Anomalous Diffusion

LSD =hµm/(2µf). With a solvent such as water, the typical values of the SD length are⇠100 nm, which renders the model applicable to all membrane proteins. However, larger inclusions such as membrane domains require the use of an extension to the SD model by Hugheset al. [105] or its approximation [106], which find an asymptotic D ⇠1/R dependence. Notably, the diffusion of lipids spanning only one leaflet is also poorly described by the SD model [107], even though some improvement is obtained by considering the contributions from interleaflet friction [108].

The SD model and its extensions have successfully described the diffusion of proteins in experiments [14, 109, 110, 111, 112] and simulations [113, 114, 115]. However, since lipids are known to diffuse together with the proteins [59], it remains unclear what the definition of the radius in the SD description is. Moreover, the parameters provided by SD model fits to experimental data might provide unphysical values [111]. Interestingly, some experiments report stronger dependence (D⇠R ¹ instead ofD⇠lnR ¹) of diffusion coefficients on protein size [116]. These observations have been associated with protein-induced deformations of the host membrane [117, 118], or explained by the limitations in the experimental setups [109].

3.2 Anomalous Diffusion

The normal Brownian diffusion only arises in case the motion is truly random, i.e. the particles move independently of each other and symmetrically along the studied dimension. Moreover, the displacements of the particle itself need to become independent at some time scale. Unfortunately, biological systems are rarely ideal, especially at all time scales.

Normal diffusion is characterized by a linear dependence between the EA-MSD and time (see Eq. (3.3)). Processes with directionality, coupling between diffusing particles, or memory effects often lead to anomalous diffusion, which instead displays

hr²i ⇠t^↵. (3.14)

Here↵is the diffusion exponent and characterizes the type of the motion. Anomalous superdiffusion (1<↵ <2) and subdiffusion (↵ <1) are both present in biological systems [63, 84, 85, 119]. Superdiffusion does not contribute to membrane dynamics to a substantial degree, whereas subdiffusion is prevalent therein. Here ↵ is often

24 3. Lateral Diffusion time-dependent, and at the time regime with the strongest subdiffusion, it reaches values of ⇠0.6–0.8 in fluid membranes [16, 120]. In the absence of confinement effects or binding events with divergent time scales (see Section 3.2.2 below), normal diffusion is usually eventually reached. In addition to long-time normal diffusion and anomalous diffusion present at intermediate times, the motion of membrane constituents over very short time intervals is ballistic, i.e. the molecules move at constant velocity (↵= 2) until they collide with their nearest neighbors.

Molecules undergoing subdiffusion cover a smaller area of the membrane in a fixed time as compared to normal diffusion, assuming equal values of the (effective) diffusion coefficient. This is highlighted in Fig. 3.1D, which demonstrates both the evolution of P(r, t) over time and the connected sub-linear scaling of the EA-MSD. Moreover, in the anomalous regime, the diffusion coefficient gets replaced by an effective one,D^⇤, which has dimensions of length²/time^↵. Importantly, the two parameters, D^⇤ and↵, describe two very different things. The former defines the rate of motion, whereas the latter represents the localization of this motion. Hence, it is possible for a molecule to rapidly sample a small region (large D^⇤ and small ↵), or to explore larger regions with a slower pace (small D^⇤ and large ↵). The heterogeneous structure and active processes present in the plasma membrane likely result in spatiotemporally varying values for these two parameters, thereby optimizing specific processes in certain environments and under certain conditions.

3.2.1 Fractional Brownian Motion

Anomalous subdiffusion can result from a multitude of mechanisms [84, 85]. Despite similar scaling of the EA-MSD with time, the mechanisms lead to distinct dynamics that uniquely manifest themselves in various physical observables.

Fractional Brownian motion (FBM), described by Mandelbrot and van Ness half a century ago [121], is a generalized case of Brownian motion whose subsequent steps are mutually correlated. Like regular Brownian motion, it is a continuous-time Gaussian process with a zero expectation value. Likely the most intuitive way to describe FBM is via the overdamped Langevin equation (compare to Eq. (3.5))

dr

dt =⇠f(t), (3.15)

where ⇠f is fractional Gaussian noise. Similar to Gaussian noise, it is normally distributed. However, it displays a power-law correlation with (compare to Eq. (3.4))

3.2. Anomalous Diffusion 25 [84, 85]

h⇠_f(t)⇠_f(t⁰)i=↵(↵ 1)D^⇤|t t⁰|^{↵ 2}. (3.16)

This same noise also drives the related fractional Langevin equation (FLE) [122]. Since FLE and FBM are seldom carefully distinguished in the literature, it is worth pointing out their difference [123] here. FLE is a generalization of the Langevin equation and hence describes the motion of a physical particle and fulfills the fluctuation–

dissipation theorem [122], whereas FBM is a generalization of the mathematical concept of Brownian motion. At short times, FLE leads to ballistic motion, whereas in the overdamped lipid it converges to FBM, just as at this limit the regular Langevin equation leads to Brownian motion. Hence, for subdiffusion in viscous membranes, FBM and FLE behave similarly and the terms are usually interchangeable.

For FBM, ↵ describes the raggedness of the motion, and larger ↵ values lead to smoother trajectories. We focus here on anomalous subdiffusion and note that with

↵ < 1, the motion is negatively correlated [84]. An example trajectory for FBM with ↵= 0.8is shown in Fig. 3.1B. Notably, depending on the value of ↵, FBM also describes normal diffusion and superdiffusion [84]. In anomalous diffusion, FBM leads to a sharper PDF (compare to Eq. (3.2) and see Fig. 3.1) [84]

P(r, t) = 1

4⇡D^⇤t^↵ ⇥exp

✓ r² 4D^⇤t^↵

◆

. (3.17)

From the position autocorrelation function, it is evident that FBM is ergodic with the EA-MSD and EA-TA-MSD scaling as hr²(t)i ⇠t^↵ and h ²( )i ⇠ ^↵, respectively [84]. Moreover, even the TA-MSD that is not averaged over many trajectories scales similarly ( ²( ) ⇠ ^↵) with a sufficiently long trajectory.

The displacement (or discretized velocity) autocorrelation function, C _t(t) = 1

( t)²h[r(t+ t) r(t)]·[r( t) r(0)]i, (3.18) has a very characteristic and identical form for both FBM and overdamped FLE that — when normalized — reads [16, 124]

C t(t)

C _t(0) = |t+ t|^↵ 2t^↵+|t t|^↵

2 t^↵ . (3.19)

26 3. Lateral Diffusion The anti-persistent nature of the fractional Gaussian noise manifests itself in the displacement autocorrelation, which displays a negative correlation and hence a backflow of diffusing particles at intermediate times. This feature is characteristic of viscoelastic materials, such as lipid membranes [125]. FBM and FLE have been associated with many dynamic processes both in the cytosol as well as in membranes [16, 84].

3.2.2 Continuous Time Random Walks

Continuous time random walks (CTRWs) [126, 127] were also introduced half a century ago by Montroll, Scher, and Weiss. They are an extension of regular random walks, where steps are taken at fixed time intervals. In CTRW, in addition to the step sizes, the waiting time between consecutive steps is a random variable and independent of earlier steps. The distribution of steps is symmetric. When the mean waiting time and the mean squared step length are finite, normal ergodic Brownian motion is recovered. However, in case the waiting time distribution has a heavy tail in the form of

(⌧)⇠⌧ ^{1 ↵}, (3.20)

and the motion is subdiffusive with ↵<1, the expectation value of the waiting time

⌧,

h⌧i= Z ₁

⌧0

⌧⁰ (⌧⁰)d⌧⁰, (3.21)

diverges. Here ⌧0 is some small value. The waiting times can be as long as the measurement meaning that subdiffusive CTRW motion displays aging. Indeed, the EA-TA-MSD scales as

h ²( )i ⇠D^⇤

⇥^{1 ↵}, (3.22)

i.e. it has an explicit dependence on measurement time ⇥ [84]. Curiously, the properties of a system displaying aging also depend on the duration between the setting up of the system and the beginning of the measurement [128]. It is also noteworthy that the time-averaged MSD, Eq. (3.22), differs from the EA-MSD in Eq. (3.14). Hence, subdiffusive CTRW presents a non-ergodic process.

3.3. Experimental Methods to Study Diffusion 27