<<results=hide,echo=FALSE>>=
require(lattice)
require(grid)
+# R in cal / mol K
+to.kcal <- function(k,temp=300) {
+ gasconst <- 1.985
+ return(gasconst*temp*log(k)/1000)
+}
@
\section{State Equation}
% double check this with the bits in the paper
\begin{equation}
- \frac{d C^{j}_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C^j_{i_\mathrm{monomer}}\right] -
- k_{bi}k_{bi\mathrm{adj}}C^j_{i_\mathrm{ves}}
+ \frac{d C_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} -
+ k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}}
\label{eq:state}
\end{equation}
+For $k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]$,
+$k_{fi}$ has units of $\frac{\mathrm{m}}{\mathrm{s}}$,
+$k_{fi\mathrm{adj}}$ and $k_{bi\mathrm{adj}}$ are unitless,
+concentration is in units of $\frac{\mathrm{n}}{\mathrm{L}}$, surface
+area is in units of $\mathrm{m}^2$, $k_{bi}$ has units of
+$\frac{1}{\mathrm{s}}$ and $C_{i_\mathrm{ves}}$ has units of
+$\mathrm{n}$, Thus, we have
+
+\begin{equation}
+ \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} -
+ \frac{1}{\mathrm{s}} \mathrm{n}
+ =
+ \frac{\mathrm{m^3}}{\mathrm{s}} \frac{\mathrm{n}}{\mathrm{L}} \frac{1000\mathrm{L}}{\mathrm{m}^3} - \frac{\mathrm{n}}{\mathrm{s}}=
+ \frac{\mathrm{n}}{\mathrm{s}} = 1000 \frac{\mathrm{n}}{\mathrm{s}} - \frac{\mathrm{n}}{\mathrm{s}}
+ \label{eq:state_units}
+\end{equation}
+
+The 1000 isn't in \fref{eq:state} above, because it is unit-dependent.
+
\subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)}
\begin{equation}
ylab="Unsaturation Forward Adjustment")
@
+<<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
+curve(to.kcal(2^x),from=0,to=sd(c(0,4)),
+ main="Unsaturation forward",
+ xlab="Standard Deviation of Unsaturation of Vesicle",
+ ylab="Unsaturation Forward (k cal)")
+@
+
+
\newpage
\subsubsection{Charge Forward}
\begin{equation}
\label{eq:charge_forward}
\end{equation}
-<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
x <- seq(-1,0,length.out=20)
y <- seq(-1,0,length.out=20)
grid <- expand.grid(x=x,y=y)
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
- xlab="Average Vesicle Charge",
- ylab="Component Charge",
- zlab="Charge Forward"))
+ xlab=list("Average Vesicle Charge",rot=30),
+ ylab=list("Component Charge",rot=-35),
+ zlab=list("Charge Forward",rot=93)))
rm(x,y,grid)
@
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+x <- seq(-1,0,length.out=20)
+y <- seq(-1,0,length.out=20)
+grid <- expand.grid(x=x,y=y)
+grid$z <- as.vector(to.kcal(60^(-outer(x,y))))
+print(wireframe(z~x*y,grid,cuts=50,
+ drape=TRUE,
+ scales=list(arrows=FALSE),
+ xlab=list("Average Vesicle Charge",rot=30),
+ ylab=list("Component Charge",rot=-35),
+ zlab=list("Charge Forward (k cal)",rot=93)))
+rm(x,y,grid)
+@
+
+
\newpage
\subsubsection{Curvature Forward}
\begin{equation}
\end{equation}
<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
-curve(10^x,from=0,to=sd(c(0.8,1.33)),
+curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
+ sd(abs(log(c(1,1.33)))),
+ sd(abs(log(c(0.8,1)))))),
main="Curvature forward",
- xlab="Standard Deviation of Curvature of Vesicle",
+ xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
ylab="Curvature Forward Adjustment")
@
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
+curve(to.kcal(10^x),from=0,to=max(c(sd(abs(log(c(0.8,1.33)))),
+ sd(abs(log(c(1,1.33)))),
+ sd(abs(log(c(0.8,1)))))),
+ main="Curvature forward",
+ xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
+ ylab="Curvature Forward Adjustment (kcal)")
+@
+
+
\newpage
\subsubsection{Length Forward}
\begin{equation}
- l_f = 3^{\mathrm{stdev}\left|\log l_\mathrm{ves}\right|}
+ l_f = 3^{\mathrm{stdev} l_\mathrm{ves}}
\label{eq:length_forward}
\end{equation}
ylab="Length Forward Adjustment")
@
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
+curve(to.kcal(3^x),from=0,to=sd(c(12,24)),
+ main="Length forward",
+ xlab="Standard Deviation of Length of Vesicle",
+ ylab="Length Forward Adjustment (kcal)")
+@
+
+
\subsubsection{Complex Formation}
\begin{equation}
CF1_f=1
\newpage
\subsubsection{Unsaturation Backward}
\begin{equation}
- un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-\left<un_\mathrm{monomer}\right>}\right|}
+ un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
\label{eq:unsaturation_backward}
\end{equation}
-<<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
grid <- expand.grid(x=seq(0,4,length.out=20),
y=seq(0,4,length.out=20))
grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
- xlab="Average Vesicle Unsaturation",
- ylab="Monomer Unsaturation",
- zlab="Unsaturation Backward"))
+ xlab=list("Average Vesicle Unsaturation",rot=30),
+ ylab=list("Monomer Unsaturation",rot=-35),
+ zlab=list("Unsaturation Backward",rot=93)))
+rm(grid)
+@
+
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+grid <- expand.grid(x=seq(0,4,length.out=20),
+ y=seq(0,4,length.out=20))
+grid$z <- to.kcal(10^(abs(3.5^-grid$x-3.5^-grid$y)))
+print(wireframe(z~x*y,grid,cuts=50,
+ drape=TRUE,
+ scales=list(arrows=FALSE),
+ xlab=list("Average Vesicle Unsaturation",rot=30),
+ ylab=list("Monomer Unsaturation",rot=-35),
+ zlab=list("Unsaturation Backward (kcal)",rot=93)))
rm(grid)
@
+
\newpage
\subsubsection{Charge Backwards}
\begin{equation}
- ch_b = 60^{-\left<{ch}_v\right> {ch}_m}
+ ch_b = 20^{\left<{ch}_v\right> {ch}_m}
\label{eq:charge_backwards}
\end{equation}
-<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
x <- seq(-1,0,length.out=20)
y <- seq(-1,0,length.out=20)
grid <- expand.grid(x=x,y=y)
-grid$z <- as.vector(60^(-outer(x,y)))
+grid$z <- as.vector(20^(outer(x,y)))
+print(wireframe(z~x*y,grid,cuts=50,
+ drape=TRUE,
+ scales=list(arrows=FALSE),
+ xlab=list("Average Vesicle Charge",rot=30),
+ ylab=list("Component Charge",rot=-35),
+ zlab=list("Charge Backwards",rot=93)))
+rm(x,y,grid)
+@
+
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+x <- seq(-1,0,length.out=20)
+y <- seq(-1,0,length.out=20)
+grid <- expand.grid(x=x,y=y)
+grid$z <- to.kcal(as.vector(20^(outer(x,y))))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
- xlab="Average Vesicle Charge",
- ylab="Component Charge",
- zlab="Charge Backwards"))
+ xlab=list("Average Vesicle Charge",rot=30),
+ ylab=list("Component Charge",rot=-35),
+ zlab=list("Charge Backwards (kcal)",rot=93)))
rm(x,y,grid)
@
\newpage
\subsubsection{Curvature Backwards}
\begin{equation}
- cu_f = 4^{\left|\left|\log cu_\mathrm{vesicle}\right|-\left|\log cu_\mathrm{monomer}\right|\right|}
+ cu_f = 7^{1-\left(20\left(\log_{e} cu_\mathrm{vesicle}-\log_{e} cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
\label{eq:curvature_backwards}
\end{equation}
-<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
y=seq(0.8,1.33,length.out=20))
-grid$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
+grid$z <- 7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1))
+print(wireframe(z~x*y,grid,cuts=50,
+ drape=TRUE,
+ scales=list(arrows=FALSE),
+ xlab=list("Vesicle Curvature",rot=30),
+ ylab=list("Monomer Curvature",rot=-35),
+ zlab=list("Curvature Backward",rot=93)))
+rm(grid)
+@
+
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+grid <- expand.grid(x=seq(0.8,1.33,length.out=20),
+ y=seq(0.8,1.33,length.out=20))
+grid$z <- to.kcal(7^(1-1/(20*(log(grid$x)-log(grid$y))^2+1)))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
- xlab="Vesicle Curvature",
- ylab="Monomer Curvature",
- zlab="Curvature Backward"))
+ xlab=list("Vesicle Curvature",rot=30),
+ ylab=list("Monomer Curvature",rot=-35),
+ zlab=list("Curvature Backward (kcal)",rot=93)))
rm(grid)
@
+
\newpage
\subsubsection{Length Backwards}
\begin{equation}
\label{eq:length_backward}
\end{equation}
-<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
grid <- expand.grid(x=seq(12,24,length.out=20),
y=seq(12,24,length.out=20))
grid$z <- 3.2^(abs(grid$x-grid$y))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
- xlab="Average Vesicle Length",
- ylab="Monomer Length",
- zlab="Length Backward"))
+ xlab=list("Average Vesicle Length",rot=30),
+ ylab=list("Monomer Length",rot=-35),
+ zlab=list("Length Backward",rot=93)))
rm(grid)
@
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+grid <- expand.grid(x=seq(12,24,length.out=20),
+ y=seq(12,24,length.out=20))
+grid$z <- to.kcal(3.2^(abs(grid$x-grid$y)))
+print(wireframe(z~x*y,grid,cuts=50,
+ drape=TRUE,
+ scales=list(arrows=FALSE),
+ xlab=list("Average Vesicle Length",rot=30),
+ ylab=list("Monomer Length",rot=-35),
+ zlab=list("Length Backward (kcal)",rot=93)))
+rm(grid)
+@
+
+
\newpage
\subsubsection{Complex Formation Backward}
\begin{equation}
\label{eq:complex_formation_backward}
\end{equation}
-<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
grid <- expand.grid(x=seq(-1,3,length.out=20),
y=seq(-1,3,length.out=20))
grid$z <- 3.2^(grid$x*grid$y-abs(grid$x*grid$y))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
- xlab="Vesicle Complex Formation",
- ylab="Monomer Complex Formation",
- zlab="Complex Formation Backward"))
+ xlab=list("Vesicle Complex Formation",rot=30),
+ ylab=list("Monomer Complex Formation",rot=-35),
+ zlab=list("Complex Formation Backward",rot=93)))
+rm(grid)
+@
+
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+grid <- expand.grid(x=seq(-1,3,length.out=20),
+ y=seq(-1,3,length.out=20))
+grid$z <- to.kcal(3.2^(grid$x*grid$y-abs(grid$x*grid$y)))
+print(wireframe(z~x*y,grid,cuts=50,
+ drape=TRUE,
+ scales=list(arrows=FALSE),
+ xlab=list("Vesicle Complex Formation",rot=30),
+ ylab=list("Monomer Complex Formation",rot=-35),
+ zlab=list("Complex Formation Backward (kcal)",rot=93)))
rm(grid)
@
+
% \bibliographystyle{plainnat}
% \bibliography{references.bib}
projects / ool / lipid_simulation_formalism.git / blobdiff