1 \documentclass[english,12pt]{article}
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55 \author{Don Armstrong}
56 \title{OOL Kinetic Formalisms}
62 <<results=hide,echo=FALSE>>=
67 \section{State Equation}
68 % double check this with the bits in the paper
70 \frac{d C_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} -
71 k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}}
75 For $k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]$,
76 $k_{fi}$ has units of $\frac{\mathrm{m}}{\mathrm{s}}$,
77 $k_{fi\mathrm{adj}}$ and $k_{bi\mathrm{adj}}$ are unitless,
78 concentration is in units of $\frac{\mathrm{n}}{\mathrm{L}}$, surface
79 area is in units of $\mathrm{m}^2$, $k_{bi}$ has units of
80 $\frac{1}{\mathrm{s}}$ and $C_{i_\mathrm{ves}}$ has units of
81 $\mathrm{n}$, Thus, we have
84 \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} -
85 \frac{1}{\mathrm{s}} \mathrm{n}
87 \frac{\mathrm{m^3}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{\mathrm{n}}{\mathrm{s}}=
88 \frac{\mathrm{n}}{\mathrm{s}} = 1000 \frac{\mathrm{n}}{\mathrm{s}} - \frac{\mathrm{n}}{\mathrm{s}}
89 \label{eq:state_units}
92 The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
94 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
97 k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
102 \subsubsection{Unsaturation Forward}
104 un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
105 \label{eq:unsaturation_forward}
108 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
109 curve(2^x,from=0,to=sd(c(0,4)),
110 main="Unsaturation forward",
111 xlab="Standard Deviation of Unsaturation of Vesicle",
112 ylab="Unsaturation Forward Adjustment")
116 \subsubsection{Charge Forward}
118 ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
119 \label{eq:charge_forward}
122 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
123 x <- seq(-1,0,length.out=20)
124 y <- seq(-1,0,length.out=20)
125 grid <- expand.grid(x=x,y=y)
126 grid$z <- as.vector(60^(-outer(x,y)))
127 print(wireframe(z~x*y,grid,cuts=50,
129 scales=list(arrows=FALSE),
130 xlab=list("Average Vesicle Charge",rot=30),
131 ylab=list("Component Charge",rot=-35),
132 zlab=list("Charge Forward",rot=93)))
137 \subsubsection{Curvature Forward}
139 cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
140 \label{eq:curvature_forward}
143 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
144 curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
145 sd(abs(log(c(1,1.33)))),
146 sd(abs(log(c(0.8,1)))))),
147 main="Curvature forward",
148 xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
149 ylab="Curvature Forward Adjustment")
153 \subsubsection{Length Forward}
155 l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
156 \label{eq:length_forward}
159 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
160 curve(3^x,from=0,to=sd(c(12,24)),
161 main="Length forward",
162 xlab="Standard Deviation of Length of Vesicle",
163 ylab="Length Forward Adjustment")
166 \subsubsection{Complex Formation}
169 \label{eq:complex_formation_forward}
172 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
175 k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
180 \subsubsection{Unsaturation Backward}
182 un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
183 \label{eq:unsaturation_backward}
186 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
187 grid <- expand.grid(x=seq(0,4,length.out=20),
188 y=seq(0,4,length.out=20))
189 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
190 print(wireframe(z~x*y,grid,cuts=50,
192 scales=list(arrows=FALSE),
193 xlab=list("Average Vesicle Unsaturation",rot=30),
194 ylab=list("Monomer Unsaturation",rot=-35),
195 zlab=list("Unsaturation Backward",rot=93)))
200 \subsubsection{Charge Backwards}
202 ch_b = 20^{\left<{ch}_v\right> {ch}_m}
203 \label{eq:charge_backwards}
206 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
207 x <- seq(-1,0,length.out=20)
208 y <- seq(-1,0,length.out=20)
209 grid <- expand.grid(x=x,y=y)
210 grid$z <- as.vector(20^(outer(x,y)))
211 print(wireframe(z~x*y,grid,cuts=50,
213 scales=list(arrows=FALSE),
214 xlab=list("Average Vesicle Charge",rot=30),
215 ylab=list("Component Charge",rot=-35),
216 zlab=list("Charge Backwards",rot=93)))
221 \subsubsection{Curvature Backwards}
223 cu_f = 7^{1-\left(20\left(\log cu_\mathrm{vesicle}-\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
224 \label{eq:curvature_backwards}
227 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
228 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
229 y=seq(0.8,1.33,length.out=20))
230 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
231 print(wireframe(z~x*y,grid,cuts=50,
233 scales=list(arrows=FALSE),
234 xlab=list("Vesicle Curvature",rot=30),
235 ylab=list("Monomer Curvature",rot=-35),
236 zlab=list("Curvature Backward",rot=93)))
241 \subsubsection{Length Backwards}
243 l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
244 \label{eq:length_backward}
247 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
248 grid <- expand.grid(x=seq(12,24,length.out=20),
249 y=seq(12,24,length.out=20))
250 grid$z <- 3.2^(abs(grid$x-grid$y))
251 print(wireframe(z~x*y,grid,cuts=50,
253 scales=list(arrows=FALSE),
254 xlab=list("Average Vesicle Length",rot=30),
255 ylab=list("Monomer Length",rot=-35),
256 zlab=list("Length Backward",rot=93)))
261 \subsubsection{Complex Formation Backward}
263 CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
264 \label{eq:complex_formation_backward}
267 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
268 grid <- expand.grid(x=seq(-1,3,length.out=20),
269 y=seq(-1,3,length.out=20))
270 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
271 print(wireframe(z~x*y,grid,cuts=50,
273 scales=list(arrows=FALSE),
274 xlab=list("Vesicle Complex Formation",rot=30),
275 ylab=list("Monomer Complex Formation",rot=-35),
276 zlab=list("Complex Formation Backward",rot=93)))
283 % \bibliographystyle{plainnat}
284 % \bibliography{references.bib}