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

Adjusting the physico-chemical properties of collagen scaffolds to accommodate primary osteoblasts and endothelial cells

Collagen-based biomaterials are used widely as tissue engineering scaffolds because of their excellent bioactivity and their similarity to the natural ECM. The regeneration of healthy bone tissue requires simultaneous support for both osteoblasts and, where angiogenesis is intended, endothelial cells. Hence it is important to tailor carefully the biochemical and structural characteristics of the scaffold to suit the needs of each cell type. This work describes for the first time a systematic study to gain insight into the cell type-specific response of primary human osteoblast (hOBs) and human dermal microvascular endothelial cells (HDMECs) to insoluble collagen-based biomaterials. The behaviour was evaluated on both 2D films and 3D scaffolds, produced using freeze-drying. The collagen was cross-linked at various EDC/NHS concentrations and mono-cultured with hOBs and HDMECs to assess the effect of architectural features and scaffold stabilization on cell behaviour. It was observed that 3D scaffolds cross-linked at 30% of the standard conditions in literature offered an optimal combination of mechanical stiffness and cellular response for both cell types, although endothelial cells were more sensitive to the degree of cross-linking than hOBs. Architectural features have a time-dependent impact on the cell migration profile, with alignment being the most influential parameter overall.

Diffusion of DNA-Binding Species in the Nucleus: A Transient Anomalous Subdiffusion Model

Single-particle tracking experiments have measured escape times of DNA-binding species diffusing in living cells: CRISPR-Cas9, TetR, and LacI. The observed distribution is a truncated power law. Working backward from the experimental results, the observed distribution appears inconsistent with a Gaussian distribution of binding energies. Working forward, the observed distribution leads to transient anomalous subdiffusion, in which diffusion is anomalous at short times and normal at long times, here only mildly anomalous. Monte Carlo simulations are used to characterize the time-dependent diffusion coefficient D(t) in terms of the anomalous exponent α, the crossover time tcross, and the limits D(0) and D(∞) and to relate these quantities to the escape time distribution. The simplest interpretations identify the escape time as the actual binding time to DNA or the period of one-dimensional diffusion on DNA in the standard model combining one-dimensional and three-dimensional search, but a more complicated interpretation may be required. The model has several implications for cell biophysics. 1) The initial anomalous regime represents the search of the DNA-binding species for its target DNA sequence. 2) Non-target DNA sites have a significant effect on search kinetics. False positives in bioinformatic searches of the genome are potentially rate-determining in vivo. For simple binding, the search would be speeded if false-positive sequences were eliminated from the genome. 3) Both binding and obstruction affect diffusion. Obstruction ought to be measured directly, using as the primary probe the DNA-binding species with the binding site inactivated and eGFP as a calibration standard among laboratories and cell types. 4) Overexpression of the DNA-binding species reduces anomalous subdiffusion because the deepest binding sites are occupied and unavailable. 5) The model provides a coarse-grained phenomenological description of diffusion of a DNA-binding species, useful in larger-scale modeling of kinetics, FCS, and FRAP.

Scale and structure dependent solute diffusivity within microporous tissue engineering scaffolds

Diffusion of nutrients to cells cultured within three-dimensional scaffolds is fundamental for cell survival during development of the tissue construct, when no vasculature is present to aid transport. Significant efforts have been made to characterize the effect of structure on solute diffusivity in nanoporous hydrogels, yet a similar thorough characterization has not been attempted for microporous scaffolds. Here, we make use of freeze-dried collagen scaffolds, possessing pore sizes in the range 150-250 μm and isotropic or aligned morphology, to study the diffusivity of fluorescent dextran molecules. Fluorescence recovery after photobleaching is used to measure the self diffusivity of the solutes within single pores, while Fickian diffusion over scales larger than the pore size is studied by assessing the solute concentration profile within the materials over time. We show that, not only do the morphological parameters of the scaffolds significantly affect the diffusivity of the solutes, but also that the assessment of such diffusivity depends on the length scale of diffusion of the molecules under investigation, with the resulting diffusion coefficients being differently affected by the scaffold structure. The results provided can guide the design of scaffolds with tailored diffusivity and nutrient concentration profiles.

MicroCT analysis of connectivity in porous structures: optimizing data acquisition and analytical methods in the context of tissue engineering

Micro-computed X-ray tomography (MicroCT) is one of the most powerful techniques available for the three-dimensional characterization of complex multi-phase or porous microarchitectures. The imaging and analysis of porous networks are of particular interest in tissue engineering due to the ability to predict various large-scale cellular phenomena through the micro-scale characterization of the structure. However, optimizing the parameters for MicroCT data capture and analyses requires a careful balance of feature resolution and computational constraints while ensuring that a structurally representative section is imaged and analysed. In this work, artificial datasets were used to evaluate the validity of current analytical methods by considering the effect of noise and pixel size arising from the data capture, and intrinsic structural anisotropy and heterogeneity. A novel ‘segmented percolation method’ was developed to exclude the effect of anomalous, non-representative features within the datasets, allowing for scale-invariant structural parameters to be obtained consistently and without manual intervention for the first time. Finally, an in-depth assessment of the imaging and analytical procedures are presented by considering percolation events such as micro-particle filtration and cell sieving within the context of tissue engineering. Along with the novel guidelines established for general pixel size selection for MicroCT, we also report our determination of 3 μm as the definitive pixel size for use in analysing connectivity for tissue engineering applications.

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.

Self-Spreading of Phospholipid Bilayer in a Patterned Framework of Polymeric Bilayer

Phospholipid bilayers spontaneously spread on a hydrophilic substrate such as glass in aqueous solution due to the energetic gain of surface wetting. This process (self-spreading) was utilized to form a patterned model biological membrane containing reconstituted membrane proteins. A mechanically stable framework of a polymerized lipid bilayer was first generated by the lithographic polymerization of a diacetylene phospholipid. Then, natural lipid membranes (fluid bilayers) were introduced into the channels between polymeric bilayers by the self-spreading from a phospholipid reservoir. The spreading velocity could be fitted into a slope of -0.5 in a double logarithmic plot versus time due to the balance between the spreading force and resistive drag. The preformed polymeric bilayer accelerated the spreading by the energetic gain of covering hydrophobic edges with a lipid bilayer. At the same time, the domains of the polymeric bilayer obstructed spreading, and the spreading velocity linearly decreased with their fractional coverage. Above the critical coverage of ca. 50%, self-spreading was completely blocked (percolation threshold) and the fluid bilayer was confined in the polymer-free regions. Nonspecific adsorption of lipids onto the surface of polymeric bilayers was negligible, which enabled a heightened signal-to-background ratio in the reconstitution and observation of membrane proteins. Self-spread bilayers had a higher density of lipids than those formed by the spontaneous rupture of vesicles (vesicle fusion), presumably due to the continual supply of lipid molecules from the reservoir. These features give the self-spreading important advantages for preparing patterned model membranes with reconstituted membrane proteins.

Oxysterol Binding Protein: Tether, Transporter… and Flux Capacitor?

The oxysterol binding protein (OSBP) is a storied protein in organelle biology. Its early roles include acting as a membrane contact site (MCS) tether as well as a lipid antiporter. A surprising new function for OSBP in MCS dynamics has now been uncovered in a recent study by Jamecna et al. (Dev. Cell 2019;49:220-234).

Anomalous transport in the soft-sphere Lorentz model

The sensitivity of anomalous transport in crowded media to the form of the inter-particle interactions is investigated through computer simulations. We extend the highly simplified Lorentz model towards realistic natural systems by modeling the interactions between the tracer and the obstacles with a smooth potential. We find that the anomalous transport at the critical point happens to be governed by the same universal exponent as for hard exclusion interactions, although the mechanism of how narrow channels are probed is rather different. The scaling behavior of simulations close to the critical point confirm this exponent. Our result indicates that the simple Lorentz model may be applicable to describing the fundamental properties of long-range transport in real crowded environments.

Percolation analysis for estimating the maximum size of particles passing through nanosphere membranes

Percolation theory can be used to study the flow-related properties of various porous systems. In particular, recently developed membranes from silica nanoparticles with surface grafted polymer brushes represent a quintessential hard-sphere soft-shell system for which fluid-flow behavior can be illuminated via a percolation framework. However, a critical parameter in membrane design involves the maximum pass-through size of particles. While percolation theory considers path connectedness of a system, little explicit consideration is given to the size of the paths that traverse the space. This paper employs a hard-sphere soft-shell percolation model to investigate maximum particle pass-through size of membranes. A pixelated (as opposed to continuous) representation of the geometry is created, and combined with readily available homology software to analyze percolation behavior. The model is validated against previously published results. For a given sphere volume fraction, the maximum diameter of a percolating path is determined by applying iterative dilations to the spheres until the percolation threshold is reached. A simple approximate relationship between maximum particle size and sphere volume fraction is derived for application to membrane design. Experimental particle cutoff size results for the polymer modified silica nanoparticle membranes were used as a partial verification of the model created in this paper. The presence of a distribution of sphere sizes (naturally created by the manufacturing process) is found to have negligible effect, compared to results for a single sphere size.

Porous Structure of Silica Colloidal Crystals

We prepared silica colloidal crystals with different pore sizes using isothermal heating evaporation-induced self-assembly in quantities suitable for nitrogen porosimetry and studied their porous structure. We observed pores of two types in agreement with the description of silica colloidal crystals as face-centered cubic packed structures containing octahedral and tetrahedral voids. We calculated the sizes of these pores using the Derjaguin-Broekhoff-de Boer theory of capillary condensation for spherical pores. We also described the pore geometry mathematically and showed that the octahedral pore radii measured experimentally matches closely the radii of the spheres of the same volume. In the case of the tetrahedral pores, the proposed approach underestimated the pore radius by ca. 40%. Overall, this simple geometrical description provides a good representation of the porous system in silica colloidal crystals.

Non-universality of the dynamic exponent in two-dimensional random media

The diffusion of solutes in two-dimensional random media is important in diverse physical situations including the dynamics of proteins in crowded cell membranes and the adsorption on nano-structured substrates. It has generally been thought that the diffusion constant, D, should display universal behavior near the percolation threshold, i.e., D ~ (ϕ − ϕ_c)^μ, where ϕ is the area fraction of the matrix, ϕ_c is the value of ϕ at the percolation threshold, and μ is the dynamic exponent. The universality of μ is important because it implies that very different processes, such as protein diffusion in membranes and the electrical conductivity in two-dimensional networks, obey similar underlying physical principles. In this work we demonstrate, using computer simulations on a model system, that the exponent μ is not universal, but depends on the microscopic nature of the dynamics. We consider a hard disc that moves via random walk in a matrix of fixed hard discs and show that μ depends on the maximum possible displacement Δ of the mobile hard disc, ranging from 1.31 at Δ ≤ 0.1 to 2.06 for relatively large values of Δ. We also show that this behavior arises from a power-law singularity in the distribution of transition rates due to a failure of the local equilibrium approximation. The non-universal value of μ obeys the prediction of the renormalization group theory. Our simulations do not, however, exclude the possibility that the non-universal values of μ might be a crossover between two different limiting values at very large and small values of Δ. The results allow one to rationalize experiments on diffusion in two-dimensional systems.

Diffusion of Proteins across Silica Colloidal Crystals

We studied the diffusion of three model proteins, lysozyme (Lz), bovine hemoglobin (BHb), and bovine serum albumin (BSA), normal to the (111) plane of sintered silica colloidal crystals with three different pore "radii" (7.5, 19, and 27 nm). We demonstrated that these colloidal crystals exhibit size selectivity when the nanopores are sufficiently small (7.5 and 19 nm). Because these nanopores are still larger than the diffusing proteins, the observed size selectivity can be attributed to the tortuosity of the colloidal nanopores. Larger (27 nm) nanopores led to higher transport rates but at the cost of selectivity. In addition to the size selectivity, we also demonstrated that 19 nm nanopores possess shape selectivity for the proteins of comparable molecular weights. We showed that the high temperature sintering required for the preparation of sintered colloidal crystals reduces the extent of interactions between the proteins and the nanopore surface, which appear to play a minor role in the diffusion, and that transport selectivity is decided solely by protein size and shape. Taken together, our observations suggest that sintered silica colloidal crystals constitute promising nanoporous membranes for protein separations, with easily controllable pore size, size and shape selectivity, and minimal surface fouling.

Determination of the Membrane Environment of CD59 in Living Cells

The organization and dynamics of proteins and lipids in the plasma membrane, and their role in membrane functionality, have been subject of a long-lasting debate. Specifically, it is unclear to what extent membrane proteins are affected by their immediate lipid environment and vice versa. Studies on model membranes and plasma membrane vesicles indicated preferences of proteins for lipid phases characterized by different acyl chain order; however, whether such phases do indeed exist in live cells is still not known. Here, we refine a previously developed micropatterning approach combined with single molecule tracking to quantify the influence of the glycosylphosphatidylinositol-anchored (GPI-anchored) protein CD59 on its molecular environment directly in the live cell plasma membrane. We find that locally enriched and immobilized CD59 presents obstacles to the diffusion of fluorescently labeled lipids with a different phase-partitioning behavior independent of cell cholesterol levels and type of lipid. Our results give no evidence for either specific binding of the lipids to CD59 or the existence of nanoscopic ordered membrane regions associated with CD59.

Subdiffusion in Membrane Permeation of Small Molecules

Within the solubility-diffusion model of passive membrane permeation of small molecules, translocation of the permeant across the biological membrane is traditionally assumed to obey the Smoluchowski diffusion equation, which is germane for classical diffusion on an inhomogeneous free-energy and diffusivity landscape. This equation, however, cannot accommodate subdiffusive regimes, which have long been recognized in lipid bilayer dynamics, notably in the lateral diffusion of individual lipids. Through extensive biased and unbiased molecular dynamics simulations, we show that one-dimensional translocation of methanol across a pure lipid membrane remains subdiffusive on timescales approaching typical permeation times. Analysis of permeant motion within the lipid bilayer reveals that, in the absence of a net force, the mean squared displacement depends on time as t0.7, in stark contrast with the conventional model, which assumes a strictly linear dependence. We further show that an alternate model using a fractional-derivative generalization of the Smoluchowski equation provides a rigorous framework for describing the motion of the permeant molecule on the pico- to nanosecond timescale. The observed subdiffusive behavior appears to emerge from a crossover between small-scale rattling of the permeant around its present position in the membrane and larger-scale displacements precipitated by the formation of transient voids.

Protein Crowding within the Postsynaptic Density Can Impede the Escape of Membrane Proteins

Mechanisms regulating lateral diffusion and positioning of glutamate receptors within the postsynaptic density (PSD) determine excitatory synaptic strength. Scaffold proteins in the PSD are abundant receptor binding partners, yet electron microscopy suggests that the PSD is highly crowded, potentially restricting the diffusion of receptors regardless of binding. However, the contribution of macromolecular crowding to receptor retention remains poorly understood. We combined experimental and computational approaches to test the effect of synaptic crowding on receptor movement and positioning in Sprague Dawley rat hippocampal neurons. We modeled AMPA receptor diffusion in synapses where the distribution of scaffold proteins was determined from photoactivated localization microscopy experiments, and receptor-scaffold association and dissociation rates were adjusted to fit single-molecule tracking and fluorescence recovery measurements. Simulations predicted that variation of receptor size strongly influences the fractional synaptic area the receptor may traverse, and the proportion that may exchange in and out of the synapse. To test the model experimentally, we designed a set of novel transmembrane (TM) probes. A single-pass TM protein with one PDZ binding motif concentrated in the synapse as do AMPARs yet was more mobile there than the much larger AMPAR. Furthermore, either the single binding motif or an increase in cytoplasmic bulk through addition of a single GFP slowed synaptic movement of a small TM protein. These results suggest that both crowding and binding limit escape of AMPARs from the synapse. Moreover, tight protein packing within the PSD may modulate the synaptic dwell time of many TM proteins important for synaptic function.

SIGNIFICANCE STATEMENT

Small alterations to the distribution within synapses of key transmembrane proteins, such as receptors, can dramatically change synaptic strength. Indeed, many diseases are thought to unbalance neural circuit function in this manner. Processes that regulate this in healthy synapses are unclear, however. By combining computer simulations with imaging methods that examined protein dynamics at multiple scales in space and time, we showed that both steric effects and protein-protein binding each regulate the mobility of receptors in the synapse. Our findings extend our knowledge of the synapse as a crowded environment that counteracts molecular diffusion, and support the idea that both molecular collisions and biochemical binding can be involved in the regulation of neural circuit performance.

Monte Carlo simulations of protein micropatterning in biomembranes: effects of immobile sticky obstacles

Single molecule trajectories of lipids and proteins can yield valuable information about the nanoscopic organization of the plasma membrane itself. The interpretation of such trajectories, however, is complicated, as the mobility of molecules can be affected by the presence of immobile obstacles, and the transient binding of the tracers to these obstacles. We have previously developed a micropatterning approach that allows for immobilizing a plasma membrane protein and probing the diffusional behavior of a putative interaction partner in living cells. Here, we provide guidelines on how this micropatterning approach can be extended to quantify interaction parameters between plasma membrane constituents in their natural environment. We simulated a patterned membrane system and evaluated the effect of different surface densities of patterned immobile obstacles on the relative mobility as well as the surface density of diffusing tracers. In the case of inert obstacles, the size of the obstacle can be assessed from its surface density at the percolation threshold, which in turn can be extracted from the diffusion behavior of the tracer. For sticky obstacles, two-dimensional dissociation constants can be determined from the tracer diffusion or surface density.

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.

Estimation of the diffusion constant from intermittent trajectories with variable position uncertainties

The movement of a particle described by Brownian motion is quantified by a single parameter, D, the diffusion constant. The estimation of D from a discrete sequence of noisy observations is a fundamental problem in biological single-particle tracking experiments since it can provide information on the environment and/or the state of the particle itself via the hydrodynamic radius. Here, we present a method to estimate D that takes into account several effects that occur in practice, important for the correct estimation of D, and that have hitherto not been combined together for an estimation of D. These effects are motion blur from the finite integration time of the camera, intermittent trajectories, and time-dependent localization uncertainty. Our estimation procedure, a maximum-likelihood estimation with an information-based confidence interval, follows directly from the likelihood expression for a discretely observed Brownian trajectory that explicitly includes these effects. We begin with the formulation of the likelihood expression and then present three methods to find the exact solution. Each method has its own advantages in either computational robustness, theoretical insight, or the estimation of hidden variables. The Fisher information for this likelihood distribution is calculated and analyzed to show that localization uncertainties impose a lower bound on the estimation of D. Confidence intervals are established and then used to evaluate our estimator on simulated data with experimentally relevant camera effects to demonstrate the benefit of incorporating variable localization errors.

Parameterizing the Transport Pathways for Cell Invasion in Complex Scaffold Architectures

Interconnecting pathways through porous tissue engineering scaffolds play a vital role in determining nutrient supply, cell invasion, and tissue ingrowth. However, the global use of the term “interconnectivity” often fails to describe the transport characteristics of these pathways, giving no clear indication of their potential to support tissue synthesis. This article uses new experimental data to provide a critical analysis of reported methods for the description of scaffold transport pathways, ranging from qualitative image analysis to thorough structural parameterization using X-ray Micro-Computed Tomography. In the collagen scaffolds tested in this study, it was found that the proportion of pore space perceived to be accessible dramatically changed depending on the chosen method of analysis. Measurements of % interconnectivity as defined in this manner varied as a function of direction and connection size, and also showed a dependence on measurement length scale. As an alternative, a method for transport pathway parameterization was investigated, using percolation theory to calculate the diameter of the largest sphere that can travel to infinite distance through a scaffold in a specified direction. As proof of principle, this approach was used to investigate the invasion behavior of primary fibroblasts in response to independent changes in pore wall alignment and pore space accessibility, parameterized using the percolation diameter. The result was that both properties played a distinct role in determining fibroblast invasion efficiency. This example therefore demonstrates the potential of the percolation diameter as a method of transport pathway parameterization, to provide key structural criteria for application-based scaffold design.

Splitting of the Universality Class of Anomalous Transport in Crowded Media

We investigate the emergence of subdiffusive transport by obstruction in continuum models for molecular crowding. While the underlying percolation transition for the accessible space displays universal behavior, the dynamic properties depend in a subtle nonuniversal way on the transport through narrow channels. At the same time, the different universality classes are robust with respect to introducing correlations in the obstacle matrix as we demonstrate for quenched hard-sphere liquids as underlying structures. Our results confirm that the microscopic dynamics can dominate the relaxational behavior even at long times, in striking contrast to glassy dynamics.

Effect of Ceramic Scaffold Architectural Parameters on Biological Response

Numerous studies have focused on the optimization of ceramic architectures to fulfill a variety of scaffold functional requirements and improve biological response. Conventional fabrication techniques, however, do not allow for the production of geometrically controlled, reproducible structures and often fail to allow the independent variation of individual geometric parameters. Current developments in additive manufacturing technologies suggest that 3D printing will allow a more controlled and systematic exploration of scaffold architectures. This more direct translation of design into structure requires a pipeline for design-driven optimization. A theoretical framework for systematic design and evaluation of architectural parameters on biological response is presented. Four levels of architecture are considered, namely (1) surface topography, (2) pore size and geometry, (3) porous networks, and (4) macroscopic pore arrangement, including the potential for spatially varied architectures. Studies exploring the effect of various parameters within these levels are reviewed. This framework will hopefully allow uncovering of new relationships between architecture and biological response in a more systematic way as well as inform future refinement of fabrication techniques to fulfill architectural necessities with a consideration of biological implications.

A Highly Sensitive and Reliable Strain Sensor Using a Hierarchical 3D and Ordered Network of Carbon Nanotubes

A 3D network of single-walled carbon nanotubes embedded in poly-(dimethylsiloxane) is presented as a promising route to the fabrication of a flexible film with ordered and interconnected single-walled carbon nanotubes. This is possible using a simple transfer method of as-grown hierarchical single-walled carbon nanotubes on a Si pillar substrate. This film is used as a highly sensitive strain gauge sensor. This type of network embedded in a polymer film should be applicable to many fields involving mechanically stable and reliable strain sensors.

Non-universal tracer diffusion in crowded media of non-inert obstacles

For tracer motion in an array of attractive obstacles we observe transient, non-ergodic anomalous diffusion depending on the obstacle density.

Wanted: scalable tracers for diffusion measurements

Scalable tracers are potentially a useful tool to examine diffusion mechanisms and to predict diffusion coefficients, particularly for hindered diffusion in complex, heterogeneous, or crowded systems. Scalable tracers are defined as a series of tracers varying in size but with the same shape, structure, surface chemistry, deformability, and diffusion mechanism. Both chemical homology and constant dynamics are required. In particular, branching must not vary with size, and there must be no transition between ordinary diffusion and reptation. Measurements using scalable tracers yield the mean diffusion coefficient as a function of size alone; measurements using nonscalable tracers yield the variation due to differences in the other properties. Candidate scalable tracers are discussed for two-dimensional (2D) diffusion in membranes and three-dimensional diffusion in aqueous solutions. Correlations to predict the mean diffusion coefficient of globular biomolecules from molecular mass are reviewed briefly. Specific suggestions for the 3D case include the use of synthetic dendrimers or random hyperbranched polymers instead of dextran and the use of core-shell quantum dots. Another useful tool would be a series of scalable tracers varying in deformability alone, prepared by varying the density of crosslinking in a polymer to make say "reinforced Ficoll" or "reinforced hyperbranched polyglycerol."

Lateral diffusion in a discrete fluid membrane with immobile particles

Due to the coupling between the plasma membrane and the actin cytoskeleton, membrane molecules such as receptor proteins can become immobilized by binding to cytoskeletal structures. We investigate the effect of immobile membrane molecules on the diffusion of mobile ones by modeling the membrane as a two-dimensional (2D) fluid composed of hard particles and performing event-driven molecular dynamics simulations at a particle density where the system is in an isotropic liquid state. We show that the diffusion coefficient sharply decreases with increasing immobile fraction, dropping by a factor of ∼ 3 as the fraction of immobile particles increases from 0 to 0.1, in a system-size dependent manner. By combining our results with earlier calculations, we estimate that a factor-of-∼ 20 reduction in diffusion coefficients in live cell membranes, a puzzling finding in cell biology, can be accounted for when less than ∼ 22% of the particles in our model system is immobilized. Furthermore, we investigate the effects of confinement induced by a correlated distribution of immobile particles by calculating the distribution of the time it takes for particles to escape from a corral. In the regime where the particles can always escape from the corral, it is found that the escape times follow an exponential distribution, and the mean escape time grows exponentially with the density of obstacles at the corral boundary, increasing by a factor of 3-5 when immobile particles cover 50% of the boundary, and is approximately proportional to the area of the corral. We believe that our findings will be useful in interpreting (1) single molecule observations of membrane molecules and (2) results of particle based simulations that explore the effect of fluid dynamics on molecular transport in a 2D fluid.

Anomalous transport in the crowded world of biological cells

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

Wanted: a positive control for anomalous subdiffusion

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

Single molecule fluorescence detection and tracking in mammalian cells: the state-of-the-art and future perspectives

Insights from single-molecule tracking in mammalian cells have the potential to greatly contribute to our understanding of the dynamic behavior of many protein families and networks which are key therapeutic targets of the pharmaceutical industry. This is particularly so at the plasma membrane, where the method has begun to elucidate the mechanisms governing the molecular interactions that underpin many fundamental processes within the cell, including signal transduction, receptor recognition, cell-cell adhesion, etc. However, despite much progress, single-molecule tracking faces challenges in mammalian samples that hinder its general application in the biomedical sciences. Much work has recently focused on improving the methods for fluorescent tagging of target molecules, detection and localization of tagged molecules, which appear as diffraction-limited spots in charge-coupled device (CCD) images, and objectively establishing the correspondence between moving particles in a sequence of image frames to follow their diffusive behavior. In this review we outline the state-of-the-art in the field and discuss the advantages and limitations of the methods available in the context of specific applications, aiming at helping researchers unfamiliar with single molecules methods to plan out their experiments.

From microscopy data to in silico environments for in vivo-oriented simulations

ABSTRACT

: In our previous study, we introduced a combination methodology of Fluorescence Correlation Spectroscopy (FCS) and Transmission Electron Microscopy (TEM), which is powerful to investigate the effect of intracellular environment to biochemical reaction processes. Now, we developed a reconstruction method of realistic simulation spaces based on our TEM images. Interactive raytracing visualization of this space allows the perception of the overall 3D structure, which is not directly accessible from 2D TEM images. Simulation results show that the diffusion in such generated structures strongly depends on image post-processing. Frayed structures corresponding to noisy images hinder the diffusion much stronger than smooth surfaces from denoised images. This means that the correct identification of noise or structure is significant to reconstruct appropriate reaction environment in silico in order to estimate realistic behaviors of reactants in vivo. Static structures lead to anomalous diffusion due to the partial confinement. In contrast, mobile crowding agents do not lead to anomalous diffusion at moderate crowding levels. By varying the mobility of these non-reactive obstacles (NRO), we estimated the relationship between NRO diffusion coefficient (Dnro) and the anomaly in the tracer diffusion (α). For Dnro=21.96 to 44.49 μm2/s, the simulation results match the anomaly obtained from FCS measurements. This range of the diffusion coefficient from simulations is compatible with the range of the diffusion coefficient of structural proteins in the cytoplasm. In addition, we investigated the relationship between the radius of NRO and anomalous diffusion coefficient of tracers by the comparison between different simulations. The radius of NRO has to be 58 nm when the polymer moves with the same diffusion speed as a reactant, which is close to the radius of functional protein complexes in a cell.

Analytical tools to distinguish the effects of localization error, confinement, and medium elasticity on the velocity autocorrelation function

Single particle tracking is a powerful technique for investigating the dynamic behavior of biological molecules. However, many of the analytical tools are prone to generate results that can lead to mistaken interpretations of the underlying transport process. Here, we explore the effects of localization error and confinement on the velocity autocorrelation function, Cυ. We show that calculation of Cυ across a range of discretizations can distinguish the effects of localization error, confinement, and medium elasticity. Thus, under certain regimes, Cυ can be used as a diagnostic tool to identify the underlying mechanism of anomalous diffusion. Finally, we apply our analysis to experimental data sets of chromosomal loci and RNA-protein particles in Escherichia coli.

Archipelago architecture of the focal adhesion: membrane molecules freely enter and exit from the focal adhesion zone

The focal adhesion (FA) is an integrin-based structure built in/on the plasma membrane, mechanically linking the extracellular matrix with the termini of actin stress fibers, providing key scaffolds for the cells to migrate in tissues. The FA was considered as a micron-scale, massive assembly of various proteins, although its formation and decomposition occur quickly in several to several 10 s of minutes. The mechanism of rapid FA regulation has been a major mystery in cell biology. Here, using fast single fluorescent-molecule imaging, we found that transferrin receptor and Thy1, non-FA membrane proteins, readily enter the FA zone, diffuse rapidly there, and exit into the bulk plasma membrane. Integrin β3 also readily enters the FA zone, and repeatedly undergoes temporary immobilization and diffusion in the FA zone, whereas approximately one-third of integrin β3 is immobilized there. These results are consistent with the archipelago architecture of the FA, which consists of many integrin islands: the membrane molecules enter the inter-island channels rather freely, and the integrins in the integrin islands can be rapidly exchanged with those in the bulk membrane. Such an archipelago architecture would allow rapid FA formation and disintegration, and might be applicable to other large protein domains in the plasma membrane.

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.

Anomalous transport of a tracer on percolating clusters

We investigate the dynamics of a single tracer exploring a course of fixed obstacles in the vicinity of the percolation transition for particles confined to the infinite cluster. The mean-square displacement displays anomalous transport, which extends to infinite times precisely at the critical obstacle density. The slowing down of the diffusion coefficient exhibits power-law behavior for densities close to the critical point and we show that the mean-square displacement fulfills a scaling hypothesis. Furthermore, we calculate the dynamic conductivity as a response to an alternating electric field. Last, we discuss the non-gaussian parameter as an indicator for heterogeneous dynamics.