From b09b39a7a1bb929470680062a8ccbab0da67590d Mon Sep 17 00:00:00 2001 From: don Date: Wed, 16 Jun 2010 23:41:06 +0000 Subject: [PATCH] update kinetic formalism git-svn-id: svn+ssh://hemlock.ucr.edu/srv/svn/misc/trunk/origins_of_life@528 25fa0111-c432-4dab-af88-9f31a2f6ac42 --- kinetic_formalism.Rnw | 163 ++++++++++++++++++++++++++++++++++-------- 1 file changed, 134 insertions(+), 29 deletions(-) diff --git a/kinetic_formalism.Rnw b/kinetic_formalism.Rnw index cedf6cf..8b387e2 100644 --- a/kinetic_formalism.Rnw +++ b/kinetic_formalism.Rnw @@ -254,11 +254,12 @@ is $10$, yielding: \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 <>= @@ -310,6 +311,12 @@ is 2, leading to: \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 +$\Sexpr{format(digits=3,to.kcal(2^(3.4)))} +\frac{\mathrm{kcal}}{\mathrm{mol}}$. + + <>= curve(2^x,from=0,to=sd(c(12,24)), main="Length forward", @@ -359,23 +366,85 @@ 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 -$x^{\left|y^{-\left}-y^{-un_\mathrm{monomer}}\right|}$ +$x^{\left| y^{-\left< un_\mathrm{ves}\right> }-y^{-un_\mathrm{monomer}} \right| }$ where $\left$ 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(2^{- \left< un_\mathrm{ves} \right> } + -2^{-un_\mathrm{monomer}}\right)^2} + \label{eq:unsaturation_backward} +\end{equation} + +The most common $\left$ 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. + + +<>= +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$y)^2) +print(wireframe(z~x*y,grid,cuts=50, + drape=TRUE, + scales=list(arrows=FALSE), + xlab=list("Average Vesicle Unsaturation",rot=30), + ylab=list("Monomer Unsaturation",rot=-35), + zlab=list("Unsaturation Backward",rot=93))) +rm(grid) +@ +<>= +grid <- expand.grid(x=seq(0,4,length.out=20), + y=seq(0,4,length.out=20)) +grid$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 $\left$ 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 = 10^{\left|3.5^{-\left}-3.5^{-un_\mathrm{monomer}}\right|} + un_b = 7^{1-\left(20\left(2^{-\left} - 2^{-un_\mathrm{monomer}} \right)^2+1\right)^{-1}} \label{eq:unsaturation_backward} \end{equation} +The most common $\left$ 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. + + <>= 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), @@ -387,7 +456,7 @@ rm(grid) <>= 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), @@ -398,18 +467,27 @@ rm(grid) @ + \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 $ch_b = x^{\left - ch_m}$ is then appropriate, and using a base of 20 for $x$ yields: +charge vesicle. An equation of the form $ch_b = a^{\left + ch_m}$ 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 $\left$ 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 $0$. + + <>= x <- seq(-1,0,length.out=20) y <- seq(-1,0,length.out=20) @@ -448,7 +526,7 @@ 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> +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 @@ -456,10 +534,19 @@ equilibrium, which is decidedly non-elegant. We have chosen bases of $a=7$ 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. + + <>= grid <- expand.grid(x=seq(0.8,1.33,length.out=20), y=seq(0.8,1.33,length.out=20)) @@ -485,26 +572,32 @@ print(wireframe(z~x*y,grid,cuts=50, 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. +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|\left-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. + + <>= grid <- expand.grid(x=seq(12,24,length.out=20), y=seq(12,24,length.out=20)) @@ -533,15 +626,27 @@ rm(grid) \newpage \subsubsection{Complex Formation Backward} + + + \begin{equation} - CF1_b=1.5^{\left CF1_\mathrm{monomer}-\left|\left CF1_\mathrm{monomer}\right|} + CF1_b=1.5^{\left CF1_\mathrm{monomer}-\left|\left CF1_\mathrm{monomer}\right|} \label{eq:complex_formation_backward} \end{equation} +The most common $\left$ 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 length $2$ to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with complex formation $0$. + + <>= 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), @@ -553,7 +658,7 @@ rm(grid) <>= 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), -- 2.39.2