From b9c7adf84db14e1d293d2aba36cbbc85f09a9e44 Mon Sep 17 00:00:00 2001 From: don Date: Fri, 11 Jun 2010 21:24:50 +0000 Subject: [PATCH] update kinetic formalism git-svn-id: svn+ssh://hemlock.ucr.edu/srv/svn/misc/trunk/origins_of_life@521 25fa0111-c432-4dab-af88-9f31a2f6ac42 --- kinetic_formalism.Rnw | 126 +++++++++++++++++++++++++++++++++++------- 1 file changed, 106 insertions(+), 20 deletions(-) diff --git a/kinetic_formalism.Rnw b/kinetic_formalism.Rnw index c77dc7b..22720d9 100644 --- a/kinetic_formalism.Rnw +++ b/kinetic_formalism.Rnw @@ -71,6 +71,20 @@ to.kcal <- function(k,temp=300) { \section{State Equation} % double check this with the bits in the paper + +Given a base forward kinetic parameter for the $i$th specie $k_{fi}$ +(which is dependent on lipid type, that is PC, PE, PS, etc.), an +adjustment parameter $k_{fi\mathrm{adj}}$ based on the vesicle and the +specific specie (length, unsaturation, etc.) (see~\fref{eq:kf_adj}), +the molar concentration of monomer of the $i$th specie +$\left[C_{i_\mathrm{monomer}}\right]$, the surface area of the vesicle +$S_\mathrm{ves}$, the base backwards kinetic parameter for the $i$th +specie $k_{bi}$ which is also dependent on lipid type, its adjustment +parameter $k_{bi\mathrm{adj}}$ (see~\fref{eq:kb_adj}), and the molar +concentration of the $i$th specie in the vesicle $C_{i_\mathrm{ves}}$, +the change in concentration of the $i$th specie in the vesicle per +change in time $\frac{d C_{i_\mathrm{ves}}}{dt}$ can be calculated: + \begin{equation} \frac{d C_{i_\mathrm{ves}}}{dt} = k_{fi}k_{fi\mathrm{adj}}\left[C_{i_\mathrm{monomer}}\right]S_\mathrm{ves} - k_{bi}k_{bi\mathrm{adj}}C_{i_\mathrm{ves}} @@ -98,6 +112,12 @@ The 1000 isn't in \fref{eq:state} above, because it is unit-dependent. \subsection{Forward adjustments ($k_{fi\mathrm{adj}}$)} +The forward rate constant adjustment, $k_{fi\mathrm{adj}}$ takes into +account unsaturation ($un_f$), charge ($ch_f$), curvature ($cu_f$), +length ($l_f$), and complex formation ($CF1_f$), each of which are +modified depending on the specific specie and the vesicle into which +the specie is entering. + \begin{equation} k_{fi\mathrm{adj}} = un_f \cdot ch_f \cdot cu_f \cdot l_f \cdot CF1_f \label{eq:kf_adj} @@ -105,18 +125,31 @@ The 1000 isn't in \fref{eq:state} above, because it is unit-dependent. \newpage \subsubsection{Unsaturation Forward} + +In order for a lipid to be inserted into a membrane, a void has to be +formed for it to fill. Voids can be generated by the combination of +unsaturated and saturated lipids forming herterogeneous domains. Void +formation is increased when the unsaturation of lipids in the vesicle +is widely distributed; in other words, the insertion of lipids into +the membrane is greater when the standard deviation of the +unsaturation is larger. Assuming that an increase in width of the +distribution linearly decreases the free energy of activation, the +$un_f$ parameter must follow +$x^{\mathrm{stdev}\left(un_\mathrm{ves}\right)}$ where $x > 1$, so a +convenient starting base for $x$ is 2: + \begin{equation} un_f = 2^{\mathrm{stdev}\left(un_\mathrm{ves}\right)} \label{eq:unsaturation_forward} \end{equation} +\setkeys{Gin}{width=3.2in} <>= curve(2^x,from=0,to=sd(c(0,4)), main="Unsaturation Forward", xlab="Standard Deviation of Unsaturation of Vesicle", ylab="Unsaturation Forward Adjustment") @ - <>= curve(to.kcal(2^x),from=0,to=sd(c(0,4)), main="Unsaturation forward", @@ -127,6 +160,22 @@ curve(to.kcal(2^x),from=0,to=sd(c(0,4)), \newpage \subsubsection{Charge Forward} + +A charged lipid such as PS approaching a vesicle with an average +charge of the same sign will experience repulsion, whereas those with +different signs will experience attraction, the degree of which is +dependent upon the charge of the monomer and the average charge of the +vesicle. If either the vesicle or the monomer has no charge, there +should be no effect of charge upon the rate. This leads us to the +following equation, $x^{-\left ch_m}$, where +$\left$ is the average charge of the vesicle, and $ch_m$ +is the charge of the monomer. If either $\left$ or $ch_m$ +is 0, the adjustment parameter is 1 (no change), whereas it decreases +if both are positive or negative, as the product of two real numbers +with the same sign is always positive. A convenient base for $x$ is +60, resulting in the following equation: + + \begin{equation} ch_f = 60^{-\left<{ch}_v\right> {ch}_m} \label{eq:charge_forward} @@ -138,36 +187,66 @@ y <- seq(-1,0,length.out=20) grid <- expand.grid(x=x,y=y) grid$z <- as.vector(60^(-outer(x,y))) print(wireframe(z~x*y,grid,cuts=50, - drape=TRUE, - scales=list(arrows=FALSE), - xlab=list("Average Vesicle Charge",rot=30), - ylab=list("Component Charge",rot=-35), - zlab=list("Charge Forward",rot=93))) + drape=TRUE, + scales=list(arrows=FALSE), + main="Charge Forward", + xlab=list("Average Vesicle Charge",rot=30), + ylab=list("Component Charge",rot=-35), + zlab=list("Charge Forward",rot=93))) rm(x,y,grid) @ - <>= x <- seq(-1,0,length.out=20) y <- seq(-1,0,length.out=20) grid <- expand.grid(x=x,y=y) grid$z <- as.vector(to.kcal(60^(-outer(x,y)))) print(wireframe(z~x*y,grid,cuts=50, - drape=TRUE, - scales=list(arrows=FALSE), - xlab=list("Average Vesicle Charge",rot=30), - ylab=list("Component Charge",rot=-35), - zlab=list("Charge Forward (kcal/mol)",rot=93))) + drape=TRUE, + scales=list(arrows=FALSE), + main="Charge Forward (kcal/mol)", + xlab=list("Average Vesicle Charge",rot=30), + ylab=list("Component Charge",rot=-35), + zlab=list("Charge Forward (kcal/mol)",rot=93))) rm(x,y,grid) @ \newpage \subsubsection{Curvature Forward} + +Curvature is a measure of the intrinsic propensity of specific lipids +to form micelles (positive curvature), inverted micelles (negative +curvature), or planar sheets (zero curvature). In this formalism, +curvature is measured as the ratio of the size of the head to that of +the base, so negative curvature is bounded by $(0,1)$, zero curvature +is 1, and positive curvature is bounded by $(1,\infty)$. The curvature +can be transformed into the typical postive/negative mapping using +$\log$, which has the additional property of making the range of +positive and negative curvature equal, and distributed about 0. + +As in the case of unsaturation, void formation is increased by the +presence of lipids with mismatched curvature. Thus, a larger +distribution of curvature in the vesicle increases the rate of lipid +insertion into the vesicle. However, a species with curvature $e^-1$ +will cancel out a species with curvature $e$, so we have to log +transform (turning these into -1 and 1), then take the absolute value +(1 and 1), and finally measure the width of the distribution. Thus, by +using the log transform to make the range of the lipid curvature equal +between positive and negative, taking the absolute value to cancel out +exactly mismatched curvatures, we come to an equation with the shape +$x^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|}$, and a +convenient base for $x$ is 10, yielding: + +{\color{red} Shouldn't a vesicle of -1,1,0 have the same activation + energy as a vesicle of 0,0,0? It doesn't currently.} + + \begin{equation} cu_f = 10^{\mathrm{stdev}\left|\log cu_\mathrm{vesicle}\right|} \label{eq:curvature_forward} \end{equation} + <>= curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))), sd(abs(log(c(1,1.33)))), @@ -176,7 +255,6 @@ curve(10^x,from=0,to=max(c(sd(abs(log(c(0.8,1.33)))), xlab="Standard Deviation of Absolute value of the Log of the Curvature of Vesicle", ylab="Curvature Forward Adjustment") @ - <>= curve(to.kcal(10^x),from=0,to=max(c(sd(abs(log(c(0.8,1.33)))), sd(abs(log(c(1,1.33)))), @@ -189,6 +267,17 @@ curve(to.kcal(10^x),from=0,to=max(c(sd(abs(log(c(0.8,1.33)))), \newpage \subsubsection{Length Forward} + +As in the case of unsaturation, void formation is easier when vesicles +are made up of components of widely different lengths. Thus, when the +width of the distribution of lengths is larger, the forward rate +should be greater as well, leading us to an equation of the form +$x^{\mathrm{stdev} l_\mathrm{ves}}$, where $\mathrm{stdev} +l_\mathrm{ves}$ is the standard deviation of the length of the +components of the vesicle, which has a maximum possible value of 8 and +a minimum of 0 in this set of experiments. A convenient base for $x$ +is 2, leading to: + \begin{equation} l_f = 2^{\mathrm{stdev} l_\mathrm{ves}} \label{eq:length_forward} @@ -200,7 +289,6 @@ curve(2^x,from=0,to=sd(c(12,24)), xlab="Standard Deviation of Length of Vesicle", ylab="Length Forward Adjustment") @ - <>= curve(to.kcal(2^x),from=0,to=sd(c(12,24)), main="Length forward", @@ -210,6 +298,9 @@ curve(to.kcal(2^x),from=0,to=sd(c(12,24)), \subsubsection{Complex Formation} +There is no contribution of complex formation to the forward reaction +rate in the current formalism. + \begin{equation} CF1_f=1 \label{eq:complex_formation_forward} @@ -219,7 +310,7 @@ curve(to.kcal(2^x),from=0,to=sd(c(12,24)), \begin{equation} k_{bi\mathrm{adj}} = un_b \cdot ch_b \cdot cu_b \cdot l_b \cdot CF1_b - \label{eq:kf_adj} + \label{eq:kb_adj} \end{equation} \newpage @@ -241,7 +332,6 @@ print(wireframe(z~x*y,grid,cuts=50, 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)) @@ -276,7 +366,6 @@ print(wireframe(z~x*y,grid,cuts=50, zlab=list("Charge Backwards",rot=93))) rm(x,y,grid) @ - <>= x <- seq(-1,0,length.out=20) y <- seq(-1,0,length.out=20) @@ -310,7 +399,6 @@ print(wireframe(z~x*y,grid,cuts=50, zlab=list("Curvature Backward",rot=93))) rm(grid) @ - <>= grid <- expand.grid(x=seq(0.8,1.33,length.out=20), y=seq(0.8,1.33,length.out=20)) @@ -344,7 +432,6 @@ print(wireframe(z~x*y,grid,cuts=50, zlab=list("Length Backward",rot=93))) rm(grid) @ - <>= grid <- expand.grid(x=seq(12,24,length.out=20), y=seq(12,24,length.out=20)) @@ -378,7 +465,6 @@ print(wireframe(z~x*y,grid,cuts=50, zlab=list("Complex Formation Backward",rot=93))) rm(grid) @ - <>= grid <- expand.grid(x=seq(-1,3,length.out=20), y=seq(-1,3,length.out=20)) -- 2.39.2