How molecular motors work in the crowded environment of living cells: coexistence and efficiency of normal and anomalous transport

Coherent light scattering from cellular dynamics in living tissues

This review examines the biological physics of intracellular transport probed by the coherent optics of dynamic light scattering from optically thick living tissues. Cells and their constituents are in constant motion, composed of a broad range of speeds spanning many orders of magnitude that reflect the wide array of functions and mechanisms that maintain cellular health. From the organelle scale of tens of nanometers and upward in size, the motion inside living tissue is actively driven rather than thermal, propelled by the hydrolysis of bioenergetic molecules and the forces of molecular motors. Active transport can mimic the random walks of thermal Brownian motion, but mean-squared displacements are far from thermal equilibrium and can display anomalous diffusion through Lévy or fractional Brownian walks. Despite the average isotropic three-dimensional environment of cells and tissues, active cellular or intracellular transport of single light-scattering objects is often pseudo-one-dimensional, for instance as organelle displacement persists along cytoskeletal tracks or as membranes displace along the normal to cell surfaces, albeit isotropically oriented in three dimensions. Coherent light scattering is a natural tool to characterize such tissue dynamics because persistent directed transport induces Doppler shifts in the scattered light. The many frequency-shifted partial waves from the complex and dynamic media interfere to produce dynamic speckle that reveals tissue-scale processes through speckle contrast imaging and fluctuation spectroscopy. Low-coherence interferometry, dynamic optical coherence tomography, diffusing-wave spectroscopy, diffuse-correlation spectroscopy, differential dynamic microscopy and digital holography offer coherent detection methods that shed light on intracellular processes. In health-care applications, altered states of cellular health and disease display altered cellular motions that imprint on the statistical fluctuations of the scattered light. For instance, the efficacy of medical therapeutics can be monitored by measuring the changes they induce in the Doppler spectra of livingex vivocancer biopsies.

Non-specific cargo-filament interactions slow down motor-driven transport

Active, motor-based cargo transport is important for many cellular functions and cellular development. However, the cell interior is complex and crowded and could have many weak, non-specific interactions with the cargo being transported. To understand how cargo-environment interactions will affect single motor cargo transport and multi-motor cargo transport, we use an artificial quantum dot cargo bound with few (~ 1) to many (~ 5-10) motors allowed to move in a dense microtubule network. We find that kinesin-driven quantum dot cargo is slower than single kinesin-1 motors. Excitingly, there is some recovery of the speed when multiple motors are attached to the cargo. To determine the possible mechanisms of both the slow down and recovery of speed, we have developed a computational model that explicitly incorporates multi-motor cargos interacting non-specifically with nearby microtubules, including, and predominantly with the microtubule on which the cargo is being transported. Our model has recovered the experimentally measured average cargo speed distribution for cargo-motor configurations with few and many motors, implying that numerous, weak, non-specific interactions can slow down cargo transport and multiple motors can reduce these interactions thereby increasing velocity.

A polymer chain with dipolar active forces in connection to spatial organization of chromatin

A living cell is an active environment where the organization and dynamics of chromatin are affected by different forms of activity. Optical experiments report that loci show subdiffusive dynamics and the chromatin fiber is seen to be coherent over micrometer-scale regions. Using a bead-spring polymer chain with dipolar active forces, we study how the subdiffusive motion of the loci generate large-scale coherent motion of the chromatin. We show that in the presence of extensile (contractile) activity, the dynamics of the loci grows faster (slower) and the spatial correlation length increases (decreases) compared to the case with no dipolar forces. Hence, both the dipolar active forces modify the elasticity of the chain. Interestingly in our model, the dynamics and organization of such dipolar active chains largely differ from the passive chain with renormalized elasticity.

Cell Responsiveness to Physical Energies: Paving the Way to Decipher a Morphogenetic Code

We discuss emerging views on the complexity of signals controlling the onset of biological shapes and functions, from the nanoarchitectonics arising from supramolecular interactions, to the cellular/multicellular tissue level, and up to the unfolding of complex anatomy. We highlight the fundamental role of physical forces in cellular decisions, stressing the intriguing similarities in early morphogenesis, tissue regeneration, and oncogenic drift. Compelling evidence is presented, showing that biological patterns are strongly embedded in the vibrational nature of the physical energies that permeate the entire universe. We describe biological dynamics as informational processes at which physics and chemistry converge, with nanomechanical motions, and electromagnetic waves, including light, forming an ensemble of vibrations, acting as a sort of control software for molecular patterning. Biomolecular recognition is approached within the establishment of coherent synchronizations among signaling players, whose physical nature can be equated to oscillators tending to the coherent synchronization of their vibrational modes. Cytoskeletal elements are now emerging as senders and receivers of physical signals, "shaping" biological identity from the cellular to the tissue/organ levels. We finally discuss the perspective of exploiting the diffusive features of physical energies to afford in situ stem/somatic cell reprogramming, and tissue regeneration, without stem cell transplantation.

Fingerprints of viscoelastic subdiffusion in random environments: Revisiting some experimental data and their interpretations

Many experimental studies revealed subdiffusion of various nanoparticles in diverse polymer and colloidal solutions, cytosol and plasma membrane of biological cells, which are viscoelastic and, at the same time, highly inhomogeneous randomly fluctuating environments. The observed subdiffusion often combines features of ergodic fractional Brownian motion (reflecting viscoelasticity) and nonergodic jumplike non-Markovian diffusional processes (reflecting disorder). Accordingly, several theories were proposed to explain puzzling experimental findings. Below we show that some of the significant and profound published experimental results are better rationalized within the viscoelastic subdiffusion approach in random environments, which is based on generalized Langevin dynamics in random potentials, than some earlier proposed theories.

An overview of diffusion models for intracellular dynamics analysis

We present an overview of diffusion models commonly used for quantifying the dynamics of intracellular particles (e.g. biomolecules) inside eukaryotic living cells. It is established that inference on the modes of mobility of molecules is central in cell biology since it reflects interactions between structures and determines functions of biomolecules in the cell. In that context, Brownian motion is a key component in short distance transportation (e.g. connectivity for signal transduction). Another dynamical process that has been heavily studied in the past decade is the motor-mediated transport (e.g. dynein, kinesin and myosin) of molecules. Primarily supported by actin filament and microtubule network, it ensures spatial organization and temporal synchronization in the intracellular mechanisms and structures. Nevertheless, the complexity of internal structures and molecular processes in the living cell influence the molecular dynamics and prevent the systematic application of pure Brownian or directed motion modeling. On the one hand, cytoskeleton density will hinder the free displacement of the particle, a phenomenon called subdiffusion. On the other hand, the cytoskeleton elasticity combined with thermal bending can contribute a phenomenon called superdiffusion. This paper discusses the basics of diffusion modes observed in eukariotic cells, by introducing the essential properties of these processes. Applications of diffusion models include protein trafficking and transport and membrane diffusion.

Collective transient ratchet transport induced by many elastically interacting particles

Several dynamical systems in nature can be maintained out-of-equilibrium, either through mutual interaction of particles or by external fields. The particle’s transport and the transient dynamics are landmarking of such systems. While single ratchet systems are genuine candidates to describe unbiased transport, we demonstrate here that coupled ratchets exhibit collective transient ratchet transport. Extensive numerical simulations for up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=1024$$\end{document}N=1024 elastically interacting ratchets establish the generation of large transient ratchet currents (RCs). The lifetimes of the transient RCs increase with N and decrease with the coupling strength between the ratchets. We demonstrate one peculiar case having a coupling-induced transient RC through the asymmetric destruction of attractors. Results suggest that physical devices built with coupled ratchet systems should present large collective transient transport of particles, whose technological applications are undoubtedly appealing and feasible.

Dynamic Dissection of Dynein and Kinesin-1 Cooperatively Mediated Intercellular Transport of Porcine Epidemic Diarrhea Coronavirus along Microtubule Using Single Virus Tracking

It is now clear that the intercellular transport on microtubules by dynein and kinesin-1 motors has an important role in the replication and spread of many viruses. Porcine epidemic diarrhea virus (PEDV) is an enveloped, single-stranded RNA virus of the Coronavirus family, which can infect swine of all ages and cause severe economic losses in the swine industry. Elucidating the molecular mechanisms of the intercellular transport of PEDV through microtubule, dynein and kinesin-1 will be crucial for understanding its pathogenesis. Here, we demonstrate that microtubule, dynein, and kinesin-1 are involved in PEDV infection and can influence PEDV fusion and accumulation in the perinuclear region but cannot affect PEDV attachment or internalization. Furthermore, we adopted a single-virus tracking technique to dynamically observe PEDV intracellular transport with five different types: unidirectional movement toward microtubule plus ends; unidirectional movement toward microtubule minus ends; bidirectional movement along the same microtubule; bidirectional movement along different microtubules and motionless state. Among these types, the functions of dynein and kinesin-1 in PEDV intercellular transport were further analyzed by single-virus tracking and found that dynein and kinesin-1 mainly transport PEDV to the minus and plus ends of the microtubules, respectively; meanwhile, they also can transport PEDV to the opposite ends of the microtubules different from their conventional transport directions and also coordinate the bidirectional movement of PEDV along the same or different microtubules through their cooperation. These results provided deep insights and references to understand the pathogenesis of PEDV as well as to develop vaccines and treatments.

Getting around the cell: physical transport in the intracellular world

Eukaryotic cells face the challenging task of transporting a variety of particles through the complex intracellular milieu in order to deliver, distribute, and mix the many components that support cell function. In this review, we explore the biological objectives and physical mechanisms of intracellular transport. Our focus is on cytoplasmic and intra-organelle transport at the whole-cell scale. We outline several key biological functions that depend on physically transporting components across the cell, including the delivery of secreted proteins, support of cell growth and repair, propagation of intracellular signals, establishment of organelle contacts, and spatial organization of metabolic gradients. We then review the three primary physical modes of transport in eukaryotic cells: diffusive motion, motor-driven transport, and advection by cytoplasmic flow. For each mechanism, we identify the main factors that determine speed and directionality. We also highlight the efficiency of each transport mode in fulfilling various key objectives of transport, such as particle mixing, directed delivery, and rapid target search. Taken together, the interplay of diffusion, molecular motors, and flows supports the intracellular transport needs that underlie a broad variety of biological phenomena.

Escape of a passive particle from an activity-induced energy landscape: emergence of slow and fast effective diffusion

Spontaneous persistent motions driven by active processes play a central role in maintaining living cells far from equilibrium. In the majority of research studies, the steady state dynamics of an active system has been described in terms of an effective temperature. By contrast, we have examined a prototype model for diffusion in an activity-induced rugged energy landscape to describe the slow dynamics of a tagged particle in a dense active environment. The expression for the mean escape time from the activity-induced rugged energy landscape holds only in the limit of low activity and the mean escape time from the rugged energy landscape increases with activity. The precise form of the active correlation will determine whether the mean escape time will depend on the persistence time or not. The activity-induced rugged energy landscape approach also allows an estimate of the non-equilibrium effective diffusivity characterizing the slow diffusive motion of the tagged particle due to activity. On the other hand, in a dilute environment, high activity augments the diffusion of the tagged particle. The enhanced diffusion can be attributed to an effective temperature higher than the ambient temperature and this is used to calculate the Kramers' mean escape time, which decreases with activity. Our results have direct relevance to recent experiments on tagged particle diffusion in condensed phases.

The Impact of Rate Formulations on Stochastic Molecular Motor Dynamics

Cells are complex structures which require considerable amounts of organization via transport of large intracellular cargo. While passive diffusion is often sufficiently fast for the transport of smaller cargo, active transport is necessary to organize large structures on the short timescales necessary for biological function. The main mechanism of this transport is by cargo attachment to motors which walk in a directed fashion along intracellular filaments. There are a number of models which seek to describe the motion of motors with attached cargo, from detailed microscopic to coarse phenomenological descriptions. We focus on the intermediate-detailed discrete stochastic hopping models, and explore how cargo transport changes depending on the number of motors, motor interaction, system constraints and rate formulations, which are derived from common thermodynamic assumptions. We find that, despite obeying the same detailed balance constraint, the choice of rate formulation considerably affects the characteristics of the overall motion of the system, with one rate formulation exhibiting novel behavior of loaded motor groups moving faster than a single unloaded motor.

Crowd Control: Effects of Physical Crowding on Cargo Movement in Healthy and Diseased Neurons

High concentration of cytoskeletal filaments, organelles, and proteins along with the space constraints due to the axon's narrow geometry lead inevitably to intracellular physical crowding along the axon of a neuron. Local cargo movement is essential for maintaining steady cargo transport in the axon, and this may be impeded by physical crowding. Molecular motors that mediate active transport share movement mechanisms that allow them to bypass physical crowding present on microtubule tracks. Many neurodegenerative diseases, irrespective of how they are initiated, show increased physical crowding owing to the greater number of stalled organelles and structural changes associated with the cytoskeleton. Increased physical crowding may be a significant factor in slowing cargo transport to synapses, contributing to disease progression and culminating in the dying back of the neuronal process. This review explores the idea that physical crowding can impede cargo movement along the neuronal process. We examine the sources of physical crowding and strategies used by molecular motors that might enable cargo to circumvent physically crowded locations. Finally, we describe sub-cellular changes in neurodegenerative diseases that may alter physical crowding and discuss the implications of such changes on cargo movement.

Antibody-Mediated Immobilization of Virions in Mucus

Antibodies have been shown to hinder the movement of herpes simplex virus virions in cervicovaginal mucus, as well as other viruses in other mucus secretions. However, it has not been possible to directly observe the mechanisms underlying this phenomenon, so the nature of virion-antibody-mucin interactions remain poorly understood. In this work, we analyzed thousands of virion traces from single particle tracking experiments to explicate how antibodies must cooperate to immobilize virions for relatively long time periods. First, using a clustering analysis, we observed a clear separation between two classes of virion behavior: freely diffusing and immobilized. While the proportion of freely diffusing virions decreased with antibody concentration, the magnitude of their diffusivity did not, implying an all-or-nothing dichotomy in the pathwise effect of the antibodies. Proceeding under the assumption that all binding events are reversible, we used a novel switch-point detection method to conclude that there are very few, if any, state switches on the experimental timescale of 20 s. To understand this slow state switching, we analyzed a recently proposed continuous-time Markov chain model for binding kinetics and virion movement. Model analysis implied that virion immobilization requires cooperation by multiple antibodies that are simultaneously bound to the virion and mucin matrix and that there is an entanglement phenomenon that accelerates antibody-mucin binding when a virion is immobilized. In addition to developing a widely applicable framework for analyzing multistate particle behavior, this work substantially enhances our mechanistic understanding of how antibodies can reinforce a mucus barrier against passive invasive species.

Heterogeneities Shape Passive Intracellular Transport

A living cell's interior is one of the most complex and intrinsically dynamic systems, providing an elaborate interplay between cytosolic crowding and ATP-driven motion that controls cellular functionality. Here, we investigated two distinct fundamental features of the merely passive, non-biomotor-shuttled material transport within the cytoplasm of Dictyostelium discoideum cells: the anomalous non-linear scaling of the mean-squared displacement of a 150-nm-diameter particle and non-Gaussian distribution of increments. Relying on single-particle tracking data of 320,000 data points, we performed a systematic analysis of four possible origins for non-Gaussian transport: 1) sample-based variability, 2) rarely occurring strong motion events, 3) ergodicity breaking/aging, and 4) spatiotemporal heterogeneities of the intracellular medium. After excluding the first three reasons, we investigated the remaining hypothesis of a heterogeneous cytoplasm as cause for non-Gaussian transport. A, to our knowledge, novel fit model with randomly distributed diffusivities implementing medium heterogeneities suits the experimental data. Strikingly, the non-Gaussian feature is independent of the cytoskeleton condition and lag time. This reveals that efficiency and consistency of passive intracellular transport and the related anomalous scaling of the mean-squared displacement are regulated by cytoskeleton components, whereas cytoplasmic heterogeneities are responsible for the generic, non-Gaussian distribution of increments.

Crowding-Activity Coupling Effect on Conformational Change of a Semi-Flexible Polymer

The behavior of a polymer in a passive crowded medium or in a very dilute active bath has been well studied, while a polymer immersed in an environment featured by both crowding and activity remains an open problem. In this paper, a systematic Langevin simulation is performed to investigate the conformational change of a semi-flexible chain in a concentrated solution packed with spherical active crowders. A very novel shrinkage-to-swelling transition is observed for a polymer with small rigidity. The underlying phase diagram is constructed in the parameter space of active force and crowder size. Moreover, the variation of the polymer gyration radius demonstrates a non-monotonic dependence on the dynamical persistence length of the active particle. Lastly, the activity-crowding coupling effect in different crowder size baths is clarified. In the case of small crowders, activity strengthens the crowding-induced shrinkage to the chain. As crowder size increases, activity turns out to be a contrasting factor to crowding, resulting in a competitive shrinkage and swelling. In the large size situation, the swelling effect arising from activity eventually becomes dominant. The present study provides a deeper understanding of the unusual behavior of a semi-flexible polymer in an active and crowded medium, associated with the nontrivial activity-crowding coupling and the cooperative crowder size effect.

Host proteins involved in microtubule-dependent HIV-1 intracellular transport and uncoating

Microtubules and microtubule-associated proteins are critical for cargo transport throughout the cell. Many viruses are able to usurp these transport systems for their own replication and spread. HIV-1 utilizes these proteins for many of its early events postentry, including transport, uncoating and reverse transcription. The molecular motor proteins dynein and kinesin-1 are the primary drivers of cargo transport, and HIV-1 utilizes these proteins for infection. In this Review, we highlight recent developments in the understanding of how HIV-1 hijacks motor transport, the key cellular and viral proteins involved, and the ways that transport influences other steps in the HIV-1 lifecycle.

Investigation of multiple-dynein transport of melanosomes by non-invasive force measurement using fluctuation unit χ

Pigment organelles known as melanosomes disperse or aggregate in a melanophore in response to hormones. These movements are mediated by the microtubule motors kinesin-2 and cytoplasmic dynein. However, the force generation mechanism of dynein, unlike that of kinesin, is not well understood. In this study, to address this issue, we investigated the dynein-mediated aggregation of melanosomes in zebrafish melanophores. We applied the fluctuation theorem of non-equilibrium statistical mechanics to estimate forces acting on melanosomes during transport by dynein, given that the energy of a system is related to its fluctuation. Our results demonstrate that multiple force-producing units cooperatively transport a single melanosome. Since the force is generated by dynein, this suggests that multiple dyneins carry a single melanosome. Cooperative transport has been reported for other organelles; thus, multiple-motor transport may be a universal mechanism for moving organelles within the cell.

High-temperature ratchets driven by deterministic and stochastic fluctuations

We consider the overdamped dynamics of a Brownian particle in an arbitrary spatial periodic and time-dependent potential on the basis of an exact solution for the probability density in the form of a power series in the inverse friction coefficient. The expression for the average velocity of a Brownian ratchet is simplified in the high-temperature consideration when only the first terms of the series can be used. For the potential of an additive-multiplicative form (a sum of a time-independent contribution and a time-dependent multiplicative perturbation), general explicit expressions are obtained which allow comparative analysis of frequency dependencies of the average velocity, implying deterministic and stochastic potential energy fluctuations. For qualitative and quantitative analysis of these dependences, we choose illustrative examples for spatial harmonic fluctuations: with deterministic time dependences of a relaxation type and stochastic time dependences describing Markovian dichotomous and harmonic noise processes. We explore the influence of fluctuation types on the ratchet effect and demonstrate its enhancement in the case of harmonic noise.

Non-Markovian intracellular transport with sub-diffusion and run-length dependent detachment rate

Intracellular transport of organelles is fundamental to cell function and health. The mounting evidence suggests that this transport is in fact anomalous. However, the reasons for the anomaly is still under debate. We examined experimental trajectories of organelles inside a living cell and propose a mathematical model that describes the previously reported transition from sub-diffusive to super-diffusive motion. In order to explain super-diffusive behaviour at long times, we introduce non-Markovian detachment kinetics of the cargo: the rate of detachment is inversely proportional to the time since the last attachment. Recently, we observed the non-Markovian detachment rate experimentally in eukaryotic cells. Here we further discuss different scenarios of how this effective non-Markovian detachment rate could arise. The non-Markovian model is successful in simultaneously describing the time averaged variance (the time averaged mean squared displacement corrected for directed motion), the mean first passage time of trajectories and the multiple peaks observed in the distributions of cargo velocities. We argue that non-Markovian kinetics could be biologically beneficial compared to the Markovian kinetics commonly used for modelling, by increasing the average distance the cargoes travel when a microtubule is blocked by other filaments. In turn, sub-diffusion allows cargoes to reach neighbouring filaments with higher probability, which promotes active motion along the microtubules.

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

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

Cancer drug therapy and stochastic modeling of "nano-motors"

Background

Controlled inhibition of kinesin motor proteins is highly desired in the field of oncology. Among other interventions, there exists “targeted chemotherapeutic regime/options” of selective Eg5 competitive and allosteric inhibitors, inducing cancer cell apoptosis and tumor regression with improved safety profiles.

Research question

Though promising, such studies are still under clinical trials, for the discovery of efficient and least harmful Eg5 inhibitors. The aim of this research was to bridge the computational modeling approach with drug design and therapy of cancer cells.

Methods

A computational model, interfaced with the clinical data of “Eg5 dynamics” and “inhibitors” via special functions, is presented in this article. Comparisons are made for the drug efficacy, and the threshold values are predicted through numerical simulations.

Results

Results are obtained to depict the dynamics induced by ispinesib, when used as an inhibitor of kinesin Eg5, on cancer cell lines.

Viscoelastic subdiffusion in a random Gaussian environment

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

Small-scale displacement fluctuations of vesicles in fibroblasts

The intracellular environment is a dynamic space filled with various organelles moving in all directions. Included in this diverse group of organelles are vesicles, which are involved in transport of molecular cargo throughout the cell. Vesicles move in either a directed or non-directed fashion, often depending on interactions with cytoskeletal proteins such as microtubules, actin filaments, and molecular motors. How these proteins affect the local fluctuations of vesicles in the cytoplasm is not clear since they have the potential to both facilitate and impede movement. Here we show that vesicle mobility is significantly affected by myosin-II, even though it is not a cargo transport motor. We find that myosin-II activity increases the effective diffusivity of vesicles and its inhibition facilitates longer states of non-directed motion. Our study suggests that altering myosin-II activity in the cytoplasm of cells can modulate the mobility of vesicles, providing a possible mechanism for cells to dynamically tune the cytoplasmic environment in space and time.

Label-free Multiscale Transport Imaging of the Living Cell

The living cell is characterized by a myriad of parallel intracellular transport processes. Simultaneously capturing their global features across multiple temporal and spatial scales is a nearly unsurmountable task. Here we present a method that enables the microscopic imaging of the entire spectrum of intracellular transport on a broad time scale without the need for prior labeling. We show that from the time-dependent fluctuation of pixel intensity, in either bright-field or phase-contrast microscopic images, a scaling factor can be derived that reflects the local Hurst coefficient (H), the value of which reveals the microscopic mechanisms of intracellular motion. The Hurst coefficient image of the interphase cell displays an unexpected, overwhelming superdiffusion (H > 0.5) in the cytoplasm and subdiffusion (H < 0.5) in the nucleus, and provides unprecedented sensitivity in detecting transport processes associated with the living state.

Retraction of rod-like mitochondria during microtubule-dependent transport

Molecular motors play relevant roles on the regulation of mitochondria size and shape, essential properties for the cell homeostasis. In this work, we tracked single rod-shaped mitochondria with nanometer precision to explore the performance of microtubule motor teams during processive anterograde and retrograde transport. We analyzed simultaneously the organelle size and verified that mitochondria retracted during retrograde transport with their leading tip moving slower in comparison with the rear tip. In contrast, mitochondria preserved their size during anterograde runs indicating a different performance of plus-end directed teams. These results were interpreted considering the different performance of dynein and kinesin teams and provide valuable information on the collective action of motors during mitochondria transport.

Motion of kinesin in a viscoelastic medium

Kinesin is a molecular motor that transports cargo along microtubules. The results of many in vitro experiments on kinesin-1 are described by kinetic models in which one transition corresponds to the forward motion and subsequent binding of the tethered motor head. We argue that in a viscoelastic medium like the cytosol of a cell this step is not Markov and has to be described by a nonexponential waiting time distribution. We introduce a semi-Markov kinetic model for kinesin that takes this effect into account. We calculate, for arbitrary waiting time distributions, the moment generating function of the number of steps made, and determine from this the average velocity and the diffusion constant of the motor. We illustrate our results for the case of a waiting time distribution that is Weibull. We find that for realistic parameter values, viscoelasticity decreases the velocity and the diffusion constant, but increases the randomness (or Fano factor).

Active and passive transport of cargo in a corrugated channel: A lattice model study

Inside cells, cargos such as vesicles and organelles are transported by molecular motors to their correct locations via active motion on cytoskeletal tracks and passive, Brownian diffusion. During the transportation of cargos, motor-cargo complexes (MCCs) navigate the confining and crowded environment of the cytoskeletal network and other macromolecules. Motivated by this, we study a minimal two-state model of motor-driven cargo transport in confinement and predict transport properties that can be tested in experiments. We assume that the motion of the MCC is directly affected by the entropic barrier due to confinement if it is in the passive, unbound state but not in the active, bound state where it moves with a constant bound velocity. We construct a lattice model based on a Fokker Planck description of the two-state system, study it using a kinetic Monte Carlo method and compare our numerical results with analytical expressions for a mean field limit. We find that the effect of confinement strongly depends on the bound velocity and the binding kinetics of the MCC. Confinement effectively reduces the effective diffusivity and average velocity, except when it results in an enhanced average binding rate and thereby leads to a larger average velocity than when unconfined.

Sensing Magnetic Fields with Magnetosensitive Ion Channels

[-15]Magnetic nanoparticles are met across many biological species ranging from magnetosensitive bacteria, fishes, bees, bats, rats, birds, to humans. They can be both of biogenetic origin and due to environmental contamination, being either in paramagnetic or ferromagnetic state. The energy of such naturally occurring single-domain magnetic nanoparticles can reach up to 10-20 room k B T in the magnetic field of the Earth, which naturally led to supposition that they can serve as sensory elements in various animals. This work explores within a stochastic modeling framework a fascinating hypothesis of magnetosensitive ion channels with magnetic nanoparticles serving as sensory elements, especially, how realistic it is given a highly dissipative viscoelastic interior of living cells and typical sizes of nanoparticles possibly involved.

A blueprint for robust crosslinking of mobile species in biogels with weakly adhesive molecular anchors

Biopolymeric matrices can impede transport of nanoparticulates and pathogens by entropic or direct adhesive interactions, or by harnessing "third-party" molecular anchors to crosslink nanoparticulates to matrix constituents. The trapping potency of anchors is dictated by association rates and affinities to both nanoparticulates and matrix; the popular dogma is that long-lived, high-affinity bonds to both species facilitate optimal trapping. Here we present a contrasting paradigm combining experimental evidence (using IgG antibodies and Matrigel®), a theoretical framework (based on multiple timescale analysis), and computational modeling. Anchors that bind and unbind rapidly from matrix accumulate on nanoparticulates much more quickly than anchors that form high-affinity, long-lived bonds with matrix, leading to markedly greater trapping potency of multiple invading species without saturating matrix trapping capacity. Our results provide a blueprint for engineering molecular anchors with finely tuned affinities to effectively enhance the barrier properties of biogels against diverse nanoparticulate species.Biological polymeric matrices often use molecular anchors, such as antibodies, to trap nanoparticulates. Here, the authors find that anchor-matrix bonds that are weak and short-lived confer superior trapping potency, contrary to the prevailing belief that effective molecular anchors should form strong bonds to both the matrix and the nanoparticulates.

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.

Run-and-pause dynamics of cytoskeletal motor proteins

Cytoskeletal motor proteins are involved in major intracellular transport processes which are vital for maintaining appropriate cellular function. When attached to cytoskeletal filaments, the motor exhibits distinct states of motility: active motion along the filaments, and pause phase in which it remains stationary for a finite time interval. The transition probabilities between motion and pause phases are asymmetric in general, and considerably affected by changes in environmental conditions which influences the efficiency of cargo delivery to specific targets. By considering the motion of individual non-interacting molecular motors on a single filament as well as a dynamic filamentous network, we present an analytical model for the dynamics of self-propelled particles which undergo frequent pause phases. The interplay between motor processivity, structural properties of filamentous network, and transition probabilities between the two states of motility drastically changes the dynamics: multiple transitions between different types of anomalous diffusive dynamics occur and the crossover time to the asymptotic diffusive or ballistic motion varies by several orders of magnitude. We map out the phase diagrams in the space of transition probabilities, and address the role of initial conditions of motion on the resulting dynamics.

Molecular machines operating on the nanoscale: from classical to quantum

The main physical features and operating principles of isothermal nanomachines in the microworld, common to both classical and quantum machines, are reviewed. Special attention is paid to the dual, constructive role of dissipation and thermal fluctuations, the fluctuation-dissipation theorem, heat losses and free energy transduction, thermodynamic efficiency, and thermodynamic efficiency at maximum power. Several basic models are considered and discussed to highlight generic physical features. This work examines some common fallacies that continue to plague the literature. In particular, the erroneous beliefs that one should minimize friction and lower the temperature for high performance of Brownian machines, and that the thermodynamic efficiency at maximum power cannot exceed one-half are discussed. The emerging topic of anomalous molecular motors operating subdiffusively but very efficiently in the viscoelastic environment of living cells is also discussed.

Non-Brownian diffusion in lipid membranes: Experiments and simulations

The dynamics of constituents and the surface response of cellular membranes-also in connection to the binding of various particles and macromolecules to the membrane-are still a matter of controversy in the membrane biophysics community, particularly with respect to crowded membranes of living biological cells. We here put into perspective recent single particle tracking experiments in the plasma membranes of living cells and supercomputing studies of lipid bilayer model membranes with and without protein crowding. Special emphasis is put on the observation of anomalous, non-Brownian diffusion of both lipid molecules and proteins embedded in the lipid bilayer. While single component, pure lipid bilayers in simulations exhibit only transient anomalous diffusion of lipid molecules on nanosecond time scales, the persistence of anomalous diffusion becomes significantly longer ranged on the addition of disorder-through the addition of cholesterol or proteins-and on passing of the membrane lipids to the gel phase. Concurrently, experiments demonstrate the anomalous diffusion of membrane embedded proteins up to macroscopic time scales in the minute time range. Particular emphasis will be put on the physical character of the anomalous diffusion, in particular, the occurrence of ageing observed in the experiments-the effective diffusivity of the measured particles is a decreasing function of time. Moreover, we present results for the time dependent local scaling exponent of the mean squared displacement of the monitored particles. Recent results finding deviations from the commonly assumed Gaussian diffusion patterns in protein crowded membranes are reported. The properties of the displacement autocorrelation function of the lipid molecules are discussed in the light of their appropriate physical anomalous diffusion models, both for non-crowded and crowded membranes. In the last part of this review we address the upcoming field of membrane distortion by elongated membrane-binding particles. We discuss how membrane compartmentalisation and the particle-membrane binding energy may impact the dynamics and response of lipid membranes. This article is part of a Special Issue entitled: Biosimulations edited by Ilpo Vattulainen and Tomasz Róg.

Models for microtubule cargo transport coupling the Langevin equation to stochastic stepping motor dynamics: Caring about fluctuations

One-dimensional models coupling a Langevin equation for the cargo position to stochastic stepping dynamics for the motors constitute a relevant framework for analyzing multiple-motor microtubule transport. In this work we explore the consistence of these models focusing on the effects of the thermal noise. We study how to define consistent stepping and detachment rates for the motors as functions of the local forces acting on them in such a way that the cargo velocity and run-time match previously specified functions of the external load, which are set on the base of experimental results. We show that due to the influence of the thermal fluctuations this is not a trivial problem, even for the single-motor case. As a solution, we propose a motor stepping dynamics which considers memory on the motor force. This model leads to better results for single-motor transport than the approaches previously considered in the literature. Moreover, it gives a much better prediction for the stall force of the two-motor case, highly compatible with the experimental findings. We also analyze the fast fluctuations of the cargo position and the influence of the viscosity, comparing the proposed model to the standard one, and we show how the differences on the single-motor dynamics propagate to the multiple motor situations. Finally, we find that the one-dimensional character of the models impede an appropriate description of the fast fluctuations of the cargo position at small loads. We show how this problem can be solved by considering two-dimensional models.

Effect of elastic colored noise in the hopping dynamics of single molecules in stretching experiments

The influence of colored noise induced by elastic fluctuations in single-molecule stretching experiments is theoretically and numerically studied. Unlike in the thermal white noise case currently considered in the literature, elastically induced hopping dynamics between folded and unfolded states is manifested through critical oscillations showing smaller end-to-end distance fluctuations (δx∼1.25nm) within the free energy wells corresponding to both states. Our results are derived by analyzing the elastic coupling between the Handle-Molecule-Handle system and the laser optical tweezers (LOT) array. It is shown that an Ornstein-Uhlenbeck process related to this elastic coupling may trigger the hopping transitions via a colored noise with an intensity proportional to the elastic constant of the LOT array. Evolution equations of the variables of the system were derived by using the irreversible thermodynamics of small systems recently proposed. Theoretical expressions for the corresponding stationary probability densities are provided and the viability of inferring the shape of the free energy from direct measurements is discussed.

Role of trapping and crowding as sources of negative differential mobility

Increasing the crowding in an environment does not necessarily trigger negative differential mobility of strongly pushed particles. Moreover, the choice of the model, in particular the kind of microscopic jump rates, may be very relevant in determining the mobility. We support these points via simple examples and we therefore address recent claims saying that crowding in an environment is likely to promote negative differential mobility. Trapping of tagged particles enhanced by increasing the force remains the mechanism determining a drift velocity not monotonous in the driving force.

Modeling magnetosensitive ion channels in the viscoelastic environment of living cells

We propose and study a model of hypothetical magnetosensitive ionic channels which are long thought to be a possible candidate to explain the influence of weak magnetic fields on living organisms ranging from magnetotactic bacteria to fishes, birds, rats, bats, and other mammals including humans. The core of the model is provided by a short chain of magnetosomes serving as a sensor, which is coupled by elastic linkers to the gating elements of ion channels forming a small cluster in the cell membrane. The magnetic sensor is fixed by one end on cytoskeleton elements attached to the membrane and is exposed to viscoelastic cytosol. Its free end can reorient stochastically and subdiffusively in viscoelastic cytosol responding to external magnetic field changes and can open the gates of coupled ion channels. The sensor dynamics is generally bistable due to bistability of the gates which can be in two states with probabilities which depend on the sensor orientation. For realistic parameters, it is shown that this model channel can operate in the magnetic field of Earth for a small number (five to seven) of single-domain magnetosomes constituting the sensor rod, each of which has a typical size found in magnetotactic bacteria and other organisms or even just one sufficiently large nanoparticle of a characteristic size also found in nature. It is shown that, due to the viscoelasticity of the medium, the bistable gating dynamics generally exhibits power law and stretched exponential distributions of the residence times of the channels in their open and closed states. This provides a generic physical mechanism for the explanation of the origin of such anomalous kinetics for other ionic channels whose sensors move in a viscoelastic environment provided by either cytosol or biological membrane, in a quite general context, beyond the fascinating hypothesis of magnetosensitive ionic channels we explore.

Distributed-order diffusion equations and multifractality: Models and solutions

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

Fractal model of anomalous diffusion

An equation of motion is derived from fractal analysis of the Brownian particle trajectory in which the asymptotic fractal dimension of the trajectory has a required value. The formula makes it possible to calculate the time dependence of the mean square displacement for both short and long periods when the molecule diffuses anomalously. The anomalous diffusion which occurs after long periods is characterized by two variables, the transport coefficient and the anomalous diffusion exponent. An explicit formula is derived for the transport coefficient, which is related to the diffusion constant, as dependent on the Brownian step time, and the anomalous diffusion exponent. The model makes it possible to deduce anomalous diffusion properties from experimental data obtained even for short time periods and to estimate the transport coefficient in systems for which the diffusion behavior has been investigated. The results were confirmed for both sub and super-diffusion.

Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion

The discovery of anomalous diffusion of larger biopolymers and submicron tracers such as endogenous granules, organelles, or virus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complex environment equally impacts the active intracellular transport of submicron cargos by molecular motors such as kinesins: does the passive anomalous diffusion of free cargo always imply its anomalously slow active transport by motors, the mean transport distance along microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashing binding potential, where the cargo particle is elastically attached to the motor. Depending on the cargo size, the loading force, the amplitude of the binding potential, the turnover frequency of the molecular motor enzyme, and the linker stiffness we demonstrate that the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highly efficient normal active transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and study the intricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomes anomalously slow and thus significantly less efficient.

Anomalous transport of subdiffusing cargos by single kinesin motors: the role of mechano-chemical coupling and anharmonicity of tether

Here we generalize our previous model of molecular motors trafficking subdiffusing cargos in viscoelastic cytosol by (i) including mechano-chemical coupling between cyclic conformational fluctuations of the motor protein driven by the reaction of ATP hydrolysis and its translational motion within the simplest two-state model of hand-over-hand motion of kinesin, and also (ii) by taking into account the anharmonicity of the tether between the motor and the cargo (its maximally possible extension length). It is shown that the major earlier results such as occurrence of normal versus anomalous transport depending on the amplitude of binding potential, cargo size and the motor turnover frequency not only survive in this more realistic model, but the results also look very similar for the correspondingly adjusted parameters. However, this more realistic model displays a substantially larger thermodynamic efficiency due to a bidirectional mechano-chemical coupling. For realistic parameters, the maximal thermodynamic efficiency can transiently be about 50% as observed for kinesins, and even larger, surprisingly also in a novel strongly anomalous (sub)transport regime, where the motor enzymatic turnovers become also anomalously slow and cannot be characterized by a turnover rate. Here anomalously slow dynamics of the cargo enforces anomalously slow cyclic kinetics of the motor protein.

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

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

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

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

Influence of molecular motors on the motion of particles in viscoelastic media

We study theoretically and by numerical simulations the motion of particles driven by molecular motors in a viscoelastic medium representing the cell cytoplasm. For this, we consider a generalized Langevin equation coupled to a stochastic stepping dynamics for the motors that takes into account the action of each motor separately. In the absence of motors, the model produces subdiffusive motion of particles characterized by a power-law scaling of the mean square displacement versus the lag time as t^{α}, with 0<α<1, similar to that observed in cells. Our results show how the action of the motors can induce a transition to a superdiffusive regime at large lag times with the characteristics of those found in experiments reported in the literature. We also show that at small lag times, the motors can act as static crosslinkers that slow down the natural subdiffusive transport. An analysis of previously reported experimental data in the relevant time scales provides evidence of this phenomenon. Finally, we study the effect of a harmonic potential representing an optical trap, and we show a way to approach to a macroscopic description of the active transport in cells. This last point stresses the relevance of the molecular motors for generating not only directed motion to specific targets, but also fast diffusivelike random motion.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.