$\Sexpr{format(digits=3,to.kcal(2^1.5))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$.
+It is not clear that the unsaturation of the inserted monomer will
+affect the rate of the insertion positively or negatively, so we do
+not include a term for it in this formalism.
+
+
\setkeys{Gin}{width=3.2in}
<<fig=TRUE,echo=FALSE,results=hide,width=5,height=5>>=
curve(2^x,from=0,to=sd(c(0,4)),
$\Sexpr{format(digits=3,to.kcal(2^(3.4)))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$.
+While it could be argued that increased length of the monomer could
+affect the rate of insertion into the membrane, it's not clear whether
+it would increase (by decreasing the number of available hydrogen
+bonds, for example) or decrease (by increasing the time taken to fully
+insert the acyl chain, for example) the rate of insertion or to what
+degree, so we do not take it into account in this formalism.
+
<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
curve(2^x,from=0,to=sd(c(12,24)),
the vesicle, and $un_\mathrm{monomer}$ is the average unsaturation. In
this equation, as the average unsaturation of the vesicle is larger,
-\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,
-
\begin{equation}
un_b = 7^{1-\left(20\left(2^{-\left<un_\mathrm{vesicle} \right>} - 2^{-un_\mathrm{monomer}} \right)^2+1\right)^{-1}}
\label{eq:unsaturation_backward}
the difference is 0, $cu_f$ needs to be one. To map negative and
positive curvature to the same range, we also need take the logarithm.
Increasing mismatches in curvature increase the rate of efflux, but
-asymptotically. \textcolor{red}{It is this property which the
- unsaturation backwards equation does \emph{not} satisfy, which I
- think it should.} An equation which satisfies this critera has the
+asymptotically. An equation which satisfies this critera has the
form $cu_f = a^{1-\left(b\left( \left< \log cu_\mathrm{vesicle} \right>
-\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}$. An
alternative form would use the aboslute value of the difference,
\newpage
\subsubsection{Complex Formation Backward}
+Complex formation describes the interaction between CHOL and PC or SM,
+where PC or SM protects the hydroxyl group of CHOL from interactions
+with water, the ``Umbrella Model''. PC ($CF1=2$) can interact with two
+CHOL, and SM ($CF1=3$) with three CHOL ($CF1=-1$). If the average of
+$CF1$ is positive (excess of SM and PC with regards to complex
+formation), species with negative $CF1$ (CHOL) will be retained. If
+average $CF1$ is negative, species with positive $CF1$ are retained.
+An equation which has this property is
+$CF1_b=a^{\left<CF1_\mathrm{ves}\right>
+ CF1_\mathrm{monomer}-\left|\left<CF1_\mathrm{ves}\right>
+ CF1_\mathrm{monomer}\right|}$, where difference of the exponent is
+zero if the average $CF1$ and the $CF1$ of the specie have the same
+sign, or double the product if the signs are different. A convenient
+base for $a$ is $1.5$.
\begin{equation}
\label{eq:complex_formation_backward}
\end{equation}
-The most common $\left<CF1_\mathrm{ves}\right>$ is around $0.925$, which leads to
-a range of $\Delta \Delta G^\ddagger$ from
+The most common $\left<CF1_\mathrm{ves}\right>$ is around $0.925$,
+which leads to a range of $\Delta \Delta G^\ddagger$ from
$\Sexpr{format(digits=3,to.kcal(1.5^(0.925*-1-abs(0.925*-1))))}
-\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex formation $-1$
-to
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex
+formation $-1$ to
$\Sexpr{format(digits=3,to.kcal(1.5^(0.925*2-abs(0.925*2))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
-for monomers with length $2$ to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex formation $0$.
+for monomers with complex formation $2$ to
+$0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex
+formation $0$.
<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
projects / ool / lipid_simulation_formalism.git / blobdiff