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

[ool/lipid_simulation_formalism.git] / kinetic_formalism.Rnw

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
-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,
@@ -583,6 +581,20 @@ rm(grid)
 \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}
@@ -590,13 +602,15 @@ rm(grid)
   \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>>=