Lateral diffusion in an archipelago. Dependence on tracer size

Contribution of different macromolecules to the diffusion of a 40 nm particle in Escherichia coli

Due to the high concentration of proteins, nucleic acids, and other macromolecules, the bacterial cytoplasm is typically described as a crowded environment. However, the extent to which each of these macromolecules individually affects the mobility of macromolecular complexes, and how this depends on growth conditions, is presently unclear. In this study, we sought to quantify the crowding experienced by an exogenous 40 nm fluorescent particle in the cytoplasm of E. coli under different growth conditions. By performing single-particle tracking measurements in cells selectively depleted of DNA and/or mRNA, we determined the contribution to crowding of mRNA, DNA, and remaining cellular components, i.e., mostly proteins and ribosomes. To estimate this contribution to crowding, we quantified the difference of the particle's diffusion coefficient in conditions with and without those macromolecules. We found that the contributions of the three classes of components were of comparable magnitude, being largest in the case of proteins and ribosomes. We further found that the contributions of mRNA and DNA to crowding were significantly larger than expected based on their volumetric fractions alone. Finally, we found that the crowding contributions change only slightly with the growth conditions. These results reveal how various cellular components partake in crowding of the cytoplasm and the consequences this has for the mobility of large macromolecular complexes.

Dynamic structure of E. coli cytoplasm: supramolecular complexes and cell aging impact spatial distribution and mobility of proteins

Protein diffusion is a critical factor governing the functioning and organization of a cell's cytoplasm. In this study, we investigate the influence of (poly)ribosome distribution, cell aging, protein aggregation, and biomolecular condensate formation on protein mobility within the E. coli cytoplasm. We employ nanoscale single-molecule displacement mapping (SMdM) to determine the spatial distribution of the proteins and to meticulously track their diffusion. We show that the distribution of polysomes does not impact the lateral diffusion coefficients of proteins. However, the degradation of mRNA induced by rifampicin treatment leads to an increase in protein mobility within the cytoplasm. Additionally, we establish a significant correlation between cell aging, the asymmetric localization of protein aggregates and reduced diffusion coefficients at the cell poles. Notably, we observe variations in the hindrance of diffusion at the poles and the central nucleoid region for small and large proteins, and we reveal differences between the old and new pole of the cell. Collectively, our research highlights cellular processes and mechanisms responsible for spatially organizing the bacterial cytoplasm into domains with different structural features and apparent viscosity.

Location of dopamine in lipid bilayers and its relevance to neuromodulator function

Dopamine (DA) is a neurotransmitter that also acts as a neuromodulator, with both functions being essential to brain function. Here, we present the first experimental measurement of DA location in lipid bilayers using x-ray diffuse scattering, solid-state deuterium NMR, and electron paramagnetic resonance. We find that the association of DA with lipid headgroups as seen in electron density profiles leads to an increase of intermembrane repulsion most likely due to electrostatic charging. DA location in the lipid headgroup region also leads to an increase of the cross-sectional area per lipid without affecting the bending rigidity significantly. The order parameters measured by solid-state deuterium NMR decrease in the presence of DA for the acyl chains of PC and PS lipids, consistent with an increase in the area per lipid due to DA. Most importantly, these results support the hypothesis that three-dimensional diffusion of DA to target membranes could be followed by relatively more efficient two-dimensional diffusion to receptors within those membranes.

Ameboid cell migration through regular arrays of micropillars under confinement

Migrating cells often encounter a wide variety of topographic features-including the presence of obstacles-when navigating through crowded biological environments. Unraveling the impact of topography and crowding on the dynamics of cells is key to better understand many essential physiological processes such as the immune response. We study the impact of geometrical cues on ameboid migration of HL-60 cells differentiated into neutrophils. A microfluidic device is designed to track the cells in confining geometries between two parallel plates with distance h, in which identical micropillars are arranged in regular pillar forests with pillar spacing e. We observe that the cells are temporarily captured near pillars, with a mean contact time that is independent of h and e. By decreasing the vertical confinement h, we find that the cell velocity is not affected, while the persistence reduces; thus, cells are able to preserve their velocity when highly squeezed but lose the ability to control their direction of motion. At a given h, we show that by decreasing the pillar spacing e in the weak lateral confinement regime, the mean escape time of cells from effective local traps between neighboring pillars grows. This effect, together with the increase of cell-pillar contact frequency, leads to the reduction of diffusion constant D. By disentangling the contributions of these two effects on D in numerical simulations, we verify that the impact of cell-pillar contacts on cell diffusivity is more pronounced at smaller pillar spacing.

Rhamnolipid Biosurfactants—Possible Natural Anticancer Agents and Autophagy Inhibitors

Background/Aim: A number of biologically active substances were proved as an alternative to conventional anticancer medicines. The aim of the study is in vitro investigation of the anticancer activity of mono- and di-Rhamnolipids (RL-1 and RL-2) against human breast cancer. Additionally, the combination with Cisplatin was analyzed. Materials and Methods: Breast cell lines (MCF-10A, MCF-7 and MDA-MB-231) were treated with RLs and in combination with Cisplatin. The viability was analyzed using MTT assay, and investigation of autophagy was performed via acridine orange staining. Results: In contrast to the healthy cells, both tested cancer lines exhibited sensitivity to RLs treatment. This effect was accompanied by an influence on the autophagy-related acidic formation process. Only for the triple-negative breast cancer cell line (MDA-MB-231) the synergistic effect of the combined treatment (10 µM Cisplatin and 1 µg/mL RL-2) was observed. Conclusion: Based on studies on the reorganization of membrane models in the presence of RL and the data about a higher amount of lipid rafts in cancer cell membranes than in non-tumorigenic, we suggest a possible mechanism of membrane remodelling by formation of endosomes. Shortly, in order to have a synergistic effect, it is necessary to have Cisplatin andRL-2 as RL2 is a molecule inducingpositive membrane curvature.

Local details versus effective medium approximation: A study of diffusion in microfluidic random networks made from Voronoi tessellations

We measured the effective diffusion coefficient in regions of microfluidic networks of controlled geometry using the fluorescence recovery after photobleaching (FRAP) technique. The geometry of the networks was based on Voronoi tessellations, and had varying characteristic length scale and porosity. For a fixed network, FRAP experiments were performed in regions of increasing size. Our results indicate that the boundary of the bleached region, and in particular the cumulative area of the channels that connect the bleached region to the rest of the network, are important in the measured value of the effective diffusion coefficient. We found that the statistical geometrical variations between different regions of the network decrease with the size of the bleached region as a power law, meaning that the statistical error of effective medium approximations decrease with the size of the studied medium with no characteristic length scale.

Molecular transport in systems containing binding obstacles

We studied the movement of particles in crowded environments by means of extensive Monte Carlo simulations. The dynamic lattice liquid model was employed for this purpose. It is based on the cooperative movement concept and allows the study of systems at high densities. The cooperative model of molecular transport is assumed: the motion of all moving particles is highly correlated. The model is supposed to mimic lateral motion in a membrane and therefore the system is two-dimensional with moving objects and traps placed on a triangular lattice. In our study the interaction (binding) of traps with moving particles was assumed. The conditions in which subdiffusive motion appeared in the system were analysed. The influence of the strength of binding on the dynamic percolation threshold was also shown. The differences in the dynamics compared to systems with impenetrable obstacles and with systems without correlation in motion were presented and discussed. It was shown that in the case of correlated motion the influence of deep traps is similar to that of impenetrable obstacles. If the traps are shallow a recovery to normal diffusion was observed for longer time periods.

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.

Self-crowding of AMPA receptors in the excitatory postsynaptic density can effectuate anomalous receptor sub-diffusion

AMPA receptors (AMPARs) and their associations with auxiliary transmembrane proteins are bulky structures with large steric-exclusion volumes. Hence, self-crowding of AMPARs, depending on the local density, may affect their lateral diffusion in the postsynaptic membrane as well as in the highly crowded postsynaptic density (PSD) at excitatory synapses. Earlier theoretical studies considered only the roles of transmembrane obstacles and the AMPAR-binding submembranous scaffold proteins in shaping receptor diffusion within PSD. Using lattice model of diffusion, the present study investigates the additional impacts of self-crowding on the anomalousity and effective diffusion coefficient (D_eff) of AMPAR diffusion. A recursive algorithm for avoiding false self-blocking during diffusion simulation is also proposed. The findings suggest that high density of AMPARs in the obstacle-free membrane itself engenders strongly anomalous diffusion and severe decline in D_eff. Adding transmembrane obstacles to the membrane accentuates the anomalousity arising from self-crowding due to the reduced free diffusion space. Contrarily, enhanced AMPAR-scaffold binding, either through increase in binding strength or scaffold density or both, ameliorates the anomalousity resulting from self-crowding. However, binding has differential impacts on D_eff depending on the receptor density. Increase in binding causes consistent decrease in D_eff for low and moderate receptor density. For high density, binding increases D_eff as long as it reduces anomalousity associated with intense self-crowding. Given a sufficiently strong binding condition when diffusion acquires normal behavior, further increase in binding causes decrease in D_eff. Supporting earlier experimental observations are mentioned and implications of present findings to the experimental observations on AMPAR diffusion are also drawn.

The transmembrane AMPA receptors (AMPARs) prominently exhibit lateral diffusion in the postsynaptic membrane at excitatory synapses. Steric obstructions to AMPAR diffusion due to the crowd of other relatively static transmembrane proteins and binding of AMPARs to the submembranous scaffold proteins in the specialized region of postsynaptic density (PSD) are well known to retard receptor diffusion, which causes receptor trapping and accumulation within PSD. However, AMPARs are significantly bulky structures and may also obstruct their own diffusion paths in the presence of their high density. It is shown here that intense self-crowding of AMPARs may lead to highly obstructed and confined receptor diffusion even in the obstacle-free medium, and the presence of other obstacles further aggravates this effect. AMPAR-scaffold binding reduces confined diffusion arising from self-crowding and strong binding engenders normal diffusion even at high receptor density. However, it overall causes reduction in the effective diffusion coefficient of the receptor diffusion.

On the critical Casimir interaction between anisotropic inclusions on a membrane

Using a lattice model and a versatile thermodynamic integration scheme, we study the critical Casimir interactions between inclusions embedded in a two-dimensional critical binary mixtures. For single-domain inclusions we demonstrate that the interactions are very long range, and their magnitudes strongly depend on the affinity of the inclusions with the species in the binary mixtures, ranging from repulsive when two inclusions have opposing affinities to attractive when they have the same affinities. When one of the inclusions has no preference for either of the species, we find negligible critical Casimir interactions. For multiple-domain inclusions, mimicking the observations that membrane proteins often have several domains with varying affinities to the surrounding lipid species, the presence of domains with opposing affinities does not cancel the interactions altogether. Instead we can observe both attractive and repulsive interactions depending on their relative orientations. With increasing number of domains per inclusion, the range and magnitude of the effective interactions decrease in a similar fashion to those of electrostatic multipoles. Finally, clusters formed by multiple-domain inclusions can result in an effective affinity patterning due to the anisotropic character of the Casimir interactions between the building blocks.

Gradient-driven diffusion and pattern formation in crowded mixtures

Gradient-driven diffusion in crowded, multicomponent mixtures is a topic of high interest because of its role in biological processes such as transport in cell membranes. In partially phase-separated solutions, gradient-driven diffusion affects microstructure, which in turn affects diffusivity; a key question is how this complex coupling controls both transport and pattern formation. To examine these mechanisms, we study a two-dimensional multicomponent lattice gas model, where "tracer" molecules diffuse between a source and a sink separated by a solution of sticky "crowder" molecules that cluster to form dynamically evolving obstacles. In the high-temperature limit, crowders and tracers are miscible, and transport may be predicted analytically. At intermediate temperatures, crowders phase separate into clusters that drift toward the tracer sink. As a result, steady-state tracer diffusivity depends nonmonotonically on both temperature and crowder density, and we observe a variety of complex microstructures. In the low-temperature limit, crowders rapidly aggregate to form obstacles that are kinetically arrested; if crowder density is near the percolation threshold, resulting tracer diffusivity shows scaling behavior with the same scaling exponent as the random resistor network model. Though highly idealized, this simple model reveals fundamental mechanisms governing coupled gradient-driven diffusion, phase separation, and microstructural evolution in crowded mixtures.

Convergence of lateral dynamic measurements in the plasma membrane of live cells from single particle tracking and STED-FCS

Fluorescence correlation spectroscopy (FCS) in combination with the super-resolution imaging method STED (STED-FCS), and single-particle tracking (SPT) are able to directly probe the lateral dynamics of lipids and proteins in the plasma membrane of live cells at spatial scales much below the diffraction limit of conventional microscopy. However, a major disparity in interpretation of data from SPT and STED-FCS remains, namely the proposed existence of a very fast (unhindered) lateral diffusion coefficient, ⩾5 µm2 s-1, in the plasma membrane of live cells at very short length scales, ≈⩽ 100 nm, and time scales, ≈1-10 ms. This fast diffusion coefficient has been advocated in several high-speed SPT studies, for lipids and membrane proteins alike, but the equivalent has not been detected in STED-FCS measurements. Resolving this ambiguity is important because the assessment of membrane dynamics currently relies heavily on SPT for the determination of heterogeneous diffusion. A possible systematic error in this approach would thus have vast implications in this field. To address this, we have re-visited the analysis procedure for SPT data with an emphasis on the measurement errors and the effect that these errors have on the measurement outputs. We subsequently demonstrate that STED-FCS and SPT data, following careful consideration of the experimental errors of the SPT data, converge to a common interpretation which for the case of a diffusing phospholipid analogue in the plasma membrane of live mouse embryo fibroblasts results in an unhindered, intra-compartment, diffusion coefficient of  ≈0.7-1.0 µm2 s-1, and a compartment size of about 100-150 nm.

Simulation of Molecular Transport in Systems Containing Mobile Obstacles

In this paper, we investigate the movement of molecules in crowded environments with obstacles undergoing Brownian motion by means of extensive Monte Carlo simulations. Our investigations were performed using the dynamic lattice liquid model, which was based on the cooperative movement concept and allowed to mimic systems at high densities where the motion of all elements (obstacles as well as moving particles) were highly correlated. The crowded environments are modeled on a two-dimensional triangular lattice containing obstacles (particles whose mobility was significantly reduced) moving by a Brownian motion. The subdiffusive motion of both elements in the system was analyzed. It was shown that the percolation transition does not exist in such systems in spite of the cooperative character of the particles' motion. The reduction of the obstacle mobility leads to the longer caging of liquid particles by mobile obstacles.

Imaging approaches for analysis of cholesterol distribution and dynamics in the plasma membrane

Cholesterol is an important lipid component of the plasma membrane (PM) of mammalian cells, where it is involved in control of many physiological processes, such as endocytosis, cell migration, cell signalling and surface ruffling. In an attempt to explain these functions of cholesterol, several models have been put forward about cholesterol's lateral and transbilayer organization in the PM. In this article, we review imaging techniques developed over the last two decades for assessing the distribution and dynamics of cholesterol in the PM of mammalian cells. Particular focus is on fluorescence techniques to study the lateral and inter-leaflet distribution of suitable cholesterol analogues in the PM of living cells. We describe also several methods for determining lateral cholesterol dynamics in the PM including fluorescence recovery after photobleaching (FRAP), fluorescence correlation spectroscopy (FCS), single particle tracking (SPT) and spot variation FCS coupled to stimulated emission depletion (STED) microscopy. For proper interpretation of such measurements, we provide some background in probe photophysics and diffusion phenomena occurring in cell membranes. In particular, we show the equivalence of the reaction-diffusion approach, as used in FRAP and FCS, and continuous time random walk (CTRW) models, as often invoked in SPT studies. We also discuss mass spectrometry (MS) based imaging of cholesterol in the PM of fixed cells and compare this method with fluorescence imaging of sterols. We conclude that evidence from many experimental techniques converges towards a model of a homogeneous distribution of cholesterol with largely free and unhindered diffusion in both leaflets of the PM.

Explicit spatiotemporal simulation of receptor-G protein coupling in rod cell disk membranes

Dim-light vision is mediated by retinal rod cells. Rhodopsin (R), a G-protein-coupled receptor, switches to its active form (R(∗)) in response to absorbing a single photon and activates multiple copies of the G-protein transducin (G) that trigger further downstream reactions of the phototransduction cascade. The classical assumption is that R and G are uniformly distributed and freely diffusing on disk membranes. Recent experimental findings have challenged this view by showing specific R architectures, including RG precomplexes, nonuniform R density, specific R arrangements, and immobile fractions of R. Here, we derive a physical model that describes the first steps of the photoactivation cascade in spatiotemporal detail and single-molecule resolution. The model was implemented in the ReaDDy software for particle-based reaction-diffusion simulations. Detailed kinetic in vitro experiments are used to parametrize the reaction rates and diffusion constants of R and G. Particle diffusion and G activation are then studied under different conditions of R-R interaction. It is found that the classical free-diffusion model is consistent with the available kinetic data. The existence of precomplexes between inactive R and G is only consistent with the data if these precomplexes are weak, with much larger dissociation rates than suggested elsewhere. Microarchitectures of R, such as dimer racks, would effectively immobilize R but have little impact on the diffusivity of G and on the overall amplification of the cascade at the level of the G protein.

Interferometric Detection of Single Gold Nanoparticles Calibrated against TEM Size Distributions

Single nanoparticle analysis: An interferometric optical approach calibrates sizes of gold nanoparticles (AuNPs) from the interference intensities by calibrating their interferometric signals against the corresponding transmission electron microscopy measurements. This method is used to investigate whether size affects the diffusion behavior of AuNPs conjugated to supported lipid bilayer membranes and to multiplex the simultaneous detection of three different AuNP labels.

GPI-anchored proteins do not reside in ordered domains in the live cell plasma membrane

The organization of proteins and lipids in the plasma membrane has been the subject of a long-lasting debate. Membrane rafts of higher lipid chain order were proposed to mediate protein interactions, but have thus far not been directly observed. Here we use protein micropatterning combined with single-molecule tracking to put current models to the test: we rearranged lipid-anchored raft proteins (glycosylphosphatidylinositol(GPI)-anchored-mGFP) directly in the live cell plasma membrane and measured the effect on the local membrane environment. Intriguingly, this treatment does neither nucleate the formation of an ordered membrane phase nor result in any enrichment of nanoscopic-ordered domains within the micropatterned regions. In contrast, we find that immobilized mGFP-GPIs behave as inert obstacles to the diffusion of other membrane constituents without influencing their membrane environment over distances beyond their physical size. Our results indicate that phase partitioning is not a fundamental element of protein organization in the plasma membrane.

Hindered nanoparticle diffusion and void accessibility in a three-dimensional porous medium

The inherent pore-scale heterogeneity of many natural and synthetic porous materials can make it difficult to model and predict porous transport because the underlying microscopic processes are often poorly understood. Here we present the results of single-particle tracking experiments in which we followed the pore-scale diffusion of individual nanoparticles, deep within a three-dimensional porous material of moderate porosity. We observed significant hydrodynamic damping of particle motion at subpore length scales, resulting in heterogeneous and spatially dependent mobility. The accessibility of the void space was strongly dependent on particle size, and related to the heterogeneous hydrodynamics. Our results suggest that pore-scale diffusion is more heterogeneous and volume accessibility more limited than previously expected. The method demonstrated here will enable studies of a broad new class of materials including porous polymers of technological interest.

The viscoelastic properties of chromatin and the nucleoplasm revealed by scale-dependent protein mobility

The eukaryotic cell nucleus harbours the DNA genome that is organized in a dynamic chromatin network and embedded in a viscous crowded fluid. This environment directly affects enzymatic reactions and target search processes that access the DNA sequence information. However, its physical properties as a reaction medium are poorly understood. Here, we exploit mobility measurements of differently sized inert green fluorescent tracer proteins to characterize the viscoelastic properties of the nuclear interior of a living human cell. We find that it resembles a viscous fluid on small and large scales but appears viscoelastic on intermediate scales that change with protein size. Our results are consistent with simulations of diffusion through polymers and suggest that chromatin forms a random obstacle network rather than a self-similar structure with fixed fractal dimensions. By calculating how long molecules remember their previous position in dependence on their size, we evaluate how the nuclear environment affects search processes of chromatin targets.

Simulation tools for particle-based reaction-diffusion dynamics in continuous space

Particle-based reaction-diffusion algorithms facilitate the modeling of the diffusional motion of individual molecules and the reactions between them in cellular environments. A physically realistic model, depending on the system at hand and the questions asked, would require different levels of modeling detail such as particle diffusion, geometrical confinement, particle volume exclusion or particle-particle interaction potentials. Higher levels of detail usually correspond to increased number of parameters and higher computational cost. Certain systems however, require these investments to be modeled adequately. Here we present a review on the current field of particle-based reaction-diffusion software packages operating on continuous space. Four nested levels of modeling detail are identified that capture incrementing amount of detail. Their applicability to different biological questions is discussed, arching from straight diffusion simulations to sophisticated and expensive models that bridge towards coarse grained molecular dynamics.

Simulation of diffusion in a crowded environment

We performed extensive and systematic simulation studies of two-dimensional fluid motion in a complex crowded environment. In contrast to other studies we focused on cooperative phenomena that occurred if the motion of particles takes place in a dense crowded system, which can be considered as a crude model of a cellular membrane. Our main goal was to answer the following question: how do the fluid molecules move in an environment with a complex structure, taking into account the fact that motions of fluid molecules are highly correlated. The dynamic lattice liquid (DLL) model, which can work at the highest fluid density, was employed. Within the frame of the DLL model we considered cooperative motion of fluid particles in an environment that contained static obstacles. The dynamic properties of the system as a function of the concentration of obstacles were studied. The subdiffusive motion of particles was found in the crowded system. The influence of hydrodynamics on the motion was investigated via analysis of the displacement in closed cooperative loops. The simulation and the analysis emphasize the influence of the movement correlation between moving particles and obstacles.

Relevance and limitations of crowding, fractal, and polymer models to describe nuclear architecture

Chromosome architecture plays an essential role for all nuclear functions, and its physical description has attracted considerable interest over the last few years among the biophysics community. These researches at the frontiers of physics and biology have been stimulated by the demand for quantitative analysis of molecular biology experiments, which provide comprehensive data on chromosome folding, or of live cell imaging experiments that enable researchers to visualize selected chromosome loci in living or fixed cells. In this review our goal is to survey several nonmutually exclusive models that have emerged to describe the folding of DNA in the nucleus, the dynamics of proteins in the nucleoplasm, or the movements of chromosome loci. We focus on three classes of models, namely molecular crowding, fractal, and polymer models, draw comparisons, and discuss their merits and limitations in the context of chromosome structure and dynamics, or nuclear protein navigation in the nucleoplasm. Finally, we identify future challenges in the roadmap to a unified model of the nuclear environment.

Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas

Recently, we measured two anomalous diffusion (AD) parameters: the spatial and the temporal AD indices, called γ and α, respectively, by using spectroscopic pulse gradient field methods. We showed that γ quantifies pseudo-superdiffusion processes, while α quantifies subdiffusion processes. Here, we propose γ and α maps obtained in a controlled heterogeneous phantom, comprised of packed micro-beads in water and in excised human meningiomas. In few words, α maps represent the multi-scale spatial distribution of the disorder degree in the system, while γ maps are influenced by local internal gradients, thus highlighting the interface between compartments characterized by different magnetic susceptibility. γ maps were already obtained by means of AD stretched exponential imaging and α-type maps have been recently achieved for fixed rat brain with the aim of highlighting the fractal dimension of specific brain regions. However, to our knowledge, the maps representative of the spatial distribution of α and γ obtained on the same controlled sample and in the same excised tissue have never been compared. Moreover, we show here, for the first time, that α maps are representative of the spatial distribution of the disorder degree of the system. In a first phase, γ and α maps of controlled phantom characterized by an ordered and a disordered rearrangement of packed micro-beads of different sizes in water and by different magnetic susceptibility (Δχ) between beads and water were obtained. In a second phase, we investigated excised human meningiomas of different consistency. Results reported here, obtained at 9.4T, show that α and γ maps are characterized by a different image contrast. Indeed, unlike γ maps, α maps are insensible to (Δχ) and they are sensible to the disorder degree of the microstructural rearrangement. These observations strongly suggest that AD indices α and γ reflect some additional microstructural information which cannot be obtained using conventional diffusion methods based on Gaussian diffusion. Moreover, α and γ maps obtained in excised meningiomas seem to provide more microstructural details above those obtained with conventional DTI analysis, which could be used to improve the classification of meningiomas based on their consistency.

A fractal model for nuclear organization: current evidence and biological implications

Chromatin is a multiscale structure on which transcription, replication, recombination and repair of the genome occur. To fully understand any of these processes at the molecular level under physiological conditions, a clear picture of the polymorphic and dynamic organization of chromatin in the eukaryotic nucleus is required. Recent studies indicate that a fractal model of chromatin architecture is consistent with both the reaction-diffusion properties of chromatin interacting proteins and with structural data on chromatin interminglement. In this study, we provide a critical overview of the experimental evidence that support a fractal organization of chromatin. On this basis, we discuss the functional implications of a fractal chromatin model for biological processes and propose future experiments to probe chromatin organization further that should allow to strongly support or invalidate the fractal hypothesis.

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

Anomalous diffusion has been observed abundantly in the plasma membrane of biological cells, but the underlying mechanisms are still unclear. In general, it has not been possible to directly image the obstacles to diffusion in membranes, which are thought to be skeleton bound proteins, protein aggregates, and lipid domains, so the dynamics of diffusing particles is used to deduce the obstacle characteristics. We present a supported lipid bilayer system in which we characterized the anomalous diffusion of lipid molecules using single molecule tracking, while at the same time imaging the obstacles to diffusion with atomic force microscopy. To explain our experimental results, we performed lattice Monte Carlo simulations of tracer diffusion in the presence of the experimentally determined obstacle configurations. We correlate the observed anomalous diffusion with obstacle area fraction, fractal dimension, and correlation length. To accurately measure an anomalous diffusion exponent, we derived an expression to account for the time-averaging inherent to all single molecule tracking experiments. We show that the length of the single molecule trajectories is critical to the determination of the anomalous diffusion exponent. We further discuss our results in the context of confinement models and the generating stochastic process.

New insights into diffusion in 3D crowded media by Monte Carlo simulations: effect of size, mobility and spatial distribution of obstacles

Particle diffusion in crowded media was studied through Monte Carlo simulations in 3D obstructed lattices. Three particular aspects affecting the diffusion, not extensively treated in a three-dimensional geometry, were analysed: the relative particle-obstacle size, the relative particle-obstacle mobility and the way of having the obstacles distributed in the simulation space (randomly or uniformly). The results are interpreted in terms of the parameters that characterize the time dependence of the diffusion coefficient: the anomalous diffusion exponent (α), the crossover time from anomalous to normal diffusion regimes (τ) and the long time diffusion coefficient (D*). Simulation results indicate that there are a more anomalous diffusion (smaller α) and a lower long time diffusion coefficient (D*) when obstacle concentration increases, and that, for a given total excluded volume and immobile obstacles, the anomalous diffusion effect is less important for bigger size obstacles. However, for the case of mobile obstacles, this size effect is inverted yielding values that are in qualitatively good agreement with in vitro experiments of protein diffusion in crowded media. These results underline that the pattern of the spatial partitioning of the obstacle excluded volume is a factor to be considered together with the value of the excluded volume itself.

Two-dimensional continuum percolation threshold for diffusing particles of nonzero radius

Lateral diffusion in the plasma membrane is obstructed by proteins bound to the cytoskeleton. The most important parameter describing obstructed diffusion is the percolation threshold. The thresholds are well known for point tracers, but for tracers of nonzero radius, the threshold depends on the excluded area, not just the obstacle concentration. Here thresholds are obtained for circular obstacles on the continuum. Random obstacle configurations are generated by Brownian dynamics or Monte Carlo methods, the obstacles are immobilized, and the percolation threshold is obtained by solving a bond percolation problem on the Voronoi diagram of the obstacles. The percolation threshold is expressed as the diameter of the largest tracer that can cross a set of immobile obstacles at a prescribed number density. For random overlapping obstacles, the results agree with the known analytical solution quantitatively. When the obstacles are soft disks with a 1/r(12) repulsion, the percolating diameter is approximately 20% lower than for overlapping obstacles. A percolation model predicts that the threshold is highly sensitive to the tracer radius. To our knowledge, such a strong dependence has so far not been reported for the plasma membrane, suggesting that percolation is not the factor controlling lateral diffusion. A definitive experiment is proposed.

Photo- and bio-physical characterization of novel violet and near-infrared lipophilic fluorophores for neuronal tracing

Lipophilic fluorescent dyes have been used to trace neuronal connections because of their ability to diffuse laterally within nerve cell membranes. Given the hundreds to thousands of connections that a typical neuron makes with its neighbours, a diffusion-matched set of spectrally distinct dyes is desirable. To extend a set of these dyes to obtain six independent labels, we have characterized the properties of novel violet and near-infrared candidates. By combining two-photon and confocal microscopy all of these candidates can be imaged using a single Titanium Sapphire laser. Here we present measurements of the two-photon action cross-sections and diffusion properties of the dyes, using either the relative diffusion distance or fluorescence recovery after photobleaching techniques, and demonstrate six-colour neuronal tracing within the spinal cord and brain tissue.

Rheological properties of a two phase lipid monolayer at the air/water interface: effect of the composition of the mixture

Many biologically relevant monolayers show coexistence of discrete domains of a long-range ordered condensed phase dispersed in a continuous, disordered, liquid-expanded phase. In this work, we determined the viscous and elastic components of the compressibility modulus and the shear viscosity of monolayers exhibiting phase coexistence with the aim at elucidating the contribution of each phase to the observed monolayer mechanical properties. To this purpose, mixed monolayers with different proportions of distearoylphosphatidylcholine (DSPC) and dimyristoylphosphatidylcholine (DMPC) were prepared and their rheological properties were analyzed. The relationship between the phase diagram of the mixture at 10 mN m(-1) and the rheological properties was studied. We found that the monolayer shear viscosity is highly dependent on the presence of domains and on the domain density. In turn, the monolayer compressibility is only influenced by the presence of domains for high domain densities. For monolayers that look homogeneous on the micrometer scale (DSPC amount lower that 23 mol %), all the analyzed rheological properties remain similar to those observed for pure DMPC monolayers, indicating that in this proportion range the DSPC molecules contribute as DMPC to the surface rheology in spite of having hydrocarbon chains four carbons longer.

Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin

The nucleus of eukaryotes is organized into functional compartments, the two most prominent being heterochromatin and nucleoli. These structures are highly enriched in DNA, proteins or RNA, and thus thought to be crowded. In vitro, molecular crowding induces volume exclusion, hinders diffusion and enhances association, but whether these effects are relevant in vivo remains unclear. Here, we establish that volume exclusion and diffusive hindrance occur in dense nuclear compartments by probing the diffusive behaviour of inert fluorescent tracers in living cells. We also demonstrate that chromatin-interacting proteins remain transiently trapped in heterochromatin due to crowding induced enhanced affinity. The kinetic signatures of these crowding consequences allow us to derive a fractal model of chromatin organization, which explains why the dynamics of soluble nuclear proteins are affected independently of their size. This model further shows that the fractal architecture differs between heterochromatin and euchromatin, and predicts that chromatin proteins use different target-search strategies in the two compartments. We propose that fractal crowding is a fundamental principle of nuclear organization, particularly of heterochromatin maintenance.

Polymerized lipid bilayers on a solid substrate: morphologies and obstruction of lateral diffusion

Substrate supported planar lipid bilayers (SPBs) are versatile models of the biological membrane in biophysical studies and biomedical applications. We previously developed a methodology for generating SPBs composed of polymeric and fluid phospholipid bilayers by using a photopolymerizable diacetylene phospholipid (DiynePC). Polymeric bilayers could be generated with micropatterns by conventional photolithography, and the degree of polymerization could be controlled by modulating UV irradiation doses. After removing nonreacted monomers, fluid lipid membranes could be integrated with polymeric bilayers. Herein, we report on a quantitative study of the morphology of polymeric bilayer domains and their obstruction toward lateral diffusion of membrane-associated molecules. Atomic force microscopy (AFM) observations revealed that polymerized DiynePC bilayers were formed as nanometer-sized domains. The ratio of polymeric and fluid bilayers could be modulated quantitatively by changing the UV irradiation dose for photopolymerization. Lateral diffusion coefficients of lipid molecules in fluid bilayers were measured by fluorescence recovery after photobleaching (FRAP) and correlated with the amount of polymeric bilayer domains on the substrate. Controlled domain structures, lipid compositions, and lateral mobility in the model membranes should allow us to fabricate model membranes that mimic complex features of biological membranes with well-defined structures and physicochemical properties.

Preparation and characterization of glycoacrylate-based polymer-tethered lipid bilayers on benzophenone-modified substrates

Polymer-tethered lipid bilayers are promising models for biological membranes as they may provide a soft, lubricating environment with sufficient spacing between the substrate and bilayer for incorporating transmembrane proteins. We present such a system that uses a glycoacrylate-based telechelic lipopolymer in combination with a lipid analogue. Characterization of the mixed monolayers of lipopolymers and free lipids at the air-water interface is used to examine the molecular organization that dictates the final assembly properties. Isotherms indicate that the source of the dominating interactions, whether polymer interactions in the subphase or alkyl chain interactions, depends on both the tethering density and area per molecule. Moreover, a critical composition exists at which the alkyl chain interactions dominate the monolayer behavior regardless of the area per molecule. Isobaric creep and hysteresis experiments suggest that permanent states due to irreversible polymer-polymer interactions are not created as the monolayer is compressed. These data, combined with theoretical polymer predictions, are used to understand the organization of the monolayers at the air-water interface and, hence, the separation distance between the bottom of the bilayer and substrate in the water-swollen state of the final bilayer assembly. Atomic force microscopy is used to confirm that the measured separation distance of 11.2 nm is on the order of what would be predicted using a theoretical analysis for a representative 5 mol % lipopolymer-tethered bilayer. Next, the homogeneity of the final bilayer is probed at multiple scales. Fluorescence microscopy is used to demonstrate that homogeneous and continuous bilayers can be formed (within the optical resolution limit of 500 nm) with all polymer tethering densities used in this study. Atomic force microscopy studies demonstrate that homogeneity comparable to that of a solid-supported lipid bilayer can be achieved for a representative 5 mol % lipopolymer-tethered bilayer. Langmuir-Blodgett transfer conditions for depositing monolayers that can be used to create homogeneous, fluid bilayers are also discussed. Finally, the distal leaflet lateral mobility is measured using fluorescence recovery after photobleaching experiments and shown to be a function of the tethering density. A possible model for the mobility data is developed in which the tethered lipids in the proximal leaflet act as immobile lipid obstacles that couple to distal leaflet lipids.

A biological interpretation of transient anomalous subdiffusion. II. Reaction kinetics

Reaction kinetics in a cell or cell membrane is modeled in terms of the first passage time for a random walker at a random initial position to reach an immobile target site in the presence of a hierarchy of nonreactive binding sites. Monte Carlo calculations are carried out for the triangular, square, and cubic lattices. The mean capture time is expressed as the product of three factors: the analytical expression of Montroll for the capture time in a system with a single target and no binding sites; an exact expression for the mean escape time from the set of lattice points; and a correction factor for the number of targets present. The correction factor, obtained from Monte Carlo calculations, is between one and two. Trapping may contribute significantly to noise in reaction rates. The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior. The distribution is analyzed using probability generating functions; this analysis resolves the contributions of the different sources of randomness to the distribution of capture times. This analysis predicts the distribution function for a lattice with perfect mixing; deviations reflect imperfect mixing in an ordinary random walk.

Long-range nonanomalous diffusion of quantum dot-labeled aquaporin-1 water channels in the cell plasma membrane

Aquaporin-1 (AQP1) is an integral membrane protein that facilitates osmotic water transport across cell plasma membranes in epithelia and endothelia. AQP1 has no known specific interactions with cytoplasmic or membrane proteins, but its recovery in a detergent-insoluble membrane fraction has suggested possible raft association. We tracked the membrane diffusion of AQP1 molecules labeled with quantum dots at an engineered external epitope at frame rates up to 91 Hz and over times up to 6 min. In transfected COS-7 cells, >75% of AQP1 molecules diffused freely over approximately 7 mum in 5 min, with diffusion coefficient, D(1-3) approximately 9 x 10(-10) cm(2)/s. In MDCK cells, approximately 60% of AQP1 diffused freely, with D(1-3) approximately 3 x 10(-10) cm(2)/s. The determinants of AQP1 diffusion were investigated by measurements of AQP1 diffusion following skeletal disruption (latrunculin B), lipid/raft perturbations (cyclodextrin and sphingomyelinase), and bleb formation. We found that cytoskeletal disruption had no effect on AQP1 diffusion in the plasma membrane, but that diffusion was increased greater than fourfold in protein de-enriched blebs. Cholesterol depletion in MDCK cells greatly restricted AQP1 diffusion, consistent with the formation of a network of solid-like barriers in the membrane. These results establish the nature and determinants of AQP1 diffusion in cell plasma membranes and demonstrate long-range nonanomalous diffusion of AQP1, challenging the prevailing view of universally anomalous diffusion of integral membrane proteins, and providing evidence against the accumulation of AQP1 in lipid rafts.

Modeling 2D and 3D diffusion

Modeling obstructed diffusion is essential to the understanding of diffusion-mediated processes in the crowded cellular environment. Simple Monte Carlo techniques for modeling obstructed random walks are explained and related to Brownian dynamics and more complicated Monte Carlo methods. Random number generation is reviewed in the context of random walk simulations. Programming techniques and event-driven algorithms are discussed as ways to speed simulations.

Kinetics of bimolecular reactions in model bilayers and biological membranes. A critical review

The quantitative study of the probability of molecular encounters giving rise to a reaction in membranes is a challenging discipline. Model systems, model in the sense that they use model bilayers and model reactants, have been widely used for this purpose, but the methodologies employed for the analysis of the results obtained in experiments, and for experimental design, are so disparate that a concerned experimentalist has difficulty in deciding about the value of each approach. This review intends to examine the several approaches that can be found in the literature showing, when feasible, the weakness, strengths and limits of application of each of them. There is not, so far, a full experimental validation of the most promising theories for the analysis of reactions in two dimensions, what leaves open a large field for new research. The major challenge resides in the time range in which the processes take place, but the possibilities of the existing techniques for these studies are far from exhausted. We review also the attempts of several authors to quantitatively analyze the kinetics of reactions in biological membranes. Especially in this field, the recently developed microspectroscopies enclose a still unexplored potential.

Dynamics and diffusion in photosynthetic membranes from rhodospirillum photometricum

Photosynthetic organisms drive their metabolism by converting light energy into an electrochemical gradient with high efficiency. This conversion depends on the diffusion of quinones within the membrane. In purple photosynthetic bacteria, quinones reduced by the reaction center (RC) diffuse to the cytochrome bc(1) complex and then return once reoxidized to the RC. In Rhodospirillum photometricum the RC-containing core complexes are found in a disordered molecular environment, with fixed light-harvesting complex/core complex ratio but without a fixed architecture, whereas additional light-harvesting complexes synthesized under low-light conditions pack into large paracrystalline antenna domains. Here, we have analyzed, using time-lapse atomic force microscopy, the dynamics of the protein complexes in the different membrane domains and find that the disordered regions are dynamic whereas ordered antennae domains are static. Based on our observations we propose, and analyze using Monte Carlo simulations, a model for quinone diffusion in photosynthetic membranes. We show that the formation of large static antennae domains may represent a strategy for increasing electron transfer rates between distant complexes within the membrane and thus be important for photosynthetic efficiency.

The influence of protein-protein interactions on the organization of proteins within thylakoid membranes

The influence of attractive protein-protein interactions on the organization of photosynthetic proteins within the thylakoid membrane was investigated. Protein-protein interactions were simulated using Monte Carlo techniques and the influence of different interaction energies was examined. It was found that weak interactions led to protein clusters whereas strong interactions led to ramified chains. An optimum curve for the relationship between interaction energy and the number of contact sites emerged. With increasing particle densities the effect decreased. In a mixture of interacting and noninteracting particles the distance between the noninteracting particles was increased and there seemed to be much more free space around them. In thylakoids, this could lead to a more homogeneous distribution of the noninteracting but rate-limiting cytochrome bf complexes. Due to the increased free space between cytochrome bf, obstruction of binding sites--occurring unavoidably in a random distribution--may be drastically reduced. Furthermore, protein-protein interactions in thylakoids may lead to a decrease in plastoquinone diffusion.

Membrane lateral mobility obstructed by polymer-tethered lipids studied at the single molecule level

Obstructed long-range lateral diffusion of phospholipids (TRITC-DHPE) and membrane proteins (bacteriorhodopsin) in a planar polymer-tethered 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine bilayer is studied using wide-field single molecule fluorescence microscopy. The obstacles are well-controlled concentrations of hydrophobic lipid-mimicking dioctadecylamine moieties in the polymer-exposed monolayer of the model membrane. Diffusion of both types of tracer molecules is well described by a percolating system with different percolation thresholds for lipids and proteins. Data analysis using a free area model of obstructed lipid diffusion indicates that phospholipids and tethered lipids interact via hard-core repulsion. A comparison to Monte Carlo lattice calculations reveals that tethered lipids act as immobile obstacles, are randomly distributed, and do not self-assemble into large-scale aggregates for low to moderate tethering concentrations. A procedure is presented to identify anomalous subdiffusion from tracking data at a single time lag. From the analysis of the cumulative distribution function of the square displacements, it was found that TRITC-DHPE and W80i show normal diffusion at lower concentrations of tethered lipids and anomalous diffusion at higher ones. This study may help improve our understanding of how lipids and proteins in biomembranes may be obstructed by very small obstacles comprising only one or very few molecules.

Diffusion in two-component lipid membranes--a fluorescence correlation spectroscopy and monte carlo simulation study

Using fluorescence correlation spectroscopy, calorimetry, and Monte Carlo simulations, we studied diffusion processes in two-component membranes close to the chain melting transition. The aim is to describe complex diffusion behavior in lipid systems in which gel and fluid domains coexist. Diffusion processes in gel membranes are significantly slower than in fluid membranes. Diffusion processes in mixed phase regions are therefore expected to be complex. Due to statistical fluctuations the gel-fluid domain patterns are not uniform in space and time. No models for such diffusion processes are available. In this article, which is both experimental and theoretical, we investigated the diffusion in DMPC-DSPC lipid mixtures as a function of temperature and composition. We then modeled the fluorescence correlation spectroscopy experiment using Monte Carlo simulations to analyze the diffusion process. It is shown that the simulations yield a very good description of the experimental diffusion processes, and that predicted autocorrelation profiles are superimposable with the experimental curves. We believe that this study adds to the discussion on the physical nature of rafts found in biomembranes.

Lateral diffusion of PEG-Lipid in magnetically aligned bicelles measured using stimulated echo pulsed field gradient 1H NMR

Lateral diffusion measurements of PEG-lipid incorporated into magnetically aligned bicelles are demonstrated using stimulated echo (STE) pulsed field gradient (PFG) proton (1H) nuclear magnetic resonance (NMR) spectroscopy. Bicelles were composed of dimyristoyl phosphatidylcholine (DMPC) plus dihexanoyl phosphatidylcholine (DHPC) (q = DMPC/DHPC molar ratio = 4.5) plus 1 mol % (relative to DMPC) dimyristoyl phosphatidylethanolamine-N-[methoxy(polyethylene glycol)-2000] (DMPE-PEG 2000) at 25 wt % lipid. 1H NMR STE spectra of perpendicular aligned bicelles contained only resonances assigned to residual HDO and to overlapping contributions from a DMPE-PEG 2000 ethoxy headgroup plus DHPC choline methyl protons. Decay of the latter's STE intensity in the STE PFG 1H NMR experiment (g(z) = 244 G cm(-1)) yielded a DMPE-PEG 2000 (1 mol %, 35 degrees C) lateral diffusion coefficient D = 1.35 x 10(-11) m2 s(-1). Hence, below the "mushroom-to-brush" transition, DMPE-PEG 2000 lateral diffusion is dictated by its DMPE hydrophobic anchor. D was independent of the diffusion time, indicating unrestricted lateral diffusion over root mean-square diffusion distances of microns, supporting the "perforated lamellae" model of bicelle structure under these conditions. Overall, the results demonstrate the feasibility of lateral diffusion measurements in magnetically aligned bicelles using the STE PFG NMR technique.