280 grid <- expand.grid(x=seq(0,max(c(sd(log(c(1,3))),
281 sd(log(c(1,0.33))),sd(log(c(0.33,3))))),length.out=20),
282 y=seq(0,max(c(mean(log(c(1,3)),
283 mean(log(c(1,0.33))),
284 mean(log(c(0.33,3)))))),length.out=20))
285 grid$z <- to.kcal(10^(grid$x*grid$y))
286 print(wireframe(z~x*y,grid,cuts=50,
288 scales=list(arrows=FALSE),
289 xlab=list("Vesicle stdev log curvature",rot=30),
290 ylab=list("Vesicle average log curvature",rot=-35),
291 zlab=list("Vesicle Curvature Forward (kcal/mol)",rot=93)))
296 \subsubsection{Length Forward}
298 As in the case of unsaturation, void formation is easier when vesicles
299 are made up of components of widely different lengths. Thus, when the
300 width of the distribution of lengths is larger, the forward rate
301 should be greater as well, leading us to an equation of the form
302 $x^{\mathrm{stdev} l_\mathrm{ves}}$, where $\mathrm{stdev}
303 l_\mathrm{ves}$ is the standard deviation of the length of the
304 components of the vesicle, which has a maximum possible value of 8 and
305 a minimum of 0 in this set of experiments. A convenient base for $x$
309 l_f = 2^{\mathrm{stdev} l_\mathrm{ves}}
310 \label{eq:length_forward}
313 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
314 curve(2^x,from=0,to=sd(c(12,24)),
315 main="Length forward",
316 xlab="Standard Deviation of Length of Vesicle",
317 ylab="Length Forward Adjustment")
319 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
320 curve(to.kcal(2^x),from=0,to=sd(c(12,24)),
321 main="Length forward",
322 xlab="Standard Deviation of Length of Vesicle",
323 ylab="Length Forward Adjustment (kcal/mol)")
327 \subsubsection{Complex Formation}
328 There is no contribution of complex formation to the forward reaction
329 rate in the current formalism.
333 \label{eq:complex_formation_forward}
336 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
338 Just as the forward rate constant adjustment $k_{fi\mathrm{adj}}$
339 does, the backwards rate constant adjustment $k_{bi\mathrm{adj}}$
340 takes into account unsaturation ($un_b$), charge ($ch_b$), curvature
341 ($cu_b$), length ($l_b$), and complex formation ($CF1_b$), each of
342 which are modified depending on the specific specie and the vesicle
343 into which the specie is entering:
347 k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
351 \subsubsection{Unsaturation Backward}
353 Unsaturation also influences the ability of a lipid molecule to leave
354 a membrane. If a molecule has an unsaturation level which is different
355 from the surrounding membrane, it will be more likely to leave the
356 membrane. The more different the unsaturation level is, the greater
357 the propensity for the lipid molecule to leave. However, a vesicle
358 with some unsaturation is more favorable for lipids with more
359 unsaturation than the equivalent amount of less unsatuturation, so the
360 difference in energy between unsaturation is not linear. Therefore, an
361 equation with the shape
362 $x^{\left|y^{-\left<un_\mathrm{ves}\right>}-y^{-un_\mathrm{monomer}}\right|}$
363 where $\left<un_\mathrm{ves}\right>$ is the average unsaturation of
364 the vesicle, and $un_\mathrm{monomer}$ is the average unsaturation. In
365 this equation, as the average unsaturation of the vesicle is larger,
367 \textcolor{red}{I don't like this equation; the explanation above
368 seems really contrived. Need to discuss.}
371 un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
372 \label{eq:unsaturation_backward}
375 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
376 grid <- expand.grid(x=seq(0,4,length.out=20),
377 y=seq(0,4,length.out=20))
378 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
379 print(wireframe(z~x*y,grid,cuts=50,
381 scales=list(arrows=FALSE),
382 xlab=list("Average Vesicle Unsaturation",rot=30),
383 ylab=list("Monomer Unsaturation",rot=-35),
384 zlab=list("Unsaturation Backward",rot=93)))
387 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
388 grid <- expand.grid(x=seq(0,4,length.out=20),
389 y=seq(0,4,length.out=20))
390 grid$z <- to.kcal(10^(abs(3.5^-grid$x-3.5^-grid$y)))
391 print(wireframe(z~x*y,grid,cuts=50,
393 scales=list(arrows=FALSE),
394 xlab=list("Average Vesicle Unsaturation",rot=30),
395 ylab=list("Monomer Unsaturation",rot=-35),
396 zlab=list("Unsaturation Backward (kcal/mol)",rot=93)))
402 \subsubsection{Charge Backwards}
403 As in the case of monomers entering a vesicle, monomers leaving a
404 vesicle leave faster if their charge has the same sign as the average
405 charge vesicle. An equation of the form $ch_b = x^{\left<ch_v\right>
406 ch_m}$ is then appropriate, and using a base of 20 for $x$ yields:
409 ch_b = 20^{\left<{ch}_v\right> {ch}_m}
410 \label{eq:charge_backwards}
413 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
414 x <- seq(-1,0,length.out=20)
415 y <- seq(-1,0,length.out=20)
416 grid <- expand.grid(x=x,y=y)
417 grid$z <- as.vector(20^(outer(x,y)))
418 print(wireframe(z~x*y,grid,cuts=50,
420 scales=list(arrows=FALSE),
421 xlab=list("Average Vesicle Charge",rot=30),
422 ylab=list("Component Charge",rot=-35),
423 zlab=list("Charge Backwards",rot=93)))
426 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
427 x <- seq(-1,0,length.out=20)
428 y <- seq(-1,0,length.out=20)
429 grid <- expand.grid(x=x,y=y)
430 grid$z <- to.kcal(as.vector(20^(outer(x,y))))
431 print(wireframe(z~x*y,grid,cuts=50,
433 scales=list(arrows=FALSE),
434 xlab=list("Average Vesicle Charge",rot=30),
435 ylab=list("Component Charge",rot=-35),
436 zlab=list("Charge Backwards (kcal/mol)",rot=93)))
441 \subsubsection{Curvature Backwards}
443 The less a monomer's intrinsic curvature matches the average curvature
444 of the vesicle in which it is in, the greater its rate of efflux. If
445 the difference is 0, $cu_f$ needs to be one. To map negative and
446 positive curvature to the same range, we also need take the logarithm.
447 Increasing mismatches in curvature increase the rate of efflux, but
448 asymptotically. \textcolor{red}{It is this property which the
449 unsaturation backwards equation does \emph{not} satisfy, which I
450 think it should.} An equation which satisfies this critera has the
451 form $cu_f = a^{1-\left(b\left(\left<\log cu_\mathrm{vesicle} \right>
452 -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}$. An
453 alternative form would use the aboslute value of the difference,
454 however, this yields a cusp and sharp increase about the curvature
455 equilibrium, which is decidedly non-elegant. We have chosen bases of
459 cu_f = 7^{1-\left(20\left(\left<\log cu_\mathrm{vesicle} \right> -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
460 \label{eq:curvature_backwards}
463 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
464 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
465 y=seq(0.8,1.33,length.out=20))
466 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
467 print(wireframe(z~x*y,grid,cuts=50,
469 scales=list(arrows=FALSE),
470 xlab=list("Vesicle Curvature",rot=30),
471 ylab=list("Monomer Curvature",rot=-35),
472 zlab=list("Curvature Backward",rot=93)))
475 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
476 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
477 y=seq(0.8,1.33,length.out=20))
478 grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)))
479 print(wireframe(z~x*y,grid,cuts=50,
481 scales=list(arrows=FALSE),
482 xlab=list("Vesicle Curvature",rot=30),
483 ylab=list("Monomer Curvature",rot=-35),
484 zlab=list("Curvature Backward (kcal/mol)",rot=93)))
490 \subsubsection{Length Backwards}
492 l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
493 \label{eq:length_backward}
496 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
497 grid <- expand.grid(x=seq(12,24,length.out=20),
498 y=seq(12,24,length.out=20))
499 grid$z <- 3.2^(abs(grid$x-grid$y))
500 print(wireframe(z~x*y,grid,cuts=50,
502 scales=list(arrows=FALSE),
503 xlab=list("Average Vesicle Length",rot=30),
504 ylab=list("Monomer Length",rot=-35),
505 zlab=list("Length Backward",rot=93)))
508 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
509 grid <- expand.grid(x=seq(12,24,length.out=20),
510 y=seq(12,24,length.out=20))
511 grid$z <- to.kcal(3.2^(abs(grid$x-grid$y)))
512 print(wireframe(z~x*y,grid,cuts=50,
514 scales=list(arrows=FALSE),
515 xlab=list("Average Vesicle Length",rot=30),
516 ylab=list("Monomer Length",rot=-35),
517 zlab=list("Length Backward (kcal/mol)",rot=93)))
523 \subsubsection{Complex Formation Backward}
525 CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
526 \label{eq:complex_formation_backward}
529 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
530 grid <- expand.grid(x=seq(-1,3,length.out=20),
531 y=seq(-1,3,length.out=20))
532 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
533 print(wireframe(z~x*y,grid,cuts=50,
535 scales=list(arrows=FALSE),
536 xlab=list("Vesicle Complex Formation",rot=30),
537 ylab=list("Monomer Complex Formation",rot=-35),
538 zlab=list("Complex Formation Backward",rot=93)))
541 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
542 grid <- expand.grid(x=seq(-1,3,length.out=20),
543 y=seq(-1,3,length.out=20))
544 grid$z <- to.kcal(3.2^(grid$x*grid$y-abs(grid$x*grid$y)))
545 print(wireframe(z~x*y,grid,cuts=50,
547 scales=list(arrows=FALSE),
548 xlab=list("Vesicle Complex Formation",rot=30),
549 ylab=list("Monomer Complex Formation",rot=-35),
550 zlab=list("Complex Formation Backward (kcal/mol)",rot=93)))
558 % \bibliographystyle{plainnat}
559 % \bibliography{references.bib}