Self-organized density patterns of molecular motors in arrays of cytoskeletal filaments

Tunable intracellular transport on converging microtubule morphologies

A common type of cytoskeletal morphology involves multiple microtubules converging with their minus ends at the microtubule organizing center (MTOC). The cargo-motor complex will experience ballistic transport when bound to microtubules or diffusive transport when unbound. This machinery allows for sequestering and subsequent dispersal of dynein-transported cargo. The general principles governing dynamics, efficiency, and tunability of such transport in the MTOC vicinity are not fully understood. To address this, we develop a one-dimensional model that includes advective transport toward an attractor (such as the MTOC) and diffusive transport that allows particles to reach absorbing boundaries (such as cellular membranes). We calculated the mean first passage time (MFPT) for cargo to reach the boundaries as a measure of the effectiveness of sequestering (large MFPT) and diffusive dispersal (low MFPT). We show that the MFPT experiences a dramatic growth, transitioning from a low to high MFPT regime (dispersal to sequestering) over a window of cargo on-/off-rates that is close to in vivo values. Furthermore, increasing either the on-rate (attachment) or off-rate (detachment) can result in optimal dispersal when the attractor is placed asymmetrically. Finally, we also describe a regime of rare events where the MFPT scales exponentially with motor velocity and the escape location becomes exponentially sensitive to the attractor positioning. Our results suggest that structures such as the MTOC allow for the sensitive control of the spatial and temporal features of transport and corresponding function under physiological conditions.

Motor Protein Transport Along Inhomogeneous Microtubules

Many cellular processes rely on the cell's ability to transport material to and from the nucleus. Networks consisting of many microtubules and actin filaments are key to this transport. Recently, the inhibition of intracellular transport has been implicated in neurodegenerative diseases such as Alzheimer's disease and Amyotrophic Lateral Sclerosis. Furthermore, microtubules may contain so-called defective regions where motor protein velocity is reduced due to accumulation of other motors and microtubule-associated proteins. In this work, we propose a new mathematical model describing the motion of motor proteins on microtubules which incorporate a defective region. We take a mean-field approach derived from a first principle lattice model to study motor protein dynamics and density profiles. In particular, given a set of model parameters we obtain a closed-form expression for the equilibrium density profile along a given microtubule. We then verify the analytic results using mathematical analysis on the discrete model and Monte Carlo simulations. This work will contribute to the fundamental understanding of inhomogeneous microtubules providing insight into microscopic interactions that may result in the onset of neurodegenerative diseases. Our results for inhomogeneous microtubules are consistent with prior work studying the homogeneous case.

Propagation Modeling and Analysis of Molecular Motors in Molecular Communication

Molecular motor networks (MMNs) are networks constructed from molecular motors to enable nanomachines to perform coordinated tasks of sensing, computing, and actuation at the nano- and micro- scales. Living cells are naturally enabled with this same mechanism to establish point-to-point communication between different locations inside the cell. Similar to a railway system, the cytoplasm contains an intricate infrastructure of tracks, named microtubules, interconnecting different internal components of the cell. Motor proteins, such as kinesin and dynein, are able to travel along these tracks directionally, carrying with them large molecules that would otherwise be unreliably transported across the cytoplasm using free diffusion. Molecular communication has been previously proposed for the design and study of MMNs. However, the topological aspects of MMNs, including the effects of branches, have been ignored in the existing studies. In this paper, a physical end-to-end model for MMNs is developed, considering the location of the transmitter node, the network topology, and the receiver nodes. The end-to-end gain and group delay are considered as the performance measures, and analytical expressions for them are derived. The analytical model is validated by Monte-Carlo simulations and the performance of MMNs is analyzed numerically. It is shown that, depending on their nature and position, MMN nodes create impedance effects that are critical for the overall performance. This model could be applied to assist the design of artificial MMNs and to study cargo transport in neurofilaments to elucidate brain diseases related to microtubule jamming.

Navigating the cell: how motors overcome roadblocks and traffic jams to efficiently transport cargo

Intracellular transport plays an essential role in maintaining the organization of cells. The importance of long-range, bi-directional transport is evidenced by the fact that its failure goes hand in hand with several diseases including neurodegenerative diseases such as Alzheimer's and Amyotrophic Lateral Sclerosis. The nanoscale cellular transport machinery consisting of cytoskeletal tracks and motor-proteins is responsible for effectively delivering important materials to specific locations inside the cell. Motor-proteins manage to overcome several challenges in the crowded cellular environment to achieve well-coordinated and effective transport. In recent years, thanks to state-of-the-art single molecule biophysical tools, we have started to gain insights into the cellular traffic rules. This perspective summarizes the challenges that motors face in navigating the complex cytoskeleton and the lessons learned about transport in crowded environments from both bottom-up in vitro studies as well as top-down in vivo studies.

Microtubule orientation and spacing within bundles is critical for long-range kinesin-1 motility

Cells rely on active transport to quickly organize cellular cargo. How cells regulate transport is not fully understood. One proposed mechanism is that motor activity could be altered through the architecture of the cytoskeleton. This mechanism is supported by the fact that the cytoskeletal network is tightly regulated in cells and filament polarity within networks dictates motor directionality. For instance, axons contain bundles of parallel microtubules and all cargos with the same motor species will move in the same direction. It is not clear how other types of networks, such as antiparallel bundles in dendrites, can regulate motor transport. To understand how the organization of microtubules within bundles can regulate transport, we studied kinesin-1 motility on three bundle types: random-polarity bundles that are close-packed, parallel polarity bundles, and antiparallel polarity bundles that are spaced apart. We find that close-packed bundles inhibit motor motion, while parallel arrays support unidirectional motion. Spacing the microtubules with microtubule-associated proteins enhances run lengths. Our results indicate that microtubule bundle architecture dictates the motion of single motors and could have effects on cargo transport. © 2014 Wiley Periodicals, Inc.

Measuring transport of motor cargos

In this chapter, we describe experimental techniques used in vitro to illuminate how small teams of motors can work to translocate cargos. We will focus on experiments utilizing in vitro reconstitution, artificial or ex vivo purified cargos, and fluorescence imaging. A number of studies have been able to recapitulate the activities of cargo transport driven by small teams of motors elucidating how multiple motors can work together to transport cargos within the cell. Here, we describe some of the methods employed and highlight important experimental details needed to perform these experiments.

Modeling cytoskeletal traffic: an interplay between passive diffusion and active transport

We introduce the totally asymmetric simple exclusion process with Langmuir kinetics on a network as a microscopic model for active motor protein transport on the cytoskeleton, immersed in the diffusive cytoplasm. We discuss how the interplay between active transport along a network and infinite diffusion in a bulk reservoir leads to a heterogeneous matter distribution on various scales: we find three regimes for steady state transport, corresponding to the scale of the network, of individual segments, or local to sites. At low exchange rates strong density heterogeneities develop between different segments in the network. In this regime one has to consider the topological complexity of the whole network to describe transport. In contrast, at moderate exchange rates the transport through the network decouples, and the physics is determined by single segments and the local topology. At last, for very high exchange rates the homogeneous Langmuir process dominates the stationary state. We introduce effective rate diagrams for the network to identify these different regimes. Based on this method we develop an intuitive but generic picture of how the stationary state of excluded volume processes on complex networks can be understood in terms of the single-segment phase diagram.

Motor transport of self-assembled cargos in crowded environments

Intracellular transport of cargo particles is performed by multiple motors working in concert. However, the mechanism of motor association to cargos is unknown. It is also unknown how long individual motors stay attached, how many are active, and how multimotor cargos would navigate a densely crowded filament with many other motors. Prior theoretical and experimental biophysical model systems of intracellular cargo have assumed fixed teams of motors transporting along bare microtubules or microtubules with fixed obstacles. Here, we investigate a regime of cargos transporting along microtubules crowded with free motors. Furthermore, we use cargos that are able to associate or dissociate motors as it translocates. We perform in vitro motility reconstitution experiments with high-resolution particle tracking. Our model system consists of a quantum dot cargo attached to kinesin motors, and additional free kinesin motors that act as traffic along the microtubule. Although high densities of kinesin motors hinder forward motion, resulting in a lower velocity, the ability to associate motors appears to enhance the run length and attachment time of the quantum dot, improving overall cargo transport. These results suggest that cargos that can associate new motors as they transport could overcome traffic jams.

Localization of cytokinesis factors to the future cell division site by microtubule-dependent transport

The mechanism by which spindle microtubules (MTs) determine the site of cell division in animal cells is still highly controversial. Putative cytokinesis "signals" have been proposed to be positioned by spindle MTs at equatorial cortical regions to increase cortical contractility and/or at polar regions to decrease contractility [Rappaport, 1986; von Dassow, 2009]. Given the relative paucity of MTs at the future division site, it has not been clear how MTs localize cytokinesis factors there. Here, we test cytokinesis models using computational and experimental approaches. We present a simple lattice-based model in which signal-kinesin complexes move by transient plus-end directed movements on MTs interspersed with occasions of uniform diffusion in the cytoplasm. In simulations, complexes distribute themselves initially at the spindle midzone and then move on astral MTs to accumulate with time at the equatorial cortex. Simulations accurately predict cleavage patterns of cells with different geometries and MT arrangements and elucidate several experimental observations that have defied easy explanation by previous models. We verify this model with experiments on indented sea urchin zygotes showing that cells often divide perpendicular to the spindle at sites distinct from the indentations. These studies support an equatorial stimulation model and provide a simple mechanism explaining how cytokinesis factors localize to the future division site.

Interplay of magnetic interactions and active movements in the formation of magnetosome chains

Magnetotactic bacteria assemble chains of magnetosomes, organelles that contain magnetic nano-crystals. A number of genetic factors involved in the controlled biomineralization of these crystals and the assembly of magnetosome chains have been identified in recent years, but how the specific biological regulation is coordinated with general physical processes such as diffusion and magnetic interactions remains unresolved. Here, these questions are addressed by simulations of different scenarios for magnetosome chain formation, in which various physical processes and interactions are either switched on or off. The simulation results indicate that purely physical processes of magnetosome diffusion, guided by their magnetic interactions, are not sufficient for the robust chain formation observed experimentally and suggest that biologically encoded active movements of magnetosomes may be required. Not surprisingly, the chain pattern is most resembling experimental results when both magnetic interactions and active movement are coordinated. We estimate that the force such active transport has to generate is compatible with forces generated by the polymerization or depolymerization of cytoskeletal filaments. The simulations suggest that the pleiotropic phenotypes of mamK deletion strains may be due to a defect in active motility of magnetosomes and that crystal formation in magneteosome vesicles is coupled to the activation of their active motility in M. gryphiswaldense, but not in M. magneticum.

Spatial modeling of vesicle transport and the cytoskeleton: the challenge of hitting the right road

The membrane trafficking machinery provides a transport and sorting system for many cellular proteins. We propose a mechanistic agent-based computer simulation to integrate and test the hypothesis of vesicle transport embedded into a detailed model cell. The method tracks both the number and location of the vesicles. Thus both the stochastic properties due to the low numbers and the spatial aspects are preserved. The underlying molecular interactions that control the vesicle actions are included in a multi-scale manner based on the model of Heinrich and Rapoport (2005). By adding motor proteins we can improve the recycling process of SNAREs and model cell polarization. Our model also predicts that coat molecules should have a high turnover at the compartment membranes, while the turnover of motor proteins has to be slow. The modular structure of the underlying model keeps it tractable despite the overall complexity of the vesicle system. We apply our model to receptor-mediated endocytosis and show how a polarized cytoskeleton structure leads to polarized distributions in the plasma membrane both of SNAREs and the Ste2p receptor in yeast. In addition, we can couple signal transduction and membrane trafficking steps in one simulation, which enables analyzing the effect of receptor-mediated endocytosis on signaling.

The RABL5 homolog IFT22 regulates the cellular pool size and the amount of IFT particles partitioned to the flagellar compartment in Chlamydomonas reinhardtii

Cilia and flagella, sensory and motile structures protruding from the cell body, rely on the continuous bidirectional traffic of intraflagellar transport (IFT) particles to ferry flagellar precursors into flagella for assembly. Cells synthesize a large pool of IFT particle proteins in the cell body, but only a small portion engages in active transport within the flagella at any given time. The atypical small G protein Rab-like 5 (RABL5) has been shown to move in an IFT-like manner in the flagella, but its function in ciliogenesis is controversial. In this report, we demonstrate that IFT22, the Chlamydomonas reinhardtii homolog of RABL5, is a bona fide IFT particle complex B subunit. Although the amount of IFT22 remains unaffected by depletion of either complex A or B, depletion of IFT22 leads to a smaller pool of both complex A and B. Strikingly, the smaller cellular pool of IFT particles does not lead to a reduced distribution of IFT particles to flagella. Instead, the amount of IFT particle proteins, including IFT22 itself, increase in the flagella. Moreover, cells over-expressing IFT22 also accumulate IFT particles in their flagella. Taken together, these data indicate that, in C. reinhardtii, IFT22 controls the cellular levels of both complex A and B, thus plays a critical role in determining the cellular availability of IFT particles. In addition, although IFT22 may not directly carry any precursors for flagellar assembly, it controls how many IFT particles participate in ferrying precursors into flagella.

COOPERATIVE TRANSPORT BY SMALL TEAMS OF MOLECULAR MOTORS

Molecular motors power directed transport of cargoes within cells. Even if a single motor is sufficient to transport a cargo, motors often cooperate in small teams. We discuss the cooperative cargo transport by several motors theoretically and explore some of its properties. In particular we emphasize how motor teams can drag cargoes through a viscous environment.

ACTIVE BIO-SYSTEMS: FROM SINGLE MOTOR MOLECULES TO COOPERATIVE CARGO TRANSPORT

Living cells contain a large number of molecular motors that convert the chemical energy released from nucleotide hydrolysis into mechanical work. This review focusses on stepping motors that move along cytoskeletal filaments. The behavior of these motors involves three distinct nonequilibrium processes that cover a wide range of length and time scales: (i) Directed stepping of single motors bound to a filament; (ii) Composite motor walks of single motors consisting of directed stepping interrupted by diffusive motion; and (iii) Cooperative transport by teams of several motors. On the molecular scale, the energy conversion of these motors leads to single steps along the filaments with a step size of about 10 nm. The corresponding chemomechanical coupling is governed by several distinct motor cycles, which represent the dominant pathways for different values of nucleotide concentrations and load force. For the kinesin motor, the competition of two such cycles determines the stall force, at which the motor velocity vanishes and the motor reverses the direction of its motion. Because of thermal noise, the stepping motors unbind from the filaments after a certain run time and run length. For kinesin, the run time is about 1 s and the run length is about 1 μm for high ATP concentration and low load force. On length scales that are large compared to the run length, a single motor undergoes composite walks consisting of directed stepping interrupted by diffusive motion. The relative importance of bound and unbound motor states depends on the binding and unbinding rates of the motors. The effective transport velocity and diffusion coefficient of the motors are determined by the geometry of the compartments, in which the motors move. The effective diffusion coefficient can be enhanced by several orders of magnitude if the motors undergo active diffusion by interacting with certain filament patterns. In vivo, stepping motors are responsible for the transport of vesicles and other types of intracellular cargo particles that shuttle between the different cell compartments. This cargo transport is usually performed by teams of motors. If all motors belong to the same molecular species, the cooperative action of the motors leads to uni-directional transport with a strongly increased run length and to a characteristic force dependence of the velocity distributions. If two antagonistic species of motors pull on the cargo, they perform a stochastic tug-of-war, which is characterized by a subtle force balance between the two motor teams and leads to seven distinct patterns of uni- and bi-directional transport. So far, all experimental observations on bi-directional transport are consistent with such a tug-of-war. Finally, the traffic of interacting motors is also briefly discussed. Depending on their mutual interactions and the compartment geometry, the motors form various spatio-temporal patterns such as traffic jams, and undergo nonequilibrium phase transitions between such transport patterns.

Loop formation of microtubules during gliding at high density

The microtubule cytoskeleton, including the associated proteins, forms a complex network essential to multiple cellular processes. Microtubule-associated motor proteins, such as kinesin-1, travel on microtubules to transport membrane bound vesicles across the crowded cell. Other motors, such as cytoplasmic dynein and kinesin-5, are used to organize the cytoskeleton during mitosis. In order to understand the self-organization processes of motors on microtubules, we performed filament-gliding assays with kinesin-1 motors bound to the cover glass with a high density of microtubules on the surface. To observe microtubule organization, 3% of the microtubules were fluorescently labeled to serve as tracers. We find that microtubules in these assays are not confined to two dimensions and can cross one other. This causes microtubules to align locally with a relatively short correlation length. At high density, this local alignment is enough to create 'intersections' of perpendicularly oriented groups of microtubules. These intersections create vortices that cause microtubules to form loops. We characterize the radius of curvature and time duration of the loops. These different behaviors give insight into how crowded conditions, such as those in the cell, might affect motor behavior and cytoskeleton organization.

Providing positional information with active transport on dynamic microtubules

Microtubules (MTs) are dynamic protein polymers that change their length by switching between growing and shrinking states in a process termed dynamic instability. It has been suggested that the dynamic properties of MTs are central to the organization of the eukaryotic intracellular space, and that they are involved in the control of cell morphology, but the actual mechanisms are not well understood. Here, we present a theoretical analysis in which we explore the possibility that a system of dynamic MTs and MT end-tracking molecular motors is providing specific positional information inside cells. We compute the MT length distribution for the case of MT-length-dependent switching between growing and shrinking states, and analyze the accumulation of molecular motors at the tips of growing MTs. Using these results, we show that a transport system consisting of dynamic MTs and associated motor proteins can deliver cargo proteins preferentially to specific positions within the cell. Comparing our results with experimental data in the model organism fission yeast, we propose that the suggested mechanisms could play important roles in setting length scales during cellular morphogenesis.

Active transport and cluster formation on 2D networks

We introduce a model for active transport on inhomogeneous networks embedded in a diffusive environment which is motivated by vesicular transport on actin filaments. In the presence of a hard-core interaction, particle clusters are observed that exhibit an algebraically decaying distribution in a large parameter regime, indicating the existence of clusters on all scales. The scale-free behavior can be understood by a mechanism promoting preferential attachment of particles to large clusters. The results are compared with a diffusion-limited aggregation model and active transport on a regular network. For both models we observe aggregation of particles to clusters which are characterized by a finite size scale if the relevant time scales and particle densities are considered.

Effects of the chemomechanical stepping cycle on the traffic of molecular motors

We discuss effects of the stepping kinetics of molecular motors on their traffic behavior on crowded filaments using a simple two-state chemomechanical cycle. While the general traffic behavior is quite robust with respect to the detailed kinetics of the step, a few observable parameters exhibit a strong dependence on these parameters. Most strikingly, the effective unbinding rate of the motors may both increase and decrease with increasing traffic density, depending on the details of the motor step. Likewise the run length either exhibits a strong decrease or almost no dependence on the traffic density. We compare our theoretical results with recent experimental observations on motor traffic.

Quantifying intermittent transport in cell cytoplasm

Active cellular transport is a fundamental mechanism for protein and vesicle delivery, cell cycle, and molecular degradation. Viruses can hijack the transport system and use it to reach the nucleus. Most transport processes consist of intermittent dynamics, where the motion of a particle, such as a virus, alternates between pure Brownian and directed movement along microtubules. In this Rapid Communication, we estimate the mean time for a particle to attach to a microtubule network. This computation leads to a coarse grained equation of the intermittent motion in radial and cylindrical geometries. Finally, by using the degradation activity inside the cytoplasm, we obtain refined asymptotic estimations for the probability and the mean time a virus reaches a small nuclear pore.

Transport of beads by several kinesin motors

The movements of beads pulled by several kinesin-1 (conventional kinesin) motors are studied both theoretically and experimentally. While the velocity is approximately independent of the number of motors pulling the beads, the walking distance or run-length is strongly increased when more motors are involved. Run-length distributions are measured for a wide range of motor concentrations and matched to theoretically calculated distributions using only two global fit parameters. In this way, the maximal number of motors pulling the beads is estimated to vary between two and seven motors for total kinesin concentrations between 0.1 and 2.5 microg/ml or between 0.27 and 6.7 nM. In the same concentration regime, the average number of pulling motors is found to lie between 1.1 and 3.2 motors.

Modeling of pattern regulation in melanophores

Melanosomes, pigment granules in melanophores, play a principal role in physiological color adaptation of fish and frog. Melanophores regulate melanosome trafficking on cytoskeletal filaments to generate a range of spatiotemporal patterns. Here, we present the first comprehensive model of spatiotemporal evolution of melanosome patterns. The model encompasses both physical and biochemical aspects of melanosome dynamics. It consists of (i) a kinetic description of biochemical reactions involved in intracellular signaling, (ii) a system of macroscopic reaction-diffusion-convection equations for melanosome concentration, and (iii) a set of constitutive relationships for coupling transport with the biochemical network. The model relates molecular-level regulatory actions to cell-level melanosome distribution, allowing unification of existing experimental observations and qualitative hypotheses into an integrated, consistent framework. The model reproduces salient features of melanosome patterns, both during transient and steady state. It gives useful insights into how cells coordinate motor-assisted transport to maintain and adapt spatial organization of intracellular organelles. In particular, we calculate the optimal transition paths from aggregation to dispersion in fish melanophores. The calculations suggest that fish melanophores optimally control intracellular signaling to maximize the efficiency of motor-assisted transport during dispersion.

Modelling microtubule patterns

The cellular cytoskeleton is well studied in terms of its biological and physical properties, making it an attractive subject for systems approaches. Here, we describe the experimental and theoretical strategies used to study the collective behaviour of microtubules and motors. We illustrate how this led to the beginning of an understanding of dynamic cellular patterns that have precise functions.

Theory of spatial patterns of intracellular organelles

Here we report on a generalized theory of spatial patterns of intracellular organelles, which are controlled by cells using cytoskeleton-based movements powered by molecular motors. The theory reveals that organelles exhibit one of the four distinct, stable patterns, namely aggregation, hyperdispersion, radial dispersion, and areal dispersion. Existence of specific patterns is determined by the contributions from three transport mechanisms, characterized by two Peclet numbers. The predicted patterns compare well with experimental data. This study provides a firm theoretical ground for classification of spatial patterns of organelles and understanding their regulation by cells.

Molecular motor traffic in a half-open tube

The traffic of molecular motors which interact through mutual exclusion is studied theoretically for half-open tube-like compartments. These half-open tubes mimic the shapes of axons. The mutual exclusion leads to traffic jams or density plateaus on the filaments. A phase transition is obtained when the motor velocity changes sign. We identify the relevant length scales and characterize the jamming behaviour using both analytical approximations and Monte Carlo simulations of lattice models.

Cooperative cargo transport by several molecular motors

The transport of cargo particles that are pulled by several molecular motors in a cooperative manner is studied theoretically in this article. The transport properties depend primarily on the maximal number N of motor molecules that may pull simultaneously on the cargo particle. Because each motor must unbind from the filament after a finite number of steps but can also rebind to it again, the actual number of pulling motors is not constant but varies with time between zero and N. An increase in the maximal number N leads to a strong increase of the average walking distance (or run length) of the cargo particle. If the cargo is pulled by up to N kinesin motors, for example, the walking distance is estimated to be 5(N-1)/N micrometers, which implies that seven or eight kinesin molecules are sufficient to attain an average walking distance in the centimeter range. If the cargo particle is pulled against an external load force, this force is shared between the motors, which provides a nontrivial motor-motor coupling and a generic mechanism for nonlinear force-velocity relationships. With increasing load force, the probability distribution of the instantaneous velocity is shifted toward smaller values, becomes broader, and develops several peaks. Our theory is consistent with available experimental data and makes quantitative predictions that are accessible to systematic in vitro experiments.