X-Git-Url: https://git.donarmstrong.com/?a=blobdiff_plain;f=kinetic_formalism_competition.Rnw;h=e2a375c65b64ebde0bae514aa9d912f9c965f49d;hb=ac8372fb657e6caba5ce032b06ab55d89fbddac6;hp=fbc030795ea1dc436e18fdd9777778c6b67a956d;hpb=f18c3e4b049a9d188c7088f5eb605176f7dcc1c9;p=ool%2Flipid_simulation_formalism.git diff --git a/kinetic_formalism_competition.Rnw b/kinetic_formalism_competition.Rnw index fbc0307..e2a375c 100644 --- a/kinetic_formalism_competition.Rnw +++ b/kinetic_formalism_competition.Rnw @@ -63,7 +63,7 @@ <>= opts_chunk$set(dev="CairoPDF",out.width="\\columnwidth",out.height="0.7\\textheight",out.extra="keepaspectratio") opts_chunk$set(cache=TRUE, autodep=TRUE) -options(scipen = -2, digits = 1) +options(scipen = -1, digits = 2) library("lattice") library("grid") library("Hmisc") @@ -120,7 +120,7 @@ to.kcal <- function(k,temp=300) { % double check this with the bits in the paper For a system with monomers $(_\mathrm{monomer})$ and a vesicle -$(_\mathrm{vesicle})$, the change in composition of the $i$th component of +$(_\mathrm{vesicle})$, the change in concentration of the $i$th component of a lipid vesicle per change in time ($d \left[C_{i_\mathrm{vesicle}}\right]/dt$) can be described by a modification of the basic mass action law: @@ -151,24 +151,27 @@ kf <- (as.numeric(kf.prime)*10^-3)/(63e-20*6.022e23) @ Each of the 5 lipid types has different kinetic parameters; where -available, these were taken from literature. - -% \begin{table} -% \centering -% \begin{tabular}{c c c c c c c c} -% \toprule -% Type & $k_\mathrm{f}$ $\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$ & $k'_\mathrm{f}$ $\left(\frac{1}{\mathrm{M} \mathrm{s}}\right)$ & $k_\mathrm{b}$ $\left(\mathrm{s}^{-1}\right)$ & Area $\left({Å}^2\right)$ & Charge & $\mathrm{CF}1$ & Curvature \\ -% \midrule -% PC & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[1]))}$ & $3.7 \times 10^6$ & $2 \times 10^{-5}$ & 63 & 0 & 2 & 0.8 \\ -% PS & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[2]))}$ & $3.7 \times 10^6$ & $1.25\times 10^{-5}$ & 54 & -1 & 0 & 1 \\ -% CHOL & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[3]))}$ & $5.1 \times 10^7$ & $2.8 \times 10^{-4}$ & 38 & 0 & -1 & 1.21 \\ -% SM & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[4]))}$ & $3.7 \times 10^6$ & $3.1 \times 10^{-3}$ & 61 & 0 & 3 & 0.8 \\ -% PE & $\Sexpr{to.latex(format(digits=3,scientific=TRUE,kf[5]))}$ & $2.3 \times 10^6$ & $1 \times 10^{-5}$ & 55 & 0 & 0 & 1.33 \\ -% \bottomrule -% \end{tabular} -% \caption{Kinetic parameters and molecular properties of lipid types} -% \label{tab:kinetic_parameters_lipid_types} -% \end{table} +available, these were taken from literature (\cref{tab:kinetic_parameters_lipid_types}). + +\begin{table} + \centering + \begin{tabular}{c c c c c c c c} + \toprule + Type & $k_\mathrm{f}$ $\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$ + & $k'_\mathrm{f}$ $\left(\frac{1}{\mathrm{M} \mathrm{s}}\right)$ + & $k_\mathrm{b}$ $\left(\mathrm{s}^{-1}\right)$ + & Area $\left({Å}^2\right)$ & Charge & $\mathrm{CF}1$ & Curvature \\ + \midrule + PC & $\Sexpr{kf[1]}$ & $3.7 \times 10^6$ & $2 \times 10^{-5}$ & 63 & 0 & 2 & 0.8 \\ + PS & $\Sexpr{kf[2]}$ & $3.7 \times 10^6$ & $1.25\times 10^{-5}$ & 54 & -1 & 0 & 1 \\ + CHOL & $\Sexpr{kf[3]}$ & $5.1 \times 10^7$ & $2.8 \times 10^{-4}$ & 38 & 0 & -1 & 1.21 \\ + SM & $\Sexpr{kf[4]}$ & $3.7 \times 10^6$ & $3.1 \times 10^{-3}$ & 61 & 0 & 3 & 0.8 \\ + PE & $\Sexpr{kf[5]}$ & $2.3 \times 10^6$ & $1 \times 10^{-5}$ & 55 & 0 & 0 & 1.33 \\ + \bottomrule + \end{tabular} + \caption{Kinetic parameters and molecular properties of lipid types} + \label{tab:kinetic_parameters_lipid_types} +\end{table} %%% \DLA{I think we may just reduce these three sections; area, $k_\mathrm{f}$ %%% and $k_\mathrm{b}$ to \cref{tab:kinetic_parameters_lipid_types} with @@ -189,7 +192,7 @@ however, \citet{Estronca2007:dhe_kinetics} measured the transfer of $\frac{1}{\mathrm{M} \mathrm{s}}$. We assume that this value is close to that of \ac{CHOL}, and use it for $k_{\mathrm{f}_\mathrm{\ac{CHOL}}}$. In the case of \ac{PE}, \citet{Abreu2004:kinetics_ld_lo} measured the association of -\ac{NBDDMPE} with \ac{POPC} \acp{LUV} found a value for $k_\mathrm{f}$ of $2.3 \times 10^{6}$~% +\ac{NBDDMPE} with \ac{POPC} \acp{LUV} and found a value for $k_\mathrm{f}$ of $2.3 \times 10^{6}$~% $\frac{1}{\mathrm{M} \mathrm{s}}$. These three authors used a slightly different kinetic formalism ($\frac{d\left[A_v\right]}{dt} = k'_\mathrm{f}[A_m][L_v] - k_\mathrm{b}[A_v]$), so we correct their values of $k_\mathrm{f}$ by @@ -541,7 +544,7 @@ unsaturated and saturated lipids forming heterogeneous 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 % +unsaturation is larger. % %%% \RZ{May need to site (at least for us) works showing %%% mismatch-dependent ``defects''}. % Assuming that an increase in width of the distribution linearly @@ -808,7 +811,7 @@ reasonable base for $x$ is 2, leading to: \end{equation} The most common $\mathrm{stdev} l_\mathrm{vesicle}$ is around $3.4$, which leads to -a range of $\Delta \Delta G^\ddagger$ of +a $\Delta \Delta G^\ddagger$ of $\Sexpr{format(digits=3,to.kcal(2^(3.4)))} \frac{\mathrm{kcal}}{\mathrm{mol}}$. @@ -896,7 +899,7 @@ Just as the forward rate constant adjustment $k_{\mathrm{f}i\mathrm{adj}}$ does, the backwards rate constant adjustment $k_{\mathrm{b}i\mathrm{adj}}$ takes into account unsaturation ($un_\mathrm{b}$), charge ($ch_\mathrm{b}$), curvature ($cu_\mathrm{b}$), length ($l_\mathrm{b}$), and complex formation ($CF1_\mathrm{b}$), each of -which are modified depending on the specific component and the vesicle +which is modified depending on the specific component and the vesicle from which the component is exiting: @@ -992,10 +995,11 @@ popViewport(2) \end{figure} \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 $ch_\mathrm{b} = a^{\left - ch_m}$ is then appropriate, and using a base of $a=20$ yields: +As in the case of monomers entering a vesicle, opposites attract. +Monomers leaving a vesicle leave faster if their charge has the same +sign as the average charge vesicle. An equation of the form +$ch_\mathrm{b} = a^{\left ch_m}$ is then appropriate, and +using a base of $a=20$ yields: \begin{equation} ch_\mathrm{b} = 20^{\left<{ch}_v\right> {ch}_m} @@ -1067,11 +1071,11 @@ popViewport(2) The less a monomer's intrinsic curvature matches the average curvature of the vesicle in which it is in, the greater its rate of efflux. If -the difference is 0, $cu_\mathrm{f}$ needs to be one. To map negative and +the curvatures match exactly, $cu_\mathrm{f}$ needs to be one. To map negative and positive curvature to the same range, we also need take the logarithm. -Positive and negative curvature lipids cancel each other out, -resulting in an average curvature of 0; the average of the log also -has this property. Increasing mismatches in curvature increase the +Positive ($cu > 1$) and negative ($0 < cu < 1$) curvature lipids cancel each other out, +resulting in an average curvature of 1; the average of the log also +has this property (average curvature of 0). Increasing mismatches in curvature increase the rate of efflux, but asymptotically. An equation which satisfies these criteria has the form $cu_\mathrm{f} = a^{1-\left(b\left( \left< \log cu_\mathrm{vesicle} \right> -\log @@ -1094,7 +1098,7 @@ $\Sexpr{format(digits=3,to.kcal(7^(1-1/(20*(-0.013-log(1.3))^2+1))))}\frac{\math for monomers with curvature 1.3 to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with curvature near 1. The full range of values possible for $cu_\mathrm{b}$ are shown in -\cref{fig:cub_graph} +\cref{fig:cub_graph}. % \RZ{What about the opposite curvatures that actually do fit to each % other?} @@ -1179,10 +1183,11 @@ will also show an increase in their dissociation constant. The most common $\left$ 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 +\frac{\mathrm{kcal}}{\mathrm{mol}}$ for monomers with length 12 +to $0\frac{\mathrm{kcal}}{\mathrm{mol}}$ +for monomers with length near 18 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 length near 18. The full range of possible values of +for monomers with length 24. The full range of possible values of $l_\mathrm{b}$ are shown in \cref{fig:lb_graph} % (for methods? From McLean84LIB: The activation free energies and free @@ -1383,15 +1388,15 @@ small number of components which are devoid of \ac{CHOL}. Once the components of the environment have been selected, their concentrations are determined. In experiments where the environmental concentration is the same across all lipid components, the -concentration is set to $10^{-10}\mathrm{M}$. In other cases, the +concentration is set to $10^{-10}$~M. In other cases, the environmental concentration is set to a random number from a gamma distribution with shape parameter 2 and an average of -$10^{-10}\mathrm{M}$. The base concentration ($10^{-10}\mathrm{M}$) +$10^{-10}$~M. The base concentration ($10^{-10}$~M) can also be altered at the initialization of the experiment to specific values for specific lipid types or components. The environment is a volume which is the maximum number of vesicles -from a single simulation (4096) times the maximum size of the vesicle +from a single simulation (4096) times the size of the vesicle (usually 10000) divided by Avagadro's number divided by the total environmental lipid concentration, or usually \Sexpr{4096*10000/6.022E23/141E-10}~L. @@ -1402,9 +1407,10 @@ Initial vesicles are seeded by selecting lipid molecules from the environment until the vesicle reaches a specific starting size. The vesicle starting size has gamma distribution with shape parameter 2 and a mean of the per-simulation specified starting size, with a -minimum of 5 lipid molecules. Lipid molecules are then selected to be -added to the lipid membrane according to four different methods. In -the constant method, molecules are added in direct proportion to their +minimum of 5 lipid molecules, or can be specified to have a precise +number of molecules. Lipid molecules are then selected to be added to +the lipid membrane according to four different methods. In the +constant method, molecules are added in direct proportion to their concentration in the environment. The uniform method adds molecules in proportion to their concentration in the environment scaled by a uniform random value, whereas the random method adds molecules in @@ -1452,7 +1458,7 @@ Determining the number of molecules to add to the lipid membrane vesicle $S_\mathrm{vesicle}$ (see \cref{sec:ves_prop}), the time interval $dt$ during which lipids are added, the base $k_{\mathrm{f}i}$, and the concentration of the monomer in the environment -$\left[C_{i\mathrm{vesicle}}\right]$ (see \cref{eq:state}). +$\left[C_{i\mathrm{monomer}}\right]$ (see \cref{eq:state}). $k_{\mathrm{f}i\mathrm{adj}}$ is calculated (see \cref{eq:kf_adj}) based on the vesicle properties and their hypothesized effect on the rate (in as many cases as possible, experimentally based)