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} \left|\log cu_\mathrm{vesicle}\right|}
254 \label{eq:curvature_forward}
257 The most common $\left|\left<\log {cu}_v\right>\right|$ is around
258 $0.013$, which with the most common $\mathrm{stdev} \log
259 cu_\mathrm{vesicle}$ of $0.213$ leads to a $\Delta \Delta G^\ddagger$
260 of $\Sexpr{format(digits=3,to.kcal(10^(0.13*0.213)))}
261 \frac{\mathrm{kcal}}{\mathrm{mol}}$. This is a consequence of the
262 relatively matched curvatures in our environment.
264 % 1.5 to 0.75 3 to 0.33
265 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
266 grid <- expand.grid(x=seq(0,max(c(sd(abs(log(c(1,3)))),
267 sd(abs(log(c(1,0.33)))),sd(abs(log(c(0.33,3)))))),length.out=20),
268 y=seq(0,max(c(mean(log(c(1,3)),
269 mean(log(c(1,0.33))),
270 mean(log(c(0.33,3)))))),length.out=20))
271 grid$z <- 10^(grid$x*grid$y)
272 print(wireframe(z~x*y,grid,cuts=50,
274 scales=list(arrows=FALSE),
275 xlab=list("Vesicle stdev log curvature",rot=30),
276 ylab=list("Vesicle average log curvature",rot=-35),
277 zlab=list("Vesicle Curvature Forward",rot=93)))
280 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
281 grid <- expand.grid(x=seq(0,max(c(sd(abs(log(c(1,3)))),
282 sd(abs(log(c(1,0.33)))),sd(abs(log(c(0.33,3)))))),length.out=20),
283 y=seq(0,max(c(mean(log(c(1,3)),
284 mean(log(c(1,0.33))),
285 mean(log(c(0.33,3)))))),length.out=20))
286 grid$z <- to.kcal(10^(grid$x*grid$y))
287 print(wireframe(z~x*y,grid,cuts=50,
289 scales=list(arrows=FALSE),
290 xlab=list("Vesicle stdev log curvature",rot=30),
291 ylab=list("Vesicle average log curvature",rot=-35),
292 zlab=list("Vesicle Curvature Forward (kcal/mol)",rot=93)))
297 \subsubsection{Length Forward}
299 As in the case of unsaturation, void formation is easier when vesicles
300 are made up of components of widely different lengths. Thus, when the
301 width of the distribution of lengths is larger, the forward rate
302 should be greater as well, leading us to an equation of the form
303 $x^{\mathrm{stdev} l_\mathrm{ves}}$, where $\mathrm{stdev}
304 l_\mathrm{ves}$ is the standard deviation of the length of the
305 components of the vesicle, which has a maximum possible value of 8 and
306 a minimum of 0 in this set of experiments. A convenient base for $x$
310 l_f = 2^{\mathrm{stdev} l_\mathrm{ves}}
311 \label{eq:length_forward}
314 The most common $\mathrm{stdev} l_\mathrm{ves}$ is around $3.4$, which leads to
315 a range of $\Delta \Delta G^\ddagger$ of
316 $\Sexpr{format(digits=3,to.kcal(2^(3.4)))}
317 \frac{\mathrm{kcal}}{\mathrm{mol}}$.
320 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
321 curve(2^x,from=0,to=sd(c(12,24)),
322 main="Length forward",
323 xlab="Standard Deviation of Length of Vesicle",
324 ylab="Length Forward Adjustment")
326 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
327 curve(to.kcal(2^x),from=0,to=sd(c(12,24)),
328 main="Length forward",
329 xlab="Standard Deviation of Length of Vesicle",
330 ylab="Length Forward Adjustment (kcal/mol)")
334 \subsubsection{Complex Formation}
335 There is no contribution of complex formation to the forward reaction
336 rate in the current formalism.
340 \label{eq:complex_formation_forward}
343 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
345 Just as the forward rate constant adjustment $k_{fi\mathrm{adj}}$
346 does, the backwards rate constant adjustment $k_{bi\mathrm{adj}}$
347 takes into account unsaturation ($un_b$), charge ($ch_b$), curvature
348 ($cu_b$), length ($l_b$), and complex formation ($CF1_b$), each of
349 which are modified depending on the specific specie and the vesicle
350 into which the specie is entering:
354 k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
358 \subsubsection{Unsaturation Backward}
360 Unsaturation also influences the ability of a lipid molecule to leave
361 a membrane. If a molecule has an unsaturation level which is different
362 from the surrounding membrane, it will be more likely to leave the
363 membrane. The more different the unsaturation level is, the greater
364 the propensity for the lipid molecule to leave. However, a vesicle
365 with some unsaturation is more favorable for lipids with more
366 unsaturation than the equivalent amount of less unsatuturation, so the
367 difference in energy between unsaturation is not linear. Therefore, an
368 equation with the shape
369 $x^{\left| y^{-\left< un_\mathrm{ves}\right> }-y^{-un_\mathrm{monomer}} \right| }$
370 where $\left<un_\mathrm{ves}\right>$ is the average unsaturation of
371 the vesicle, and $un_\mathrm{monomer}$ is the average unsaturation. In
372 this equation, as the average unsaturation of the vesicle is larger,
375 un_b = 7^{1-\left(20\left(2^{-\left<un_\mathrm{vesicle} \right>} - 2^{-un_\mathrm{monomer}} \right)^2+1\right)^{-1}}
376 \label{eq:unsaturation_backward}
379 The most common $\left<un_\mathrm{ves}\right>$ is around $1.7$, which leads to
380 a range of $\Delta \Delta G^\ddagger$ from
381 $\Sexpr{format(digits=3,to.kcal(7^(1-1/(5*(2^-1.7-2^-0)^2+1))))}
382 \frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with 0 unsaturation
384 $\Sexpr{format(digits=3,to.kcal(7^(1-1/(5*(2^-1.7-2^-4)^2+1))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
385 for monomers with 4 unsaturations.
388 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
389 grid <- expand.grid(x=seq(0,4,length.out=20),
390 y=seq(0,4,length.out=20))
391 grid$z <- (7^(1-1/(5*(2^-grid$x-2^-grid$y)^2+1)))
392 print(wireframe(z~x*y,grid,cuts=50,
394 scales=list(arrows=FALSE),
395 xlab=list("Average Vesicle Unsaturation",rot=30),
396 ylab=list("Monomer Unsaturation",rot=-35),
397 zlab=list("Unsaturation Backward",rot=93)))
400 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
401 grid <- expand.grid(x=seq(0,4,length.out=20),
402 y=seq(0,4,length.out=20))
403 grid$z <- to.kcal((7^(1-1/(5*(2^-grid$x-2^-grid$y)^2+1))))
404 print(wireframe(z~x*y,grid,cuts=50,
406 scales=list(arrows=FALSE),
407 xlab=list("Average Vesicle Unsaturation",rot=30),
408 ylab=list("Monomer Unsaturation",rot=-35),
409 zlab=list("Unsaturation Backward (kcal/mol)",rot=93)))
416 \subsubsection{Charge Backwards}
417 As in the case of monomers entering a vesicle, monomers leaving a
418 vesicle leave faster if their charge has the same sign as the average
419 charge vesicle. An equation of the form $ch_b = a^{\left<ch_v\right>
420 ch_m}$ is then appropriate, and using a base of $a=20$ yields:
423 ch_b = 20^{\left<{ch}_v\right> {ch}_m}
424 \label{eq:charge_backwards}
427 The most common $\left<ch_v\right>$ is around $-0.164$, which leads to
428 a range of $\Delta \Delta G^\ddagger$ from
429 $\Sexpr{format(digits=3,to.kcal(20^(-.164*-1)))}
430 \frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with charge $-1$ to
431 $0\frac{\mathrm{kcal}}{\mathrm{mol}}$
432 for monomers with charge $0$.
435 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
436 x <- seq(-1,0,length.out=20)
437 y <- seq(-1,0,length.out=20)
438 grid <- expand.grid(x=x,y=y)
439 grid$z <- as.vector(20^(outer(x,y)))
440 print(wireframe(z~x*y,grid,cuts=50,
442 scales=list(arrows=FALSE),
443 xlab=list("Average Vesicle Charge",rot=30),
444 ylab=list("Component Charge",rot=-35),
445 zlab=list("Charge Backwards",rot=93)))
448 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
449 x <- seq(-1,0,length.out=20)
450 y <- seq(-1,0,length.out=20)
451 grid <- expand.grid(x=x,y=y)
452 grid$z <- to.kcal(as.vector(20^(outer(x,y))))
453 print(wireframe(z~x*y,grid,cuts=50,
455 scales=list(arrows=FALSE),
456 xlab=list("Average Vesicle Charge",rot=30),
457 ylab=list("Component Charge",rot=-35),
458 zlab=list("Charge Backwards (kcal/mol)",rot=93)))
463 \subsubsection{Curvature Backwards}
465 The less a monomer's intrinsic curvature matches the average curvature
466 of the vesicle in which it is in, the greater its rate of efflux. If
467 the difference is 0, $cu_f$ needs to be one. To map negative and
468 positive curvature to the same range, we also need take the logarithm.
469 Increasing mismatches in curvature increase the rate of efflux, but
470 asymptotically. \textcolor{red}{It is this property which the
471 unsaturation backwards equation does \emph{not} satisfy, which I
472 think it should.} An equation which satisfies this critera has the
473 form $cu_f = a^{1-\left(b\left( \left< \log cu_\mathrm{vesicle} \right>
474 -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}$. An
475 alternative form would use the aboslute value of the difference,
476 however, this yields a cusp and sharp increase about the curvature
477 equilibrium, which is decidedly non-elegant. We have chosen bases of
481 cu_b = 7^{1-\left(20\left(\left< \log cu_\mathrm{vesicle} \right> -\log cu_\mathrm{monomer} \right)^2+1\right)^{-1}}
482 \label{eq:curvature_backwards}
485 The most common $\left<\log cu_\mathrm{ves}\right>$ is around $-0.013$, which leads to
486 a range of $\Delta \Delta G^\ddagger$ from
487 $\Sexpr{format(digits=3,to.kcal(7^(1-1/(20*(-0.013-log(0.8))^2+1))))}
488 \frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature 0.8
490 $\Sexpr{format(digits=3,to.kcal(7^(1-1/(20*(-0.013-log(1.3))^2+1))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
491 for monomers with curvature 1.3 to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature near 1.
494 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
495 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
496 y=seq(0.8,1.33,length.out=20))
497 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
498 print(wireframe(z~x*y,grid,cuts=50,
500 scales=list(arrows=FALSE),
501 xlab=list("Vesicle Curvature",rot=30),
502 ylab=list("Monomer Curvature",rot=-35),
503 zlab=list("Curvature Backward",rot=93)))
506 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
507 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
508 y=seq(0.8,1.33,length.out=20))
509 grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)))
510 print(wireframe(z~x*y,grid,cuts=50,
512 scales=list(arrows=FALSE),
513 xlab=list("Vesicle Curvature",rot=30),
514 ylab=list("Monomer Curvature",rot=-35),
515 zlab=list("Curvature Backward (kcal/mol)",rot=93)))
520 \subsubsection{Length Backwards}
522 In a model membrane, the dissociation constant increases by a factor
523 of approximately 3.2 per carbon decrease in acyl chain length (Nichols
524 1985). Unfortunatly, the experimental data known to us only measures
525 chain length less than or equal to the bulk lipid, and does not exceed
526 it, and is only known for one bulk lipid species (DOPC). We assume
527 then, that the increase is in relationship to the average vesicle, and
528 that lipids with larger acyl chain length will also show an increase
529 in their dissociation constant.
532 l_b = 3.2^{\left|\left<l_\mathrm{ves}\right>-l_\mathrm{monomer}\right|}
533 \label{eq:length_backward}
536 The most common $\left<\log l_\mathrm{ves}\right>$ is around $17.75$, which leads to
537 a range of $\Delta \Delta G^\ddagger$ from
538 $\Sexpr{format(digits=3,to.kcal(3.2^abs(12-17.75)))}
539 \frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with length 12
541 $\Sexpr{format(digits=3,to.kcal(3.2^abs(24-17.75)))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
542 for monomers with length 24 to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature near 18.
545 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
546 grid <- expand.grid(x=seq(12,24,length.out=20),
547 y=seq(12,24,length.out=20))
548 grid$z <- 3.2^(abs(grid$x-grid$y))
549 print(wireframe(z~x*y,grid,cuts=50,
551 scales=list(arrows=FALSE),
552 xlab=list("Average Vesicle Length",rot=30),
553 ylab=list("Monomer Length",rot=-35),
554 zlab=list("Length Backward",rot=93)))
557 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
558 grid <- expand.grid(x=seq(12,24,length.out=20),
559 y=seq(12,24,length.out=20))
560 grid$z <- to.kcal(3.2^(abs(grid$x-grid$y)))
561 print(wireframe(z~x*y,grid,cuts=50,
563 scales=list(arrows=FALSE),
564 xlab=list("Average Vesicle Length",rot=30),
565 ylab=list("Monomer Length",rot=-35),
566 zlab=list("Length Backward (kcal/mol)",rot=93)))
572 \subsubsection{Complex Formation Backward}
577 CF1_b=1.5^{\left<CF1_\mathrm{ves}\right> CF1_\mathrm{monomer}-\left|\left<CF1_\mathrm{ves}\right> CF1_\mathrm{monomer}\right|}
578 \label{eq:complex_formation_backward}
581 The most common $\left<CF1_\mathrm{ves}\right>$ is around $0.925$, which leads to
582 a range of $\Delta \Delta G^\ddagger$ from
583 $\Sexpr{format(digits=3,to.kcal(1.5^(0.925*-1-abs(0.925*-1))))}
584 \frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex formation $-1$
586 $\Sexpr{format(digits=3,to.kcal(1.5^(0.925*2-abs(0.925*2))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
587 for monomers with length $2$ to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex formation $0$.
590 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
591 grid <- expand.grid(x=seq(-1,3,length.out=20),
592 y=seq(-1,3,length.out=20))
593 grid$z <- 1.5^(grid$x*grid$y-abs(grid$x*grid$y))
594 print(wireframe(z~x*y,grid,cuts=50,
596 scales=list(arrows=FALSE),
597 xlab=list("Vesicle Complex Formation",rot=30),
598 ylab=list("Monomer Complex Formation",rot=-35),
599 zlab=list("Complex Formation Backward",rot=93)))
602 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
603 grid <- expand.grid(x=seq(-1,3,length.out=20),
604 y=seq(-1,3,length.out=20))
605 grid$z <- to.kcal(1.5^(grid$x*grid$y-abs(grid$x*grid$y)))
606 print(wireframe(z~x*y,grid,cuts=50,
608 scales=list(arrows=FALSE),
609 xlab=list("Vesicle Complex Formation",rot=30),
610 ylab=list("Monomer Complex Formation",rot=-35),
611 zlab=list("Complex Formation Backward (kcal/mol)",rot=93)))
619 % \bibliographystyle{plainnat}
620 % \bibliography{references.bib}