From 6366fa5af62fa94f9628aa73ac345d2711bcc945 Mon Sep 17 00:00:00 2001 From: don Date: Sun, 20 Jun 2010 18:24:27 +0000 Subject: [PATCH] fix up CF1 explanation and length issues; remove complaint about unsaturation git-svn-id: svn+ssh://hemlock.ucr.edu/srv/svn/misc/trunk/origins_of_life@542 25fa0111-c432-4dab-af88-9f31a2f6ac42 --- kinetic_formalism.Rnw | 30 ++++++++++++++++++++++-------- 1 file changed, 22 insertions(+), 8 deletions(-) diff --git a/kinetic_formalism.Rnw b/kinetic_formalism.Rnw index e52b838..afd5984 100644 --- 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 +$CF1_b=a^{\left + CF1_\mathrm{monomer}-\left|\left + CF1_\mathrm{monomer}\right|}$, 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} @@ -590,13 +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 $0$. <>= -- 2.39.2