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\begin{document} |
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%\renewcommand{\thefootnote}{\fnsymbol{footnote}} |
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%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} |
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\centering |
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\includegraphics[width=4in]{2lipidModel} |
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\caption{The parameters defining the behavior of the lipid |
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models. $l / d$ is the ratio of the head group to body diameter. |
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models. $\sigma_h / d$ is the ratio of the head group to body diameter. |
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Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
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was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
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used in other coarse-grained (DPD) simulations. The dipolar strength |
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used in other coarse-grained simulations. The dipolar strength |
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(and the temperature and pressure) were the only other parameters that |
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were varied systematically.\label{fig:lipidModel}} |
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\end{figure} |
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R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
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}}{4}}. |
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\end{equation} |
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Here, $d_i$ and $d_j$ are effect charge separation distances |
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associated with each of the two dipolar sites. This approximation to |
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the multipole expansion maintains the fast fall-off of the multipole |
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potentials but lacks the normal divergences when two polar groups get |
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close to one another. |
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Here, $d_i$ and $d_j$ are charge separation distances associated with |
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each of the two dipolar sites. This approximation to the multipole |
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expansion maintains the fast fall-off of the multipole potentials but |
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lacks the normal divergences when two polar groups get close to one |
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another. |
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|
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For the interaction between nonequivalent uniaxial ellipsoids (in this |
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case, between spheres and ellipsoids), the spheres are treated as |
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replicate the dielectric properties of water. Note that although we |
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are using larger cutoff and switching radii than Marrink {\it et al.}, |
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our solvent density at 300 K remains at 0.944 g cm$^{-3}$, and the |
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solvent diffuses at 0.43 $\AA^2 ps^{-1}$ (roughly twice as fast as |
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liquid water). |
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solvent diffuses at 0.43 $\AA^2 ps^{-1}$ (only twice as fast as liquid |
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water). |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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The parameters that were systematically varied in this study were the |
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size of the head group ($\sigma_h$), the strength of the dipole moment |
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($\mu$), and the temperature of the system. Values for $\sigma_h$ |
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ranged from 5.5 \AA\ to 6.5 \AA\ . If the width of the tails is taken |
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ranged from 5.5 \AA\ to 6.5 \AA. If the width of the tails is taken |
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to be the unit of length, these head groups correspond to a range from |
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$1.2 d$ to $1.41 d$. Since the solvent beads are nearly identical in |
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diameter to the tail ellipsoids, all distances that follow will be |
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entirely on the size of the head bead relative to the molecular body. |
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These values are tabulated in table \ref{tab:property}. Kucera {\it |
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et al.} have measured values for the head group spacings for a number |
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of PC lipid bilayers that range from 30.8 \AA (DLPC) to 37.8 (DPPC). |
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of PC lipid bilayers that range from 30.8 \AA\ (DLPC) to 37.8 \AA\ (DPPC). |
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They have also measured values for the area per lipid that range from |
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60.6 |
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\AA$^2$ (DMPC) to 64.2 \AA$^2$ |
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We have computed translational diffusion constants for lipid molecules |
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from the mean-square displacement, |
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\begin{equation} |
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D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
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D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
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\end{equation} |
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of the lipid bodies. Translational diffusion constants for the |
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different head-to-tail size ratios (all at 300 K) are shown in table |
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polynomial correlation function, |
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\begin{equation} |
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C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
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\mu}_{i}(0) \right) |
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\mu}_{i}(0) \right) \rangle |
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\end{equation} |
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of the head group dipoles. The orientational correlation functions |
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appear to have multiple components in their decay: a fast ($12 \pm 2$ |
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\caption{Fit values for the rotational correlation times for the head |
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groups ($\tau^h$) and molecular bodies ($\tau^b$) as well as the |
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translational diffusion constants for the molecule as a function of |
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the head-to-body width ratio (all at 300 K). In all of the phases, |
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the head group correlation functions decay with an fast librational |
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contribution ($12 \pm 1$ ps). There are additional moderate |
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($\tau^h_{\rm mid}$) and slow $\tau^h_{\rm slow}$ contributions to |
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orientational decay that depend strongly on the phase exhibited by the |
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lipids. The symmetric ripple phase ($\sigma_h / d = 1.35$) appears to |
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exhibit the slowest molecular reorientation.} |
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the head-to-body width ratio. All correlation functions and transport |
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coefficients were computed from microcanonical simulations with an |
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average temperture of 300 K. In all of the phases, the head group |
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correlation functions decay with an fast librational contribution ($12 |
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\pm 1$ ps). There are additional moderate ($\tau^h_{\rm mid}$) and |
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slow $\tau^h_{\rm slow}$ contributions to orientational decay that |
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depend strongly on the phase exhibited by the lipids. The symmetric |
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ripple phase ($\sigma_h / d = 1.35$) appears to exhibit the slowest |
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molecular reorientation.} |
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\begin{tabular}{lcccc} |
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\hline |
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$\sigma_h / d$ & $\tau^h_{\rm mid} (ns)$ & $\tau^h_{\rm |
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slow} (\mu s)$ & $\tau_b (\mu s)$ & $D (\times 10^{-11} m^2 s^{-1})$ \\ |
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slow} (\mu s)$ & $\tau^b (\mu s)$ & $D (\times 10^{-11} m^2 s^{-1})$ \\ |
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\hline |
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1.20 & $0.4$ & $9.6$ & $9.5$ & $0.43(1)$ \\ |
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1.28 & $2.0$ & $13.5$ & $3.0$ & $5.91(3)$ \\ |
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computable for the all-atom and coarse-grained simulations that have |
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been published in the literature.\cite{deVries05} |
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|
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Experimental verification of our predictions of dipolar orientation |
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correlating with the ripple direction would require knowing both the |
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local orientation of a rippled region of the membrane (available via |
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AFM studies of supported bilayers) as well as the local ordering of |
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the membrane dipoles. Obtaining information about the local |
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orientations of the membrane dipoles may be available from |
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fluorescence detected linear dichroism (LD). Benninger {\it et al.} |
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have recently used axially-specific chromophores |
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2-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-phospocholine |
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($\beta$-BODIPY FL C5-HPC or BODIPY-PC) and 3,3' |
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dioctadecyloxacarbocyanine perchlorate (DiO) in their |
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fluorescence-detected linear dichroism (LD) studies of plasma |
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membranes of living cells.\cite{Benninger:2005qy} The DiO dye aligns |
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its transition moment perpendicular to the membrane normal, while the |
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BODIPY-PC transition dipole is parallel with the membrane normal. |
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Without a doubt, using fluorescence detection of linear dichroism in |
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concert with AFM surface scanning would be difficult experiments to |
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carry out. However, there is some hope of performing experiments to |
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either verify or falsify the predictions of our simulations. |
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Although our model is simple, it exhibits some rich and unexpected |
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behaviors. It would clearly be a closer approximation to reality if |
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we allowed bending motions between the dipoles and the molecular |
