1 \documentclass[english,12pt]{article}
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18 \usepackage[x11names,svgnames]{xcolor}
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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 \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}}
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
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}
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}
97 The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
99 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
102 k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
107 \subsubsection{Unsaturation Forward}
109 un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
110 \label{eq:unsaturation_forward}
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")
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 (k cal)")
129 \subsubsection{Charge Forward}
131 ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
132 \label{eq:charge_forward}
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$z <- as.vector(60^(-outer(x,y)))
140 print(wireframe(z~x*y,grid,cuts=50,
142 scales=list(arrows=FALSE),
143 xlab=list("Average Vesicle Charge",rot=30),
144 ylab=list("Component Charge",rot=-35),
145 zlab=list("Charge Forward",rot=93)))
149 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
150 x <- seq(-1,0,length.out=20)
151 y <- seq(-1,0,length.out=20)
152 grid <- expand.grid(x=x,y=y)
153 grid$z <- as.vector(to.kcal(60^(-outer(x,y))))
154 print(wireframe(z~x*y,grid,cuts=50,
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 (k cal)",rot=93)))
165 \subsubsection{Curvature Forward}
167 cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
168 \label{eq:curvature_forward}
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")
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)")
191 \subsubsection{Length Forward}
193 l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
194 \label{eq:length_forward}
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")
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)")
212 \subsubsection{Complex Formation}
215 \label{eq:complex_formation_forward}
218 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
221 k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
226 \subsubsection{Unsaturation Backward}
228 un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
229 \label{eq:unsaturation_backward}
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$y))
236 print(wireframe(z~x*y,grid,cuts=50,
238 scales=list(arrows=FALSE),
239 xlab=list("Average Vesicle Unsaturation",rot=30),
240 ylab=list("Monomer Unsaturation",rot=-35),
241 zlab=list("Unsaturation Backward",rot=93)))
245 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
246 grid <- expand.grid(x=seq(0,4,length.out=20),
247 y=seq(0,4,length.out=20))
248 grid$z <- to.kcal(10^(abs(3.5^-grid$x-3.5^-grid$y)))
249 print(wireframe(z~x*y,grid,cuts=50,
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)",rot=93)))
260 \subsubsection{Charge Backwards}
262 ch_b = 20^{\left<{ch}_v\right> {ch}_m}
263 \label{eq:charge_backwards}
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$z <- as.vector(20^(outer(x,y)))
271 print(wireframe(z~x*y,grid,cuts=50,
273 scales=list(arrows=FALSE),
274 xlab=list("Average Vesicle Charge",rot=30),
275 ylab=list("Component Charge",rot=-35),
276 zlab=list("Charge Backwards",rot=93)))
280 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
281 x <- seq(-1,0,length.out=20)
282 y <- seq(-1,0,length.out=20)
283 grid <- expand.grid(x=x,y=y)
284 grid$z <- to.kcal(as.vector(20^(outer(x,y))))
285 print(wireframe(z~x*y,grid,cuts=50,
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)",rot=93)))
295 \subsubsection{Curvature Backwards}
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}
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$y))^2+1))
305 print(wireframe(z~x*y,grid,cuts=50,
307 scales=list(arrows=FALSE),
308 xlab=list("Vesicle Curvature",rot=30),
309 ylab=list("Monomer Curvature",rot=-35),
310 zlab=list("Curvature Backward",rot=93)))
314 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
315 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
316 y=seq(0.8,1.33,length.out=20))
317 grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)))
318 print(wireframe(z~x*y,grid,cuts=50,
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)",rot=93)))
329 \subsubsection{Length Backwards}
331 l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
332 \label{eq:length_backward}
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$y))
339 print(wireframe(z~x*y,grid,cuts=50,
341 scales=list(arrows=FALSE),
342 xlab=list("Average Vesicle Length",rot=30),
343 ylab=list("Monomer Length",rot=-35),
344 zlab=list("Length Backward",rot=93)))
348 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
349 grid <- expand.grid(x=seq(12,24,length.out=20),
350 y=seq(12,24,length.out=20))
351 grid$z <- to.kcal(3.2^(abs(grid$x-grid$y)))
352 print(wireframe(z~x*y,grid,cuts=50,
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)",rot=93)))
363 \subsubsection{Complex Formation Backward}
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}
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$y))
373 print(wireframe(z~x*y,grid,cuts=50,
375 scales=list(arrows=FALSE),
376 xlab=list("Vesicle Complex Formation",rot=30),
377 ylab=list("Monomer Complex Formation",rot=-35),
378 zlab=list("Complex Formation Backward",rot=93)))
382 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
383 grid <- expand.grid(x=seq(-1,3,length.out=20),
384 y=seq(-1,3,length.out=20))
385 grid$z <- to.kcal(3.2^(grid$x*grid$y-abs(grid$x*grid$y)))
386 print(wireframe(z~x*y,grid,cuts=50,
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)",rot=93)))
399 % \bibliographystyle{plainnat}
400 % \bibliography{references.bib}