X-Git-Url: https://git.donarmstrong.com/?a=blobdiff_plain;ds=sidebyside;f=kinetic_formalism.Rnw;h=afd5984a9bc79ceee84952b0f991e12ac80d50ee;hb=6366fa5af62fa94f9628aa73ac345d2711bcc945;hp=8b387e28d27c3ae07b3de4e4235f084dab454267;hpb=b09b39a7a1bb929470680062a8ccbab0da67590d;p=ool%2Flipid_simulation_formalism.git diff --git a/kinetic_formalism.Rnw b/kinetic_formalism.Rnw index 8b387e2..afd5984 100644 --- a/kinetic_formalism.Rnw +++ b/kinetic_formalism.Rnw @@ -148,6 +148,11 @@ $1.5$, which leads to a $\Delta \Delta G^\ddagger$ of $\Sexpr{format(digits=3,to.kcal(2^1.5))} \frac{\mathrm{kcal}}{\mathrm{mol}}$. +It is not clear that the unsaturation of the inserted monomer will +affect the rate of the insertion positively or negatively, so we do +not include a term for it in this formalism. + + \setkeys{Gin}{width=3.2in} <>= curve(2^x,from=0,to=sd(c(0,4)), @@ -316,6 +321,13 @@ a range of $\Delta \Delta G^\ddagger$ of $\Sexpr{format(digits=3,to.kcal(2^(3.4)))} \frac{\mathrm{kcal}}{\mathrm{mol}}$. +While it could be argued that increased length of the monomer could +affect the rate of insertion into the membrane, it's not clear whether +it would increase (by decreasing the number of available hydrogen +bonds, for example) or decrease (by increasing the time taken to fully +insert the acyl chain, for example) the rate of insertion or to what +degree, so we do not take it into account in this formalism. + <>= curve(2^x,from=0,to=sd(c(12,24)), @@ -371,62 +383,6 @@ 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(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 = 7^{1-\left(20\left(2^{-\left} - 2^{-un_\mathrm{monomer}} \right)^2+1\right)^{-1}} \label{eq:unsaturation_backward} @@ -523,9 +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, @@ -627,6 +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 +$CF1_b=a^{\left + CF1_\mathrm{monomer}-\left|\left + CF1_\mathrm{monomer}\right|}$, 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} @@ -634,13 +602,15 @@ rm(grid) \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 +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 +\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 $0$. <>=