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>>=
67 \section{State Equation}
68 % double check this with the bits in the paper
70 \frac{d C^{j}_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C^j_{i_\mathrm{monomer}}\right] -
71 k_{bi}k_{bi\mathrm{adj}}C^j_{i_\mathrm{ves}}
75 \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
78 k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f
83 \subsubsection{Unsaturation Forward}
85 un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}
86 \label{eq:unsaturation_forward}
89 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
90 curve(2^x,from=0,to=sd(c(0,4)),
91 main="Unsaturation forward",
92 xlab="Standard Deviation of Unsaturation of Vesicle",
93 ylab="Unsaturation Forward Adjustment")
97 \subsubsection{Charge Forward}
99 ch_f = 60^{-\left<{ch}_v\right> {ch}_m}
100 \label{eq:charge_forward}
103 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
104 x <- seq(-1,0,length.out=20)
105 y <- seq(-1,0,length.out=20)
106 grid <- expand.grid(x=x,y=y)
107 grid$z <- as.vector(60^(-outer(x,y)))
108 print(wireframe(z~x*y,grid,cuts=50,
110 scales=list(arrows=FALSE),
111 xlab="Average Vesicle Charge",
112 ylab="Component Charge",
113 zlab="Charge Forward"))
118 \subsubsection{Curvature Forward}
120 cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
121 \label{eq:curvature_forward}
124 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
125 curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
126 sd(abs(log(c(1,1.33)))),
127 sd(abs(log(c(0.8,1)))))),
128 main="Curvature forward",
129 xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
130 ylab="Curvature Forward Adjustment")
134 \subsubsection{Length Forward}
136 l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
137 \label{eq:length_forward}
140 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
141 curve(3^x,from=0,to=sd(c(12,24)),
142 main="Length forward",
143 xlab="Standard Deviation of Length of Vesicle",
144 ylab="Length Forward Adjustment")
147 \subsubsection{Complex Formation}
150 \label{eq:complex_formation_forward}
153 \subsection{Backward adjustments ($k_{bi\mathrm{adj}}$)}
156 k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b
161 \subsubsection{Unsaturation Backward}
163 un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-\left<un_\mathrm{monomer}\right>}\right|}
164 \label{eq:unsaturation_backward}
167 <<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
168 grid <- expand.grid(x=seq(0,4,length.out=20),
169 y=seq(0,4,length.out=20))
170 grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
171 print(wireframe(z~x*y,grid,cuts=50,
173 scales=list(arrows=FALSE),
174 xlab="Average Vesicle Unsaturation",
175 ylab="Monomer Unsaturation",
176 zlab="Unsaturation Backward"))
181 \subsubsection{Charge Backwards}
183 ch_b = 20^{\left<{ch}_v\right> {ch}_m}
184 \label{eq:charge_backwards}
187 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
188 x <- seq(-1,0,length.out=20)
189 y <- seq(-1,0,length.out=20)
190 grid <- expand.grid(x=x,y=y)
191 grid$z <- as.vector(20^(outer(x,y)))
192 print(wireframe(z~x*y,grid,cuts=50,
194 scales=list(arrows=FALSE),
195 xlab="Average Vesicle Charge",
196 ylab="Component Charge",
197 zlab="Charge Backwards"))
202 \subsubsection{Curvature Backwards}
204 cu_f = 7^{1-\left(20\left(\log cu_\mathrm{vesicle}-\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
205 \label{eq:curvature_backwards}
208 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
209 grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
210 y=seq(0.8,1.33,length.out=20))
211 grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
212 print(wireframe(z~x*y,grid,cuts=50,
214 scales=list(arrows=FALSE),
215 xlab="Vesicle Curvature",
216 ylab="Monomer Curvature",
217 zlab="Curvature Backward"))
222 \subsubsection{Length Backwards}
224 l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
225 \label{eq:length_backward}
228 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
229 grid <- expand.grid(x=seq(12,24,length.out=20),
230 y=seq(12,24,length.out=20))
231 grid$z <- 3.2^(abs(grid$x-grid$y))
232 print(wireframe(z~x*y,grid,cuts=50,
234 scales=list(arrows=FALSE),
235 xlab="Average Vesicle Length",
236 ylab="Monomer Length",
237 zlab="Length Backward"))
242 \subsubsection{Complex Formation Backward}
244 CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
245 \label{eq:complex_formation_backward}
248 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
249 grid <- expand.grid(x=seq(-1,3,length.out=20),
250 y=seq(-1,3,length.out=20))
251 grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
252 print(wireframe(z~x*y,grid,cuts=50,
254 scales=list(arrows=FALSE),
255 xlab="Vesicle Complex Formation",
256 ylab="Monomer Complex Formation",
257 zlab="Complex Formation Backward"))
264 % \bibliographystyle{plainnat}
265 % \bibliography{references.bib}