Models for Normal Diffusion

5.2 Models for Normal Diffusion

In this Section, our work considering models describing normal diffusion is described.

The validity of these models, introduced in Sections 3.1.2 and 3.1.3, is evaluated in biologically relevant conditions. Publications III and IV consider this work.

5.2.1 The Free Area Model Provides Nonphysical Parameters

The free area (FA) model [10] (see Section 3.1.2) has been successfully adapted to lipid bilayers, where it captures the temperature dependence of diffusion coefficients.

Furthermore, it has explained the decrease in lipid diffusivities upon the addition of cholesterol by the reduction in the available free area [10]. However, there is still debate on the validity of the model and especially on the underlying diffusion mechanism [11, 100, 101]. We put the FA model to the test in a different setting.

Lipid monolayers and their dynamics are crucial for lung function [81]. Moreover, monolayers allow for a systematic probing of diffusion as a function of both lipid area and temperature in both simulations and experiments. Also, with the use of simulations, the parameters provided by the FA model fits can be directly compared to the corresponding physical properties of the monolayer. We chose this approach and simulated protein-free pulmonary surfactant monolayers at different APLs, as described in Section 4.3.

Visualization of the lipid motion (see Fig. 5.2A already reveals flow-like patterns and suggests that hopping between vacant sites is not the correct physical description of diffusion. However, as Fig. 5.2B demonstrates, the lipid diffusion coefficients as a function of APL are reasonably well fitted with the FA model (Eq. (3.12), solid line), in line with experiments [99]. Here, data for DPPC — the most abundant lipid in the mixture — are shown. The fit provides activation energies E_a of ⇠14 kJ/mol for all lipid species, whereas for the close-packed areas of the lipids, a₀, we obtain values of ⇠40 Ų, again similar across all lipid moieties.

The activation energy can also be extracted via an Arrhenius analysis, assuming that diffusion is an activated process. Following the Arrhenius equation,

D=D0⇥exp

✓ EArrh

RT

◆

, (5.1)

58 5. New Insights and Advancements Provided by This Thesis

Figure 5.2 Key results from Publication III: A) Typical flow patterns of lipids in the monolayer with an APL of 48 Ų over a 10 ns interval. B) Diffusion coefficients of DPPC as a function of APL (markers) together with a FA model fit (solid line). C) Diffusion coefficients (markers) in systems with APLs of 48 Ų and 68 Ų plotted in the Arrhenius manner (see Eq. (5.1)). Solid lines show the linear fits. D) Profiles of close-packed cross-sectional areas of DPPC (solid lines). The dashed curves show the free volume profiles.

Data are displayed for systems with APLs of 48 Ų (red) and 68 Ų (blue).

a plot of lnD against T ¹ provides a slope of E_Arrh/R,i.e. the Arrhenius activation energy divided by the universal gas constant. Hence, to extractE_Arrh, we simulated monolayers with APLs equal to 48 and 68 Ų at different temperatures. These data, shown in Fig. 5.2C for DPPC in monolayers with APLs of 48 Ų (red) and 68 Ų (blue), provide activation energies of 31 and 25 kJ/mol, respectively. While these values are in line with the values measured for liquid-disordered bilayers [239], they are approximately twice as large as the values predicted by the FA model fits. This discrepancy between E_Arrh andE_a has also been demonstrated for lipid bilayers [101].

The close-packed cross-sectional area profiles for DPPC are shown by solid curves in Fig. 5.2D for systems with APLs equal to 48 Ų (red) and 68 Ų (blue). For lipids

5.2. Models for Normal Diffusion 59 (solid lines), the curves reach maximum values of ⇠25 and ⇠35 Ų, respectively, only somewhat smaller than the value of⇠40 Ų obtained from the FA model fits. It must be kept in mind, though, that a lipid is not a rigid rod and its cross-sectional area varies drastically along its length [100]. Therefore — considering the inaccuracies of our calculation — the values of a0 seem to be in the right ballpark. However, if the values of a0 given by the FA model are considered quantitatively, the situation is more complicated. The area parameter a0 should describe the cross-sectional close-packed area of a lipid (see Eq. (3.12)), but the deviation from the true cross-sectional close-packed area we determined from the data is of the order of 50 %. The obvious question is whether a0 describes any physical parameter realistically, or whether it serves just as a fitting parameter.

5.2.2 Crowding Breaks Down The Saffman–Delbrück Model

The Saffman–Delbrück (SD) model [12, 13] (see Section 3.1.3) has been successfully applied to describe the diffusion of membrane proteins in the protein-poor limit [14, 109, 110, 111, 112]. The central feature of the SD model is that it predicts a weak logarithmic size-dependence for the diffusion coefficients of proteins (D ⇠ lnR ¹ with protein radii R). Notably, the SD model is a fundamental concept for explaining protein motion and hence diffusion-controlled reactions in biomembranes. However, even though cellular membranes are incredibly crowded with proteins, the SD model has not been put to the test under such conditions, neither experimentally nor in simulations. To fill this gap, we simulated CG membranes with different concentrations of a polydisperse set of proteins as well as 2DLJ fluids with embedded disks of different radii. Moreover, we also considered crowded membranes with monodisperse sets of proteins. All these models are described in Section 4.3.

The protein diffusion coefficients, shown in Fig. 5.3A, display a nonlinear decrease upon protein crowding, and this decrease is the most drastic for the smallest proteins.

For the estimation of protein radii, we performed single-protein simulations and analyzed the dynamics of lipids in the vicinity of the proteins. It is established that membrane proteins diffuse together with a tightly-bound lipid shell [59], which needs to be taken into account when considering the protein’s hydrodynamic radius. Such

“effective” radii (Re↵) are shown in Fig. 5.3B together with the radii of the 2DLJ disks. Notably, the CG proteins span the typical size range of known membrane proteins.

60 5. New Insights and Advancements Provided by This Thesis

Figure 5.3 Key results from Publication IV. A) The diffusion coefficients of the proteins at different LP ratios. B) The R_e↵ of the proteins in the CG simulations (main plot) and the radii of the 2DLJ disks (inset). “Indiv.” stands for an individual (free) 2DLJ particle.

C) Diffusion coefficients as a function of protein size in dilute systems. Data are shown in reduced units for a direct comparison between the CG and 2DLJ systems. The insets show data in logarithmic scale. The solid and dashed lines indicate fits of the SD model and Stokes-like scaling, respectively. D) Diffusion coefficients as a function of protein size in crowded systems.

The protein diffusion coefficients are shown as a function of Re↵ in Figs. 5.3C and 5.3D for dilute and crowded conditions, respectively. In the CG systems, these correspond to LP = 400 and LP = 50, respectively, whereas in the 2DLJ systems there are 1000 and 300 free particles per disk. The use of reduced units reveals a striking similarity between the behaviors of the CG (colored markers) and 2DLJ (black markers) systems. Here, the diffusion coefficients and radii of the proteins and the disks are scaled by the diffusion coefficients (D0) and radii (R0) of the lipids and the free LJ particles, respectively. Importantly, while the SD-like weak dependence D ⇠ lnR ¹ (solid line) fits the data under dilute conditions well, a stronger Stokes-like lawD⇠R ¹ (dashed line) arises under crowding. This crossover is most convincingly demonstrated in the logarithmic scale, as shown in the insets in

5.3. Anomalous Diffusion in Membranes 61