fix up CF1 explanation and length issues; remove complaint about unsaturation

git-svn-id: svn+ssh://hemlock.ucr.edu/srv/svn/misc/trunk/origins_of_life@542 25fa0111-c432-4dab-af88-9f31a2f6ac42

diff --git a/kinetic_formalism.Rnw b/kinetic_formalism.Rnw
index e52b8380ddbf4395af33f19581ad84dbb344070c..afd5984a9bc79ceee84952b0f991e12ac80d50ee 100644 (file)
--- a/kinetic_formalism.Rnw
+++ b/kinetic_formalism.Rnw
@@ -479,9 +479,7 @@ of the vesicle in which it is in, the greater its rate of efflux. If
 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
 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,
 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,


@@ -583,6 +581,20 @@ rm(grid)
 \newpage
 \subsubsection{Complex Formation Backward}
 
 \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}
 
 
 \begin{equation}


@@ -590,13 +602,15 @@ rm(grid)
   \label{eq:complex_formation_backward}
 \end{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))))}
 $\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

 $\Sexpr{format(digits=3,to.kcal(1.5^(0.925*2-abs(0.925*2))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$
 $\Sexpr{format(digits=3,to.kcal(1.5^(0.925*2-abs(0.925*2))))}\frac{\mathrm{kcal}}{\mathrm{mol}}$

-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>>=
 
 
 <<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=