A biological interpretation of transient anomalous subdiffusion. I. Qualitative model

Two-photon microscopy of acoustofluidic trapping for highly sensitive cell analysis

We combine two-photon-excited fluorescence microscopy and acoustofluidic trapping in a spherical microchamber to in vitro study cells and cell clusters three-dimensionally close to in vivo conditions. The two-photon microscopy provides the in-depth 3D analysis of the spherical microchamber dimensions as well as the positions of trapped samples therein with high spatial precision and high temporal resolution enabling even tracking of the fast moving particles. Furthermore, optical sectioning allows to gather information of individual cells in trapped cell clusters inside the chamber. We demonstrate real-time monitoring of osmosis in A549 lung cells and red blood cells as one possible biomedical application. The observed osmosis reduced the cell membrane diameter by approximately 4 μm in the A549 cells and by approximately 2 μm in the red blood cells. Our approach provides an important optical tool for future investigations of cell functions and cell-cell interactions avoiding wall-contact inside an acoustofluidic device.

Fundamental Dynamics of Popularity-Similarity Trajectories in Real Networks

Real networks are complex dynamical systems, evolving over time with the addition and deletion of nodes and links. Currently, there exists no principled mathematical theory for their dynamics-a grand-challenge open problem. Here, we show that the popularity and similarity trajectories of nodes in hyperbolic embeddings of different real networks manifest universal self-similar properties with typical Hurst exponents H≪0.5. This means that the trajectories are predictable, displaying antipersistent or "mean-reverting" behavior, and they can be adequately captured by a fractional Brownian motion process. The observed behavior can be qualitatively reproduced in synthetic networks that possess a latent geometric space, but not in networks that lack such space, suggesting that the observed subdiffusive dynamics are inherently linked to the hidden geometry of real networks. These results set the foundations for rigorous mathematical machinery for describing and predicting real network dynamics.

Diffusive lensing as a mechanism of intracellular transport and compartmentalization

While inhomogeneous diffusivity has been identified as a ubiquitous feature of the cellular interior, its implications for particle mobility and concentration at different length scales remain largely unexplored. In this work, we use agent-based simulations of diffusion to investigate how heterogeneous diffusivity affects the movement and concentration of diffusing particles. We propose that a nonequilibrium mode of membrane-less compartmentalization arising from the convergence of diffusive trajectories into low-diffusive sinks, which we call 'diffusive lensing,' is relevant for living systems. Our work highlights the phenomenon of diffusive lensing as a potentially key driver of mesoscale dynamics in the cytoplasm, with possible far-reaching implications for biochemical processes.

Activity-induced diffusion recovery in crowded colloidal suspensions

We show that the forces generated by active enzyme molecules are strong enough to influence the dynamics of their surroundings under artificial crowded environments. We measured the behavior of polymer microparticles in a quasi-two-dimensional system under aqueous environment, at various area fraction values of particles. In the presence of enzymatic activity, not only was the diffusion of the suspended particles enhanced at shorter time-scales, but the system also showed a transition from subdiffusive to diffusive dynamics at longer time-scale limits. Similar observations were also recorded with enzyme-functionalized microparticles. Brownian dynamics simulations have been performed to support the experimental observations.

Quantification of cell shape, intracellular flows and transport based on DIC object detection and tracking

Computational image analysis combined with label-free imaging has helped maintain its relevance for cell biology, despite the rapid technical improvements in fluorescence microscopy with the molecular specificity of tags. Here, we discuss some computational tools developed in our lab and their application to quantify cell shape, intracellular organelle movement and bead transport in vitro, using differential interference contrast (DIC) microscopy data as inputs. The focus of these methods is image filtering to enhance image gradients, and combining them with segmentation and single particle tracking (SPT). We demonstrate the application of these methods to Escherichia coli cell length estimation and tracking of densely packed lipid granules in Caenorhabditis elegans one-celled embryos, diffusing beads in solutions of different viscosities and kinesin-driven transport on microtubules. These approaches demonstrate how improvements to low-level image analysis methods can help obtain insights through quantitative cellular and subcellular microscopy.

Macromolecular interactions and geometrical confinement determine the 3D diffusion of ribosome-sized particles in live Escherichia coli cells

The crowded bacterial cytoplasm is comprised of biomolecules that span several orders of magnitude in size and electrical charge. This complexity has been proposed as the source of the rich spatial organization and apparent anomalous diffusion of intracellular components, although this has not been tested directly. Here, we use biplane microscopy to track the 3D motion of self-assembled bacterial Genetically Encoded Multimeric nanoparticles (bGEMs) with tunable size (20 to 50 nm) and charge (-2160 to +1800 e) in live Escherichia coli cells. To probe intermolecular details at spatial and temporal resolutions beyond experimental limits, we also developed a colloidal whole-cell model that explicitly represents the size and charge of cytoplasmic macromolecules and the porous structure of the bacterial nucleoid. Combining these techniques, we show that bGEMs spatially segregate by size, with small 20-nm particles enriched inside the nucleoid, and larger and/or positively charged particles excluded from this region. Localization is driven by entropic and electrostatic forces arising from cytoplasmic polydispersity, nucleoid structure, geometrical confinement, and interactions with other biomolecules including ribosomes and DNA. We observe that at the timescales of traditional single molecule tracking experiments, motion appears sub-diffusive for all particle sizes and charges. However, using computer simulations with higher temporal resolution, we find that the apparent anomalous exponents are governed by the region of the cell in which bGEMs are located. Molecular motion does not display anomalous diffusion on short time scales and the apparent sub-diffusion arises from geometrical confinement within the nucleoid and by the cell boundary.

Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Anomalous diffusion, non-Gaussianity, nonergodicity, and confinement in stochastic-scaled Brownian motion with diffusing diffusivity dynamics

Scaled Brownian motions (SBMs) with power-law time-dependent diffusivity have been used to describe various types of anomalous diffusion yet Gaussian observed in granular gases kinetics, turbulent diffusion, and molecules mobility in cells, to name a few. However, some of these systems may exhibit non-Gaussian behavior which can be described by SBM with diffusing diffusivity (DD-SBM). Here, we numerically investigate both free and confined DD-SBM models characterized by fixed or stochastic scaling exponent of time-dependent diffusivity. The effects of distributed scaling exponent, random diffusivity, and confinement are considered. Different regimes of ultraslow diffusion, subdiffusion, normal diffusion, and superdiffusion are observed. In addition, weak ergodic and non-Gaussian behaviors are also detected. These results provide insights into diffusion in time-fluctuating diffusivity landscapes with potential applications to time-dependent temperature systems spreading in heterogeneous environments.

Simulation study on the effect of polydisperse nanoparticles on polymer diffusion in crowded environments

The diffusion of polymer chains in a crowded environment with large and small immobile, attractive nanoparticles (NPs) is studied using Langevin dynamics simulations. For orderly distributed NPs on the simple cubic lattice, our results show that the diffusion of polymer chains is dependent on the NP-NP distance or lattice distance d. At low d where NPs are placed closely, subdiffusion occurs at a sufficiently high polydispersity of NPs, PD. Both the apparent diffusion coefficient and subdiffusion exponent of polymer chains decrease with increasing PD, attributed to the adsorption of polymers on NP clusters formed by larger NPs. At large d, normal diffusion is always observed, and the diffusion coefficient increases with increasing PD. The reason is that, at high PD, the difference between single large NP adsorption and double large NP adsorption is reduced, which increases the exchange of a polymer between the two adsorption states. Finally, the impact of size polydispersity of NPs on the diffusion of polymer chains in a crowded environment with randomly distributed NPs is also investigated. The results show that the position disorder of NPs enhances the subdiffusion of the system.

Computational mesoscale framework for biological clustering and fractal aggregation

Hierarchical clustering due to diffusion and reaction is a widespread occurrence in natural phenomena, displaying fractal behavior with non-integer size scaling. The study of this phenomenon has garnered interest in both biological systems such as morphogenesis and blood clotting, and synthetic systems such as colloids and polymers. The modeling of biological clustering can be difficult, as it can occur on a variety of scales and involve multiple mechanisms, necessitating the use of various methods to capture its behavior. Here, we propose a novel framework, the generalized-mesoscale-clustering (GMC), for the study of complex hierarchical clustering phenomena in biological systems. The GMC framework incorporates the effects of hydrodynamic interactions, bonding, and surface tension, and allows for the analysis of both static and dynamic states of cluster development. The framework is applied to a range of biological clustering mechanisms, with a focus on blood-related clustering from fibrin network formation to platelet aggregation. Our study highlights the importance of a comprehensive characterization of the structural properties of the cluster, including fractal dimension, pore-scale diffusion, initiation time, and consolidation time, in fully understanding the behavior of biological clustering systems. The GMC framework also provides the potential to investigate the temporal evolution and mechanical properties of the clusters by tracking bond density and including hydrodynamic interactions.

Simulation Study on the Mechanism of Intermediate Subdiffusion of Diffusive Particles in Crowded Systems

The intermediate subdiffusion of diffusive particles in crowded systems is studied for two model systems: the continuous time random walk (CTRW) model and the obstruction-binding model. For the CTRW model with an arbitrarily given longest waiting time τmax, we find that the diffusive particle exhibits subdiffusion below τmax and recovers normal diffusion above τmax. For the obstruction-binding model with randomly distributed attractive obstacles, the diffusion of the diffusive particle is dependent on the binding energy and the density of obstacles. Interestingly, diffusion curves for different binding strengths can be overlapped by rescaling the simulation time, indicating that the diffusive particle in the obstruction-binding model can change from the intermediate subdiffusion to the normal diffusion at a long-term simulation scale. The results of the two model systems show that the diffusive particles only exhibit intermediate subdiffusion below the longest waiting time. Therefore, long timescale subdiffusion would only be observed in the CTRW model with an infinitely long waiting time and in the obstruction-binding model with an infinitely large binding strength.

Exploring non-equilibrium processes and spatio-temporal scaling laws in heated egg yolk using coherent X-rays

The soft-grainy microstructure of cooked egg yolk is the result of a series of out-of-equilibrium processes of its protein-lipid contents; however, it is unclear how egg yolk constituents contribute to these processes to create the desired microstructure. By employing X-ray photon correlation spectroscopy, we investigate the functional contribution of egg yolk constituents: proteins, low-density lipoproteins (LDLs), and yolk-granules to the development of grainy-gel microstructure and microscopic dynamics during cooking. We find that the viscosity of the heated egg yolk is solely determined by the degree of protein gelation, whereas the grainy-gel microstructure is controlled by the extent of LDL aggregation. Overall, protein denaturation-aggregation-gelation and LDL-aggregation follows Arrhenius-type time-temperature superposition (TTS), indicating an identical mechanism with a temperature-dependent reaction rate. However, above 75 °C TTS breaks down and temperature-independent gelation dynamics is observed, demonstrating that the temperature can no longer accelerate certain non-equilibrium processes above a threshold value.

Increased intracellular crowding during hyperosmotic stress

Hyperosmotic stress activates in live cells numerous processes and also promotes intracellular protein/RNA aggregation and phase separation. However, the time course and the extent of these changes remain largely uncharacterized. To investigate dynamic changes in intracellular macromolecular crowding (MMC) induced by hyperosmotic stress in live cells, we used fluorescence lifetime imaging microscopy and fluorescence correlation spectroscopy (FCS) to quantify changes in the local environment by measuring the fluorescence lifetime and the diffusion of the monomeric enhanced green fluorescent protein (eGFP), respectively. Real-time monitoring of eGFP fluorescence lifetime showed that a faster response to environmental changes due to MMC is observed than when measuring the acceptor/donor emission ratio using the MMC-sensitive Förster resonance energy transfer sensor (GimRET). This suggests that eGFP molecular electronic states and/or collision frequency are affected by changes in the immediate surroundings due to MMC without requiring conformational changes as is the case for the GimRET sensor. Furthermore, eGFP diffusion assessed by FCS indicated higher intracellular viscosity due to increased MMC during hyperosmotic stress. Our findings reveal that changes in eGFP fluorescence lifetime and diffusion are early indicators of elevated intracellular MMC. Our approach can therefore be used to reveal in live cells short-lived transient states through which MMC builds over time, which could not be observed when measuring changes in other physical properties that occur at slower time scales.

The Role of Polymer Chain Stiffness and Guest Nanoparticle Loading in Improving the Glass Transition Temperature of Polymer Nanocomposites

The impact of polymer chain stiffness characterized by the bending modulus (k_θ) on the glass transition temperature (T_g) of pure polymer systems, as well as polymer nanocomposites (PNCs), is investigated using molecular dynamics simulations. At small k_θ values, the pure polymer system and respective PNCs are in an amorphous state, whereas at large k_θ values, both systems are in a semicrystalline state with a glass transition at low temperature. For the pure polymer system, T_g initially increases with k_θ and does not change obviously at large k_θ. However, the T_g of PNCs shows interesting behaviors with the increasing volume fraction of nanoparticles (f_NP) at different k_θ values. T_g tends to increase with f_NP at small k_θ, whereas it becomes suppressed at large k_θ.

Polarity and mixed-mode oscillations may underlie different patterns of cellular migration

In mesenchymal cell motility, several migration patterns have been observed, including directional, exploratory and stationary. Two key members of the Rho-family of GTPases, Rac and Rho, along with an adaptor protein called paxillin, have been particularly implicated in the formation of such migration patterns and in regulating adhesion dynamics. Together, they form a key regulatory network that involves the mutual inhibition exerted by Rac and Rho on each other and the promotion of Rac activation by phosphorylated paxillin. Although this interaction is sufficient in generating wave-pinning that underscores cellular polarization comprised of cellular front (high active Rac) and back (high active Rho), it remains unclear how they interact collectively to induce other modes of migration detected in Chinese hamster Ovary (CHO-K1) cells. We previously developed a six-variable (6V) reaction-diffusion model describing the interactions of these three proteins (in their active/phosphorylated and inactive/unphosphorylated forms) along with other auxiliary proteins, to decipher their role in generating wave-pinning. In this study, we explored, through computational modeling and image analysis, how differences in timescales within this molecular network can potentially produce the migration patterns in CHO-K1 cells and how switching between migration modes could occur. To do so, the 6V model was reduced to an excitable 4V spatiotemporal model possessing three different timescales. The model produced not only wave-pinning in the presence of diffusion, but also mixed-mode oscillations (MMOs) and relaxation oscillations (ROs). Implementing the model using the Cellular Potts Model (CPM) produced outcomes in which protrusions in the cell membrane changed Rac-Rho localization, resulting in membrane oscillations and fast directionality variations similar to those observed experimentally in CHO-K1 cells. The latter was assessed by comparing the migration patterns of experimental with CPM cells using four metrics: instantaneous cell speed, exponent of mean-square displacement ([Formula: see text]-value), directionality ratio and protrusion rate. Variations in migration patterns induced by mutating paxillin's serine 273 residue were also captured by the model and detected by a machine classifier, revealing that this mutation alters the dynamics of the system from MMOs to ROs or nonoscillatory behaviour through variation in the scaled concentration of an active form of an adhesion protein called p21-Activated Kinase 1 (PAK). These results thus suggest that MMOs and adhesion dynamics are the key mechanisms regulating CHO-K1 cell motility.

Effect of detachment of motor protein from track on its transport

The transportation of the cargoes in biological cells is primarily driven by the motor proteins on filamentous protein tracks. The stochastic nature of the motion of motor protein often leads to its spontaneous detachment from the track. We formulate a mathematical model to study the effect of the detachment of motor protein on its track bound transport. We calculate two quantities: the distance traveled by the motor protein in given time, and the average time taken by a single motor protein to reach a target distance. Expectedly, both of these quantities decrease with the increasing detachment rate if the motor velocity is kept fixed. However, the existing experimental data suggest that a change in the detachment rate also affects the velocity of the motor protein. This relation between motor protein speed and its detachment rate results in a non-monotonic dependence on the distance traveled in fixed time and transport rate to a fixed distance. Therefore, we demonstrate that optimal motor speeds can be identified for the time and distance controlled conditions.

Temperature and friction fluctuations inside a harmonic potential

In this article we study the trapped motion of a molecule undergoing diffusivity fluctuations inside a harmonic potential. For the same diffusing-diffusivity process, we investigate two possible interpretations. Depending on whether diffusivity fluctuations are interpreted as temperature or friction fluctuations, we show that they display drastically different statistical properties inside the harmonic potential. We compute the characteristic function of the process under both types of interpretations and analyze their limit behavior. Based on the integral representations of the processes we compute the mean-squared displacement and the normalized excess kurtosis. In the long-time limit, we show for friction fluctuations that the probability density function (PDF) always converges to a Gaussian whereas in the case of temperature fluctuations the stationary PDF can display either Gaussian distribution or generalized Laplace (Bessel) distribution depending on the ratio between diffusivity and positional correlation times.

Fractional Integrable Nonlinear Soliton Equations

Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg-deVries and nonlinear Schrödinger equations. These equations are shown to predict superdispersive transport of nondissipative solitons in fractional media.

Transient Anomalous Diffusion MRI Measurement Discriminates Porous Polymeric Matrices Characterized by Different Sub-Microstructures and Fractal Dimension

Considering the current development of new nanostructured and complex materials and gels, it is critical to develop a sub-micro-scale sensitivity tool to quantify experimentally new parameters describing sub-microstructured porous systems. Diffusion NMR, based on the measurement of endogenous water's diffusion displacement, offers unique information on the structural features of materials and tissues. In this paper, we applied anomalous diffusion NMR protocols to quantify the subdiffusion of water and to measure, in an alternative, non-destructive and non-invasive modality, the fractal dimension d_w of systems characterized by micro and sub-micro geometrical structures. To this end, three highly heterogeneous porous-polymeric matrices were studied. All the three matrices composed of glycidylmethacrylate-divynilbenzene porous monoliths obtained through the High Internal Phase Emulsion technique were characterized by pores of approximately spherical symmetry, with diameters in the range of 2-10 μm. Pores were interconnected by a plurality of window holes present on pore walls, which were characterized by size coverings in the range of 0.5-2 μm. The walls were characterized by a different degree of surface roughness. Moreover, complementary techniques, namely Field Emission Scanning Electron Microscopy (FE-SEM) and dielectric spectroscopy, were used to corroborate the NMR results. The experimental results showed that the anomalous diffusion α parameter that quantifies subdiffusion and d_w = 2/α changed in parallel to the specific surface area S (or the surface roughness) of the porous matrices, showing a submicroscopic sensitivity. The results reported here suggest that the anomalous diffusion NMR method tested may be a valid experimental tool to corroborate theoretical and simulation results developed and performed for describing highly heterogeneous and complex systems. On the other hand, non-invasive and non-destructive anomalous subdiffusion NMR may be a useful tool to study the characteristic features of new highly heterogeneous nanostructured and complex functional materials and gels useful in cultural heritage applications, as well as scaffolds useful in tissue engineering.

Dependence of diffusion in Escherichia coli cytoplasm on protein size, environmental conditions, and cell growth

Inside prokaryotic cells, passive translational diffusion typically limits the rates with which cytoplasmic proteins can reach their locations. Diffusion is thus fundamental to most cellular processes, but the understanding of protein mobility in the highly crowded and non-homogeneous environment of a bacterial cell is still limited. Here, we investigated the mobility of a large set of proteins in the cytoplasm of Escherichia coli, by employing fluorescence correlation spectroscopy (FCS) combined with simulations and theoretical modeling. We conclude that cytoplasmic protein mobility could be well described by Brownian diffusion in the confined geometry of the bacterial cell and at the high viscosity imposed by macromolecular crowding. We observed similar size dependence of protein diffusion for the majority of tested proteins, whether native or foreign to E. coli. For the faster-diffusing proteins, this size dependence is well consistent with the Stokes-Einstein relation once taking into account the specific dumbbell shape of protein fusions. Pronounced subdiffusion and hindered mobility are only observed for proteins with extensive interactions within the cytoplasm. Finally, while protein diffusion becomes markedly faster in actively growing cells, at high temperature, or upon treatment with rifampicin, and slower at high osmolarity, all of these perturbations affect proteins of different sizes in the same proportions, which could thus be described as changes of a well-defined cytoplasmic viscosity.

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

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

Diffusion in Heterogenous Media and Sorption—Desorption Processes

We investigate particle diffusion in a heterogeneous medium limited by a surface where sorption–desorption processes are governed by a kinetic equation. We consider that the dynamics of the particles present in the medium are governed by a diffusion equation with a spatial dependence on the diffusion coefficient, i.e., K(x) = D|x|−η, with −1 < η and D = const, respectively. This system is analyzed in a semi-infinity region, i.e., the system is defined in the interval [0,∞) for an arbitrary initial condition. The solutions are obtained and display anomalous spreading, that is, the dynamics may be viewed as anomalous diffusion, which in turn is related, and hence, the model can be directly applied to several complex systems ranging from biological fluids to electrolytic cells.

Mechanics of 3D Cell-Hydrogel Interactions: Experiments, Models, and Mechanisms

Hydrogels are highly water-swollen molecular networks that are ideal platforms to create tissue mimetics owing to their vast and tunable properties. As such, hydrogels are promising cell-delivery vehicles for applications in tissue engineering and have also emerged as an important base for ex vivo models to study healthy and pathophysiological events in a carefully controlled three-dimensional environment. Cells are readily encapsulated in hydrogels resulting in a plethora of biochemical and mechanical communication mechanisms, which recapitulates the natural cell and extracellular matrix interaction in tissues. These interactions are complex, with multiple events that are invariably coupled and spanning multiple length and time scales. To study and identify the underlying mechanisms involved, an integrated experimental and computational approach is ideally needed. This review discusses the state of our knowledge on cell-hydrogel interactions, with a focus on mechanics and transport, and in this context, highlights recent advancements in experiments, mathematical and computational modeling. The review begins with a background on the thermodynamics and physics fundamentals that govern hydrogel mechanics and transport. The review focuses on two main classes of hydrogels, described as semiflexible polymer networks that represent physically cross-linked fibrous hydrogels and flexible polymer networks representing the chemically cross-linked synthetic and natural hydrogels. In this review, we highlight five main cell-hydrogel interactions that involve key cellular functions related to communication, mechanosensing, migration, growth, and tissue deposition and elaboration. For each of these cellular functions, recent experiments and the most up to date modeling strategies are discussed and then followed by a summary of how to tune hydrogel properties to achieve a desired functional cellular outcome. We conclude with a summary linking these advancements and make the case for the need to integrate experiments and modeling to advance our fundamental understanding of cell-matrix interactions that will ultimately help identify new therapeutic approaches and enable successful tissue engineering.

TRAIT2D: a Software for Quantitative Analysis of Single Particle Diffusion Data

Single particle tracking (SPT) is one of the most widely used tools in optical microscopy to evaluate particle mobility in a variety of situations, including cellular and model membrane dynamics. Recent technological developments, such as Interferometric Scattering microscopy, have allowed recording of long, uninterrupted single particle trajectories at kilohertz framerates. The resulting data, where particles are continuously detected and do not displace much between observations, thereby do not require complex linking algorithms. Moreover, while these measurements offer more details into the short-term diffusion behaviour of the tracked particles, they are also subject to the influence of localisation uncertainties, which are often underestimated by conventional analysis pipelines. we thus developed a Python library, under the name of TRAIT2D (Tracking Analysis Toolbox - 2D version), in order to track particle diffusion at high sampling rates, and analyse the resulting trajectories with an innovative approach. The data analysis pipeline introduced is more localisation-uncertainty aware, and also selects the most appropriate diffusion model for the data provided on a statistical basis. A trajectory simulation platform also allows the user to handily generate trajectories and even synthetic time-lapses to test alternative tracking algorithms and data analysis approaches. A high degree of customisation for the analysis pipeline, for example with the introduction of different diffusion modes, is possible from the source code. Finally, the presence of graphical user interfaces lowers the access barrier for users with little to no programming experience.

Membrane Nanoscopic Organization of D2L Dopamine Receptor Probed by Quantum Dot Tracking

The role of lateral mobility and nanodomain organization of G protein-coupled receptors in modulating subcellular signaling has been under increasing scrutiny. Investigation of D2 dopamine receptor diffusion dynamics is of particular interest, as these receptors have been linked to altered neurotransmission in affective disorders and represent the primary target for commonly prescribed antipsychotics. Here, we applied our single quantum dot tracking approach to decipher intrinsic diffusion patterns of the wild-type long isoform of the D2 dopamine receptor and its genetic variants previously identified in several cohorts of schizophrenia patients. We identified a subtle decrease in the diffusion rate of the Val96Ala mutant that parallels its previously reported reduced affinity for potent neuroleptics clozapine and chlorpromazine. Slower Val96Ala variant diffusion was not accompanied by a change in receptor-receptor transient interactions as defined by the diffraction-limited quantum dot colocalization events. In addition, we implemented a Voronoї tessellation-based algorithm to compare nanoclustering of the D2 dopamine receptor to the dominant anionic phospholipid phosphatidylinositol 4,5-bisphosphate in the plasma membrane of live cells.

Concepts to Reveal Parvovirus-Nucleus Interactions

Parvoviruses are small single-stranded (ss) DNA viruses, which replicate in the nucleoplasm and affect both the structure and function of the nucleus. The nuclear stage of the parvovirus life cycle starts at the nuclear entry of incoming capsids and culminates in the successful passage of progeny capsids out of the nucleus. In this review, we will present past, current, and future microscopy and biochemical techniques and demonstrate their potential in revealing the dynamics and molecular interactions in the intranuclear processes of parvovirus infection. In particular, a number of advanced techniques will be presented for the detection of infection-induced changes, such as DNA modification and damage, as well as protein-chromatin interactions.

Single Quantum Dot Tracking Unravels Agonist Effects on the Dopamine Receptor Dynamics

D2 dopamine receptors (DRD2s) belong to a family of G protein-coupled receptors that modulate synaptic dopaminergic tone via regulation of dopamine synthesis, storage, and synaptic release. DRD2s are the primary target for traditional antipsychotic medications; dysfunctional DRD2 signaling has been linked to major depressive disorder, attention-deficit hyperactivity disorder, addiction, Parkinson's, and schizophrenia. DRD2 lateral diffusion appears to be an important post-translational regulatory mechanism; however, the dynamic response of DRD2s to ligand-induced activation is poorly understood. Dynamic imaging of the long isoform of DRD2 (D2L) fused to an N-terminal antihemagglutinin (HA) epitope and transiently expressed in HEK-293 cells was achieved through a combination of a high-affinity biotinylated anti-HA antigen-binding fragment (Fab) and streptavidin-conjugated quantum dots (QD). Significant reduction (∼40%) in the rate of lateral diffusion of QD-tagged D2L proteins was observed under agonist (quinpirole; QN)-stimulated conditions compared to basal conditions. QN-induced diffusional slowing was accompanied by an increase in frequency, lifetime, and confinement of temporary arrest of lateral diffusion (TALL), an intrinsic property of single receptor lateral motion. The role of the actin cytoskeleton in QN-induced diffusional slowing of D2L was also explored. The observed dynamic changes appear to be a sensitive indicator of the receptor activity status and might also spatially and temporally shape the receptor-mediated downstream signaling. This dynamic information could potentially be useful in informing drug discovery efforts based on single-molecule pharmacology.

A role for Dynlt3 in melanosome movement, distribution, acidity and transfer

Skin pigmentation is dependent on cellular processes including melanosome biogenesis, transport, maturation and transfer to keratinocytes. However, how the cells finely control these processes in space and time to ensure proper pigmentation remains unclear. Here, we show that a component of the cytoplasmic dynein complex, Dynlt3, is required for efficient melanosome transport, acidity and transfer. In Mus musculus melanocytes with decreased levels of Dynlt3, pigmented melanosomes undergo a more directional motion, leading to their peripheral location in the cell. Stage IV melanosomes are more acidic, but still heavily pigmented, resulting in a less efficient melanosome transfer. Finally, the level of Dynlt3 is dependent on β-catenin activity, revealing a function of the Wnt/β-catenin signalling pathway during melanocyte and skin pigmentation, by coupling the transport, positioning and acidity of melanosomes required for their transfer.

Visualization of subdiffusive sites in a live single cell

We measured anomalous diffusion in human prostate cancer cells which were transfected with the Alexa633 fluorescent RNA probe and co-transfected with enhanced green fluorescent protein-labeled argonaute2 protein by laser scanning microscopy. The image analysis arose from diffusion based on a “two-level system”. A trap was an interaction site where the diffusive motion was slowed down. Anomalous subdiffusive spreading occurred at cellular traps. The cellular traps were not immobile. We showed how the novel analysis method of imaging data resulted in new information about the number of traps in the crowded and heterogeneous environment of a single human prostate cancer cell. The imaging data were consistent with and explained by our modern ideas of anomalous diffusion of mixed origins in live cells. Our original research presented in this study is significant as we obtained a complex diffusion mechanism in live single cells.

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.

Accelerating and retarding anomalous diffusion: A Bernstein function approach

We have discovered here a duality relation between infinitely divisible subordinators which can produce both retarding and accelerating anomalous diffusion in the framework of the special Bernstein function approach. As a consequence, we show that conjugate pairs of Bernstein functions taken as Laplace exponents can produce in a natural way both retarding and accelerating anomalous diffusion (either subdiffusion or superdiffusion). This provides a unified way to control the dynamics of complex biological processes leading to transient anomalous diffusion in single-particle tracking experiments. Moreover, this permits one to explain better the relaxation diagram positioning two different power laws of relaxation, including the celebrated Havriliak-Negami law.

Stochastic bond dynamics facilitates alignment of malaria parasite at erythrocyte membrane upon invasion

Malaria parasites invade healthy red blood cells (RBCs) during the blood stage of the disease. Even though parasites initially adhere to RBCs with a random orientation, they need to align their apex toward the membrane in order to start the invasion process. Using hydrodynamic simulations of a RBC and parasite, where both interact through discrete stochastic bonds, we show that parasite alignment is governed by the combination of RBC membrane deformability and dynamics of adhesion bonds. The stochastic nature of bond-based interactions facilitates a diffusive-like re-orientation of the parasite at the RBC membrane, while RBC deformation aids in the establishment of apex-membrane contact through partial parasite wrapping by the membrane. This bond-based model for parasite adhesion quantitatively captures alignment times measured experimentally and demonstrates that alignment times increase drastically with increasing rigidity of the RBC membrane. Our results suggest that the alignment process is mediated simply by passive parasite adhesion.

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.

Dynamic heterogeneity and non-Gaussian statistics for ganglioside GM1s and acetylcholine receptors on live cell membrane

We have carried out a comparative study of the lateral motion of ganglioside GM1, which is a glycosphingolipid residing on the outer leaflet of the plasma membrane, and acetylcholine receptor (AChR), which is a well-characterized ion channel. Both the lipid molecules and the transmembrane proteins reside on the plasma membranes of live Xenopus muscle cells. From a thorough analysis of a large volume of individual molecular trajectories obtained from more than 300 live cells over a wide range of sampling rates and long durations, we find that the GM1s and AChRs share the same dynamic heterogeneity and non-Gaussian statistics. Our measurements with the ATP-depleted cells reveal that the diffusion dynamics of the GM1s and AChRs is uniformly affected by the intracellular ATP level of the living muscle cells, further demonstrating that membrane diffusion is strongly coupled to the dynamics of the underlying cortical actin network, as predicted by the dynamic picket-fence model.

Unraveling Dynamic Transitions in Time-Resolved Biomolecular Motions by A Dressed Diffusion Model

Recent experimental data reveal the complexity of diffusion dynamics beyond the scope of classical Brownian dynamics. The particles exhibit diverse diffusive motions from the anomalous toward classical diffusion over a wide range of temporal scales. Here a dressed diffusion model is developed to account for non-Brownian phenomena. By coupling the particle dynamics with a local field, the dressed diffusion model generalizes the Langevin equation through coupled damping kernels and generates the salient feature of time-dependent diffusion dynamics reported in the experimental measurements of biomolecules. The dressed diffusion model provides one quantitative aspect for future endeavors in this rapid-growing field.

Non-Gaussian subdiffusion of single-molecule tracers in a hydrated polymer network

Single molecule tracking experiments inside a hydrated polymer network have shown that the tracer motion is subdiffusive due to the viscoelastic environment inside the gel-like network. This property can be related to the negative autocorrelation of the instantaneous displacements at short times. Although the displacements of the individual tracers exhibit Gaussian statistics, the displacement distribution of all the trajectories combined from different spatial locations of the polymer network exhibits a non-Gaussian distribution. Here, we analyze many individual tracer trajectories to show that the central portion of the non-Gaussian distribution can be well approximated by an exponential distribution that spreads sublinearly with time. We explain all these features seen in the experiment by a generalized Langevin model for an overdamped particle with algebraically decaying correlations. We show that the degree of non-Gaussianity can change with the extent of heterogeneity, which is controlled in our model by the experimentally observed distributions of the motion parameters.

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.

Guided nuclear exploration increases CTCF target search efficiency

The enormous size of mammalian genomes means that for a DNA-binding protein, the number of non-specific, off-target sites vastly exceeds the number of specific, cognate sites. How mammalian DNA-binding proteins overcome this challenge to efficiently locate their target sites is not known. Here through live-cell single-molecule tracking, we show that CCCTC-binding factor, CTCF, is repeatedly trapped in small zones that likely correspond to CTCF clusters, in a manner that is largely dependent on an internal RNA-binding region (RBR_i). We develop a new theoretical model, Anisotropic Diffusion through transient Trapping in Zones (ADTZ), to explain CTCF dynamics. Functionally, transient RBR_i-mediated trapping increases the efficiency of CTCF target search by ~2.5 fold. Overall, our results suggest a “guided” mechanism where CTCF clusters concentrate diffusing CTCF proteins near cognate binding sites, thus increasing the local ON-rate. We suggest that local guiding may allow DNA-binding proteins to more efficiently locate their target sites.

Mechanical Interaction between Cells Facilitates Molecular Transport

In vivo, eukaryotic cells are embedded in a matrix environment, where they grow and develop. Generally, this extracellular matrix (ECM) is an anisotropic fibrous structure, through which macromolecules and biochemical signaling molecules at the nanometer scale diffuse. The ECM is continuously remodeled by cells, via mechanical interactions, which lead to a potential link between biomechanical and biochemical cell-cell interactions. Here, it is studied how cell-induced forces applied on the ECM impact the biochemical transport of molecules between distant cells. It is experimentally observed that cells remodel the ECM by increasing fiber alignment and density of the matrix between them over time. Using random walk simulations on a 3D lattice, elongated fixed obstacles are implemented that mimic the fibrous ECM structure. Both diffusion of a tracer molecule and the mean first-passage time a molecule secreted from one cell takes to reach another cell are measured. The model predicts that cell-induced remodeling can lead to a dramatic speedup in the transport of molecules between cells. Fiber alignment and densification cause reduction of the transport dimensionality from a 3D to a much more rapid 1D process. Thus, a novel mechanism of mechano-biochemical feedback in the regulation of long-range cell-cell communication is suggested.

Tracking Single Molecules in Biomembranes: Is Seeing Always Believing?

The spatial organization of molecules in cell membranes and their dynamic interactions play a central role in regulating cell functions. Single-particle tracking (SPT), a technique in which single molecules are imaged and tracked in real time, has led to breakthrough discoveries regarding these spatiotemporal complexities of cell membranes. There are, however, emerging concerns about factors that might produce misleading interpretations of SPT results. Here, we briefly review the application of SPT to understanding the nanoscale heterogeneities of plasma membranes, with a focus on the unique challenges, pitfalls, and limitations that confront the use of nanoparticles as imaging probes for tracking the dynamics of single molecules in cell membranes.

HEXIM1 Diffusion in the Nucleus Is Regulated by Its Interactions with Both 7SK and P-TEFb

How nuclear proteins diffuse and find their targets remains a key question in the transcription field. Dynamic proteins in the nucleus are classically subdiffusive and undergo anomalous diffusion, yet the underlying physical mechanisms are still debated. In this study, we explore the contribution of interactions to the generation of anomalous diffusion by the means of fluorescence spectroscopy and simulation. Using interaction-deficient mutants, our study indicates that HEXIM1 interactions with both 7SK RNA and positive transcription elongation factor b are critical for HEXIM1 subdiffusion and thus provides evidence of the effects of protein-RNA interaction on molecular diffusion. Numerical simulations allowed us to establish that the proportions of distinct oligomeric HEXIM1 subpopulations define the apparent anomaly parameter of the whole population. Slight changes in the proportions of these oligomers can lead to significant shifts in the diffusive features and recapitulate the modifications observed in cells with the various interaction-deficient mutants. By combining simulations and experiments, our work opens new prospects in which the anomaly α coefficient in diffusion becomes a helpful tool to infer alterations in molecular interactions.

Analytic model for transient anomalous diffusion with highly persistent correlations

In recent decades, many stochastic processes have been proposed as models for real world time series data with anomalous spreading, highly persistent correlations, and transient distributional characteristics. We introduce the higher order fractional tempered stable motion as the stochastic integral of the tempered stable motion with respect to a generalized higher order moving average kernel, which provides an analytic model for stochastic processes possessing these characteristics. This stochastic process provides a mathematical model for anomalous diffusion with a transient distribution resembling higher order fractional stable motion on short timescales and higher order fractional Brownian motion in the long run. The specifics of the crossover dynamics from the Lévy stable anomalous diffusion to the Gaussian anomalous diffusion are controlled by explicit parameter values that correspond to physical attributes of the process. It is well suited to modeling anomalous diffusion of any "type" (sub-, super-, regular, or hyperdiffusion) under appropriate parametrizations due to its power-law scaling of variance with respect to time. It is also a useful model for position-velocity-acceleration triples due to its convenient path differentiability and integrability properties. To highlight the potential physical relevance of this model for real world data, we outline its key statistical properties including its covariance structure, memory, and second order self-similarity. We also give an easy to implement elementary method for sample path generation which may be used as a basis for simulation and Monte Carlo studies.

Bacterial hopping and trapping in porous media

Diverse processes-e.g. bioremediation, biofertilization, and microbial drug delivery-rely on bacterial migration in disordered, three-dimensional (3D) porous media. However, how pore-scale confinement alters bacterial motility is unknown due to the opacity of typical 3D media. As a result, models of migration are limited and often employ ad hoc assumptions. Here we reveal that the paradigm of run-and-tumble motility is dramatically altered in a porous medium. By directly visualizing individual Escherichia coli, we find that the cells are intermittently and transiently trapped as they navigate the pore space, exhibiting diffusive behavior at long time scales. The trapping durations and the lengths of "hops" between traps are broadly distributed, reminiscent of transport in diverse other disordered systems; nevertheless, we show that these quantities can together predict the long-time bacterial translational diffusivity. Our work thus provides a revised picture of bacterial motility in complex media and yields principles for predicting cellular migration.

Modeling of the axon plasma membrane structure and its effects on protein diffusion

The axon plasma membrane consists of the membrane skeleton, which comprises ring-like actin filaments connected to each other by spectrin tetramers, and the lipid bilayer, which is tethered to the skeleton via, at least, ankyrin. Currently it is unknown whether this unique axon plasma membrane skeleton (APMS) sets the diffusion rules of lipids and proteins in the axon. To answer this question, we developed a coarse-grain molecular dynamics model for the axon that includes the APMS, the phospholipid bilayer, transmembrane proteins (TMPs), and integral monotopic proteins (IMPs) in both the inner and outer lipid layers. We first showed that actin rings limit the longitudinal diffusion of TMPs and the IMPs of the inner leaflet but not of the IMPs of the outer leaflet. To reconcile the experimental observations, which show restricted diffusion of IMPs of the outer leaflet, with our simulations, we conjectured the existence of actin-anchored proteins that form a fence which restricts the longitudinal diffusion of IMPs of the outer leaflet. We also showed that spectrin filaments could modify transverse diffusion of TMPs and IMPs of the inner leaflet, depending on the strength of the association between lipids and spectrin. For instance, in areas where spectrin binds to the lipid bilayer, spectrin filaments would restrict diffusion of proteins within the skeleton corrals. In contrast, in areas where spectrin and lipids are not associated, spectrin modifies the diffusion of TMPs and IMPs of the inner leaflet from normal to confined-hop diffusion. Overall, we showed that diffusion of axon plasma membrane proteins is deeply anisotropic, as longitudinal diffusion is of different type than transverse diffusion. Finally, we investigated how accumulation of TMPs affects diffusion of TMPs and IMPs of both the inner and outer leaflets by changing the density of TMPs. We showed that the APMS structure acts as a fence that restricts the diffusion of TMPs and IMPs of the inner leaflet within the membrane skeleton corrals. Our findings provide insight into how the axon skeleton acts as diffusion barrier and maintains neuronal polarity.

The axon plasma membrane skeleton consists of repeated periodic actin ring-like structures along its length connected via spectrin tetramers and anchored to the lipid bilayer at least via ankyrin. However, it is currently unclear whether this structure controls diffusion of lipids and proteins in the axon. Here, we developed a coarse-grain molecular dynamics computational model for the axon plasma membrane that comprises minimal representations for the APMS and the lipid bilayer. In a departure from current models, we found that actin rings limit diffusion of proteins only in the inner membrane leaflet. Then, we showed that actin anchored proteins likely act as “fences” confining diffusion of proteins in the outer leaflet. Our simulations, unexpectedly, also revealed that spectrin filaments could impede transverse diffusion in the inner leaflet of the axon and in some conditions modify diffusion from normal to abnormal. We predicted that diffusion of axon plasma membrane proteins is anisotropic as longitudinal diffusion is of different type than transverse (azimuthal) diffusion. We conclude that the periodic structure of the axon plays a critical role in controlling diffusion of proteins and lipids in the axon plasma membrane.

Collective effects of yeast cytoplasmic dynein based microtubule transport

Teams of cortically anchored dyneins pulling at microtubules (MTs) are known to be essential for aster, spindle and nuclear positioning during cell division and fertilization. While the single-molecule basis of dynein processivity is now better understood, the effect of increasing numbers of motors on transport is not clear. Here, we examine the collective transport properties of a Saccharomyces cerevisiae cytoplasmic dynein fragment, widely used as a minimal model, by a combination of quantitative MT gliding assays and stochastic simulations. We find both MT lengths and motor densities qualitatively affect the degree of randomness of MT transport. However, the directionality and velocity of MTs increase above a threshold number of motors (N) interacting with a filament. To better understand this behavior, we simulate a gliding assay based on a model of uniformly distributed immobilized motors transporting semi-flexible MTs. Each dynein dimer is modeled as an effective stochastic stepper with asymmetric force dependent detachment dynamics, based on single-molecule experiments. Simulations predict increasing numbers of motors (N) result in a threshold dependent transition in directionality and transport velocity and a monotonic decrease in effective diffusivity. Thus both experiment and theory show qualitative agreement in the emergence of coordination in transport above a threshold number of motor heads. We hypothesize that the phase-transition like property of this dynein could play a role in vivo during yeast mitosis, when this dynein localizes to the cortex and pulls astral MTs of increasing length, resulting in correct positioning and orientation of the nucleus at the bud-neck.

Controlling Anomalous Diffusion in Lipid Membranes

Diffusion in cell membranes is not just simple two-dimensional Brownian motion but typically depends on the timescale of the observation. The physical origins of this anomalous subdiffusion are unresolved, and model systems capable of quantitative and reproducible control of membrane diffusion have been recognized as a key experimental bottleneck. Here, we control anomalous diffusion using supported lipid bilayers containing lipids derivatized with polyethylene glycol (PEG) headgroups. Bilayers with specific excluded area fractions are formed by control of PEG lipid mole fraction. These bilayers exhibit a switch in diffusive behavior, becoming anomalous as bilayer continuity is disrupted. Using a combination of single-molecule fluorescence and interferometric imaging, we measure the anomalous behavior in this model over four orders of magnitude in time. Diffusion in these bilayers is well described by a power-law dependence of the mean-square displacement with observation time. Anomaleity in this system can be tailored by simply controlling the mole fraction of PEG lipid, producing bilayers with diffusion parameters similar to those observed for anomalous diffusion in biological membranes.