Distribution of cholesterol in asymmetric membranes driven by composition and differential stress

Abstract

Many lipid membranes of eukaryotic cells are asymmetric, which means the two leaflets differ in at least one physical property, such as lipid composition or lateral stress. Maintaining this asymmetry is helped by the fact that ordinary phospholipids rarely transition between leaflets, but cholesterol is an exception: its flip-flop times are in the microsecond range, so that its distribution between leaflets is determined by a chemical equilibrium. In particular, preferential partitioning can draw cholesterol into a more saturated leaflet, and phospholipid number asymmetry can force it out of a compressed leaflet. Combining highly coarse-grained membrane simulations with theoretical modeling, we investigate how these two driving forces play against each other until cholesterol's chemical potential is equilibrated. The theory includes two coupled elastic sheets and a Flory-Huggins mixing free energy with a χ parameter. We obtain a relationship between χ and the interaction strength between cholesterol and lipids in either of the two leaflets, and we find that it depends, albeit weakly, on lipid number asymmetry. The differential stress measurements under various asymmetry conditions agree with our theoretical predictions. Using the two kinds of asymmetries in combination, we find that it is possible to counteract the phospholipid number bias, and the resultant stress in the membrane, via the control of cholesterol mixing in the leaflets.

Figure 1

Cholesterol asymmetry $\delta c$ as a function of the mixing parameter difference $\delta \chi$ (assuming also that $\overline{\chi }=0$). The individual curves correspond to lipid asymmetries $\delta \ell$ between 0% and 5%, as indicated in the boxed labels. For this figure, the cholesterol mole fraction is $\varphi =10\%$, and the scaled stretching modulus is ${\tilde{K}}_A=20$. To see this figure in color, go online.

Figure 2

Cholesterol asymmetry $\delta c$ as a function of cholesterol mole fraction $\varphi$ for different values of leaflet mixing parameter differences $\delta \chi \in \{-1,-0.9,\dots ,0\}$ as indicated in the boxed labels (while $\overline{\chi }=0$). The red dashed curve for $\delta \chi \approx -0.327$ is not exactly horizontal ($\delta c$ varies approximately between $-0.016$ and $-0.012$), but this small $\varphi$ dependence is invisible at this scale. For this figure, the asymmetry $\delta \ell =2\%$, and the scaled stretching modulus is again ${\tilde{K}}_A=20$. To see this figure in color, go online.

Figure 3

Probability $P\left(n\right)$ of $n$$\oplus$-type lipids venturing into the $\ominus$ leaflet, for a symmetric membrane containing 512 lipids in total (but no additional cholesterol). The solid curve is a fit to Eq. 31, giving ${\epsilon }^{\ast }\approx 7k_{\text{B}}T$. Error bars show the error of the mean. To see this figure in color, go online.

Figure 4

Difference between the lateral stress profiles of a completely symmetric membrane and an asymmetric membrane with $\delta \ell =2\%$, as a function of the distance from the mid-plane. Neither membrane contains any cholesterol. To see this figure in color, go online.

Figure 5

Differential stress $\Delta {\Sigma }_{\text{0}}$ as a function of cholesterol fraction $\varphi$ for an asymmetric membrane at $\delta \ell =2\%$. The orange points are directly measured from the stress profile. The purple triangles are inferred from the measured cholesterol asymmetry $\delta c$ via Eq. 8, while the green curve instead predicts $\delta c$ via Eq. 18, where ${\chi }_{+}$ is the only free parameter. The blue vertical dashed line represents the critical $\varphi$ value past which $\Delta {\Sigma }_{\text{0}}=0$ if the cholesterol distribution were exclusively determined by differential stress reduction—see Eq. 35b. Error bars show the error of the mean. To see this figure in color, go online.

Figure 6

Differential stress $\Delta {\Sigma }_{\text{0}}$ as a function of cholesterol concentration $\varphi$, under conditions of a partitioning bias—cholesterol prefers to be in the $\oplus$ leaflet—while the number asymmetry $\delta \ell =0$. The orange points are directly measured from the stress profile. The purple triangles are inferred from the measured cholesterol asymmetry $\delta c$ via Eq. 8, while the green curve instead predicts $\delta c$ via Eq. 18, where ${\chi }_{+}$ is the only free parameter. Error bars show the error of the mean. To see this figure in color, go online.

Figure 7

Plotting cholesterol asymmetry $\delta c$ against $\varphi$ for fixed values of $w_{\text{c,}\pm }$ and $\delta \ell$ lets us obtain a fit (in green) for the mixing parameter $\chi$ using Eq. 18. This is analogous to the theoretical plots in Fig. 2. This particular plot is for $w_{\text{c,}-}=w_{\text{c,0}}$, $w_{\text{c,}+}=1.675\sigma$, and $\delta \ell =0$, and it leads to ${\chi }_{+}(\delta \ell =0)=-1.24$. Error bars show the error of the mean. To see this figure in color, go online.

Figure 8

${\chi }_{+}$ values for a range of interaction strengths $w_{\text{c,}+}$, at $\delta \ell =0\%$ (blue dots) and $\delta \ell =2\%$ (orange triangles). The data points and the corresponding fits are shown in the same color for clarity. The residuals for the cubic polynomial fit (bottom left inset) indicate that there is no other underlying trend. The parametric plot (top right inset) tells us that there is a relatively constant positive offset for $\delta \ell =2\%$. Error bars show the error of the mean. To see this figure in color, go online.

Figure 9

Cholesterol asymmetry $\delta c$ as a function of $\delta \chi$, for phospholipid asymmetry $\delta \ell =0\%$ (blue) and $2\%$ (green). The prediction from our theory is shown as a solid line against the simulated data points. These plots mirror the ones in Fig. 1. Error bars show the error of the mean. To see this figure in color, go online.

Figure 10

Differential stress $\Delta {\Sigma }_{\text{0}}$ versus cholesterol asymmetry $\delta c$, for a system in which all lipids interact identically $(w_{\text{c,}\pm }=w_{\text{c,0}})$ but there is a number asymmetry $\delta \ell =2\%$. Measurements are shown in orange with horizontal and vertical uncertainties calculated from blocking and bootstrapping, respectively. Purple triangles are predictions from our theory using Eq. 29, while the green line is a prediction that ignores the small contribution from a residual mixing parameter $\delta \chi$. To see this figure in color, go online.

Figure 11

Differential stress and cholesterol asymmetry as a function of phospholipid asymmetry. (A) The latter given as $\delta \ell$ (lower common axis) or $L_{\text{cyto}}/L_{\text{exo}}$ (upper common axis). Stress is either measured in dimensionless form (${\tilde{K}}_A$, left axis) or in units of mN/m (right axis), while cholesterol asymmetry is given as $\delta c$ (left axis) or as a fraction on the exoplasmic side, $C_{\text{exo}}/C_{\text{total}}$ (right axis). The blue-to-red lines correspond to a set of non-ideal mixing parameters $\delta \chi \in \{-2,-1.5,\dots ,+2\}$ as indicated in the boxed labels. The green iso-lines in (B) indicate the exoplasmic cholesterol mol fraction $C_{\text{exo}}/(C_{\text{exo}}+L_{\text{exo}})$, with values given in the boxed labels. The dashed gray line in (B) is the cholesterol asymmetry $\delta c_{\infty }$ for the hypothetical case in which only area-matching matters; see also Eq. 36. To see this figure in color, go online.

Figure 12

Schematic illustrations of asymmetric red blood cell bilayers with a phospholipid imbalance $L_{\text{cyto}}/L_{\text{exo}}$ as indicated next to each image. Lipid types represent different degrees of saturation, as measured by the number of double bonds: 0 (dark blue), 1 (blue), 2 (purple), and $\ge 2$ (red). Cholesterol is indicated as shorter gray sticks. The relative abundances of phospholipids in each leaflet approximately reflect those given in Fig. 1 b of ref. (7), while the cholesterol distribution is calculated from our model, using $\delta \chi =-1.5$. To see this figure in color, go online.