Review article: Anomalous diffusion in lipid bilayer membranes

The dynamics of constituents and the surface response of cellular membranes—also in connection to the binding of various particles and macromolecules to the membrane—are still a matter of controversy in the membrane biophysics community, particularly with respect to crowded membranes of living biological cells. We here put into perspective recent single particle tracking experiments in the plasma membranes of living cells and supercomputing studies of lipid bilayer model membranes with and without protein crowding. Special emphasis is put on the observation of anomalous, non-Brownian diffusion of both lipid molecules and proteins embedded in the lipid bilayer. While single component, pure lipid bilayers in simulations exhibit only transient anomalous diffusion of lipid molecules on nanosecond time scales, the persistence of anomalous diffusion becomes significantly longer ranged on the addition of disorder—through the addition of cholesterol or proteins—and on passing of the membrane lipids to the gel phase. Concurrently, experiments demonstrate the anomalous diffusion of membrane embedded proteins up to macroscopic time scales in the minute time range. Particular emphasis will be put on the physical character of the anomalous diffusion, in particular, the occurrence of ageing observed in the experiments—the effective diffusivity of the measured particles is a decreasing function of time. Moreover, we present results for the time dependent local scaling exponent of the mean squared displacement of the monitored particles. Recent results finding deviations from the commonly assumed Gaussian diffusion patterns in protein crowded membranes are reported. The properties of the displacement autocorrelation function of the lipid molecules are discussed in the light of their appropriate physical anomalous diffusion models, both for non-crowded and crowded membranes. In the last part of this review we address the upcoming field of membrane distortion by elongated membrane-binding particles. We discuss how membrane compartmentalisation and the particle–membrane binding energy may impact the dynamics and response of lipid membranes.
R. Metzler, J.-H. Jeon, and A. G. Cherstvy, Biochimica et Biophysica Acta - Biomembranes 1858, 2451 (2016)

The most likely is different from the mean II: the few encounter limit

When a particle attains a certain distance from its origin for the first time, scientists call this event a first passage event. The theory of first passage is an important concept in any field in which a stochastically moving particle reaching a threshold value is of note. A paramount example is gene regulation in living biological cells in which diffusing proteins binding to a specific target on the genome initiate important follow-up reactions. Here we present, for the first time, a full and asymptotically exact analysis of the distribution of first passage times in a finite volume.
We confirm our analytical results by computer simulations of spatially confined systems, and we measure tens of thousands of realizations. In addition to unifying the first-passage universality classes known in the literature, one of our central results is the proximity effect dominating all kinetics in the few-encounters limit: Whenever only a few particles need to arrive at their target, first-passage events in which particles move straight toward their target are decisive (i.e., all other paths are statistically less relevant). We prove that this situation is in fact a universal result for a variety of stochastic processes, even in the presence of external forcing. In other words, the proximity effect is independent of the details of the transport. Our findings shed light on how both the speed and precision of the target search process can be optimized.
We expect that our results will be relevant to fields ranging from biological physics to geophysics to econophysics in which first passage is important.
A. Godec and R. Metzler, Phys. Rev. X 6, 041037 (2016)

Overdamping transition delayed

It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases.
A. Bodrova, A. V. Chechkin, A. G. Cherstvy, H. Safdari, I. M. Sokolov, and R. Metzler, Sci. Rep. 6, 30520 (2016)

The most likely is different from the mean I

The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.
A. Godec and R. Metzler, Sci. Rep. 6, 20349 (2016)

Thirteen decades of ageing dynamics, in a single molecule

Single-molecule techniques have long given us insight into the motion and interactions of individual molecules. But simulations now show that the dynamics inside single proteins is not as simple as we thought — and that proteins are forever changing. News & Views article.
Nature Physics 12, 113 (2016)

Non-Gaussian anomalous diffusion in protein crowded lipid bilayer membranes

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.
J.-H. Jeon, M. Javanainen, H. Martinez-Seara, R. Metzler, and I. Vattulainen, Phys. Rev. X 6, 021006 (2016)