+\begin{equation}
+ un_b = 10^{\left(2^{- \left< un_\mathrm{ves} \right> }
+ -2^{-un_\mathrm{monomer}}\right)^2}
+ \label{eq:unsaturation_backward}
+\end{equation}
+
+The most common $\left<un_\mathrm{ves}\right>$ is around $1.7$, which leads to
+a range of $\Delta \Delta G^\ddagger$ from
+$\Sexpr{format(digits=3,to.kcal(10^((2^-1.7-2^-0)^2)))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with 0 unsaturation
+to
+$\Sexpr{format(digits=3,to.kcal(10^((2^-1.7-2^-4)^2)))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
+for monomers with 4 unsaturations.
+
+
+<<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$z <- 10^((2^-grid$x-2^-grid$y)^2)
+print(wireframe(z~x*y,grid,cuts=50,
+ drape=TRUE,
+ scales=list(arrows=FALSE),
+ xlab=list("Average Vesicle Unsaturation",rot=30),
+ ylab=list("Monomer Unsaturation",rot=-35),
+ zlab=list("Unsaturation Backward",rot=93)))
+rm(grid)
+@
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+grid <- expand.grid(x=seq(0,4,length.out=20),
+ y=seq(0,4,length.out=20))
+grid$z <- to.kcal(10^((2^-grid$x-2^-grid$y)^2))
+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/mol)",rot=93)))
+rm(grid)
+@
+
+\subsubsection{Unsaturation Backward II}
+
+Unsaturation also influences the ability of a lipid molecule to leave
+a membrane. If a molecule has an unsaturation level which is different
+from the surrounding membrane, it will be more likely to leave the
+membrane. The more different the unsaturation level is, the greater
+the propensity for the lipid molecule to leave. However, a vesicle
+with some unsaturation is more favorable for lipids with more
+unsaturation than the equivalent amount of less unsatuturation, so the
+difference in energy between unsaturation is not linear. Therefore, an
+equation with the shape
+$x^{\left| y^{-\left< un_\mathrm{ves}\right> }-y^{-un_\mathrm{monomer}} \right| }$
+where $\left<un_\mathrm{ves}\right>$ is the average unsaturation of
+the vesicle, and $un_\mathrm{monomer}$ is the average unsaturation. In
+this equation, as the average unsaturation of the vesicle is larger,