Anomalous Diffusion in Membranes

membranes with a monodisperse set of proteins, shown in Fig. 4.4 (see Publication IV for details).

5.3 Anomalous Diffusion in Membranes

In this Section, our research that examined the nature of lipid and protein diffusion in crowded membranes is described. Publications V and VI consider this work.

5.3.1 Lipid Packing Promotes Anomalous Diffusion

As explained in Section 2.2, lipid monolayers do not have an equilibrium area.

Instead, their area and hence the level of lipid packing can be varied continuously by the applied pressure. Indeed, pulmonary surfactant monolayers undergo repeated compression–expansion cycles due to breathing. The surfactant forms a dynamic system that arranges itself into domains that promote lipid–protein interactions and folds away from the interface into lipid reservoirs during exhaling [81]. Therefore, lipid dynamics might prove to be a fundamental aspect of lung function. To study the coupling between monolayer packing and dynamics, we exploited the same monolayer models as in the research considering the validity of the free area model.

As demonstrated in Fig. 5.4A, the MSD curves reveal the different diffusion modes when plotted on the logarithmic scale. Here, data are shown in blue and red for monolayers with APLs equal to 44 Ų and 68 Ų, corresponding to the Lc and Le

phases, respectively. Fig. 5.4B shows the diffusion exponent ↵ as a function of lag time. Here, the short-time ballistic behavior is likely not entirely captured due to the limited sampling rate of the simulation. At intermediate lag times between 1 ps and 1 µs, lipid motion is subdiffusive with↵ values as small as 0.3 for the Lc

phase monolayer. Finally, normal diffusion is reached at a 100 ns to 1 µs timescale, depending on the level of packing.

5.3.2 Crowding Extends the Anomalous Diffusion Regime

FCS experiments have revealed that protein crowding induces anomalous subdiffusion in model membranes [15]. These experiments consider↵to be constant over the whole

62 5. New Insights and Advancements Provided by This Thesis

Figure 5.4 Key results for lipid monolayers from Publication V: A) The MSD versus lag time shown on the logarithmic scale. Data for systems with APL equal to 44 Ų and 68 Ų are shown in blue and red, respectively. The dashed lines highlight a slope of 1, corresponding to normal diffusion, whereas the dotted line shows a slope of 2, corresponding to ballistic motion. B) Evolution of↵ versus lag time. Coloring as in panel A. The minimum values for both curves are shown, and the colored dashed lines extrapolate the curves to both ballistic (↵ = 2) and normal (↵ = 1) regimes. Note that lag time on the abscissa is given in the logarithmic scale.

extended fitting interval, which seems like a bold assumption based on the monolayer data above. Due to this experimental limitation, the (lag) time dependence of ↵ on protein crowding has remained unknown. To gain this knowledge, we simulated a set of membranes with different degrees of crowding. Moreover, the proteins in these membranes displayed strong (SA) or weak (WA) aggregation tendencies (see Section 4.3).

The diffusion coefficients of lipids and proteins in the SA (red) and WA (blue) systems are shown as a function of LP in Figs. 5.5A and 5.5B, respectively. In line with experiments [14], the diffusion coefficients show a decrease upon protein crowding.

This decrease is linear for lipids, yet steeper for proteins. In general, the values measured for the WA systems are somewhat larger, as expected. The dependence of ↵ on lag time is shown for lipids in WA and SA systems in Fig. 5.5C. Along with the behavior observed for monolayers (see Fig. 5.4), diffusion is anomalous at short lag times. However, in LP = INF membranes normal diffusion is reached at the ⇠100 ns timescale, in line with the behavior of the Le phase monolayer (see Fig. 5.4B). However, at around this timescale, a new subdiffusive regime emerges in the crowded membranes. This regime ranges from ⇠100 ns up to macroscopic time scales in systems with a realistic level of crowding [4]. In the crowded SA membranes, the values of↵decrease to ⇠0.6 before they begin their recovery towards

5.3. Anomalous Diffusion in Membranes 63 one and hence normal diffusion. The values of ↵ observed in the WA membranes are somewhat larger. The extent of this second subdiffusive regime apparently depends on the level of crowding, as well as the aggregation tendency of the proteins.

0

Figure 5.5 Key results for crowded membranes from Publication V: A & B) Diffusion coefficients of A) lipids and B) proteins as a function of LP. Data are shown for red) SA and blue) WA systems. C)↵ as a function of lag time in WA and SA systems with LP = 50 and LP = INF. The dashed gray line highlights normal diffusion (↵= 1). The dotted lines estimate where this is reached by the calculated curves or their extrapolations (dashed line).

D) The formation of protein clusters in top) SA or bottom) WA LP = 50 system. Each row corresponds to a protein, and the coloring of each cell shows the number of other proteins in contact with it at a given time. E) Sample trajectory of a lipid in the SA LP = 50 system.

The red arrow points to an escape event from confinement.

This membrane-dependent tendency of the proteins to aggregate is demonstrated in Fig. 5.5D. The data are shown for top) SA and bottom) WA systems with LP = 50.

Each line provides data for a single protein, and the coloring reveals the number of other proteins bound to it. In the SA system, each protein rapidly associates with

64 5. New Insights and Advancements Provided by This Thesis 2–4 others, whereas in the WA system the clusters are somewhat smaller and their formation takes substantially longer. This observation suggests that the substantial dip in↵ in the SA systems could originate from protein aggregates that restrict lipids into pools. These lipids need to escape this confinement before the regime of normal diffusion is reached. This behavior is indeed observed in sample lipid trajectories, such as that shown for the SA system with LP = 50 in Fig. 5.5E. Here, the red arrow highlights an event where the tracked lipid escapes confinement.

5.3.3 Protein Crowding Changes the Subdiffusion Mechanism

In our earlier study [16], we discovered that anomalous subdiffusion in protein-free bilayers follows the FBM/FLE mechanism (see Section 3.2.1). Notably, this feature holds throughout Ld, Lo, and even gel phases, and manifests itself in numerous ways:

the subdiffusion is ergodic and Gaussian and it does not display aging [16]. Moreover, the calculated displacement autocorrelation function matches precisely that predicted for FBM (see Eq. (3.18)). These findings conclusively showed that subdiffusion in protein-free lipid membranes follows the FBM/FLE description. However, with the drastic effects of protein crowding on membrane dynamics, it is not evident that this mechanism also applies to subdiffusion in crowded membranes. To clarify this issue, we further analyzed the dynamics in the crowded (LP = 50) and dilute (LP = INF) WA and SA membranes shown in Figs. 4.5A–4.5C and discussed above.

The simulations were extended to collect reliable statistics of protein diffusion, and the CG simulations were complemented by 2-dimensional Lennard-Jones (2DLJ) systems with obstacles, depicted in Figs. 4.5D–4.5F.

The radial PDFs, P(r, ), of lipids are shown Figs. 5.6A and 5.6B for the SA systems with LP = INF and LP = 50, respectively. Lipids in the dilute system follow a Gaussian distribution (P(r, )⇠exp( r²)), highlighted by the gray dashed line.

However, in the case of the crowded membrane, the Gaussian fit fails and gets replaced by a sum of two terms of a more general form (P(r, ) ⇠ P

exp( r^⇣i)) with the values of ⇣i highlighted in Fig. 5.6B. This fit is shown by a solid line. The data for lipids in the crowded WA membrane as well as proteins in WA and SA membranes follow the same behavior (data not shown here). The non-Gaussianity already suggests that diffusion in crowded membranes does not follow the FBM/FLE mechanism, so we pursued indicators of alternative descriptions (see Sections 3.2.2 and 3.2.3). Characteristic examples of the TA-MSD curves of lipids with a fixed lag time ( =100 ns) are shown for the dilute LP = INF and crowded LP = 50 SA

5.3. Anomalous Diffusion in Membranes 65 membranes in Figs. 5.6C and 5.6D, respectively. In the dilute system, all trajectories rapidly converge to the mean value h ( )i, shown by a black dashed line. Hence, the dilute system does not display aging. Moreover, the corresponding ergodicity breaking parameter (see Eq. (3.9)), plotted in Fig. 5.7A, shows the typical ⇠⇥ ¹ convergence of FBM/FLE [84] with both = 100 ns (blue) and = 5 µs (red), respectively.

The TA-MSD data of lipids for the crowded membrane displays drastically different behavior (see Fig. 5.6D). While most trajectories converge to their mean value (examples given by reddish curves), some show a constant decline (bluish curves), a rapid increase (greenish curves), or even a combination of both (orangish curves).

The decline events likely initiate when a lipid gets confined, whereas their release from such confinement leads to a rapid increase in the MSD. Despite this heterogeneity, h ( )i is constant over measurement time, indicating that no aging takes place.

Hence, CTRW is ruled out as a possible subdiffusion mechanism.

0 10 20 30 40

logPr(r,100ns) SA, LP = INF Single Gaussian

Figure 5.6 First set of results for coarse-grained simulations in Publication VI. A & B) Radial PDFs for lipids in the A) dilute (LP = INF) and B) crowded (LP = 50) SA systems over a lag time of 100 ns. Dashed lines show Gaussian fits, while the solid lines are fits to the non-Gaussian exponents (see text for details). C & D) Examples of TA-MSD curves for C) dilute LP = INF and D) crowded LP = 50 SA systems.

The ergodicity breaking parameters of lipids for the crowded LP = 50 case, shown

66 5. New Insights and Advancements Provided by This Thesis

Figure 5.7 Second set of results for coarse-grained simulations in Publication VI. A &

B) Ergodicity breaking parameter for lipids with = 100 ns (blue) and = 5 µs (red) in A) dilute LP = INF and B) crowded LP = 50 SA systems. C & D) Example maps of MSD over a 10 ns time interval in the C) LP = 50 SA and the D) LP = INF SA (I) systems. E) Normalized displacement autocorrelation of lipids in the crowded (LP = 50) SA system.

in Fig. 5.7B for both = 100 ns (blue) and = 5 µs (red), converge towards zero, although slower than what is predicted for FBM/FLE. The displacement autocorrelation of lipids (Eq. (3.18)), shown in Fig. 5.7E for the crowded LP = 50 SA membrane, displays the negative dip characteristic for FMB/FLE, yet does not entirely follow the theoretical prediction. Nevertheless, the CTRW model (Eq. (3.23)) does not explain the data either. Figs. 5.7C and 5.7D, show examples of the spatial variation of the MSD of lipids over a time interval of 1 ns in the C) SA LP = 50 and D) LP = INF systems. For the crowded case, protein arrangement effects a heterogeneous diffusivity landscape that also varies with time, whereas in the dilute membrane the variations are milder. Further modeling work (see Publication VI) suggests that this spatiotemporal heterogeneity is the reason behind the observed non-Gaussian yet ergodic subdiffusion.

Central results for the 2DLJ systems (see Fig. 4.5D–F) are shown in Fig. 5.8. The EA-TA-MSD curves are shown in Fig. 5.8A. The differences in the curves are best

5.3. Anomalous Diffusion in Membranes 67

Figure 5.8 Key results for 2DLJ simulations in Publication VI. A) TA-EA-MSD shown in double logarithmic scale. B) Diffusion exponent ↵ as a function of lag time. C & D) 1D displacement distributions over a 5 ns time interval for the C) 2DLJ-FREE and D) 2DLJ-SC systems.

highlighted by plotting the time evolution of ↵ in lag time, shown in Fig. 5.8B.

Notably, Fig. 5.8B shows that the convergence of ↵ to one is observed in all systems.

However, the SC system shows a dip in ↵, similar to what is observed in the LP = 50 SA system above (compare Figs. 5.8B and 5.5C). In the WC system, the obstacles somewhat extend the anomalous diffusion regime, yet do not lead to a similar dip in

↵. As shown in Fig. 5.8C the particles in the 2DLJ-FREE system show Gaussian diffusion. However, in the 2DLJ-WC (not shown here) and 2DLJ-SC system (see Fig. 5.8D), the obstacles effect a non-Gaussian displacement distribution described by two generalized Gaussians, in line with the behavior of the crowded CG systems.

68 5. New Insights and Advancements Provided by This Thesis