Correlating anomalous diffusion with lipid bilayer membrane structure using single molecule tracking and atomic force microscopy

Improved Highly Mobile Membrane Mimetic Model for Investigating Protein-Cholesterol Interactions

Cholesterol (CHL) plays an integral role in modulating the function and activity of various mammalian membrane proteins. Due to the slow dynamics of lipids, conventional computational studies of protein-CHL interactions rely on either long-time scale atomistic simulations or coarse-grained approximations to sample the process. A highly mobile membrane mimetic (HMMM) has been developed to enhance lipid diffusion and thus used to facilitate the investigation of lipid interactions with peripheral membrane proteins and, with customized in silico solvents to replace phospholipid tails, with integral membrane proteins. Here, we report an updated HMMM model that is able to include CHL, a nonphospholipid component of the membrane, henceforth called HMMM-CHL. To this end, we had to optimize the effect of the customized solvents on CHL behavior in the membrane. Furthermore, the new solvent is compatible with simulations using force-based switching protocols. In the HMMM-CHL, both improved CHL dynamics and accelerated lipid diffusion are integrated. To test the updated model, we have applied it to the characterization of protein-CHL interactions in two membrane protein systems, the human β2-adrenergic receptor (β2AR) and the mitochondrial voltage-dependent anion channel 1 (VDAC-1). Our HMMM-CHL simulations successfully identified CHL binding sites and captured detailed CHL interactions in excellent consistency with experimental data as well as other simulation results, indicating the utility of the improved model in applications where an enhanced sampling of protein-CHL interactions is desired.

Subdiffusive-Brownian crossover in membrane proteins: a generalized Langevin equation-based approach

In this work, we propose a generalized Langevin equation-based model to describe the lateral diffusion of a protein in a lipid bilayer. The memory kernel is represented in terms of a viscous (instantaneous) and an elastic (noninstantaneous) component modeled through a Dirac δ function and a three-parameter Mittag-Leffler type function, respectively. By imposing a specific relationship between the parameters of the three-parameter Mittag-Leffler function, the different dynamical regimes-namely ballistic, subdiffusive, and Brownian, as well as the crossover from one regime to another-are retrieved. Within this approach, the transition time from the ballistic to the subdiffusive regime and the spectrum of relaxation times underlying the transition from the subdiffusive to the Brownian regime are given. The reliability of the model is tested by comparing the mean-square displacement derived in the framework of this model and the mean-square displacement of a protein diffusing in a membrane calculated through molecular dynamics simulations.

Physical Properties and Reactivity of Microdomains in Phosphatidylinositol-Containing Supported Lipid Bilayer

We characterized the size, distribution, and fluidity of microdomains in a lipid bilayer containing phosphatidylinositol (PI) and revealed their roles during the two-dimensional assembly of a membrane deformation protein (FBP17). The morphology of the supported lipid bilayer (SLB) consisting of PI and phosphatidylcholine (PC) on a mica substrate was observed with atomic force microscope (AFM). Single particle tracking (SPT) was performed for the PI+PC-SLB on the mica substrate by using the diagonal illumination setup. The AFM topography showed that PI-derived submicron domains existed in the PI+PC-SLB. The spatiotemporal dependence of the lateral lipid diffusion obtained by SPT showed that the microdomain had lower fluidity than the surrounding region and worked as the obstacles for the lipid diffusion. We observed the two-dimensional assembly of FBP17, which is one of F-BAR family proteins included in endocytosis processes and has the function generating lipid bilayer tubules in vitro. At the initial stage of the FBP17 assembly, the PI-derived microdomain worked as a scaffold for the FBP17 adsorption, and the fluid surrounding region supplied FBP17 to grow the FBP17 domain via the lateral molecular diffusion. This study demonstrated an example clearly revealing the roles of two lipid microregions during the protein reaction on a lipid bilayer.

Time-fractional characterization of brine reaction and precipitation in porous media

Brine reaction and precipitation in porous media sometimes occur in the presence of a strong fluid flowing field, which induces the mobilization of the precipitated salts and distorts their spatial distribution. It is interesting to investigate how the distribution responds to such mobilization. We view these precipitates as random walkers in the complex inner space of the porous media, where they make stochastic jumps among locations and possibly wait between successive transitions. In consideration of related experimental results, the waiting time of the precipitates at a particular position is allowed to range widely from short sojourn to permanent residence. Through the model of a continuous-time random walk, a class of time-fractional equations for the precipitate's concentration profile is derived, including that in the Riemann-Liouville formalism and the Prabhakar formalism. The solutions to these equations show the general pattern of the precipitate's spatiotemporal evolution: a coupling of mass accumulation and mass transport. And the degree to which the mass is mobilized turns out to be monotonically correlated to the fractional exponent α. Moreover, to keep the completeness of the model, we further discuss how the interaction among the precipitates influences the precipitation process. In doing so, a time-fractional non-linear Fokker-Planck equation with source term is introduced and solved. It is shown that the interaction among the precipitates slightly perturbs their spatial distribution. This distribution is largely dominated by the brine reaction itself and the interaction between the precipitates and the porous media.

Scaling relationships for translational diffusion constants applied to membrane domain dissolution and growth

We compare the way that relationships for diffusion constants scale with the size of diffusing membrane domains and the geometry of their environments. Then, we review our experimental work on the dynamics of dissolution/growth of membrane domains in crowding induced mixing, phase separation, and Ostwald ripening in a highly confined environment. Overall, the scaling relationships applied to diffusion constants obtained by fits to our dynamic data indicate that dissolution and growth is influenced by the diffusion of clusters or small domains of lipids, in addition to kinetic processes and geometrical constraints.

Fractal analysis of lateral movement in biomembranes

Lateral movement of a molecule in a biomembrane containing small compartments (0.23-μm diameter) and large ones (0.75 μm) is analyzed using a fractal description of its walk. The early time dependence of the mean square displacement varies from linear due to the contribution of ballistic motion. In small compartments, walking molecules do not have sufficient time or space to develop an asymptotic relation and the diffusion coefficient deduced from the experimental records is lower than that measured without restrictions. The model makes it possible to deduce the molecule step parameters, namely the step length and time, from data concerning confined and unrestricted diffusion coefficients. This is also possible using experimental results for sub-diffusive transport. The transition from normal to anomalous diffusion does not affect the molecule step parameters. The experimental literature data on molecular trajectories recorded at a high time resolution appear to confirm the modeled value of the mean free path length of DOPE for Brownian and anomalous diffusion. Although the step length and time give the proper values of diffusion coefficient, the DOPE speed calculated as their quotient is several orders of magnitude lower than the thermal speed. This is interpreted as a result of intermolecular interactions, as confirmed by lateral diffusion of other molecules in different membranes. The molecule step parameters are then utilized to analyze the problem of multiple visits in small compartments. The modeling of the diffusion exponent results in a smooth transition to normal diffusion on entering a large compartment, as observed in experiments.

Non-Fickian diffusion of methanol in mesoporous media: Geometrical restrictions or adsorption-induced?

The methanol mass transfer in the mesoporous silica and alumina/zeolite H-ZSM-5 grains has been studied. We demonstrate that the methanol diffusion is characterized as a time-fractional for both solids. Methanol transport occurs in the super-diffusive regime, which is faster comparing to the Fickian diffusion. We show that the fractional exponents defining the regime of transport are different for each porous grain. The difference between the values of the fractional exponents is associated with a difference in the energetic strength of the active sites of the surface of the media of different chemical nature as well as the geometrical restrictions of the porous media. Increasing by six-fold, the pore diameter leads to a 1.1 fold increase of the fractional exponent. Decreasing by three-fold, the methanol desorption energy results into the same increasing the fractional exponent. Our findings support that mainly the adsorption process, which is defined by the energetic disorder of the corresponding surface active sites, is likely to be the driving force of the abnormality of the mass transfer in the porous media. Therefore, the fractional exponent is a fundamental characteristic which is individual for each combination of the porous solid and diffusing species.

Using Diffusion To Characterize Interfacial Heterogeneity

We report on the use of molecular diffusional motion over a range of length scales to characterize compositional heterogeneity in monolayer structures. This work focuses on the diffusional motion of perylene in two types of films supported on functionalized silica surfaces: single-component (stearic acid) and two-component (hydrocarbon/fluorocarbon) films. Langmuir-Blodgett (LB) monolayers were deposited directly on silica or were bound to surface-modified silica by means of metal ion complexation. The LB films were characterized by their π-A isotherms and by Brewster angle microscopy (BAM) during formation and deposition. Chromophore mobility and monolayer structural heterogeneity were evaluated by comparing rotational diffusion data (fluorescence anisotropy decay imaging) and translational diffusion data (fluorescence recovery after photobleaching) on the same LB films. Our results indicate that the mobility of the chromophore depends sensitively on both metal ion identity and film composition.

An analytical correlated random walk model and its application to understand subdiffusion in crowded environment

Subdiffusion in crowded environment such as movement of macromolecule in a living cell has often been observed experimentally. The primary reason for subdiffusion is volume exclusion by the crowder molecules. However, other effects such as hydrodynamic interaction may also play an important role. Although there are a large number of computer simulation studies on understanding molecular crowding, there is a lack of theoretical models that can be connected to both experiment and simulation. In the current work, we have formulated a one-dimensional correlated random walk model by connecting this to the motion in a crowded environment. We have found the exact solution of the probability distribution function of the model by solving it analytically. The parameters of our model can be obtained either from simulation or experiment. It has been shown that this analytical model captures some of the general features of diffusion in crowded environment as given in the previous literature and its prediction for transient subdiffusion closely matches the observations of a previous study of computer simulation of Escherichia coli cytoplasm. It is likely that this model will open up more development of theoretical models in this area.

Fractal model of anomalous diffusion

An equation of motion is derived from fractal analysis of the Brownian particle trajectory in which the asymptotic fractal dimension of the trajectory has a required value. The formula makes it possible to calculate the time dependence of the mean square displacement for both short and long periods when the molecule diffuses anomalously. The anomalous diffusion which occurs after long periods is characterized by two variables, the transport coefficient and the anomalous diffusion exponent. An explicit formula is derived for the transport coefficient, which is related to the diffusion constant, as dependent on the Brownian step time, and the anomalous diffusion exponent. The model makes it possible to deduce anomalous diffusion properties from experimental data obtained even for short time periods and to estimate the transport coefficient in systems for which the diffusion behavior has been investigated. The results were confirmed for both sub and super-diffusion.

Preparation of mica supported lipid bilayers for high resolution optical microscopy imaging

Supported lipid bilayers (SLBs) are widely used as a model for studying membrane properties (phase separation, clustering, dynamics) and its interaction with other compounds, such as drugs or peptides. However SLB characteristics differ depending on the support used. Commonly used techniques for SLB imaging and measurements are single molecule fluorescence microscopy, FCS and atomic force microscopy (AFM). Because most optical imaging studies are carried out on a glass support, while AFM requires an extremely flat surface (generally mica), results from these techniques cannot be compared directly, since the charge and smoothness properties of these materials strongly influence diffusion. Unfortunately, the high level of manual dexterity required for the cutting and gluing thin slices of mica to the glass slide presents a hurdle to routine use of mica for SLB preparation. Although this would be the method of choice, such prepared mica surfaces often end up being uneven (wavy) and difficult to image, especially with small working distance, high numerical aperture lenses. Here we present a simple and reproducible method for preparing thin, flat mica surfaces for lipid vesicle deposition and SLB preparation. Additionally, our custom made chamber requires only very small volumes of vesicles for SLB formation. The overall procedure results in the efficient, simple and inexpensive production of high quality lipid bilayer surfaces that are directly comparable to those used in AFM studies.

Thermal annealing triggers collapse of biphasic supported lipid bilayers into multilayer islands

The collapse of phase-separating single, supported lipid bilayers, consisting of mixtures of a zwitterionic phospholipid (POPC) and an anionic lipid (DPPA) upon thermal annealing in the presence of ions is examined using a combination of scanning probe, epifluorescence, and ellipsometric microscopies. We find that thermal annealing in the presence of ions in the bathing medium induces an irreversible transition from domain-textured, single supported bilayers to one comprising islands of multibilayer stacks, whose lateral area decays with lamellarity, producing pyramidal staircase "mesa" topography. The higher order lamellae are almost invariably localized above the anionic-lipid rich, gel-phase domains in the parent bilayer and depends on the ions in the bathing medium. The collapse mechanism appears to involve synergistic influences of two independent mechanisms: (1) stabilization of the incipient headgroup-headgroup interface in the emergent multibilayer configuration facilitated by ions in the bath and (2) domain-boundary templated folding. This collapse mechanism is consistent with previous theoretical predictions of topography-induced rippling instability in collapsing lipid monolayers and suggests the role of the mismatch in height and/or spontaneous curvature at domain boundaries in the collapse of phase-separated single supported bilayers.

TrackArt: the user friendly interface for single molecule tracking data analysis and simulation applied to complex diffusion in mica supported lipid bilayers

Background

Single molecule tracking (SMT) analysis of fluorescently tagged lipid and protein probes is an attractive alternative to ensemble averaged methods such as fluorescence correlation spectroscopy (FCS) or fluorescence recovery after photobleaching (FRAP) for measuring diffusion in artificial and plasma membranes. The meaningful estimation of diffusion coefficients and their errors is however not straightforward, and is heavily dependent on sample type, acquisition method, and equipment used. Many approaches require advanced computing and programming skills for their implementation.

Findings

Here we present TrackArt software, an accessible graphic interface for simulation and complex analysis of multiple particle paths. Imported trajectories can be filtered to eliminate spurious or corrupted tracks, and are then analyzed using several previously described methodologies, to yield single or multiple diffusion coefficients, their population fractions, and estimated errors. We use TrackArt to analyze the single-molecule diffusion behavior of a sphingolipid analog SM-Atto647N, in mica supported DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine) bilayers. Fitting with a two-component diffusion model confirms the existence of two separate populations of diffusing particles in these bilayers on mica. As a demonstration of the TrackArt workflow, we characterize and discuss the effective activation energies required to increase the diffusion rates of these populations, obtained from Arrhenius plots of temperature-dependent diffusion. Finally, TrackArt provides a simulation module, allowing the user to generate models with multiple particle trajectories, diffusing with different characteristics. Maps of domains, acting as impermeable or permeable obstacles for particles diffusing with given rate constants and diffusion coefficients, can be simulated or imported from an image. Importantly, this allows one to use simulated data with a known diffusion behavior as a comparison for results acquired using particular algorithms on actual, “natural” samples whose diffusion behavior is to be extracted. It can also serve as a tool for demonstrating diffusion principles.

Conclusions

TrackArt is an open source, platform-independent, Matlab-based graphical user interface, and is easy to use even for those unfamiliar with the Matlab programming environment. TrackArt can be used for accurate simulation and analysis of complex diffusion data, such as diffusion in lipid bilayers, providing publication-quality formatted results.

Shot-noise limited localization of single 20 nm gold particles with nanometer spatial precision within microseconds

Single-particle tracking (SPT) is a powerful approach to investigate dynamics without ensemble average. Continuing effort has been made to track smaller particles with better spatial precision at higher speed. In this work, we demonstrate SPT of 20 nm gold nanoparticle (GNP) with 2 nm spatial precision up to 500 kHz by using microsecond interferometric scattering (μs-iSCAT) microscopy. The linear scattering signal from single GNPs is detected by a high-speed CMOS camera via interference. Through this homodyne detection, shot-noise limited sensitivity, and therefore optimal localization precision are achieved at high speed where considerable electronic noise is present. Using μs-iSCAT microscopy, we observe anomalous diffusion of GNPs labeled to lipid molecules in a supported bilayer membrane prepared on a glass substrate. The combination of nanometer spatial precision and microsecond temporal resolution provides the opportunity to study rapid motions of nano-objects on molecular scale with unprecedented clarity.

Anomalous diffusion of proteins in sheared lipid membranes

We use coarse grained molecular dynamics simulations to investigate diffusion properties of sheared lipid membranes with embedded transmembrane proteins. In membranes without proteins, we find normal in-plane diffusion of lipids in all flow conditions. Protein embedded membranes behave quite differently: by imposing a simple shear flow and sliding the monolayers of the membrane over each other, the motion of protein clusters becomes strongly superdiffusive in the shear direction. In such a circumstance, the subdiffusion regime is predominant perpendicular to the flow. We show that superdiffusion is a result of accelerated chaotic motions of protein-lipid complexes within the membrane voids, which are generated by hydrophobic mismatch or the transport of lipids by proteins.

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-Gaussian distributions of spatial displacements, heterogeneous diffusion and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

Test for determining a subdiffusive model in ergodic systems from single trajectories

Experiments on particle motion show that it is often subdiffusive. This subdiffusion may be due to trapping, percolationlike structures, or viscoelastic behavior of the medium. While the models based on trapping (leading to continuous-time random walks) can easily be distinguished from the rest by testing their nonergodicity, the latter two cases are harder to distinguish. We propose a statistical test for distinguishing between these two based on the space-filling properties of trajectories, and prove its feasibility and specificity using synthetic data. We moreover present a flow chart for making a decision on a type of subdiffusion for a broader class of models.

Wanted: a positive control for anomalous subdiffusion

Anomalous subdiffusion in cells and model systems is an active area of research. The main questions are whether diffusion is anomalous or normal, and if it is anomalous, its mechanism. The subject is controversial, especially the hypothesis that crowding causes anomalous subdiffusion. Anomalous subdiffusion measurements would be strengthened by an experimental standard, particularly one able to cross-calibrate the different types of measurements. Criteria for a calibration standard are proposed. First, diffusion must be anomalous over the length and timescales of the different measurements. The length-scale is fundamental; the time scale can be adjusted through the viscosity of the medium. Second, the standard must be theoretically well understood, with a known anomalous subdiffusion exponent, ideally readily tunable. Third, the standard must be simple, reproducible, and independently characterizable (by, for example, electron microscopy for nanostructures). Candidate experimental standards are evaluated, including obstructed lipid bilayers; aqueous systems obstructed by nanopillars; a continuum percolation system in which a prescribed fraction of randomly chosen obstacles in a regular array is ablated; single-file diffusion in pores; transient anomalous subdiffusion due to binding of particles in arrays such as transcription factors in randomized DNA arrays; and computer-generated physical trajectories.