Protein Crowding in Lipid Bilayers Gives Rise to Non-Gaussian Anomalous Lateral Diffusion of Phospholipids and Proteins

Biomembranes are exceptionally crowded with proteins with typical protein-to-lipid ratios being around $1:50-1:100$. Protein crowding has a decisive role in lateral membrane dynamics as shown by recent experimental and computational studies that have reported anomalous lateral diffusion of phospholipids and membrane proteins in crowded lipid membranes. Based on extensive simulations and stochastic modeling of the simulated trajectories, we here investigate in detail how increasing crowding by membrane proteins reshapes the stochastic characteristics of the anomalous lateral diffusion in lipid membranes. We observe that correlated Gaussian processes of the fractional Langevin equation type, identified as the stochastic mechanism behind lipid motion in noncrowded bilayer, no longer adequately describe the lipid and protein motion in crowded but otherwise identical membranes. It turns out that protein crowding gives rise to a multifractal, non-Gaussian, and spatiotemporally heterogeneous anomalous lateral diffusion on time scales from nanoseconds to, at least, tens of microseconds. Our investigation strongly suggests that the macromolecular complexity and spatiotemporal membrane heterogeneity in cellular membranes play critical roles in determining the stochastic nature of the lateral diffusion and, consequently, the associated dynamic phenomena within membranes. Clarifying the exact stochastic mechanism for various kinds of biological membranes is an important step towards a quantitative understanding of numerous intramembrane dynamic phenomena.

Popular Summary

Cell membranes comprised largely of lipids and proteins are exceptionally crowded and typically contain 50–100 lipids per protein. However, the role of protein crowding on lateral membrane dynamics has received very little attention, which is quite surprising given that numerous studies have recently identified crowding to play an important role in multiple biological phenomena such as protein stability, signaling, and gene transcription. Furthermore, since lateral protein and lipid diffusion are the key processes driving a number of dynamical processes in cell membranes, the importance of understanding the role of crowding in lateral membrane dynamics is crucial. Here, we show that the theoretical paradigms established and validated for protein-poor membranes no longer hold true in protein-rich membranes.

We use simulations and stochastic modeling on microsecond time scales to show that protein crowding gives rise to non-Gaussian anomalous diffusion. By studying lipid and protein trajectories and analyzing how diffusivity varies spatially in the membrane plane, we show that protein crowding significantly changes membrane dynamics. In particular, lipids undergo diffusion that strongly depends on the local environment around the lipid, and the motion of lipids and proteins becomes strongly correlated for increasing crowding. Our results strongly suggest that, because of protein crowding, macromolecular complexity and spatiotemporal membrane heterogeneity play critical roles in determining the stochastic nature of lateral diffusion and other associated dynamic membrane phenomena. In particular, our findings highlight that correlated molecular motion associated with anomalous protein and lipid diffusion fosters the formation of functional units in cell membranes.

We expect that our findings will bring us a significant step closer to understanding the dynamics of biological membranes.

Figure 1

Snapshots of the membrane systems at the end of the respective simulations runs. From left to right: (i) protein-poor membrane composed of 2045 DPPC phospholipids and a single NaK channel protein; (ii) protein-rich membrane system composed of 1600 DPPC lipids and 16 NaK proteins: aggregating system; (iii) protein-rich membrane system composed of 1600 DLPC lipids and 16 NaK proteins: nonaggregating system; (iv) schematic structures of a DPPC phospholipid and a NaK channel protein employed in our coarse-grained simulations. For both DPPC and the NaK channel, the transparent coarse-grained structure is shown on top of the atomistic representation.

Figure 2

Time-averaged MSD traces $\overline{{\delta }^2(\mathrm{\Delta })}$ for both individual lipids (gray thin lines) and NaK proteins (red thick lines) in protein-poor DPPC (left), protein-rich DPPC (middle), and protein-rich DLPC (right) membranes. In each panel, the blue thick line represents the mean time-averaged MSD $⟨\overline{{\delta }^2(\mathrm{\Delta })}⟩$ for all lipids in the membrane. The upper panels represent the variation of the scaling exponent $\alpha$ versus the lag time $\mathrm{\Delta }$ for lipids (blue line) and proteins (red line) in each membrane system from fits to the corresponding mean time-averaged MSDs. In this plot, the offset $\mathrm{\Delta }=10\mathrm{ns}$ is due to the finite interval in fitting the instantaneous slope of the time-averaged MSD. Note the different scales of the ordinates in the different panels. We check that the change of the fit range does not seriously alter the $\alpha (\mathrm{\Delta })$ curves. Note that, in general, the scaling exponents $\alpha$ of the proteins exhibit larger fluctuation than those of the lipids. In particular, its sudden decrease for $\mathrm{\Delta }>5\mu \mathrm{s}$ shown in the noncrowded membrane reflects the statistically insignificant fluctuation of a single time-averaged MSD curve.

Figure 3

Cumulative distribution $\mathrm{\Pi }(r^2,\mathrm{\Delta })$ for squared displacements $r^2=[\mathbf{r}(t+\mathrm{\Delta })-\mathbf{r}(t){]}^2$ of individual lipids. The value $-\log [1-\mathrm{\Pi }(r^2,\mathrm{\Delta })]$ is plotted against $r^2$ at lag times $\mathrm{\Delta }=1\mathrm{ns}$ (red), 10 ns (green), 100 ns (blue), 1000 ns (brown), and 5000 ns (magenta), from left to right. For a Gaussian process the scaling of $-\log (1-\mathrm{\Pi })\sim r^2$ is obtained [94] as indicated with dashed lines in the plot. Data are for the noncrowded DPPC (top), aggregating protein-crowded DPPC (middle), and nonaggregating protein-crowded DLPC (bottom) systems. For the latter two cases the exponent deviates substantially from a value of 2, as seen from the Gaussian guide lines. The solid lines depict the theoretical curves [Eq. (5)] of $-\log (1-\mathrm{\Pi })$ expected from the non-Gaussian propagator $P$ fitting to the simulation data shown in Fig. 4.

Figure 4

Radial propagator $P_r(r,\mathrm{\Delta })$ for DPPC lipids in protein-poor and protein-rich membranes. In all cases the dashed and solid lines, respectively, represent the fitting curve to the simulation result: (dashed) Gaussian propagator $F_{\mathrm{G}}(r)=c_1\exp [-(r/2{\sigma }_1{)}^2]$, (solid) non-Gaussian propagator $F_{\mathrm{NG}}(r)=c_1\exp [-(r/2{\sigma }_1{)}^{{\delta }_1}]+c_2\exp [-(r/2{\sigma }_2{)}^{{\delta }_2}]$, where the annotated $\delta$ values are the fit values. (Top left) The protein-poor case with $\mathrm{\Delta }=1\mathrm{ns}$ (red), 10 ns (green), and 100 ns (blue). The other panels are for the protein-rich case with $\mathrm{\Delta }=1\mathrm{ns}$ (top right), 100 ns (bottom left), and 1000 ns (bottom right). The two-component Gaussian fit to the $P$ is supplemented in Fig. S7 in Supplemental Material [80].

Figure 5

Cumulative distribution $\mathrm{\Pi }(r^2,\mathrm{\Delta })$ and propagator $P_r(r,\mathrm{\Delta })$ for proteins: (top) Protein-poor DPPC, (middle) protein-rich DPPC, and (bottom) protein-rich DLPC membranes. For all panels the color code denotes the results at $\mathrm{\Delta }=1\mathrm{ns}$ (red), 10 ns (green), 100 ns (blue), 1000 ns (brown), and 5000 ns (magenta). As in Fig. 4, the dashed and solid lines in $\log P_r(r,\mathrm{\Delta })$ represent the fit curve using the Gaussian and the non-Gaussian propagators, respectively. The annotated $\delta$ values are the estimated non-Gaussian exponents $\delta$ from fitting.

Figure 6

Moment ratio of lipid molecules, $⟨{\mathbf{r}}^4⟩/⟨{\mathbf{r}}^2{⟩}^2$ (regular) and $⟨{\mathbf{r}}_{\mathrm{MME}}^4⟩/⟨{\mathbf{r}}_{\mathrm{MME}}^2{⟩}^2$ (mean maximal excursion). (Top left) DPPCs in the protein-poor membrane, (top right) DPPCs in the protein-rich membrane, (bottom left) DLPCs in the protein-rich membrane. Here, the MME distance ${\mathbf{r}}_{\mathrm{MME}}(t)$ refers to the maximum distance reached from the origin until time $t$ [102]. For two-dimensional FLE motion with $0<\alpha <1$, the moment ratio is 2 for regular and $<1.49$ (dashed line) for MME. (Bottom right) Velocity autocorrelation function $C(\mathrm{\Delta })=⟨{\mathbf{v}}_{\delta t}(t+\mathrm{\Delta })·{\mathbf{v}}_{\delta t}(t)⟩$ for the average velocity ${\mathbf{v}}_{\delta t}(t)=[\mathbf{r}(t+\delta t)-\mathbf{r}(t)]/\delta t$ with the chosen time interval $\delta t=100(\mathrm{ns})$. The result for DPPCs in the protein-rich membrane is compared to the corresponding theoretical curve of FLE [53].

Figure 7

Diffusion maps of DPPC lipid molecules in protein-poor (top) and protein-rich (bottom) membranes. In each case the four plots depict the diffusion maps at $t=1$, 10, 100, and 1000 ns from left to right. Blue regions are the unoccupied space by the lipid at the given time, while the gray dashed circles illustrate the position of the proteins. Axis coordinates are in nanometers.

Figure 8

Spatiotemporal fluctuations of the lipid diffusion in noncrowded (top) and crowded (bottom) DPPC membranes. (Left) Individual time-averaged MSD curves are plotted as a function of measurement time $T$ at fixed $\mathrm{\Delta }=100\mathrm{ns}$. The thick gray lines represent the mean time-averaged MSD $⟨\overline{{\delta }^2(T)}⟩$. (Right) Ergodicity breaking parameter $\mathrm{EB}(T)$, Eq. (7), depicted for $\mathrm{\Delta }=100\mathrm{ns}$ (red circle) and 5000 ns (blue squares).

Figure 9

(Left) Temporal fluctuation of the diffusivity $K_{\alpha }(t)$ for two chosen individual lipid molecules in the protein-crowded DPPC membrane. The diffusivity $K_{\alpha }$ is given in units of $\mu {\mathrm{m}}^2/{\mathrm{ns}}^{\alpha }$ and the temporal resolution is 100 ns. (Right) Probability density $\mathcal{P}(K_{\alpha })$ for the obtained $K_{\alpha }$ value from all DPPC lipid molecules in the protein-crowded membrane. The dashed line represents a double-Gaussian fit to the data. Plots corresponding to the protein-poor system and nonaggregating systems can be found in Figs. S11 and S10 in Supplemental Material [80].

Figure 10

(a)–(c) The Lennard-Jones (LJ) argon fluid systems simulated in our study. (a) The 2D argon fluid without obstacles. (b) The argon fluid particles with an obstacle arrangement that leads to weak confinement. (c) The argon fluid with an obstacle arrangement that gives rise to strong confinement. Note that the effect of confinement in (b) and (c) is transient. (d) The averaged time-averaged MSD curves for the argon particles in (a)–(c) systems. (e) The anomalous diffusion exponent $\alpha$ versus lag time $\mathrm{\Delta }$ estimated from the mean time-averaged MSDs in (d). (f) The $x$ component propagator $P(x,\mathrm{\Delta }=1\mathrm{ns})$ with a single-Gaussian fit (dashed line) for the obstacle-free case. (g) The propagator $P$ for the weakly confined argon particles with a single-Gaussian (dashed line) and double-non-Gaussian (solid line) fits. (h) The propagator $P$ for the strongly confined argons with a single-Gaussian (dashed line) and double-non-Gaussian (solid line) fits.