Superdiffusive motion of membrane-targeting C2 domains

Reorganization of the Flagellum Scaffolding Induces a Sperm Standstill During Fertilization

Mammalian sperm delve into the female reproductive tract to fertilize the female gamete. The available information about how sperm regulate their motility during the final journey to the fertilization site is extremely limited. In this work, we investigated the structural and functional changes in the sperm flagellum after AE and during the interaction with the eggs. The evidence demonstrates that the double helix actin network surrounding the mitochondrial sheath of the midpiece undergoes structural changes prior to the motility cessation. This structural modification is accompanied by a decrease in diameter of the midpiece and is driven by intracellular calcium changes that occur concomitant with a reorganization of the actin helicoidal cortex. Midpiece contraction occurs in a subset of cells that undergo AE, live-cell imaging during in vitro fertilization showed that the midpiece contraction is required for motility cessation after fusion is initiated. These findings provide the first evidence of the F-actin network's role in regulating sperm motility, adapting its function to meet specific cellular requirements during fertilization, and highlighting the broader significance of understanding sperm motility.

Significant statement

In this work, we demonstrate that the helical structure of polymerized actin in the flagellum undergoes a rearrangement at the time of sperm-egg fusion. This process is driven by intracellular calcium and promotes a decrease in the sperm midpiece diameter as well as the arrest in motility, which is observed after the fusion process is initiated.

Non-Markovian Diffusion and Adsorption-Desorption Dynamics: Analytical and Numerical Results

The interplay of diffusion with phenomena like stochastic adsorption-desorption, absorption, and reaction-diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, we investigate the diffusion process in heterogeneous media, focusing on non-Markovian diffusion. This process is limited by a surface interaction with the bulk, described by a specific boundary condition relevant to systems such as living cells and biomaterials. The surface can adsorb and desorb particles, and the adsorbed particles may undergo lateral diffusion before returning to the bulk. Different behaviors of the system are identified through analytical and numerical approaches.

HSAFM-MIREBA - Methodology for Inferring REsolution in biological applications

Due to a lack of requirement for any direct labelling of the target molecule, high speed atomic force microscopy (HS-AFM) is a potentially powerful procedure for the assessment of biological processes involving macromolecules. When the sample is static the AFM device can be purposefully setup to recover high-resolution information about the feature in question. However, when the feature to be studied moves an appreciable amount during the course of the measurement, the obtained image will be blurred. Encountering such blurred observations prompts the experimenter to sacrifice higher resolution images for higher scanning speeds by tuning available experimental parameters (such as the scanned image area, the image pixel size, the resonance frequency of the cantilever and/or the diameter of the AFM tip). The present work describes a software tool, HSAFM-MIREBA (High Speed Atomic Force Microscopy - Methodology for Inferring REsolution in Biological Applications) that allows for pre-experimental optimization of such parameters through iterative rounds of simulation of both the dynamic surface process and the HS-AFM measurement (based on the particular set of governing parameters). A representative set of five dynamic biological processes that describe a range of diffusive and directed motions (which can themselves be tuned by altering characteristic governing parameter sets) are provided.

Heterogeneous anomalous transport in cellular and molecular biology

It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.

Protein-coated nanoparticles exhibit Lévy flights on a suspended lipid bilayer

Lateral diffusion of nano-objects on lipid membranes is a crucial process in cell biology. Recent studies indicate that nanoparticle lateral diffusion is affected by the presence of membrane proteins and deviates from Brownian motion. Gold nanoparticles (Au NPs) stabilized by short thiol ligands were dispersed near a free-standing bilayer formed in a 3D microfluidic chip. Using dark-field microscopy, the position of single NPs at the bilayer surface was tracked over time. Numerical analysis of the NP trajectories shows that NP diffusion on the bilayer surface corresponds to Brownian motion. The addition of bovine serum albumin (BSA) protein to the solution led to the formation of a protein corona on the NP surface. We found that protein-coated NPs show anomalous superdiffusion and that the distribution of their relative displacement obeys Lévy flight statistics. This superdiffusive motion is attributed to a drastic reduction in adhesive energies between the NPs and the bilayer in the presence of the protein corona. This hypothesis was confirmed by numerical simulations mimicking the random walk of a single particle near a weakly adhesive surface. These results may be generalized to other classes of nano-objects that experience adsorption-desorption behaviour with a weakly adhesive surface.

Rotational Dynamics of The Transmembrane Domains Play an Important Role in Peptide Dynamics of Viral Fusion and Ion Channel Forming Proteins—A Molecular Dynamics Simulation Study

Focusing on the transmembrane domains (TMDs) of viral fusion and channel-forming proteins (VCPs), experimentally available and newly generated peptides in an ideal conformation of the S and E proteins of severe acute respiratory syndrome coronavirus type 2 (SARS-CoV-2) and SARS-CoV, gp41 and Vpu, both of human immunodeficiency virus type 1 (HIV-1), haemagglutinin and M2 of influenza A, as well as gB of herpes simplex virus (HSV), are embedded in a fully hydrated lipid bilayer and used in multi-nanosecond molecular dynamics simulations. It is aimed to identify differences in the dynamics of the individual TMDs of the two types of viral membrane proteins. The assumption is made that the dynamics of the individual TMDs are decoupled from their extra-membrane domains, and that the mechanics of the TMDs are distinct from each other due to the different mechanism of function of the two types of proteins. The diffusivity coefficient (DC) of the translational and rotational diffusion is decreased in the oligomeric state of the TMDs compared to those values when calculated from simulations in their monomeric state. When comparing the calculations for two different lengths of the TMD, a longer full peptide and a shorter purely TMD stretch, (i) the difference of the calculated DCs begins to level out when the difference exceeds approximately 15 amino acids per peptide chain, and (ii) the channel protein rotational DC is the most affected diffusion parameter. The rotational dynamics of the individual amino acids within the middle section of the TMDs of the fusion peptides remain high upon oligomerization, but decrease for the channel peptides, with an increasing number of monomers forming the oligomeric state, suggesting an entropic penalty on oligomerization for the latter.

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

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

Cdc42 localized in the CatSper signaling complex regulates cAMP-dependent pathways in mouse sperm

Sperm acquire the ability to fertilize in a process called capacitation and undergo hyperactivation, a change in the motility pattern, which depends on Ca²⁺ transport by CatSper channels. CatSper is essential for fertilization and it is subjected to a complex regulation that is not fully understood. Here, we report that similar to CatSper, Cdc42 distribution in the principal piece is confined to four linear domains and this localization is disrupted in CatSper1-null sperm. Cdc42 inhibition impaired CatSper activity and other Ca²⁺ -dependent downstream events resulting in a severe compromise of the sperm fertilizing potential. We also demonstrate that Cdc42 is essential for CatSper function by modulating cAMP production by soluble adenylate cyclase (sAC), providing a new regulatory mechanism for the stimulation of CatSper by the cAMP-dependent pathway. These results reveal a broad mechanistic insight into the regulation of Ca²⁺ in mammalian sperm, a matter of critical importance in male infertility as well as in contraception.

Molecular motion and tridimensional nanoscale localization of kindlin control integrin activation in focal adhesions

Focal adhesions (FAs) initiate chemical and mechanical signals involved in cell polarity, migration, proliferation and differentiation. Super-resolution microscopy revealed that FAs are organized at the nanoscale into functional layers from the lower plasma membrane to the upper actin cytoskeleton. Yet, how FAs proteins are guided into specific nano-layers to promote interaction with given targets is unknown. Using single protein tracking, super-resolution microscopy and functional assays, we link the molecular behavior and 3D nanoscale localization of kindlin with its function in integrin activation inside FAs. We show that immobilization of integrins in FAs depends on interaction with kindlin. Unlike talin, kindlin displays free diffusion along the plasma membrane outside and inside FAs. We demonstrate that the kindlin Pleckstrin Homology domain promotes membrane diffusion and localization to the membrane-proximal integrin nano-layer, necessary for kindlin enrichment and function in FAs. Using kindlin-deficient cells, we show that kindlin membrane localization and diffusion are crucial for integrin activation, cell spreading and FAs formation. Thus, kindlin uses a different route than talin to reach and activate integrins, providing a possible molecular basis for their complementarity during integrin activation.

Dynamics of long-term protein aggregation on low-fouling surfaces

Understanding the mechanisms of protein interactions with solid surfaces is critical to predict how proteins affect the performance of materials in biological environments. Low-fouling and ultra-low fouling surfaces are often evaluated in short-term protein adsorption experiments, where 'short-term' is defined as the time required to reach an initial apparent or pseudo-equilibrium, which is usually less than 600 s. However, it has long been recognized that these short-term observations fail to predict protein adsorption behavior in the long-term, characterized by irreversible accumulation of protein on the surface. This important long-term behavior is frequently ignored or attributed to slow changes in surface chemistry over time-such as oxidation-often with little or no experimental evidence. Here, we report experiments measuring protein adsorption on "low-fouling" and "ultralow-fouling" surfaces using single-molecule localization microscopy to directly probe protein adsorption and desorption. The experiments detect protein adsorption for thousands of seconds, enabling direct observation of both short-term (reversible adsorption) and long-term (irreversible adsorption leading to accumulation) protein-surface interactions. By bridging the gap between these two time scales in a single experiment, this work enables us to develop a single mathematical model that predicts behavior in both temporal regimes. The experimental data in combination with the resulting model provide several important insights: (1) short-term measurements of protein adsorption using ensemble-averaging methods may not be sufficient for designing antifouling materials; (2) all investigated surfaces eventually foul when in long-term contact with protein solutions; (3) fouling can occur through surface-induced oligomerization of proteins which may be a distinct step from irreversible adsorption; and (4) surfaces can be designed to reduce oligomerization or the adsorption of oligomers, to prevent or delay fouling.

Non-Normalizable Quasi-Equilibrium Solution of the Fokker-Planck Equation for Nonconfining Fields

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker–Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker–Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.

Testing of Multifractional Brownian Motion

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

Transient dynamics in a nonequilibrium superdiffusive reaction-diffusion process: Nonequilibrium random search as a case study

Transient regimes, often difficult to characterize, can be fundamental in establishing final steady states features of reaction-diffusion phenomena. This is particularly true in ecological problems. Here, through both numerical simulations and an analytic approximation, we analyze the transient of a nonequilibrium superdiffusive random search when the targets are created at a certain rate and annihilated upon encounters (a key dynamics, e.g., in biological foraging). The steady state is achieved when the number of targets stabilizes to a constant value. Our results unveil how key features of the steady state are closely associated to the particularities of the initial evolution. The searching efficiency variation in time is also obtained. It presents a rather surprising universal behavior at the asymptotic limit. These analyses shed some light into the general relevance of transients in reaction-diffusion systems.

Elucidating the Origin of Heterogeneous Anomalous Diffusion in the Cytoplasm of Mammalian Cells

Diffusion of tracer particles in the cytoplasm of mammalian cells is often anomalous with a marked heterogeneity even within individual particle trajectories. Despite considerable efforts, the mechanisms behind these observations have remained largely elusive. To tackle this problem, we performed extensive single-particle tracking experiments on quantum dots in the cytoplasm of living mammalian cells at varying conditions. Analyses of the trajectories reveal a strong, microtubule-dependent subdiffusion with antipersistent increments and a substantial heterogeneity. Furthermore, particles stochastically switch between different mobility states, most likely due to transient associations with the cytoskeleton-shaken endoplasmic reticulum network. Comparison to simulations highlight that all experimental observations can be fully described by an intermittent fractional Brownian motion, alternating between two states of different mobility.

Anomalous protein kinetics on low-fouling surfaces

In this work, protein-surface interactions were probed in terms of adsorption and desorption of a model protein, bovine serum albumin, on a low-fouling surface with single-molecule localization microscopy. Single-molecule experiments enable precise determination of both adsorption and desorption rates. Strikingly the experimental data show anomalous desorption kinetics, evident as a surface dwell time that exhibits a power-law distribution, i.e. a heavy-tailed rather than the expected exponential distribution. As a direct consequence of this heavy-tailed distribution, the average desorption rate depends upon the time scale of the experiment and the protein surface concentration does not reach equilibrium. Further analysis reveals that the observed anomalous desorption emerges due to the reversible formation of a small fraction of soluble protein multimers (small oligomers), such that each one desorbs from the surface with a different rate. The overall kinetics can be described by a series of elementary reactions, yielding simple scaling relations that predict experimental observations. This work reveals a mechanistic origin for anomalous desorption kinetics that can be employed to interpret observations where low-protein fouling surfaces eventually foul when in long-term contact with protein solutions. The work also provides new insights that can be used to define design principles for non-fouling surfaces and to predict their performance.

Self-assembly of lipid rafts revealed by fluorescence correlation spectroscopy in living breast cancer cells

Lipids and proteins in the plasma membrane are laterally heterogeneous and formalised as lipid rafts featuring unique biophysical properties. However, the self-assembly mechanism of lipid raft cannot be revealed even its physical properties and components were determined in specific physiological processes. In this study, two-photon generalised polarisation imaging and fluorescence correlation spectroscopy were used to study the fusion of lipid rafts through the membrane phase and the lateral diffusion of lipids in living breast cancer cells. A self-assembly model of lipid rafts associated with lipid diffusion and membrane phase was proposed to demonstrate the lipid sorting ability of lipid rafts in the plasma membrane. The results showed that the increased proportion of slow subdiffusion of G_M1 -binding cholera toxin B-subunit (CT-B) was accompanied with an increased liquid-ordered domain during the β-estradiol-induced fusion of lipid rafts. And slow subdiffusion of CT-B was vanished with the depletion of lipid rafts. Whereas the dialkylindocarbocyanine (DiIC₁₈ ) diffusion was not specifically regulated by lipid rafts. This study will open up a new insight for uncovering the self-assembly of lipid rafts in specific pathophysiological processes.

Protein adsorption measurements on low fouling and ultralow fouling surfaces: A critical comparison of surface characterization techniques

Ultralow protein fouling behavior is a common target for new high-performance materials. Ultralow fouling is often defined based on the amount of irreversibly adsorbed protein (< 5 ng cm^-2) measured by a surface ensemble averaging method. However, protein adsorption at solid interfaces is a dynamic process involving multiple steps, which may include adsorption, desorption, and irreversible protein denaturation. In order to better optimize the performance of antifouling surfaces, it is imperative to fully understand how proteins interact with surfaces, including kinetics of adsorption and desorption, conformation, stability, and amount of adsorbed proteins. Defining ultralow fouling surfaces based on a measurement at or near the limit of detection of a surface-averaged measurement may not capture all of this behavior. Single-molecule microscopy techniques can resolve individual protein-surface interactions with high temporal and spatial resolution. This information can be used to tune the properties of surfaces to better resist protein adsorption. In this work, we demonstrate how combining surface plasmon resonance, X-ray photoelectron spectroscopy, atomic force microscopy, and single-molecule localization microscopy provides a more complete picture of protein adsorption on low fouling and ultralow fouling polyelectrolyte multilayer and polymer brush surfaces, over different regimes of protein concentration. In this case, comparing the surfaces using surface plasmon resonance alone is insufficient to rank their resistance to protein adsorption. Our results suggest a revision of the accepted definition of ultralow fouling surfaces is timely: with the advent of time-resolved studies of protein adsorption kinetics at the single-molecule level, it is neither necessary nor sufficient to rely on a surface averaging techniques to qualify ultralow fouling surfaces. Since protein adsorption is a dynamic process, understanding how surface properties affect the kinetics of protein adsorption will enable the design of future generations of advanced antifouling materials. STATEMENT OF SIGNIFICANCE: The design of ultralow fouling surfaces is often optimized based on a single surface-averaging technique measuring the amount of irreversibly adsorbed protein. This work provides a critical comparison of alternative techniques for evaluating protein adsorption on low fouling and ultralow fouling surfaces, and demonstrates how additional information about the dynamics of protein-surface interactions at the interface can be obtained by application of single-molecule microscopy. This approach could be used to better elucidate mechanisms of protein resistance and design principles for advanced ultralow fouling materials.

Statistical testing approach for fractional anomalous diffusion classification

Taking advantage of recent single-particle tracking data, we compare the popular standard mean-squared displacement method with a statistical testing hypothesis procedure for three testing statistics and for two particle types: membrane receptors and the G proteins coupled to them. Each method results in different classifications. For this reason, more rigorous statistical tests are analyzed here in detail. The main conclusion is that the statistical testing approaches might provide good results even for short trajectories, but none of the proposed methods is "the best" for all considered examples; in other words, one needs to combine different approaches to get a reliable classification.

From Non-Normalizable Boltzmann-Gibbs Statistics to Infinite-Ergodic Theory

We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as 1/x, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a mean-squared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.

Identifying diffusive motions in single-particle trajectories on the plasma membrane via fractional time-series models

In this paper we show that an autoregressive fractionally integrated moving average time-series model can identify two types of motion of membrane proteins on the surface of mammalian cells. Specifically we analyze the motion of the voltage-gated sodium channel Nav1.6 and beta-2 adrenergic receptors. We find that the autoregressive (AR) part models well the confined dynamics whereas the fractionally integrated moving average (FIMA) model describes the nonconfined periods of the trajectories. Since the Ornstein-Uhlenbeck process is a continuous counterpart of the AR model, we are also able to calculate its physical parameters and show their biological relevance. The fitted FIMA and AR parameters show marked differences in the dynamics of the two studied molecules.

Nanostructured Surfaces That Mimic the Vascular Endothelial Glycocalyx Reduce Blood Protein Adsorption and Prevent Fibrin Network Formation

Blood-contacting materials are critical in many applications where long-term performance is desired. However, there are currently no engineered materials used in cardiovascular implants and devices that completely prevent clotting when in long-term contact with whole blood. The most common approach to developing next-generation blood-compatible materials is to design surface chemistries and structures that reduce or eliminate protein adsorption to prevent blood clotting. This work proposes a new paradigm for controlling protein-surface interactions by strategically mimicking key features of the glycocalyx lining the interior surfaces of blood vessels: negatively charged glycosaminoglycans organized into a polymer brush with nanoscale domains. The interactions of two important proteins from blood (albumin and fibrinogen) with these new glycocalyx mimics are revealed in detail using surface plasmon resonance and single-molecule microscopy. Surface plasmon resonance shows that these blood proteins interact reversibly with the glycocalyx mimics, but have no irreversible adsorption above the limit of detection. Single-molecule microscopy is used to compare albumin and fibrinogen interactions on surfaces with and without glycocalyx-mimetic nanostructures. Microscopy videos reveal a new mechanism whereby the glycocalyx-mimetic nanostructures eliminate the formation of fibrin networks on the surfaces. This approach shows for the first time that the nanoscale structure and organization of glycosaminoglycans in the glycocalyx are essential to (i) reduce protein adsorption, (ii) reversibly bind fibrin(ogen), and (iii) inhibit fibrin network formation on surfaces. The insights gained from this work suggest new design principles for blood-compatible surfaces. New surfaces developed using these design principles could reduce risk of catastrophic failures of blood-contacting medical devices.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Single-molecule force spectroscopy of protein-membrane interactions

Many biological processes rely on protein–membrane interactions in the presence of mechanical forces, yet high resolution methods to quantify such interactions are lacking. Here, we describe a single-molecule force spectroscopy approach to quantify membrane binding of C2 domains in Synaptotagmin-1 (Syt1) and Extended Synaptotagmin-2 (E-Syt2). Syts and E-Syts bind the plasma membrane via multiple C2 domains, bridging the plasma membrane with synaptic vesicles or endoplasmic reticulum to regulate membrane fusion or lipid exchange, respectively. In our approach, single proteins attached to membranes supported on silica beads are pulled by optical tweezers, allowing membrane binding and unbinding transitions to be measured with unprecedented spatiotemporal resolution. C2 domains from either protein resisted unbinding forces of 2–7 pN and had binding energies of 4–14 k_BT per C2 domain. Regulation by bilayer composition or Ca²⁺ recapitulated known properties of both proteins. The method can be widely applied to study protein–membrane interactions.

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

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

Ergodicity breaking on the neuronal surface emerges from random switching between diffusive states

Stochastic motion on the surface of living cells is critical to promote molecular encounters that are necessary for multiple cellular processes. Often the complexity of the cell membranes leads to anomalous diffusion, which under certain conditions it is accompanied by non-ergodic dynamics. Here, we unravel two manifestations of ergodicity breaking in the dynamics of membrane proteins in the somatic surface of hippocampal neurons. Three different tagged molecules are studied on the surface of the soma: the voltage-gated potassium and sodium channels Kv1.4 and Nav1.6 and the glycoprotein CD4. In these three molecules ergodicity breaking is unveiled by the confidence interval of the mean square displacement and by the dynamical functional estimator. Ergodicity breaking is found to take place due to transient confinement effects since the molecules alternate between free diffusion and confined motion.

Quantifying non-ergodicity of anomalous diffusion with higher order moments

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

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

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

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

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

Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length

In this paper, we study ergodic properties of α-stable autoregressive fractionally integrated moving average (ARFIMA) processes which form a large class of anomalous diffusions. A crucial practical question is how long trajectories one needs to observe in an experiment in order to claim that the analyzed data are ergodic or not. This will be solved by checking the asymptotic convergence to 0 of the empirical estimator F(n) for the dynamical functional D(n) defined as a Fourier transform of the n-lag increments of the ARFIMA process. Moreover, we introduce more flexible concept of the ε-ergodicity.

Strange kinetics of bulk-mediated diffusion on lipid bilayers

Diffusion at solid-liquid interfaces is crucial in many technological and biophysical processes. Although its behavior seems to be deceivingly simple, recent studies showing passive superdiffusive transport suggest that diffusion on surfaces may hide rich complexities. In particular, bulk-mediated diffusion occurs when molecules are transiently released from the surface to perform three-dimensional excursions into the liquid bulk. This phenomenon bears the dichotomy where a molecule always return to the surface but the mean jump length is infinite. Such behavior is associated with a breakdown of the central limit theorem and weak ergodicity breaking. Here, we use single-particle tracking to study the statistics of bulk-mediated diffusion on a supported lipid bilayer. We find that the time-averaged mean square displacement (MSD) of individual trajectories, the archetypal measure in diffusion processes, does not converge to the ensemble MSD but it remains a random variable, even in the long observation-time limit. The distribution of time averages is shown to agree with a Lévy flight model. Our results also unravel intriguing anomalies in the statistics of displacements. The time-averaged MSD is shown to depend on experimental time and investigations of fractional moments show a scaling 〈|r(t)|(q)〉∼t(qν(q)) with non-linear exponents, i.e. ν(q) ≠ const. This type of behavior is termed strong anomalous diffusion and is rare among experimental observations.