Force production by depolymerizing microtubules: a theoretical study

Responsive Supramolecular Polymers for Diagnosis and Treatment

Stimuli-responsive supramolecular polymers are ordered nanosized materials that are held together by non-covalent interactions (hydrogen-bonding, metal-ligand coordination, π-stacking and, host-guest interactions) and can reversibly undergo self-assembly. Their non-covalent nature endows supramolecular polymers with the ability to respond to external stimuli (temperature, light, ultrasound, electric/magnetic field) or environmental changes (temperature, pH, redox potential, enzyme activity), making them attractive candidates for a variety of biomedical applications. To date, supramolecular research has largely evolved in the development of smart water-soluble self-assemblies with the aim of mimicking the biological function of natural supramolecular systems. Indeed, there is a wide variety of synthetic biomaterials formulated with responsiveness to control and trigger, or not to trigger, aqueous self-assembly. The design of responsive supramolecular polymers ranges from the use of hydrophobic cores (i.e., benzene-1,3,5-tricarboxamide) to the introduction of macrocyclic hosts (i.e., cyclodextrins). In this review, we summarize the most relevant advances achieved in the design of stimuli-responsive supramolecular systems used to control transport and release of both diagnosis agents and therapeutic drugs in order to prevent, diagnose, and treat human diseases.

Measuring and modeling forces generated by microtubules

Tubulins are essential proteins, which are conserved across all eukaryotic species. They polymerize to form microtubules, cytoskeletal components of paramount importance for cellular mechanics. The microtubules combine an extraordinarily high flexural rigidity and a non-equilibrium behavior, manifested in their intermittent assembly and disassembly. These chemically fueled dynamics allow microtubules to generate significant pushing and pulling forces at their ends to reposition intracellular organelles, remodel membranes, bear compressive forces, and transport chromosomes during cell division. In this article, we review classical and recent studies, which have allowed the quantification of microtubule-generated forces. The measurements, to which we owe most of the quantitative information about microtubule forces, were carried out in biochemically reconstituted systems in vitro. We also discuss how mathematical and computational modeling has contributed to the interpretations of these results and shaped our understanding of the mechanisms of force production by tubulin polymerization and depolymerization.

Alternative Approaches to Understand Microtubule Cap Morphology and Function

Microtubules (MTs) are essential cellular machines built from concatenated αβ-tubulin heterodimers. They are responsible for two central and opposite functions from the dynamic point of view: scaffolding (static filaments) and force generation (dynamic MTs). These roles engage multiple physiological processes, including cell shape, polarization, division and movement, and intracellular long-distance transport. At the most basic level, the MT regulation is chemical because GTP binding and hydrolysis have the ability to promote assembly and disassembly in the absence of any other constraint. Due to the stochastic GTP hydrolysis, a chemical gradient from GTP-bound to GDP-bound tubulin is created at the MT growing end (GTP cap), which is translated into a cascade of structural regulatory changes known as MT maturation. This is an area of intense research, and several models have been proposed based on information mostly gathered from macromolecular crystallography and cryo-electron microscopy studies. However, these classical structural biology methods lack temporal resolution and can be complemented, as shown in this mini-review, by other approaches such as time-resolved fiber diffraction and computational modeling. Together with studies on structurally similar tubulins from the prokaryotic world, these inputs can provide novel insights on MT assembly, dynamics, and the GTP cap.

Working strokes produced by curling protofilaments at disassembling microtubule tips can be biochemically tuned and vary with species

The disassembly of microtubules can generate force and drive intracellular motility. During mitosis, for example, chromosomes remain persistently attached via kinetochores to the tips of disassembling microtubules, which pull the sister chromatids apart. According to the conformational wave hypothesis, such force generation requires that protofilaments curl outward from the disassembling tips to exert pulling force directly on kinetochores. Rigorously testing this idea will require modifying the mechanical and energetic properties of curling protofilaments, but no way to do so has yet been described. Here, by direct measurement of working strokes generated in vitro by curling protofilaments, we show that their mechanical energy output can be increased by adding magnesium, and that yeast microtubules generate larger and more energetic working strokes than bovine microtubules. Both the magnesium and species-dependent increases in work output can be explained by lengthening the protofilament curls, without any change in their bending stiffness or intrinsic curvature. These observations demonstrate how work output from curling protofilaments can be tuned and suggest evolutionary conservation of the amount of curvature strain energy stored in the microtubule lattice.

Controlling Exchange Pathways in Dynamic Supramolecular Polymers by Controlling Defects

Supramolecular fibers composed of monomers that self-assemble directionally via noncovalent interactions are ubiquitous in nature, and of great interest in chemistry. In these structures, the constitutive monomers continuously exchange in-and-out the assembly according to a well-defined supramolecular equilibrium. However, unraveling the exchange pathways and their molecular determinants constitutes a nontrivial challenge. Here, we combine coarse-grained modeling, enhanced sampling, and machine learning to investigate the key factors controlling the monomer exchange pathways in synthetic supramolecular polymers having an intrinsic dynamic behavior. We demonstrate how the competition of directional vs. nondirectional interactions between the monomers controls the creation/annihilation of defects in the supramolecular polymers, from where monomers exchange proceeds. This competition determines the exchange pathway, dictating whether a fiber statistically swaps monomers from the tips or from all along its length. Finally, thanks to their generality, our models allow the investigation of molecular approaches to control the exchange pathways in these dynamic assemblies.

Microtubules pull the strings: disordered sequences as efficient couplers of microtubule-generated force

Microtubules are dynamic polymers that grow and shrink through addition or loss of tubulin subunits at their ends. Microtubule ends generate mechanical force that moves chromosomes and cellular organelles, and provides mechanical tension. Recent literature describes a number of proteins and protein complexes that couple dynamics of microtubule ends to movements of their cellular cargoes. These 'couplers' are quite diverse in their microtubule-binding domains (MTBDs), while sharing similarity in function, but a systematic understanding of the principles underlying their activity is missing. Here, I review various types of microtubule couplers, focusing on their essential activities: ability to follow microtubule ends and capture microtubule-generated force. Most of the couplers require presence of unstructured positively charged sequences and multivalency in their microtubule-binding sites to efficiently convert the microtubule-generated force into useful connection to a cargo. An overview of the microtubule features supporting end-tracking and force-coupling, and the experimental methods to assess force-coupling properties is also provided.

Mechanisms of microtubule dynamics and force generation examined with computational modeling and electron cryotomography

Microtubules are dynamic tubulin polymers responsible for many cellular processes, including the capture and segregation of chromosomes during mitosis. In contrast to textbook models of tubulin self-assembly, we have recently demonstrated that microtubules elongate by addition of bent guanosine triphosphate tubulin to the tips of curving protofilaments. Here we explore this mechanism of microtubule growth using Brownian dynamics modeling and electron cryotomography. The previously described flaring shapes of growing microtubule tips are remarkably consistent under various assembly conditions, including different tubulin concentrations, the presence or absence of a polymerization catalyst or tubulin-binding drugs. Simulations indicate that development of substantial forces during microtubule growth and shortening requires a high activation energy barrier in lateral tubulin-tubulin interactions. Modeling offers a mechanism to explain kinetochore coupling to growing microtubule tips under assisting force, and it predicts a load-dependent acceleration of microtubule assembly, providing a role for the flared morphology of growing microtubule ends.

Mechanics and kinetics of dynamic instability

During dynamic instability, self-assembling microtubules (MTs) stochastically alternate between phases of growth and shrinkage. This process is driven by the presence of two distinct states of MT subunits, GTP- and GDP-bound tubulin dimers, that have different structural properties. Here, we use a combination of analysis and computer simulations to study the mechanical and kinetic regulation of dynamic instability in three-dimensional (3D) self-assembling MTs. Our model quantifies how the 3D structure and kinetics of the distinct states of tubulin dimers determine the mechanical stability of MTs. We further show that dynamic instability is influenced by the presence of quenched disorder in the state of the tubulin subunit as reflected in the fraction of non-hydrolysed tubulin. Our results connect the 3D geometry, kinetics and statistical mechanics of these tubular assemblies within a single framework, and may be applicable to other self-assembled systems where these same processes are at play.

The Mechanical Role of Microtubules in Tissue Remodeling

During morphogenesis, tissues undergo extensive remodeling to get their final shape. Such precise sculpting requires the application of forces generated within cells by the cytoskeleton and transmission of these forces through adhesion molecules within and between neighboring cells. Within individual cells, microtubules together with actomyosin filaments and intermediate filaments form the composite cytoskeleton that controls cell mechanics during tissue rearrangements. While studies have established the importance of actin-based mechanical forces that are coupled via intercellular junctions, relatively little is known about the contribution of other cytoskeletal components such as microtubules to cell mechanics during morphogenesis. In this review the focus is on recent findings, highlighting the direct mechanical role of microtubules beyond its well-established role in trafficking and signaling during tissue formation.

Mechanisms of Motor-Independent Membrane Remodeling Driven by Dynamic Microtubules

Microtubule-dependent organization of membranous organelles occurs through motor-based pulling and by coupling microtubule dynamics to membrane remodeling. For example, tubules of endoplasmic reticulum (ER) can be extended by kinesin- and dynein-mediated transport and through the association with the tips of dynamic microtubules. The binding between ER and growing microtubule plus ends requires End Binding (EB) proteins and the transmembrane protein STIM1, which form a tip-attachment complex (TAC), but it is unknown whether these proteins are sufficient for membrane remodeling. Furthermore, EBs and their partners undergo rapid turnover at microtubule ends, and it is unclear how highly transient protein-protein interactions can induce load-bearing processive motion. Here, we reconstituted membrane tubulation in a minimal system with giant unilamellar vesicles, dynamic microtubules, an EB protein, and a membrane-bound protein that can interact with EBs and microtubules. We showed that these components are sufficient to drive membrane remodeling by three mechanisms: membrane tubulation induced by growing microtubule ends, motor-independent membrane sliding along microtubule shafts, and membrane pulling by shrinking microtubules. Experiments and modeling demonstrated that the first two mechanisms can be explained by adhesion-driven biased membrane spreading on microtubules. Optical trapping revealed that growing and shrinking microtubule ends can exert forces of ∼0.5 and ∼5 pN, respectively, through attached proteins. Rapidly exchanging molecules that connect membranes to dynamic microtubules can thus bear a sufficient load to induce membrane deformation and motility. Furthermore, combining TAC components and a membrane-attached kinesin in the same in vitro assays demonstrated that they can cooperate in promoting membrane tubule extension.

Structural effects of cap, crack, and intrinsic curvature on the microtubule catastrophe kinetics

Microtubules (MTs) experience an effect called "catastrophe," which is the transition from the MT growth to a sudden dramatic shrinkage in length. The straight guanosine triphosphate (GTP)-tubulin cap at the filament tip and the intrinsic curvature of guanosine diphosphate (GDP)-tubulins are known to be the key thermodynamic factors that determine MT catastrophe, while the hydrolysis of this GTP-cap acts as the kinetic control of the process. Although several theoretical models have been developed, assuming the catastrophe occurs when the GTP-cap shrinks to a minimal stabilizing size, the structural effect of the GTP-cap and GDP-curvature is not explicitly included; thus, their influence on catastrophe kinetics remains less understood. To investigate this structural effect, we apply a single-protofilament model with one GTP-cap while assuming a random hydrolysis mechanism and take the occurrence of a crack in the lateral bonds between neighboring protofilaments as the onset of the catastrophe. Therein, we find the effective potential of the tip along the peel-off direction and formulate the catastrophe kinetics as a mean first-passage time problem, subject to thermal fluctuations. We consider cases with and without a compressive force on the MT tip, both of which give a quadratic effective potential, making MT catastrophe an Ornstein-Uhlenbeck process in our formalism. In the free-standing case, the mean catastrophe time has a sensitive tubulin-concentration dependence, similar to a double-exponential function, and agrees well with the experiment. For a compressed MT, we find a modified exponential function of force that shortens the catastrophe time.

How Defects Control the Out-of-Equilibrium Dissipative Evolution of a Supramolecular Tubule

Supramolecular architectures that work out-of-equilibrium or that can change in specific ways when absorbing external energy are ubiquitous in nature. Gaining the ability to create via self-assembly artificial materials possessing such fascinating behaviors would have a major impact in many fields. However, the rational design of similar dynamic structures requires to understand and, even more challenging, to learn how to master the molecular mechanisms governing how the assembled systems evolve far from the equilibrium. Typically, this represents a daunting challenge due to the limited molecular insight that can be obtained by the experiments or by classical modeling approaches. Here we combine coarse-grained molecular models and advanced simulation approaches to study at submolecular (<5 Å) resolution a supramolecular tubule, which breaks and disassembles upon absorption of light energy triggering isomerization of its azobenzene-containing monomers. Our approach allows us to investigate the molecular mechanism of monomer transition in the assembly and to elucidate the kinetic process for the accumulation of the transitions in the system. Despite the stochastic nature of the excitation process, we demonstrate how these tubules preferentially dissipate the absorbed energy locally via the amplification of defects in their supramolecular structure. We find that this constitutes the best kinetic pathway for accumulating monomer transitions in the system, which determines the dynamic evolution out-of-equilibrium and the brittle behavior of the assembly under perturbed conditions. Thanks to the flexibility of our models, we finally come out with a general principle, where defects explain and control the brittle/soft behavior of such light-responsive assemblies.

Damage and Failure of Axonal Microtubule under Extreme High Strain Rate: An In-Silico Molecular Dynamics Study

As a major cytoskeleton element of the axon, the breaking of microtubules (MTs) has been considered as a major cause of the axon degeneration. High strain rate loading is considered as one of the key factors in microtubule breaking. Due to the small size of microtubule, the real-time behavior of microtubule breaking is hard to capture. This study employs fully-atomistic molecular dynamics (MD) simulation to determine the failure modes of microtubule under different loadings conditions such as, unidirectional stretching, bending and hydrostatic expansion. For each loading conditions, MT is subjected to extreme high strain rate (108-109 s-1) loading. We argue that such level of high strain rate may be realized during cavitation bubble implosion. For each loading type, we have determined the critical energy for MT rupture. The associated rupture mechanisms are also discussed. We observed that the stretching has the lowest energy barrier to break the MT at the nanosecond time scale. Moreover, the breakage between the dimers starts at ~16% of total strain when stretched, which is much smaller compared to the reported strain-at-failure (50%) for lower strain rate loading. It suggests that MT fails at a significantly smaller strain states when loaded at higher strain rates.

Multivalency of NDC80 in the outer kinetochore is essential to track shortening microtubules and generate forces

Presence of multiple copies of the microtubule-binding NDC80 complex is an evolutionary conserved feature of kinetochores, points of attachment of chromosomes to spindle microtubules. This may enable multivalent attachments to microtubules, with implications that remain unexplored. Using recombinant human kinetochore components, we show that while single NDC80 complexes do not track depolymerizing microtubules, reconstituted particles containing the NDC80 receptor CENP-T bound to three or more NDC80 complexes do so effectively, as expected for a kinetochore force coupler. To study multivalency systematically, we engineered modules allowing incremental addition of NDC80 complexes. The modules’ residence time on microtubules increased exponentially with the number of NDC80 complexes. Modules with two or more complexes tracked depolymerizing microtubules with increasing efficiencies, and stalled and rescued microtubule depolymerization in a force-dependent manner when conjugated to cargo. Our observations indicate that NDC80, rather than through biased diffusion, tracks depolymerizing microtubules by harnessing force generated during microtubule disassembly.

Before a cell divides, its genome duplicates so that each copy can be given to the daughter cells. In a dividing cell, the chromosomes – the structures that store genetic information – look like an ‘X’. This is because each chromosome is formed of two identical, rod-like, ‘sister chromatids’ which are attached by their middle. Each daughter cell should inherit one of the chromatids.

As division progresses, both sister chromatids in a pair fasten to ‘microtubules’, string-like structures made of a large number of identical proteins stacked together. These strings attach each chromatids to opposite sides of the cell. Then, the ends of the microtubules that bind to a chromatid start to peel off and disassemble. The microtubules get shorter and shorter, which creates a force that pulls the chromatids apart.

Microtubules latch on a chromatid via a large structure known as the kinetochore, which has tether-like protein complexes called NDC80 at its surface. NDC80 links the kinetochore with the microtubules, yet little is known about this connection. In particular, it is unclear how this complex relays the forces from the shortening microtubules to the chromatids, and how many NDC80 complexes are required for this process.

To study how these proteins interact without any molecular background ‘noise’ from the cell, Volkov, Huis in ‘t Veld et al. engineered simplified versions of the microtubule-kinetochore-NDC80 connection using components of human kinetochores. These versions, named ‘modules’, contained different numbers of NDC80 complexes, from one to four copies.

Volkov, Huis in ‘t Veld et al. found that single NDC80 complexes did not follow the microtubules as they shortened, while the connections with two or more NDC80 complexes did. When a few modules, each with two or three NDC80s, were closeby, they also bound to the end of the same shortening microtubule, and captured more force as a team. NDC80 complexes therefore work together to connect to microtubule ends and harness their energy.

The artificial kinetochore-microtubule-NDC80 connections developed by Volkov, Huis in ‘t Veld et al. provides a new method to study how cells divide, and it could reveal how other proteins and biological processes participate in this mechanism. It could also help understand how chromatids are kept from separating incorrectly during division, which is an error that could be fatal for the cell.

Dynamics of microtubules: highlights of recent computational and experimental investigations

Microtubules are found in most eukaryotic cells, with homologs in eubacteria and archea, and they have functional roles in mitosis, cell motility, intracellular transport, and the maintenance of cell shape. Numerous efforts have been expended over the last two decades to characterize the interactions between microtubules and the wide variety of microtubule associated proteins that control their dynamic behavior in cells resulting in microtubules being assembled and disassembled where and when they are required by the cell. We present the main findings regarding microtubule polymerization and depolymerization and review recent work about the molecular motors that modulate microtubule dynamics by inducing either microtubule depolymerization or severing. We also discuss the main experimental and computational approaches used to quantify the thermodynamics and mechanics of microtubule filaments.

Molecular photoswitches mediating the strain-driven disassembly of supramolecular tubules

Chemists have created molecular machines and switches with specific mechanical responses that were typically demonstrated in solution, where mechanically relevant motion is dissipated in the Brownian storm. The next challenge consists of designing specific mechanisms through which the action of individual molecules is transmitted to a supramolecular architecture, with a sense of directionality. Cellular microtubules are capable of meeting such a challenge. While their capacity to generate pushing forces by ratcheting growth is well known, conversely these versatile machines can also pull microscopic objects apart through a burst of their rigid tubular structure. One essential feature of this disassembling mechanism is the accumulation of strain in the tubules, which develops when tubulin dimers change shape, triggered by a hydrolysis event. We envision a strategy toward supramolecular machines generating directional pulling forces by harnessing the mechanically purposeful motion of molecular switches in supramolecular tubules. Here, we report on wholly synthetic, water-soluble, and chiral tubules that incorporate photoswitchable building blocks in their supramolecular architecture. Under illumination, these tubules display a nonlinear operation mode, by which light is transformed into units of strain by the shape changes of individual switches, until a threshold is reached and the tubules unleash the strain energy. The operation of this wholly synthetic and stripped-down system compares to the conformational wave by which cellular microtubules disassemble. Additionally, atomistic simulations provide molecular insight into how strain accumulates to induce destabilization. Our findings pave the way toward supramolecular machines that would photogenerate pulling forces, at the nanoscale and beyond.

Deformation pattern in vibrating microtubule: Structural mechanics study based on an atomistic approach

The mechanical properties of microtubules are of great importance for understanding their biological function and for applications in artificial devices. Although microtubule mechanics has been extensively studied both theoretically and experimentally, the relation to its molecular structure is understood only partially. Here, we report on the structural analysis of microtubule vibration modes calculated by an atomistic approach. Molecular dynamics was applied to refine the atomic structure of a microtubule and a C_α elastic network model was analyzed for its normal modes. We mapped fluctuations and local deformations up to the level of individual aminoacid residues. The deformation is mode-shape dependent and principally different in α-tubulins and β-tubulins. Parts of the tubulin dimer sequence responding specifically to longitudinal and radial stress are identified. We show that substantial strain within a microtubule is located both in the regions of contact between adjacent dimers and in the body of tubulins. Our results provide supportive evidence for the generally accepted assumption that the mechanics of microtubules, including its anisotropy, is determined by the bonds between tubulins.

Direct measurement of conformational strain energy in protofilaments curling outward from disassembling microtubule tips

Disassembling microtubules can generate movement independently of motor enzymes, especially at kinetochores where they drive chromosome motility. A popular explanation is the 'conformational wave' model, in which protofilaments pull on the kinetochore as they curl outward from a disassembling tip. But whether protofilaments can work efficiently via this spring-like mechanism has been unclear. By modifying a previous assay to use recombinant tubulin and feedback-controlled laser trapping, we directly demonstrate the spring-like elasticity of curling protofilaments. Measuring their mechanical work output suggests they carry ~25% of the energy of GTP hydrolysis as bending strain, enabling them to drive movement with efficiency similar to conventional motors. Surprisingly, a β-tubulin mutant that dramatically slows disassembly has no effect on work output, indicating an uncoupling of disassembly speed from protofilament strain. These results show the wave mechanism can make a major contribution to kinetochore motility and establish a direct approach for measuring tubulin mechano-chemistry.

Mechanism of Ska Recruitment by Ndc80 Complexes to Kinetochores

Yeast use the ring-shaped Dam1 complex to slide down depolymerizing microtubules to move chromosomes, but current models suggest that other eukaryotes do not have a sliding ring. We visualized Ndc80 and Ska complexes on microtubules by electron microscopic tomography to identify the structure of the human kinetochore-microtubule attachment. Ndc80 recruits the Ska complex so that the V shape of the Ska dimer interacts along protofilaments. We identify a mutant of the Ndc80 tail that is deficient in Ska recruitment to kinetochores and in orienting Ska along protofilaments in vitro. This mutant Ndc80 binds microtubules with normal affinity but is deficient in clustering along protofilaments. We propose that Ska is recruited to kinetochores by clusters of Ndc80 proteins and that our structure of Ndc80 and Ska complexes on microtubules suggests a mechanism for metazoan kinetochores to couple the depolymerization of microtubules to power the movement of chromosomes.

Biophysics of Microtubule End Coupling at the Kinetochore

The main physiological function of mitotic kinetochores is to provide durable attachment to spindle microtubules, which segregate chromosomes in order to partition them equally between the two daughter cells. Numerous kinetochore components that can bind directly to microtubules have been identified, including ATP-dependent motors and various microtubule-associated proteins with no motor activity. A major challenge facing the field is to explain chromosome motions based on the biochemical and structural properties of these individual kinetochore components and their assemblies. This chapter reviews the molecular mechanisms responsible for the motions associated with dynamic microtubule tips at the single-molecule level, as well as the activities of multimolecular ensembles called couplers. These couplers enable persistent kinetochore motion even under load, but their exact composition and structure remain unknown. Because no natural or artificial macro-machines function in an analogous manner to these molecular nano-devices, understanding their underlying biophysical mechanisms will require conceptual advances.

Electrostatic forces drive poleward chromosome motions at kinetochores

BACKGROUND

Recent experiments regarding Ndc80/Hec1 in force generation at kinetochores for chromosome motions have prompted speculation about possible models for interactions between positively charged molecules at kinetochores and negative charge at and near the plus ends of microtubules.

DISCUSSION

A clear picture of how kinetochores and centrosomes establish and maintain a dynamic coupling to microtubules for force generation during the complex motions of mitosis remains elusive. The current paradigm of molecular cell biology requires that specific molecules, or molecular geometries, for force generation be identified. However, it is possible to explain several different mitotic motions-including poleward force production at kinetochores-within a classical electrostatics approach in terms of experimentally known charge distributions, modeled as surface and volume bound charges interacting over nanometer distances.

CONCLUSION

We propose here that implicating Ndc80/Hec1 as a bound volume positive charge distribution in electrostatic generation of poleward force at kinetochores is most consistent with a wide range of experimental observations on mitotic motions, including polar production of poleward force and chromosome congression.

Multi scale modeling of 2450MHz electric field effects on microtubule mechanical properties

Microtubule (MT) rigidity and response to 2450MHz electric fields were investigated, via multi scale modeling approach. For this purpose, six systems were designed and simulated to consider all types of feasible interactions between α and β monomers in MT, by using all atom molecular dynamics method. Subsequently, coarse grain modeling was used to design different lengths of MT. Investigation of effects of external 2450MHz electric field on MT showed MT less rigidity in the presence of such field, which may perturb its functions. Moreover, an additional computational setup was designed to study effects of 2450MHz field on MT response to AFM tip. It was found, more tip velocity led to MT faster transformation and less time was required to change MT elastic response to plastic one, applying constant radius. Moreover it was observed smaller tip caused to increase required time to change MT elastic response to plastic one, considering constant velocity. Furthermore, exposing MT to 2450MHz field led to no significant changes in MT response to AFM tip, but quick change in MT elastic response to plastic one.

Molecular and Mechanical Causes of Microtubule Catastrophe and Aging

Tubulin polymers, microtubules, can switch abruptly from the assembly to shortening. These infrequent transitions, termed "catastrophes", affect numerous cellular processes but the underlying mechanisms are elusive. We approached this complex stochastic system using advanced coarse-grained molecular dynamics modeling of tubulin-tubulin interactions. Unlike in previous simplified models of dynamic microtubules, the catastrophes in this model arise owing to fluctuations in the composition and conformation of a growing microtubule tip, most notably in the number of protofilament curls. In our model, dynamic evolution of the stochastic microtubule tip configurations over a long timescale, known as the system's "aging", gives rise to the nonexponential distribution of microtubule lifetimes, consistent with experiment. We show that aging takes place in the absence of visible changes in the microtubule wall or tip, as this complex molecular-mechanical system evolves slowly and asymptotically toward the steady-state level of the catastrophe-promoting configurations. This new, to our knowledge, theoretical basis will assist detailed mechanistic investigations of the mechanisms of action of different microtubule-binding proteins and drugs, thereby enabling accurate control over the microtubule dynamics to treat various pathologies.

Microtubule-based force generation

Microtubules are vital to many important cell processes, such as cell division, transport of cellular cargo, organelle positioning, and cell migration. Owing to their diverse functions, understanding microtubule function is an important part of cell biological research that can help in combating various diseases. For example, microtubules are an important target of chemotherapeutic drugs such as paclitaxel because of their pivotal role in cell division. Many functions of microtubules relate to the generation of mechanical forces. These forces are generally either a direct result of microtubule polymerization/depolymerization or generated by motor proteins that move processively along microtubules. In this review, we summarize recent efforts to quantify and model force generation by microtubules in the context of microtubule function. WIREs Nanomed Nanobiotechnol 2017, 9:e1428. doi: 10.1002/wnan.1428 For further resources related to this article, please visit the WIREs website.

Conformational mechanism for the stability of microtubule-kinetochore attachments

Regulating the stability of microtubule (MT)-kinetochore attachments is fundamental to avoiding mitotic errors and ensuring proper chromosome segregation during cell division. Although biochemical factors involved in this process have been identified, their mechanics still need to be better understood. Here we introduce and simulate a mechanical model of MT-kinetochore interactions in which the stability of the attachment is ruled by the geometrical conformations of curling MT-protofilaments entangled in kinetochore fibrils. The model allows us to reproduce, with good accuracy, in vitro experimental measurements of the detachment times of yeast kinetochores from MTs under external pulling forces. Numerical simulations suggest that geometrical features of MT-protofilaments may play an important role in the switch between stable and unstable attachments.

Statistical mechanics provides novel insights into microtubule stability and mechanism of shrinkage

Microtubules are nano-machines that grow and shrink stochastically, making use of the coupling between chemical kinetics and mechanics of its constituent protofilaments (PFs). We investigate the stability and shrinkage of microtubules taking into account inter-protofilament interactions and bending interactions of intrinsically curved PFs. Computing the free energy as a function of PF tip position, we show that the competition between curvature energy, inter-PF interaction energy and entropy leads to a rich landscape with a series of minima that repeat over a length-scale determined by the intrinsic curvature. Computing Langevin dynamics of the tip through the landscape and accounting for depolymerization, we calculate the average unzippering and shrinkage velocities of GDP protofilaments and compare them with the experimentally known results. Our analysis predicts that the strength of the inter-PF interaction (E(s)(m)) has to be comparable to the strength of the curvature energy (E(b)(m)) such that E(s)(m) - E(b)(m) ≈ 1kBT, and questions the prevalent notion that unzippering results from the domination of bending energy of curved GDP PFs. Our work demonstrates how the shape of the free energy landscape is crucial in explaining the mechanism of MT shrinkage where the unzippered PFs will fluctuate in a set of partially peeled off states and subunit dissociation will reduce the length.

Tubulin bond energies and microtubule biomechanics determined from nanoindentation in silico

Microtubules, the primary components of the chromosome segregation machinery, are stabilized by longitudinal and lateral noncovalent bonds between the tubulin subunits. However, the thermodynamics of these bonds and the microtubule physicochemical properties are poorly understood. Here, we explore the biomechanics of microtubule polymers using multiscale computational modeling and nanoindentations in silico of a contiguous microtubule fragment. A close match between the simulated and experimental force–deformation spectra enabled us to correlate the microtubule biomechanics with dynamic structural transitions at the nanoscale. Our mechanical testing revealed that the compressed MT behaves as a system of rigid elements interconnected through a network of lateral and longitudinal elastic bonds. The initial regime of continuous elastic deformation of the microtubule is followed by the transition regime, during which the microtubule lattice undergoes discrete structural changes, which include first the reversible dissociation of lateral bonds followed by irreversible dissociation of the longitudinal bonds. We have determined the free energies of dissociation of the lateral (6.9 ± 0.4 kcal/mol) and longitudinal (14.9 ± 1.5 kcal/mol) tubulin–tubulin bonds. These values in conjunction with the large flexural rigidity of tubulin protofilaments obtained (18,000–26,000 pN·nm²) support the idea that the disassembling microtubule is capable of generating a large mechanical force to move chromosomes during cell division. Our computational modeling offers a comprehensive quantitative platform to link molecular tubulin characteristics with the physiological behavior of microtubules. The developed in silico nanoindentation method provides a powerful tool for the exploration of biomechanical properties of other cytoskeletal and multiprotein assemblies.

A mathematical model of force generation by flexible kinetochore-microtubule attachments

Important mechanical events during mitosis are facilitated by the generation of force by chromosomal kinetochore sites that attach to dynamic microtubule tips. Several theoretical models have been proposed for how these sites generate force, and molecular diffusion of kinetochore components has been proposed as a key component that facilitates kinetochore function. However, these models do not explicitly take into account the recently observed flexibility of kinetochore components and variations in microtubule shape under load. In this paper, we develop a mathematical model for kinetochore-microtubule connections that directly incorporates these two important components, namely, flexible kinetochore binder elements, and the effects of tension load on the shape of shortening microtubule tips. We compare our results with existing biased diffusion models and explore the role of protein flexibility inforce generation at the kinetochore-microtubule junctions. Our model results suggest that kinetochore component flexibility and microtubule shape variation under load significantly diminish the need for high diffusivity (or weak specific binding) of kinetochore components; optimal kinetochore binder stiffness regimes are predicted by our model. Based on our model results, we suggest that the underlying principles of biased diffusion paradigm need to be reinterpreted.

Kinetochores require oligomerization of Dam1 complex to maintain microtubule attachments against tension and promote biorientation

Kinetochores assemble on centromeric DNA and present arrays of proteins that attach directly to the dynamic ends of microtubules. Kinetochore proteins coordinate at the microtubule interface through oligomerization, but how oligomerization contributes to kinetochore function has remained unclear. Here, using a combination of biophysical assays and live-cell imaging, we find that oligomerization of the Dam1 kinetochore complex is required for its ability to form microtubule attachments that are robust against tension in vitro and in vivo. An oligomerization-deficient Dam1 complex that retains wild-type microtubule binding activity is primarily defective in coupling to disassembling microtubule ends under mechanical loads applied by a laser trap in vitro. In cells, the oligomerization-deficient Dam1 complex is unable to support stable bipolar alignment of sister chromatids, indicating failure of kinetochore-microtubule attachments under tension. We propose that oligomerization is an essential and conserved feature of kinetochore components that is required for accurate chromosome segregation during mitosis.

Effects of microtubule mechanics on hydrolysis and catastrophes

We introduce a model for microtubule (MT) mechanics containing lateral bonds between dimers in neighboring protofilaments, bending rigidity of dimers, and repulsive interactions between protofilaments modeling steric constraints to investigate the influence of mechanical forces on hydrolysis and catastrophes. We use the allosteric dimer model, where tubulin dimers are characterized by an equilibrium bending angle, which changes from 0° to 22° by hydrolysis of a dimer. This also affects the lateral interaction and bending energies and, thus, the mechanical equilibrium state of the MT. As hydrolysis gives rise to conformational changes in dimers, mechanical forces also influence the hydrolysis rates by mechanical energy changes modulating the hydrolysis rate. The interaction via the MT mechanics then gives rise to correlation effects in the hydrolysis dynamics, which have not been taken into account before. Assuming a dominant influence of mechanical energies on hydrolysis rates, we investigate the most probable hydrolysis pathways both for vectorial and random hydrolysis. Investigating the stability with respect to lateral bond rupture, we identify initiation configurations for catastrophes along the hydrolysis pathways and values for a lateral bond rupture force. If we allow for rupturing of lateral bonds between dimers in neighboring protofilaments above this threshold force, our model exhibits avalanche-like catastrophe events.

The composition, functions, and regulation of the budding yeast kinetochore

The propagation of all organisms depends on the accurate and orderly segregation of chromosomes in mitosis and meiosis. Budding yeast has long served as an outstanding model organism to identify the components and underlying mechanisms that regulate chromosome segregation. This review focuses on the kinetochore, the macromolecular protein complex that assembles on centromeric chromatin and maintains persistent load-bearing attachments to the dynamic tips of spindle microtubules. The kinetochore also serves as a regulatory hub for the spindle checkpoint, ensuring that cell cycle progression is coupled to the achievement of proper microtubule-kinetochore attachments. Progress in understanding the composition and overall architecture of the kinetochore, as well as its properties in making and regulating microtubule attachments and the spindle checkpoint, is discussed.

Forces due to curving protofilaments in microtubules

Microtubules consist of 13 protofilaments arranged in the form of a cylinder. The protofilaments are composed of longitudinally attached tubulin dimers that can exist in either a less curved state [GTP-bound tubulin (T)] or a more curved state [GDP-bound tubulin (D)]. Hydrolysis of T into D leaves the straight and laterally attached protofilaments of the microtubule in a mechanically stressed state, thus leading to their unzipping. The elastic energy in the unzipping protofilaments can be harnessed by a force transducer such as the Dam1-kinetochore ring complex in order to exert pulling force on chromosomes during cell division. In the present paper we develop a simple continuum model to obtain this pulling force as a function of the mechanical properties of protofilaments and the size of the Dam1-kinetochore ring. We also extend this model to investigate the role played by the T subunits found at the plus end of the microtubule (the T cap) on the mechanical stability of microtubules.

Highly Transient Molecular Interactions Underlie the Stability of Kinetochore-Microtubule Attachment During Cell Division

Chromosome segregation during mitosis is mediated by spindle microtubules that attach to chromosomal kinetochores with strong yet labile links. The exact molecular composition of the kinetochore-microtubule interface is not known but microtubules are thought to bind to kinetochores via the specialized microtubule-binding sites, which contain multiple microtubule-binding proteins. During prometaphase the lifetime of microtubule attachments is short but in metaphase it increases 3-fold, presumably owing to dephosphorylation of the microtubule-binding proteins that increases their affinity. Here, we use mathematical modeling to examine in quantitative and systematic manner the general relationships between the molecular properties of microtubule-binding proteins and the resulting stability of microtubule attachment to the protein-containing kinetochore site. We show that when the protein connections are stochastic, the physiological rate of microtubule turnover is achieved only if these molecular interactions are very transient, each lasting fraction of a second. This "microscopic" time is almost four orders of magnitude shorter than the characteristic time of kinetochore-microtubule attachment. Cooperativity of the microtubule-binding events further increases the disparity of these time scales. Furthermore, for all values of kinetic parameters the microtubule stability is very sensitive to the minor changes in the molecular constants. Such sensitivity of the lifetime of microtubule attachment to the kinetics and cooperativity of molecular interactions at the microtubule-binding site may hinder the accurate regulation of kinetochore-microtubule stability during mitotic progression, and it necessitates detailed experimental examination of the microtubule-binding properties of kinetochore-localized proteins.

Microtubule catastrophe from protofilament dynamics

The disappearance of the guanosine triphosphate- (GTP) tubulin cap is widely believed to be the forerunner event for the growth-shrinkage transition ("catastrophe") in microtubule filaments in eukaryotic cells. We study a discrete version of a stochastic model of the GTP cap dynamics, originally proposed by Flyvbjerg, Holy, and Leibler [Phys. Rev. Lett. 73, 2372 (1994)]. Our model includes both spontaneous and vectorial hydrolysis, as well as dissociation of a nonhydrolyzed dimer from the filament after incorporation. In the first part of the paper, we apply this model to a single protofilament of a microtubule. A catastrophe transition is defined for each protofilament, similarly to the earlier one-dimensional models, the frequency of occurrence of which is then calculated under various conditions but without explicit assumption of steady-state conditions. Using a perturbative approach, we show that the leading asymptotic behavior of the protofilament catastrophe in the limit of large growth velocities is remarkably similar across different models. In the second part of the paper, we extend our analysis to the entire filament by making a conjecture that a minimum number of such transitions are required to occur for the onset of microtubule catastrophe. The frequency of microtubule catastrophe is then determined using numerical simulations and compared with analytical and semianalytical estimates made under steady-state and quasi-steady-state assumptions, respectively, for the protofilament dynamics. A few relevant experimental results are analyzed in detail and compared with predictions from the model. Our results indicate that loss of GTP cap in two to three protofilaments is necessary to trigger catastrophe in a microtubule.

Comparative studies of microtubule mechanics with two competing models suggest functional roles of alternative tubulin lateral interactions

The dynamic assembly and disassembly of microtubules and the mechanical properties of these polymers are essential for many key cellular processes. Mathematical and computational modeling, especially coupled mechanochemical modeling, has contributed significantly to our understanding of microtubule dynamics. However, critical discrepancies exist between experimental observations and modeling results that need to be resolved before further progress toward a complete model can be made. Open sheet structures ranging in length from several hundred nanometers to one micron have often been observed at the growing ends of microtubules in in vitro studies. Existing modeling studies predict these sheet structures to be short and rare intermediates of microtubule disassembly rather than important components of the assembly process. Atomic force microscopy (AFM) studies also reveal interesting step-like gaps of the force-indentation curve that cannot yet be explained by existing theoretical models. We have carried out computational studies to compare the mechanical properties of two alternative models: a more conventional model where tubulin dimers are added directly into a microtubule lattice, and one that considers an additional type of tubulin lateral interaction proposed to exist in intermediate sheet structures during the microtubule assembly process. The first model involves a single type of lateral interactions between tubulin subunits, whereas the latter considers a second type that can convert to the canonical lateral contact during microtubule closure into a cylinder. Our analysis shows that only the second model can reproduce the AFM results over a broad parameter range. We propose additional studies using different sizes of AFM tips that would allow to unambiguously distinguish the relative validity of the two models.

A mathematical model of ParA filament-mediated chromosome movement in Caulobacter crescentus

Caulobacter crescentus uses the dynamic interactions between ParA and ParB proteins to segregate copies of its circular chromosome. In this paper, we develop two mathematical models of the movement of the circular chromosome of this bacterium during division. In the first model, posed as a set of stochastic differential equations (SDE), we propose that a simple biased diffusion mechanism for ParB/ParA interactions can reproduce the observed patterns of ParB and ParA localization in the cell. The second model, posed as a set of nonlinear partial differential equations, is a continuous treatment of the problem where we use results from the SDE model to describe ParB/ParA interactions and we also track ParA monomer dynamics in the cytoplasm. For both models, we show that if ParB complexes bind weakly and nonspecifically to ParA filaments, then they can closely track and move with the edge of a shrinking ParA filament bundle. Unidirectional chromosome movement occurs when ParB complexes have a passive role in depolymerizing ParA filaments. Finally, we show that tight control of ParA filament dynamics is essential for proper segregation.

Biophysics of mitosis

Mitosis is the process by which eukaryotic cells organize and segregate their chromosomes in preparation for cell division. It is accomplished by a cellular machine composed largely of microtubules (MTs) and their associated proteins. This article reviews literature on mitosis from a biophysical point of view, drawing attention to the assembly and motility processes required to do this complex job with precision. Work from both the recent and the older literature is integrated into a description of relevant biological events and the experiments that probe their mechanisms. Theoretical work on specific subprocesses is also reviewed. Our goal is to provide a document that will expose biophysicists to the fascination of this quite amazing process and provide them with a good background from which they can pursue their own research interests in the subject.

Why is the microtubule lattice helical?

Microtubules polymerize from identical tubulin heterodimers, which form a helical lattice pattern that is the microtubule. This pattern always has left-handed chirality, but it is not known why. But as tubulin, similar to other proteins, evolved for a purpose, the question of the title of this artcile appears to be meaningful. In a computer simulation that explores the 'counterfactual biology' of microtubules without helicity, we demonstrate that these have the same mechanical properties as Nature's microtubules with helicity. Thus only a dynamical reason for helicity is left as potential explanation. We find that helicity solves 'the problem of the blind mason', i.e. how to correctly build a structure, guided only by the shape of the bricks. This answer in turn raises some new questions for researchers to address.

Growth and shortening of microtubules: a two-state model approach

In this study, a two-state mechanochemical model is presented to describe the dynamic instability of microtubules (MTs) in cells. The MT switches between two states, the assembly and disassembly states. In assembly state, the growth of MTs includes two processes: free GTP-tubulin binding to the tip of protofilament (PF) and conformation change of PF, during which the first tubulin unit that curls outwards is rearranged onto the MT surface, using the energy released from the hydrolysis of GTP in the penultimate tubulin unit. In the disassembly state, the shortening of MTs also includes two processes, the release of GDP-tubulin from the tip of PF and the curling of one new tubulin unit out of the MT surface. Switches between these two states, which are usually called rescue and catastrophe, happen stochastically with external force-dependent rates. Using this two-state model with parameters obtained by fitting the recent experimental data, detailed properties of MT growth are obtained. I find that MT is mainly in the assembly state, its mean growth velocity increases with both the external force and the GTP-tubulin concentration, and an MT will shorten on average without an external force. To know more about the external force and GTP-tubulin concentration-dependent properties of MT growth, and for future experimental verification of this two-state model, 11 critical forces are defined and discussed numerically.

Filament depolymerization can explain chromosome pulling during bacterial mitosis

Chromosome segregation is fundamental to all cells, but the force-generating mechanisms underlying chromosome translocation in bacteria remain mysterious. Caulobacter crescentus utilizes a depolymerization-driven process in which a ParA protein structure elongates from the new cell pole, binds to a ParB-decorated chromosome, and then retracts via disassembly, pulling the chromosome across the cell. This poses the question of how a depolymerizing structure can robustly pull the chromosome that disassembles it. We perform Brownian dynamics simulations with a simple, physically consistent model of the ParABS system. The simulations suggest that the mechanism of translocation is “self-diffusiophoretic”: by disassembling ParA, ParB generates a ParA concentration gradient so that the ParA concentration is higher in front of the chromosome than behind it. Since the chromosome is attracted to ParA via ParB, it moves up the ParA gradient and across the cell. We find that translocation is most robust when ParB binds side-on to ParA filaments. In this case, robust translocation occurs over a wide parameter range and is controlled by a single dimensionless quantity: the product of the rate of ParA disassembly and a characteristic relaxation time of the chromosome. This time scale measures the time it takes for the chromosome to recover its average shape after it is has been pulled. Our results suggest explanations for observed phenomena such as segregation failure, filament-length-dependent translocation velocity, and chromosomal compaction.

Reliable chromosome segregation is crucial to all dividing cells. In some bacteria, segregation has been found to occur in a rather counterintuitive way: the chromosome attaches to a filament bundle and erodes it by causing depolymerization of the filaments. Moreover, unlike eukaryotic cells, bacteria do not use molecular motors and/or macromolecular tethers to position their chromosomes. This raises the general question of how depolymerizing filaments alone can continuously and robustly pull cargo as the filaments themselves are falling apart. In this work, we introduce the first quantitative physical model for depolymerization-driven translocation in a many-filament system. Our simulations of this model suggest a novel underlying mechanism for robust translocation, namely self-diffusiophoresis, motion of an object in a self-generated concentration gradient in a viscous environment. In this case, the cargo generates and sustains a concentration gradient of filaments by inducing them to depolymerize. We demonstrate that our model agrees well with existing experimental observations such as segregation failure, filament-length-dependent translocation velocity, and chromosomal compaction. In addition, we make several predictions–including predictions for the specific modes by which the chromosome binds to the filament structure and triggers its disassembly–that can be tested experimentally.

ENGINEERING AND THERAPEUTIC APPLICATIONS OF MICROTUBULES

Living organisms are fascinating systems. The macromolecules that make up a living cell possess equally astounding structural and functional characteristics. By taking simple cues from how these biopolymers organize and work inside the cell, one can draw inspiration to utilize them outside their natural environment for several purposes. Microtubules are example of biopolymers that demonstrate extraordinary properties of hierarchical self-organization, dynamic remodeling and mechanical rigidity. Mimicking the principles and properties of microtubules and improving them have opened novel engineering avenues. In addition, due to the functions that microtubules perform during cell division, they are excellent therapeutic drug targets for anticancer agents. In this work, we describe the biological properties and functions of microtubules, and discuss their engineering and therapeutic applications.

Tubulin depolymerization may be an ancient biological motor

The motions of mitotic chromosomes are complex and show considerable variety across species. A wealth of evidence supports the idea that microtubule-dependent motor enzymes contribute to this variation and are important both for spindle formation and for the accurate completion of chromosome segregation. Motors that walk towards the spindle pole are, however, dispensable for at least some poleward movements of chromosomes in yeasts, suggesting that depolymerizing spindle microtubules can generate mitotic forces in vivo. Tubulin protofilaments that flare outward in association with microtubule shortening may be the origin of such forces, because they can move objects that are appropriately attached to a microtubule wall. For example, some kinetochore-associated proteins can couple experimental objects, such as microspheres, to shortening microtubules in vitro, moving them over many micrometers. Here, we review recent evidence about such phenomena, highlighting the force-generation mechanisms and different coupling strategies. We also consider bending filaments of the tubulin-like protein FtsZ, which form rings girding bacteria at their sites of cytokinesis. Mechanical similarities between these force-generation systems suggest a deep phylogenetic relationship between tubulin depolymerization in eukaryotic mitosis and FtsZ-mediated ring contraction in bacteria.

Chromosome size in diploid eukaryotic species centers on the average length with a conserved boundary

Understanding genome and chromosome evolution is important for understanding genetic inheritance and evolution. Universal events comprising DNA replication, transcription, repair, mobile genetic element transposition, chromosome rearrangements, mitosis, and meiosis underlie inheritance and variation of living organisms. Although the genome of a species as a whole is important, chromosomes are the basic units subjected to genetic events that coin evolution to a large extent. Now many complete genome sequences are available, we can address evolution and variation of individual chromosomes across species. For example, “How are the repeat and nonrepeat proportions of genetic codes distributed among different chromosomes in a multichromosome species?” “Is there a general rule behind the intuitive observation that chromosome lengths tend to be similar in a species, and if so, can we generalize any findings in chromosome content and size across different taxonomic groups?” Here, we show that chromosomes within a species do not show dramatic fluctuation in their content of mobile genetic elements as the proliferation of these elements increases from unicellular eukaryotes to vertebrates. Furthermore, we demonstrate that, notwithstanding the remarkable plasticity, there is an upper limit to chromosome-size variation in diploid eukaryotes with linear chromosomes. Strikingly, variation in chromosome size for 886 chromosomes in 68 eukaryotic genomes (including 22 human autosomes) can be viably captured by a single model, which predicts that the vast majority of the chromosomes in a species are expected to have a base pair length between 0.4035 and 1.8626 times the average chromosome length. This conserved boundary of chromosome-size variation, which prevails across a wide taxonomic range with few exceptions, indicates that cellular, molecular, and evolutionary mechanisms, possibly together, confine the chromosome lengths around a species-specific average chromosome length.

Kinetochores' gripping feat: conformational wave or biased diffusion?

Climbing up a cliff while the rope unravels underneath your fingers does not sound like a well-planned adventure. Yet chromosomes face a similar challenge during each cell division. Their alignment and accurate segregation depends on staying attached to the assembling and disassembling tips of microtubule fibers. This coupling is mediated by kinetochores, intricate machines that attach chromosomes to an ever-changing microtubule substrate. Two models for kinetochore-microtubule coupling were proposed a quarter century ago: conformational wave and biased diffusion. These models differ in their predictions for how coupling is performed and regulated. The availability of purified kinetochore proteins has enabled biochemical and biophysical analyses of the kinetochore-microtubule interface. Here, we discuss what these studies reveal about the contributions of each model.

A coordinated molecular 'fishing' mechanism in heterodimeric kinesin

Kar3 is a kinesin motor that facilitates chromosome segregation during cell division. Unlike many members of the kinesin superfamily, Kar3 forms a heterodimer with non-motor protein Vik1 or Cik1 in vivo. The heterodimers show ATP-driven minus-end directed motility along a microtubule (MT) lattice, and also serve as depolymerase at the MT ends. The molecular mechanisms behind this dual functionality remain mysterious. Here, a molecular mechanical model for the Kar3/Vik1 heterodimer based on structural, kinetic and motility data reveals a long-range chemomechanical transmission mechanism that resembles a familiar fishing tactic. By this molecular 'fishing', ATP-binding to Kar3 dissociates catalytically inactive Vik1 off MT to facilitate minus-end sliding of the dimer on the MT lattice. When the dimer binds the frayed ends of MT, the fishing channels ATP hydrolysis energy into MT depolymerization by a mechanochemical effect. The molecular fishing thus provides a unified mechanistic ground for Kar3's dual functionality. The fishing-promoted depolymerization differs from the depolymerase mechanisms found in homodimeric kinesins. The fishing also enables intermolecular coordination with a chemomechanical coupling feature different from the paradigmatic pattern of homodimeric motors. This study rationalizes some puzzling experimental observation, and suggests new experiments for further elucidation of the fishing mechanism.

In vitro assays to study the tracking of shortening microtubule ends and to measure associated forces

Accurate segregation of mitotic chromosomes relies in part on a strong linkage between the kinetochores and the plus ends of spindle microtubules (MTs). These attachments are maintained even as the MTs shorten from their kinetochore-associated ends, and despite the large variability in the magnitude of load from the chromosomal "cargo." Analysis of the underlying mechanisms has recently been facilitated by the identification and purification of various kinetochore complexes. In this chapter we review some existing approaches to study the interaction of these protein complexes with the ends of shortening MTs in vitro. Specifically, we describe the application of a "segmented" MT technique, which allows quantitative characterization of the tracking of the shortening MT ends by fluorescent proteins and protein-coated beads, as well as controlled measurement of the associated forces. There is a marked similarity between these methods and the approaches that are used to study the motions and forces produced by ATP-dependent motor enzymes walking on coverslip-attached, stable MTs. However, optical resolution at the shortening ends of coverslip-tethered MTs is not as good and the thermal noise is high. Furthermore, there are significant differences in the mechanisms of motions of microbeads driven by motors and by MT depolymerization, as well as in the interpretation of the resulting forces. Clearly, the depolymerization-driven motions are difficult to study and the corresponding phenomenology and theories are more complex than in the motors field. We hope, however, that the relatively straightforward assays based on "segmented" MTs, which are described below, will become a routine methodology, thereby helping to advance the studies of the MT-depolymerization-dependent motility.

Force transduction by the microtubule-bound Dam1 ring

The coupling between the depolymerization of microtubules (MTs) and the motion of the Dam1 ring complex is now thought to play an important role in the generation of forces during mitosis. Our current understanding of this motion is based on a number of detailed computational models. Although these models realize possible mechanisms for force transduction, they can be extended by variation of any of a large number of poorly measured parameters and there is no clear strategy for determining how they might be distinguished experimentally. Here we seek to identify and analyze two distinct mechanisms present in the computational models. In the first, the splayed protofilaments at the end of the depolymerizing MT physically prevent the Dam1 ring from falling off the end, and in the other, an attractive binding secures the ring to the microtubule. Based on this analysis, we discuss how to distinguish between competing models that seek to explain how the Dam1 ring stays on the MT. We propose novel experimental approaches that could resolve these models for the first time, either by changing the diffusion constant of the Dam1 ring (e.g., by tethering a long polymer to it) or by using a time-varying load.

The perpetual movements of anaphase

One of the most extraordinary events in the lifetime of a cell is the coordinated separation of sister chromatids during cell division. This is truly the essence of the entire mitotic process and the reason for the most profound morphological changes in cytoskeleton and nuclear organization that a cell may ever experience. It all occurs within a very short time window known as "anaphase", as if the cell had spent the rest of its existence getting ready for this moment in an ultimate act of survival. And there is a good reason for this: no space for mistakes. Problems in the distribution of chromosomes during cell division have been correlated with aneuploidy, a common feature observed in cancers and several birth defects, and the main cause of spontaneous abortion in humans. In this paper, we critically review the mechanisms of anaphase chromosome motion that resisted the scrutiny of more than 100 years of research, as part of a tribute to the pioneering work of Miguel Mota.