tweak length and complex formation formalisms

[ool/lipid_simulation_formalism.git] / kinetic_formalism.Rnw

diff --git a/kinetic_formalism.Rnw b/kinetic_formalism.Rnw
index c36f6c9ffcc4010ab01984184b3127bc3facca92..cedf6cfdd8f55a3a42eafba12c228523b85e663e 100644 (file)
--- a/kinetic_formalism.Rnw
+++ b/kinetic_formalism.Rnw
@@ -250,34 +250,48 @@ is $10$, yielding:
 
 \begin{equation}
  % cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}
-  cu_f = 10^{\left<\log cu_\mathrm{vesicle} \right>}
+  cu_f = 10^{\left|\left<\log cu_\mathrm{vesicle} \right>\right|\mathrm{stdev} \left|\log cu_\mathrm{vesicle}\right|}
   \label{eq:curvature_forward}
 \end{equation}
 
-The most common $\left<\log {cu}_v\right>$ is around $-0.165$, which leads to
-a range of $\Delta \Delta G^\ddagger$ from
-$\Sexpr{format(digits=3,to.kcal(60^(-.165*-1)))}
-\frac{\mathrm{kcal}}{\mathrm{mol}}$ to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$.
+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
+$\Sexpr{format(digits=3,to.kcal(10^(0.13*0.213)))}
+\frac{\mathrm{kcal}}{\mathrm{mol}}$
 
 % 1.5 to 0.75 3 to 0.33
-<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
-curve(10^x,from=0,to=max(abs(c(mean(log(c(0.8,1.33))),
-                    mean(log(c(1,1.33))),
-                    mean(log(c(0.8,1)))))),
-      main="Curvature forward",
-      xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
-      ylab="Curvature Forward Adjustment")
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+grid <- expand.grid(x=seq(0,max(c(sd(abs(log(c(1,3)))),
+                      sd(abs(log(c(1,0.33)))),sd(abs(log(c(0.33,3)))))),length.out=20),
+                    y=seq(0,max(c(mean(log(c(1,3)),
+                      mean(log(c(1,0.33))),
+                      mean(log(c(0.33,3)))))),length.out=20))
+grid$z <- 10^(grid$x*grid$y)
+print(wireframe(z~x*y,grid,cuts=50,
+          drape=TRUE,
+          scales=list(arrows=FALSE),
+          xlab=list("Vesicle stdev log curvature",rot=30),
+          ylab=list("Vesicle average log curvature",rot=-35),
+          zlab=list("Vesicle Curvature Forward",rot=93)))
+rm(grid)
 @ 
-<<fig=TRUE,echo=FALSE,results=hide,width=7,height=5>>=
-curve(to.kcal(10^(x^2)),from=0,to=max(abs(c(mean(log(c(0.8,1.33))),
-                                 mean(log(c(1,1.33))),
-                                 mean(log(c(0.8,1)))))),
-      main="Curvature forward",
-      xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle",
-      ylab="Curvature Forward Adjustment (kcal/mol)")
+<<fig=TRUE,echo=FALSE,results=hide,width=7,height=7>>=
+grid <- expand.grid(x=seq(0,max(c(sd(abs(log(c(1,3)))),
+                      sd(abs(log(c(1,0.33)))),sd(abs(log(c(0.33,3)))))),length.out=20),
+                    y=seq(0,max(c(mean(log(c(1,3)),
+                      mean(log(c(1,0.33))),
+                      mean(log(c(0.33,3)))))),length.out=20))
+grid$z <- to.kcal(10^(grid$x*grid$y))
+print(wireframe(z~x*y,grid,cuts=50,
+          drape=TRUE,
+          scales=list(arrows=FALSE),
+          xlab=list("Vesicle stdev log curvature",rot=30),
+          ylab=list("Vesicle average log curvature",rot=-35),
+          zlab=list("Vesicle Curvature Forward (kcal/mol)",rot=93)))
+rm(grid)
 @ 
 
-
 \newpage
 \subsubsection{Length Forward}
 
@@ -474,8 +488,20 @@ rm(grid)
 
 \newpage
 \subsubsection{Length Backwards}
+
+In a model membrane, the dissociation constant decreases by a factor
+of approximately 3.2 per carbon increase in acyl chain length (Nichols
+1985). Unfortunatly, the known experimental data 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).
+
+
+The dissociation constant decreases by approximately 3.2 per carbon
+increase in acyl chain length (Nichols 1985). We assume that this
+decrease is in relationship to the average vesicle length.
+
 \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}
 
@@ -508,7 +534,7 @@ rm(grid)
 \newpage
 \subsubsection{Complex Formation Backward}
 \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}