$\Sexpr{format(digits=3,to.kcal(2^1.5))}
\frac{\mathrm{kcal}}{\mathrm{mol}}$.
+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)),
\label{eq:curvature_forward}
\end{equation}
-The most common $\left|\left<\log {cu}_v\right>\right|$ is around $0.013$, which
-with the most common $\mathrm{stdev} \log cu_\mathrm{vesicle}$ of
-$0.213$ leads to a $\Delta \Delta G^\ddagger$ of
-$\Sexpr{format(digits=3,to.kcal(10^(0.13*0.213)))}
-\frac{\mathrm{kcal}}{\mathrm{mol}}$
+The most common $\left|\left<\log {cu}_v\right>\right|$ is around
+$0.013$, which with the most common $\mathrm{stdev} \log
+cu_\mathrm{vesicle}$ of $0.213$ leads to a $\Delta \Delta G^\ddagger$
+of $\Sexpr{format(digits=3,to.kcal(10^(0.13*0.213)))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$. This is a consequence of the
+relatively matched curvatures in our environment.
% 1.5 to 0.75 3 to 0.33
<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
\label{eq:length_forward}
\end{equation}
+The most common $\mathrm{stdev} l_\mathrm{ves}$ is around $3.4$, which leads to
+a range of $\Delta \Delta G^\ddagger$ of
+$\Sexpr{format(digits=3,to.kcal(2^(3.4)))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$.
+
+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)),
main="Length forward",
unsaturation than the equivalent amount of less unsatuturation, so the
difference in energy between unsaturation is not linear. Therefore, an
equation with the shape
-$x^{\left|y^{-\left<un_\mathrm{ves}\right>}-y^{-un_\mathrm{monomer}}\right|}$
+$x^{\left| y^{-\left< un_\mathrm{ves}\right> }-y^{-un_\mathrm{monomer}} \right| }$
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,
-\textcolor{red}{I don't like this equation; the explanation above
- seems really contrived. Need to discuss.}
-
\begin{equation}
- un_b = 10^{\left|3.5^{-\left<un_\mathrm{ves}\right>}-3.5^{-un_\mathrm{monomer}}\right|}
+ 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}
\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
+$\Sexpr{format(digits=3,to.kcal(7^(1-1/(5*(2^-1.7-2^-0)^2+1))))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with 0 unsaturation
+to
+$\Sexpr{format(digits=3,to.kcal(7^(1-1/(5*(2^-1.7-2^-4)^2+1))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
+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$z <- 10^(abs(3.5^-grid$x-3.5^-grid$y))
+grid$z <- (7^(1-1/(5*(2^-grid$x-2^-grid$y)^2+1)))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
<<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$y)))
+grid$z <- to.kcal((7^(1-1/(5*(2^-grid$x-2^-grid$y)^2+1))))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
@
+
\newpage
\subsubsection{Charge Backwards}
As in the case of monomers entering a vesicle, monomers leaving a
vesicle leave faster if their charge has the same sign as the average
-charge vesicle. An equation of the form $ch_b = x^{\left<ch_v\right>
- ch_m}$ is then appropriate, and using a base of 20 for $x$ yields:
+charge vesicle. An equation of the form $ch_b = a^{\left<ch_v\right>
+ ch_m}$ is then appropriate, and using a base of $a=20$ yields:
\begin{equation}
ch_b = 20^{\left<{ch}_v\right> {ch}_m}
\label{eq:charge_backwards}
\end{equation}
+The most common $\left<ch_v\right>$ is around $-0.164$, which leads to
+a range of $\Delta \Delta G^\ddagger$ from
+$\Sexpr{format(digits=3,to.kcal(20^(-.164*-1)))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with charge $-1$ to
+$0\frac{\mathrm{kcal}}{\mathrm{mol}}$
+for monomers with charge $0$.
+
+
<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
x <- seq(-1,0,length.out=20)
y <- seq(-1,0,length.out=20)
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
-form $cu_f = a^{1-\left(b\left(\left<\log cu_\mathrm{vesicle} \right>
+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,
however, this yields a cusp and sharp increase about the curvature
$a=7$ and $b=20$.
\begin{equation}
- cu_f = 7^{1-\left(20\left(\left<\log cu_\mathrm{vesicle} \right> -\log cu_\mathrm{monomer}\right)^2+1\right)^{-1}}
+ cu_b = 7^{1-\left(20\left(\left< \log cu_\mathrm{vesicle} \right> -\log cu_\mathrm{monomer} \right)^2+1\right)^{-1}}
\label{eq:curvature_backwards}
\end{equation}
+The most common $\left<\log cu_\mathrm{ves}\right>$ is around $-0.013$, which leads to
+a range of $\Delta \Delta G^\ddagger$ from
+$\Sexpr{format(digits=3,to.kcal(7^(1-1/(20*(-0.013-log(0.8))^2+1))))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature 0.8
+to
+$\Sexpr{format(digits=3,to.kcal(7^(1-1/(20*(-0.013-log(1.3))^2+1))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
+for monomers with curvature 1.3 to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature near 1.
+
+
<<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))
rm(grid)
@
-
\newpage
\subsubsection{Length Backwards}
+
+In a model membrane, the dissociation constant increases by a factor
+of approximately 3.2 per carbon decrease in acyl chain length (Nichols
+1985). Unfortunatly, the experimental data known to us only measures
+chain length less than or equal to the bulk lipid, and does not exceed
+it, and is only known for one bulk lipid species (DOPC). We assume
+then, that the increase is in relationship to the average vesicle, and
+that lipids with larger acyl chain length will also show an increase
+in their dissociation constant.
+
\begin{equation}
- l_b = 3.2^{\left|l_\mathrm{ves}-l_\mathrm{monomer}\right|}
+ l_b = 3.2^{\left|\left<l_\mathrm{ves}\right>-l_\mathrm{monomer}\right|}
\label{eq:length_backward}
\end{equation}
+The most common $\left<\log l_\mathrm{ves}\right>$ is around $17.75$, which leads to
+a range of $\Delta \Delta G^\ddagger$ from
+$\Sexpr{format(digits=3,to.kcal(3.2^abs(12-17.75)))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with length 12
+to
+$\Sexpr{format(digits=3,to.kcal(3.2^abs(24-17.75)))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
+for monomers with length 24 to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature near 18.
+
+
<<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))
\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}
- CF1_b=1.5^{CF1_\mathrm{ves} CF1_\mathrm{monomer}-\left|CF1_\mathrm{ves} CF1_\mathrm{monomer}\right|}
+ CF1_b=1.5^{\left<CF1_\mathrm{ves}\right> CF1_\mathrm{monomer}-\left|\left<CF1_\mathrm{ves}\right> CF1_\mathrm{monomer}\right|}
\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
+$\Sexpr{format(digits=3,to.kcal(1.5^(0.925*-1-abs(0.925*-1))))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex
+formation $-1$ to
+$\Sexpr{format(digits=3,to.kcal(1.5^(0.925*2-abs(0.925*2))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
+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>>=
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$y))
+grid$z <- 1.5^(grid$x*grid$y-abs(grid$x*grid$y))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
<<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$y)))
+grid$z <- to.kcal(1.5^(grid$x*grid$y-abs(grid$x*grid$y)))
print(wireframe(z~x*y,grid,cuts=50,
drape=TRUE,
scales=list(arrows=FALSE),
projects / ool / lipid_simulation_formalism.git / blobdiff