Communication: Consistent picture of lateral subdiffusion in lipid bilayers: Molecular dynamics simulation and exact results

Multiple diffusive behaviors of the random walk in inhomogeneous environments

Anomalous diffusive behaviors are observed in highly inhomogeneous but relatively stable environments such as intracellular media and are increasingly attracting attention. In this paper we develop a coupled continuous-time random walk model in which the waiting time is power-law coupled with the local environmental diffusion coefficient. We provide two forms of the waiting time density, namely, a heavy-tailed density and an exponential density. For different waiting time densities, anomalous diffusions with the diffusion exponent between 0 and 2 and Brownian yet non-Gaussian diffusion can be realized within the present model. The diffusive behaviors are analyzed and discussed by deriving the mean-squared displacement and probability density function. In addition we derive the effective jump length density corresponding to the decoupled form to help distinguish the diffusion types. Our model unifies two kinds of anomalous diffusive behavior with different characteristics in the same inhomogeneous environment into a theoretical framework. The model interprets the random motion of particles in a complex inhomogeneous environment and reproduces the experimental results of different biological and physical systems.

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.

Molecular Dynamics Simulation of 2-Benzimidazolyl-Urea with DPPC Lipid Membrane and Comparison with a Copper(II) Complex Derivative

Benzimidazole derivatives have gained attention recently due to their wide pharmacological activity acting as anti-inflammatory, hypotensive, analgesic, and anti-aggregatory agents. They are also common ligands in transition metal coordination chemistry, forming complex compounds with enhanced biological properties, especially in targeted cancer therapy. A key issue to understand anti-tumour effects is drug permeability through cellular membranes, as poor permeability outcomes can avert further futile drug development. In this work, we conducted atomistic molecular dynamics (MD) simulations and biased MD simulations to explore the interactions of 2-benzimidazolyl-urea with a phospholipid bilayer (dipalmitoylphosphatidylcholine, DPPC) together with a previously synthesized copper(II) complex compound. The aim was to study the permeability of these compounds by assessing their free energy profile along the bilayer normal. The simulations indicated that both the ligand (2-benzimidazolyl-urea, BZIMU) and the complex show a similar behaviour, yielding high energy barriers for the permeation process. However, with increasing concentration of BZIMU, the molecules tend to aggregate and form a cluster, leading to the formation of a pore. Clustering and pore formation can possibly explain the previously observed cytotoxicity of the BZIMU molecule via membrane damage.

Anomalous Diffusion of Peripheral Membrane Signaling Proteins from All-Atom Molecular Dynamics Simulations

Peripheral membrane proteins bind transiently to membrane surfaces as part of many signaling pathways. The bound proteins perform two-dimensional (2-D) diffusion on the membrane surface during the recruitment function. To better understand the interplay between the 2-D diffusion of these protein domains and their membrane binding modes, we performed multimicrosecond all-atom molecular dynamics simulations of two regulatory domains, a C2 domain and a pleckstrin homology (PH) domain, in their experimentally determined bound configuration to a lipid bilayer. The protein bound configurations are preserved throughout the simulation trajectories. Both protein domains exhibit anomalous diffusion with distinct features in their dynamics that reflect their different modes of binding. An analysis of their diffusive behavior reveals common features with the diffusion of lipid molecules in lipid bilayers, suggesting that the 2-D motion of protein domains bound to the membrane surface is modulated by the viscoelastic nature of the lipid bilayer.

On the Structure and Flip-flop of Free Docosahexaenoic Acid in a Model Human Brain Membrane

Among the omega-3 fatty acids, docosahexaenoic acid (DHA, sn22:6) is particularly vital in human brain cell membranes. There is considerable interest in DHA because low-level DHA has been associated with declined cognitive function and poor visual acuity. In this work, atomistic molecular dynamics simulations were used to investigate the effects of free protonated DHA (DHAP) in molar fractions of 0, 17, 30, and 38% in a realistic model of a healthy brain cell membrane comprising 26 lipid types. Numerous flip-flop events of DHAP were observed and categorized as successful or aborted. Novel use of the machine learning technique, density-based spatial clustering of applications with noise (DBSCAN), effectively identified flip-flop events by way of clustering. Our data show that increasing amounts of DHAP in the membrane disorder the bilayer packing, fluidize the membrane, and increase the rates of successful flip-flop from k = 0.2 μs^-1 (17% DHAP) to 0.8 μs^-1 (30% DHAP) and to 1.3 μs^-1 (38% DHAP). In addition, we also provided a detailed understanding of the flip-flop mechanism of DHAP across this complex membrane. Interestingly, we noted the role of hydrogen bonds in two distinct coordinated flip-flop phenomena between two DHAP molecules: double flip-flop and assisted flip-flop. Understanding the effects of various concentrations of DHAP on the dynamics within a lipid membrane and the resulting structural properties of the membrane are important when considering the use of DHAP as a dietary supplement or as a potential therapeutic.

Anomalous diffusion in a bench-scale pulsed fluidized bed

We present our analysis on microrheology of a bench-scale pulsed fluidized bed, which represents a weakly confined system. Nonlinear gas-particle and particle-particle interactions resulting from pulsed flow are associated with harmonic and subharmonic modes. While periodic structured bubble patterns are observed at the mesoscale, particle-scale measurements reveal anomalous diffusion in the driven granular medium. We use single-particle tracks to analyze ergodicity and ageing properties at two pulsing frequencies having remarkably different mesoscale features. The scaling of ensemble-averaged mean-squared displacement is not unique. The distribution of time-averaged mean-squared displacements is non-Gaussian, asymmetric, and has a finite trivial contribution from particles in crowded quasistatic surroundings. Results indicate weak ergodicity breaking, which along with ageing characterizes the nonstationary and out-of-equilibrium dynamics.

Effects of lipid heterogeneity on model human brain lipid membranes

Cell membranes naturally contain a heterogeneous lipid distribution. However, homogeneous bilayers are commonly preferred and utilised in computer simulations due to their relative simplicity, and the availability of lipid force field parameters. Recently, experimental lipidomics data for the human brain cell membranes under healthy and Alzheimer's disease (AD) conditions were investigated, since disruption to the lipid composition has been implicated in neurodegenerative disorders, including AD [R. B. Chan et al., J. Biol. Chem., 2012, 287, 2678-2688]. In order to observe the effects of lipid complexity on the various bilayer properties, molecular dynamics simulations were used to study four membranes with increasing heterogeneity: a pure POPC membrane, a POPC and cholesterol membrane in a 1 : 1 ratio (POPC-CHOL), and to our knowledge, the first realistic models of a healthy brain membrane and an Alzheimer's diseased brain membrane. Numerous structural, interfacial, and dynamical properties, including the area per lipid, interdigitation, dipole potential, and lateral diffusion of the two simple models, POPC and POPC-CHOL, were analysed and compared to those of the complex brain models consisting of 27 lipid components. As the membranes gain heterogeneity, a number of alterations were found in the structural and dynamical properties, and more significant differences were observed in the lateral diffusion. Additionally, we observed snorkeling behaviour of the lipid tails that may play a role in the permeation of small molecules across biological membranes. In this work, atomistic description of realistic brain membrane models is provided, which can add insight towards the permeability and transport pathways of small molecules across these membrane barriers.

Testing of Multifractional Brownian Motion

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index 0<H<1. In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena.

Interactions between Beta-2-Glycoprotein-1 and Phospholipid Bilayer-A Molecular Dynamic Study

This study aims to investigate the interactions appearing when the beta-2-glycoprotein-1 binds to a lipid bilayer. The inter- and intra-molecular forces acting between the two macromolecular systems have been investigated using a molecular dynamics simulation method. The importance of water bridges has also been addressed. Additionally, the viscoelastic response of the bilayer has been studied. In detail, the (saturated-chain) 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) and (unsaturated-chain) 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) bilayers have been chosen to test their behavior near the protein. Both of the lipids have a polar head but different chemical structures and are similar to the main phospholipids present in the synovial fluid. This study is meaningful for further explaining the worsening friction properties in articular cartilage, as the inactivation of phospholipid bilayers by beta-2-glycoprotein-1 is believed to be a cause of the destruction of cartilage in most rheumatic diseases and osteoarthritis. It was found that the protein binds stronger to the DPPC bilayer than to the POPE, but in both cases, it has the potential to change the local bilayer stability. Nevertheless, the binding forces are placed within a small area (only a few lipids contribute to the binding, creating many interactions). However, together, they are not stronger than the covalent bonds between C-O, thus, potentially, it is possible to push the lipids into the bilayer but detaching the lipids' heads from the tail is not possible. Additionally, the protein causes water displacement from the vicinity of the bilayer, and this may be a contributor to the instability of the bilayer (disrupting the water bridges needed for the stabilization of the bilayer, especially in the case of DPPC where the heads are not so well stabilized by H-bonds as they are in POPE). Moreover, it was found that the diffusivity of lipids in the DPPC bilayer bound to the protein is significantly different from the diffusivity of the ones which are not in contact with the protein. The POPE bilayer is stiffer due to intramolecular interactions, which are stronger than in the DPPC; thus, the viscous to elastic effects in the POPE case are more significant than in the case of the DPPC. It is, therefore, harder to destabilize the POPE bilayer than the DPPC one.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

Structural dynamics of pentapeptide repeat proteins

Pentapeptide repeat proteins (PRPs) represent a large superfamily with more than 38 000 sequences in nearly 3500 species, the majority belonging to cyanobacteria but represented among all branches of life. PRPs contain at least eight consecutive pentapeptide repeats with the consensus (A/C/S/V/T/L/I)(D/N/S/K/E/I/R)(L/F)(S/T/R/E/Q/K/V/D)(G/D/E/N/R/Q/K). PRPs fold into right-handed quadrilateral β helices, also known as repeat-five-residue (Rfr)-folds, with four consecutive pentapeptide repeats comprising a single coil, the ~90° change in polypeptide direction in square-shaped coils achieved by type I, II and IV β turns, and hydrogen bonds between coils establishing β ladders on each Rfr-fold face. PRPs are broadly categorized into group 1 and 2 involved in antibiotic resistance and group 3 currently having unknown functions. Motivated by their intriguing structures, we are investigating PRP biophysical characteristics, including Rfr-fold thermal stability, β turn and β ladder hydrogen bond amide exchange rates and backbone dynamics. Here, we present analysis of 20 ns molecular dynamics (MD) simulations and all atom normal mode analysis (aaNMA) calculations for four group 1 and group 2 and four group 3 PRPs whose structures have been determined by X-ray crystallography. The MD cross-correlation matrices and aaNMA indicated strong correlated motion between adjacent coils and weak coupled motion between coils separated by one or more intervening coils. Slow anticorrelated motions were detected between adjacent coils in aaNMA modes that we hypothesize are requisite to access exchange-competent states necessary to permit solvent exchange of amide hydrogens involved in β-ladder and β-turns hydrogen bonds, which can have lifetimes on the order of months.

Lévy walk dynamics in an external harmonic potential

Lévy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs-their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some longstanding puzzles around LWs.

Assessing the extent of the structural and dynamic modulation of membrane lipids due to pore forming toxins: insights from molecular dynamics simulations

Infections caused by many virulent bacterial strains are triggered by the release of pore forming toxins (PFTs), which form oligomeric transmembrane pore complexes on the target plasma membrane. The spatial extent of the perturbation to the surrounding lipids during pore formation is relatively unexplored. Using all-atom molecular dynamics simulations, we investigate the changes in the structure and dynamics of lipids in a 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) lipid bilayer in the presence of contrasting PFTs. Cytolysin A (ClyA), an α toxin with its inserted wedge shaped bundle of inserted α helices, induces significant asymmetry across the membrane leaflets in comparison with α hemolysin (AHL), a β toxin. Despite the differences in hydrophobic mismatch and uniquely different topologies of the two oligomers, perturbations to lipid order as reflected in the tilt angle and order parameters and membrane thinning are short ranged, lying within ∼2.5 nm from the periphery of either pore complex, and commensurate with distances typically associated with van der Waals forces. In contrast, the spatial extent of perturbations to the lipid dynamics extends outward to at least 4 nm for both proteins, and the continuous survival probabilities reveal the presence of a tightly bound shell of lipids in this region. Displacement probability distributions show long tails and the distinctly non-Gaussian features reflect the induced dynamic heterogeneity. A detailed profiling of the protein-lipid contacts with tyrosine, tryptophan, lysine and arginine residues shows increased non-polar contacts in the cytoplasmic leaflet for both PFTs, with a higher number of atomic contacts in the case of AHL in the extracellular leaflet due to the mushroom-like topology of the pore complex. The short ranged nature of the perturbations observed in this simple one component membrane suggests inherent plasticity of membrane lipids enabling the recovery of the structure and membrane fluidity even in the presence of these large oligomeric transmembrane protein assemblies. This observation has implications in membrane repair processes such as budding or vesicle fusion events used to mitigate PFT virulence, where the underlying lipid dynamics and fluidity in the vicinity of the pore complex are expected to play an important role.

Carotenoids promote lateral packing and condensation of lipid membranes

Carotenoids are pigment molecules that protect biomembranes against degradation and may be involved in the formation of functional bacterial membrane microdomains. Little is known on whether different types of carotenoids have different effects on the membrane or if there is any concentration dependence of these effects. In this work, we present results from molecular dynamics simulations of phospholipid bilayers containing different amounts of either β-carotene or zeaxanthin. Both β-carotene and zeaxanthin show the ability to laterally condense the membrane lipids and reduce their inter-leaflet interactions. With increasing concentrations, both carotenoids increase the bilayer thickness and rigidity. The results reveal that carotenoids have similar effects to cholesterol on regulating the behavior of fluid-phase membranes, suggesting that they could function as sterol substitutes and confirming their potential role in the formation of functional membrane domains.

Diffusive transport of nanoscale objects through cell membranes: a computational perspective

Diffusion is an essential and fundamental means of transport of substances on cell membranes, and the dynamics of biomembranes plays a crucial role in the regulation of numerous cellular processes. The understanding of the complex mechanisms and the nature of particle diffusion have a bearing on establishing guidelines for the design of efficient transport materials and unique therapeutic approaches. Herein, this review article highlights the most recent advances in investigating diffusion dynamics of nanoscale objects on biological membranes, focusing on the approaches of tailored computer simulations and theoretical analysis. Due to the presence of the complicated and heterogeneous environment on native cell membranes, the diffusive transport behaviors of nanoparticles exhibit unique and variable characteristics. The general aspects and basic theories of normal diffusion and anomalous diffusion have been introduced. In addition, the influence of a series of external and internal factors on the diffusion behaviors is discussed, including particle size, membrane curvature, particle-membrane interactions or particle-inclusion, and the crowding degree of membranes. Finally, we seek to identify open problems in the existing experimental, simulation, and theoretical research studies, and to propose challenges for future development.

Regulation of migration of chemotactic tumor cells by the spatial distribution of collagen fiber orientation

Collagen fibers, an important component of the extracellular matrix (ECM), can both inhibit and promote cellular migration. In vitro studies have revealed that the fibers' orientations are crucial to cellular invasion, while in vivo investigations have led to the development of tumor-associated collagen signatures (TACS) as an important prognostic factor. Studying biophysical regulation of cell invasion and the effect of the fibers' orientation not only deepens our understanding of the phenomenon, but also helps classify the TACSs precisely, which is currently lacking. We present a stochastic model for random or chemotactic migration of cells in fibrous ECM, and study the role of the various factors in it. The model provides a framework for quantitative classification of the TACSs, and reproduces quantitatively recent experimental data for cell motility. It also indicates that the spatial distribution of the fibers' orientations and extended correlations between them, hitherto ignored, as well as dynamics of cellular motion all contribute to regulation of the cells' invasion length, which represents a measure of metastatic risk. Although the fibers' orientations trivially affect randomly moving cells, their effect on chemotactic cells is completely nontrivial and unexplored, which we study in this paper.

Influence of Bilayer Size and Number in Multi-Bilayer DOPC Simulations at Full and Low Hydration

Biophysical studies of model cell membranes at full and low hydration are usually carried out using scattering measurements on multi-bilayer systems. Molecular simulations of lipid bilayers aimed at reproducing those experimental conditions are usually conducted using single bilayers with different amounts of water. These simulation conditions may lead to artifacts arising from size effects and self-interactions because of periodic boundary conditions. We have tested the influence of the size and number of bilayers on membrane properties using the Lipid14 force field for lipids in molecular dynamics simulations of 1,2-dioleoyl- sn-glycero-3-phosphocholine bilayers at full hydration (44 water molecules per lipid), low hydration (18 water molecules per lipid), and dehydration (9 water molecules per lipid). A number of additional simulations were conducted with the Slipids force field for comparison. We have found that the average area per lipid (APL), thickness, mass density profiles, and acyl tail order parameters are insensitive to the size and the number of bilayers for all hydration states. The Lipid14 force field can also successfully reproduce the experimentally observed decrease in APL and corresponding increase in bilayer thickness upon dehydration, reflecting the increase in ordering as the system becomes more gel-like. Additionally, decreasing hydration levels were associated with a trend away from normal lateral diffusion and toward more subdiffusive regimes across both force fields. In summary, at least for the Lipid14 force field, the use of a single bilayer with 128 phospholipid molecules provides an adequate representation of multi-bilayer systems at varying levels of hydration.

Perfect anomalous transport of subdiffusive cargos by molecular motors in viscoelastic cytosol

Multiple experiments show that various submicron particles such as magnetosomes, RNA messengers, viruses, and even much smaller nanoparticles such as globular proteins diffuse anomalously slow in viscoelastic cytosol of living cells. Hence, their sufficiently fast directional transport by molecular motors such as kinesins is crucial for the cell operation. It has been shown recently that the traditional flashing Brownian ratchet models of molecular motors are capable to describe both normal and anomalous transport of such subdiffusing cargos by molecular motors with a very high efficiency. This work elucidates further an important role of mechanochemical coupling in such an anomalous transport. It shows a natural emergence of a perfect subdiffusive ratchet regime due to allosteric effects, where the random rotations of a "catalytic wheel" at the heart of the motor operation become perfectly synchronized with the random stepping of a heavily loaded motor, so that only one ATP molecule is consumed on average at each motor step along microtubule. However, the number of rotations made by the catalytic engine and the traveling distance both scale sublinearly in time. Nevertheless, this anomalous transport can be very fast in absolute terms.

Viscoelastic subdiffusion in a random Gaussian environment

Viscoelastic subdiffusion governed by a fractional Langevin equation is studied numerically in a random Gaussian environment modeled by stationary Gaussian potentials with decaying spatial correlations. This anomalous diffusion is archetypal for living cells, where cytoplasm is known to be viscoelastic and a spatial disorder also naturally emerges. We obtain some first important insights into it within a model one-dimensional study. Two basic types of potential correlations are studied: short-range exponentially decaying and algebraically slow decaying with an infinite correlation length, both for a moderate (several k_BT, in the units of thermal energy), and strong (5-10k_BT) disorder. For a moderate disorder, it is shown that on the ensemble level viscoelastic subdiffusion can easily overcome the medium's disorder. Asymptotically, it is not distinguishable from the disorder-free subdiffusion. However, a strong scatter in single-trajectory averages is nevertheless seen even for a moderate disorder. It features a weak ergodicity breaking, which occurs on a very long yet transient time scale. Furthermore, for a strong disorder, a very long transient regime of logarithmic, Sinai-type diffusion emerges. It can last longer and be faster in the absolute terms for weakly decaying correlations as compared with the short-range correlations. Residence time distributions in a finite spatial domain are of a generalized log-normal type and are reminiscent also of a stretched exponential distribution. They can be easily confused for power-law distributions in view of the observed weak ergodicity breaking. This suggests a revision of some experimental data and their interpretation.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Diffusion of small particles in polymer films

The motion of small probe molecules in a two-dimensional system containing frozen polymer chains was studied by means of Monte Carlo simulations. The model macromolecules were coarse-grained and restricted to vertices of a triangular lattice. The cooperative motion algorithm was used to generate representative configurations of macromolecular systems of different polymer concentrations. The remaining unoccupied lattice sites of the system were filled with small molecules. The structure of the polymer film, especially near the percolation threshold, was determined. The dynamic lattice liquid algorithm was then employed for studies of the dynamics of small objects in the polymer matrix. The influence of chain length and polymer concentration on the mobility and the character of motion of small molecules were studied. Short- and long-time dynamic behaviors of solvent molecules were also described. Conditions of anomalous diffusions' appearance in such systems are discussed. The influence of the structure of the matrix of obstacles on the molecular transport was discussed.

Multiscale molecular dynamics simulations of lipid interactions with P-glycoprotein in a complex membrane

P-glycoprotein (P-gp) can transport a wide range of very different hydrophobic organic molecules across the membrane. Its ability to extrude molecules from the cell creates delivery problems for drugs that target proteins in the central nervous system (CNS) and also causes drug-resistance in many forms of cancer. Whether a drug will be susceptible to export by P-gp is difficult to predict and currently this is usually assessed with empirical and/or animal models. Thus, there is a need to better understand how P-gp works at the molecular level in order to fulfil the 3Rs: Refinement, reduction and replacement of animals in research. As structural information increasingly becomes available, our understanding at the molecular level improves. Proteins like P-gp are however very dynamic entities and thus one of the most appropriate ways to study them is with molecular dynamics simulations, especially as this can capture the influence of the surrounding environment. Recent parameterization developments have meant that it is now possible to simulate lipid bilayers that more closely resemble in vivo membranes in terms of their composition. In this report we construct a complex lipid bilayer that mimics the composition of brain epithelial cells and examine the interactions of it with P-gp. We find that the negatively charged phosphatidylserine lipids in the inner leaflet of the membrane tend to form an annulus around P-gp. We also observed the interaction of cholesterol with three distinct areas of the P-gp. Potential of mean force (PMF) calculations suggest that a crevice between transmembrane helices 10 and 12 has particularly favourable interaction energy for cholesterol.

Computational and theoretical approaches for studies of a lipid recognition protein on biological membranes

Many cellular functions, including cell signaling and related events, are regulated by the association of peripheral membrane proteins (PMPs) with biological membranes containing anionic lipids, e.g., phosphatidylinositol phosphate (PIP). This association is often mediated by lipid recognition modules present in many PMPs. Here, I summarize computational and theoretical approaches to investigate the molecular details of the interactions and dynamics of a lipid recognition module, the pleckstrin homology (PH) domain, on biological membranes. Multiscale molecular dynamics simulations using combinations of atomistic and coarse-grained models yielded results comparable to those of actual experiments and could be used to elucidate the molecular mechanisms of the formation of protein/lipid complexes on membrane surfaces, which are often difficult to obtain using experimental techniques. Simulations revealed some modes of membrane localization and interactions of PH domains with membranes in addition to the canonical binding mode. In the last part of this review, I address the dynamics of PH domains on the membrane surface. Local PIP clusters formed around the proteins exhibit anomalous fluctuations. This dynamic change in protein-lipid interactions cause temporally fluctuating diffusivity of proteins, i.e., the short-term diffusivity of the bound protein changes substantially with time, and may in turn contribute to the formation/dissolution of protein complexes in membranes.

Multiscale molecular dynamics simulations of lipid interactions with P-glycoprotein in a complex membrane

P-glycoprotein (P-gp) can transport a wide range of very different hydrophobic organic molecules across the membrane. Its ability to extrude molecules from the cell creates delivery problems for drugs that target proteins in the central nervous system (CNS) and also causes drug-resistance in many forms of cancer. Whether a drug will be susceptible to export by P-gp is difficult to predict and currently this is usually assessed with empirical and/or animal models. Thus, there is a need to better understand how P-gp works at the molecular level in order to fulfil the 3Rs: Refinement, reduction and replacement of animals in research. As structural information increasingly becomes available, our understanding at the molecular level improves. Proteins like P-gp are however very dynamic entities and thus one of the most appropriate ways to study them is with molecular dynamics simulations, especially as this can capture the influence of the surrounding environment. Recent parameterization developments have meant that it is now possible to simulate lipid bilayers that more closely resemble in vivo membranes in terms of their composition. In this report we construct a complex lipid bilayer that mimics the composition of brain epithelial cells and examine the interactions of it with P-gp. We find that the negatively charged phosphatidylserine lipids in the inner leaflet of the membrane tend to form an annulus around P-gp. We also observed the interaction of cholesterol with three distinct areas of the P-gp. Potential of mean force (PMF) calculations suggest that a crevice between transmembrane helices 10 and 12 has particularly favourable interaction energy for cholesterol.

Quantifying non-ergodicity of anomalous diffusion with higher order moments

Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes.

Breathing modes of Kolumbo submarine volcano (Santorini, Greece)

Submarine volcanoes, such as Kolumbo (Santorini, Greece) are natural laboratories for fostering multidisciplinary studies. Their investigation requires the most innovative marine technology together with advanced data analysis. Conductivity and temperature of seawater were recorded directly above Kolumbo’s hydrothermal vent system. The respective time series have been analyzed in terms of non–equilibrium techniques. The energy dissipation of the volcanic activity is monitored by the temperature variations of seawater. The venting dynamics of chemical products is monitored by water conductivity. The analysis of the time series in terms of stochastic processes delivers scaling exponents with turning points between consecutive regimes for both conductivity and temperature. Changes of conductivity are shown to behave as a universal multifractal and their variance is subdiffusive as the scaling exponents indicate. Temperature is constant over volcanic rest periods and a universal multifractal behavior describes its changes in line with a subdiffusive character otherwise. The universal multifractal description illustrates the presence of non–conservative conductivity and temperature fields showing that the system never retains a real equilibrium state. The existence of a repeated pattern of the combined effect of both seawater and volcanic activity is predicted. The findings can shed light on the dynamics of chemical products emitted from the vents and point to the presence of underlying mechanisms that govern potentially hazardous, underwater volcanic environments.

Communication: A multiscale Bayesian inference approach to analyzing subdiffusion in particle trajectories

Anomalous diffusion is characterized by its asymptotic behavior for t → ∞. This makes it difficult to detect and describe in particle trajectories from experiments or computer simulations, which are necessarily of finite length. We propose a new approach using Bayesian inference applied directly to the observed trajectories sampled at different time scales. We illustrate the performance of this approach using random trajectories with known statistical properties and then use it for analyzing the motion of lipid molecules in the plane of a lipid bilayer.

Biomimetic Phospholipid Membrane Organization on Graphene and Graphene Oxide Surfaces: A Molecular Dynamics Simulation Study

Supported phospholipid membrane patches stabilized on graphene surfaces have shown potential in sensor device functionalization, including biosensors and biocatalysis. Lipid dip-pen nanolithography (L-DPN) is a method useful in generating supported membrane structures that maintain lipid functionality, such as exhibiting specific interactions with protein molecules. Here, we have integrated L-DPN, atomic force microscopy, and coarse-grained molecular dynamics simulation methods to characterize the molecular properties of supported lipid membranes (SLMs) on graphene and graphene oxide supports. We observed substantial differences in the topologies of the stabilized lipid structures depending on the nature of the surface (polar graphene oxide vs nonpolar graphene). Furthermore, the addition of water to SLM systems resulted in large-scale reorganization of the lipid structures, with measurable effects on lipid lateral mobility within the supported membranes. We also observed reduced lipid ordering within the supported structures relative to free-standing lipid bilayers, attributed to the strong hydrophobic interactions between the lipids and support. Together, our results provide insight into the molecular effects of graphene and graphene oxide surfaces on lipid bilayer membranes. This will be important in the design of these surfaces for applications such as biosensor devices.

Manipulation and Motion of Organelles and Single Molecules in Living Cells

The biomolecule is among the most important building blocks of biological systems, and a full understanding of its function forms the scaffold for describing the mechanisms of higher order structures as organelles and cells. Force is a fundamental regulatory mechanism of biomolecular interactions driving many cellular processes. The forces on a molecular scale are exactly in the range that can be manipulated and probed with single molecule force spectroscopy. The natural environment of a biomolecule is inside a living cell, hence, this is the most relevant environment for probing their function. In vivo studies are, however, challenged by the complexity of the cell. In this review, we start with presenting relevant theoretical tools for analyzing single molecule data obtained in intracellular environments followed by a description of state-of-the art visualization techniques. The most commonly used force spectroscopy techniques, namely optical tweezers, magnetic tweezers, and atomic force microscopy, are described in detail, and their strength and limitations related to in vivo experiments are discussed. Finally, recent exciting discoveries within the field of in vivo manipulation and dynamics of single molecule and organelles are reviewed.

Dynamic interactions between a membrane binding protein and lipids induce fluctuating diffusivity

Pleckstrin homology (PH) domains are membrane-binding lipid recognition proteins that interact with phosphatidylinositol phosphate (PIP) molecules in eukaryotic cell membranes. Diffusion of PH domains plays a critical role in biological reactions on membrane surfaces. Although diffusivity can be estimated by long-time measurements, it lacks information on the short-time diffusive nature. We reveal two diffusive properties of a PH domain bound to the surface of a PIP-containing membrane using molecular dynamics simulations. One is fractional Brownian motion, attributed to the motion of the lipids with which the PH domain interacts. The other is temporally fluctuating diffusivity; that is, the short-time diffusivity of the bound protein changes substantially with time. Moreover, the diffusivity for short-time measurements is intrinsically different from that for long-time measurements. This fluctuating diffusivity results from dynamic changes in interactions between the PH domain and PIP molecules. Our results provide evidence that the complexity of protein-lipid interactions plays a crucial role in the diffusion of proteins on biological membrane surfaces. Changes in the diffusivity of PH domains and related membrane-bound proteins may in turn contribute to the formation/dissolution of protein complexes in membranes.

The role of Tat peptide self-aggregation in membrane pore stabilization: insights from a computational study

Role of Tat peptide self-aggregation to direct transduction in cells is highlighted in a computational study of dimer versus monomer.

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Assessing gastric toxicity of xanthone derivatives of anti-inflammatory activity using simulation and experimental approaches

Xanthones are tricyclic compounds of natural or synthetic origin exhibiting a broad spectrum of therapeutic activities. Three synthetic xanthone derivatives (KS1, KS2, and KS3) with properties typical for nonsteroidal anti-inflammatory drugs (NSAID) were objects of the presented model study. NSAIDs are in common use however; several of them exhibit gastric toxicity predominantly resulting from their direct interactions with the outermost lipid layer of the gastric mucosa that impair its hydrophobic barrier property. Among the studied xanthones, gastric toxicity of only KS2 has been determined in previous pharmacological studies, and it is low. In this study, carried out using X-ray diffraction and computer simulation, a palmitoyloleoylphosphatidylcholine-cholesterol bilayer (POPC-Chol) was used as a model of a hydrophobic layer of lipids protecting gastric mucosa as POPC and Chol are the main lipids in human mucus. X-ray diffraction data were used to validate the computer model. The aim of the study was to assess potential gastric toxicity of the xanthones by analysing their atomic level interactions with lipids, ions, and water in the lipid bilayer and their effect on the bilayer physicochemical properties. The results show that xanthones have small effect on the bilayer properties except for its rigidity whereas their interactions with water, ions, and lipids depend on their protonation state and for a given state, are similar for all the xanthones. As gastric toxicity of KS2 is low, based on MD simulations one can predict that toxicity of KS1 and KS3 is also low.

Interactions of rod-like particles on responsive elastic sheets

What are the physical laws of the mutual interactions of objects bound to cell membranes, such as various membrane proteins or elongated virus particles? To rationalise this, we here investigate by extensive computer simulations mutual interactions of rod-like particles adsorbed on the surface of responsive elastic two-dimensional sheets. Specifically, we quantify sheet deformations as a response to adhesion of such filamentous particles. We demonstrate that tip-to-tip contacts of rods are favoured for relatively soft sheets, while side-by-side contacts are preferred for stiffer elastic substrates. These attractive orientation-dependent substrate-mediated interactions between the rod-like particles on responsive sheets can drive their aggregation and self-assembly. The optimal orientation of the membrane-bound rods is established via responding to the elastic energy profiles created around the particles. We unveil the phase diagramme of attractive-repulsive rod-rod interactions in the plane of their separation and mutual orientation. Applications of our results to other systems featuring membrane-associated particles are also discussed.

Influence of external potentials on heterogeneous diffusion processes

In this paper we consider heterogeneous diffusion processes with the power-law dependence of the diffusion coefficient on the position and investigate the influence of external forces on the resulting anomalous diffusion. The heterogeneous diffusion processes can yield subdiffusion as well as superdiffusion, depending on the behavior of the diffusion coefficient. We assume that not only the diffusion coefficient but also the external force has a power-law dependence on the position. We obtain analytic expressions for the transition probability in two cases: when the power-law exponent in the external force is equal to 2η-1, where 2η is the power-law exponent in the dependence of the diffusion coefficient on the position, and when the external force has a linear dependence on the position. We found that the power-law exponent in the dependence of the mean square displacement on time does not depend on the external force; this force changes only the anomalous diffusion coefficient. In addition, the external force having the power-law exponent different from 2η-1 limits the time interval where the anomalous diffusion occurs. We expect that the results obtained in this paper may be relevant for a more complete understanding of anomalous diffusion processes.

Membrane Compartmentalization Reducing the Mobility of Lipids and Proteins within a Model Plasma Membrane

The cytoskeleton underlying cell membranes may influence the dynamic organization of proteins and lipids within the bilayer by immobilizing certain transmembrane (TM) proteins and forming corrals within the membrane. Here, we present coarse-grained resolution simulations of a biologically realistic membrane model of asymmetrically organized lipids and TM proteins. We determine the effects of a model of cytoskeletal immobilization of selected membrane proteins using long time scale coarse-grained molecular dynamics simulations. By introducing compartments with varying degrees of restraints within the membrane models, we are able to reveal how compartmentalization caused by cytoskeletal immobilization leads to reduced and anomalous diffusional mobility of both proteins and lipids. This in turn results in a reduced rate of protein dimerization within the membrane and of hopping of membrane proteins between compartments. These simulations provide a molecular realization of hierarchical models often invoked to explain single-molecule imaging studies of membrane proteins.

Asymptotic neutron scattering laws for anomalously diffusing quantum particles

The paper deals with a model-free approach to the analysis of quasielastic neutron scattering intensities from anomalously diffusing quantum particles. All quantities are inferred from the asymptotic form of their time-dependent mean square displacements which grow ∝t(α), with 0 ≤ α < 2. Confined diffusion (α = 0) is here explicitly included. We discuss in particular the intermediate scattering function for long times and the Fourier spectrum of the velocity autocorrelation function for small frequencies. Quantum effects enter in both cases through the general symmetry properties of quantum time correlation functions. It is shown that the fractional diffusion constant can be expressed by a Green-Kubo type relation involving the real part of the velocity autocorrelation function. The theory is exact in the diffusive regime and at moderate momentum transfers.

Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion

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.

Direct comparison of elastic incoherent neutron scattering experiments with molecular dynamics simulations of DMPC phase transitions

Neutron scattering techniques have been employed to investigate 1,2-dimyristoyl-sn -glycero-3-phosphocholine (DMPC) membranes in the form of multilamellar vesicles (MLVs) and deposited, stacked multilamellar-bilayers (MLBs), covering transitions from the gel to the liquid phase. Neutron diffraction was used to characterise the samples in terms of transition temperatures, whereas elastic incoherent neutron scattering (EINS) demonstrates that the dynamics on the sub-macromolecular length-scale and pico- to nano-second time-scale are correlated with the structural transitions through a discontinuity in the observed elastic intensities and the derived mean square displacements. Molecular dynamics simulations have been performed in parallel focussing on the length-, time- and temperature-scales of the neutron experiments. They correctly reproduce the structural features of the main gel-liquid phase transition. Particular emphasis is placed on the dynamical amplitudes derived from experiment and simulations. Two methods are used to analyse the experimental data and mean square displacements. They agree within a factor of 2 irrespective of the probed time-scale, i.e. the instrument utilized. Mean square displacements computed from simulations show a comparable level of agreement with the experimental values, albeit, the best match with the two methods varies for the two instruments. Consequently, experiments and simulations together give a consistent picture of the structural and dynamical aspects of the main lipid transition and provide a basis for future, theoretical modelling of dynamics and phase behaviour in membranes. The need for more detailed analytical models is pointed out by the remaining variation of the dynamical amplitudes derived in two different ways from experiments on the one hand and simulations on the other.

Ergodicity testing for anomalous diffusion: small sample statistics

The analysis of trajectories recorded in experiments often requires calculating time averages instead of ensemble averages. According to the Boltzmann hypothesis, they are equivalent only under the assumption of ergodicity. In this paper, we implement tools that allow to study ergodic properties. This analysis is conducted in two classes of anomalous diffusion processes: fractional Brownian motion and subordinated Ornstein-Uhlenbeck process. We show that only first of them is ergodic. We demonstrate this by applying rigorous statistical methods: mean square displacement, confidence intervals, and dynamical functional test. Our methodology is universal and can be implemented for analysis of many experimental data not only if a large sample is available but also when there are only few trajectories recorded.

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form D(x) ∼ D0|x|(α0) in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent α in time and space. In the case of annealed disorder, for which effectively α0 = α0(t), we show how the long time scaling of the ensemble mean squared displacement (MSD) and the amplitude variation of individual realizations of the time averaged MSD are affected by the disorder strength. For the case of quenched disorder, the long time behavior becomes effectively Brownian after a number of jumps between the domains of a stratified medium. In the latter situation, the averages are taken over both an ensemble of particles and different realizations of the disorder. As physical observables, we analyze in detail the ensemble and time averaged MSDs, the ergodicity breaking parameter, and higher order moments of the time averages.

Non-Brownian diffusion in lipid membranes: Experiments and simulations

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. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

The Role of the Membrane in the Structure and Biophysical Robustness of the Dengue Virion Envelope

The dengue virion is surrounded by an envelope of membrane proteins surrounding a lipid bilayer. We have combined the cryoelectron microscopy structures of the membrane proteins (PDB: 3J27) with a lipid bilayer whose composition is based on lipidomics data for insect cell membranes, to obtain a near-atomic resolution computational model of the envelope of the dengue virion. A coarse-grained molecular dynamics simulation on the microsecond timescale enables analysis of key biophysical properties of the dengue outer envelope. Properties analyzed include area per lipid values (for a spherical virion with a mixed lipid composition), bilayer thickness, and lipid diffusion coefficients. Despite the absence of cholesterol from the lipid bilayer, the virion exhibits biophysical robustness (slow lipid diffusion alongside stable bilayer thickness, virion diameter, and shape) that matches the cholesterol-rich membrane of influenza A, with similarly anomalous diffusion of lipids. Biophysical robustness of the envelope may confer resilience to environmental perturbations.

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The dengue virus envelope is a lipid bilayer plus an outer layer of membrane proteins

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The structures of the proteins plus lipidomics data were used to model the envelope

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Microsecond MD simulations revealed the dynamic behavior of the lipid bilayer

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Protein interactions confer “raft-like” robustness on a cholesterol-free membrane

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes

We investigate the diffusive and ergodic properties of massive and confined particles in a model disordered medium, in which the local diffusivity fluctuates in time while its mean has a power law dependence on the diffusion time.

Anomalous Dynamics of a Lipid Recognition Protein on a Membrane Surface

Pleckstrin homology (PH) domains are lipid-binding modules present in peripheral membrane proteins which interact with phosphatidyl-inositol phosphates (PIPs) in cell membranes. We use multiscale molecular dynamics simulations to characterize the localization and anomalous dynamics of the DAPP1 PH domain on the surface of a PIP-containing lipid bilayer. Both translational and rotational diffusion of the PH domain on the lipid membrane surface exhibit transient subdiffusion, with an exponent α ≈ 0.5 for times of less than 10 ns. In addition to a PIP₃ molecule at the canonical binding site of the PH domain, we observe additional PIP molecules in contact with the protein. Fluctuations in the number of PIPs associated with the PH domain exhibit 1/f noise. We suggest that the anomalous diffusion and long-term correlated interaction of the PH domain with the membrane may contribute to an enhanced probability of encounter with target complexes on cell membrane surfaces.