Cell biology: powerful curves

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.

Dynamical Majorana edge modes in a broad class of topological mechanical systems

Mechanical systems can display topological characteristics similar to that of topological insulators. Here we report a large class of topological mechanical systems related to the BDI symmetry class. These are self-assembled chains of rigid bodies with an inversion centre and no reflection planes. The particle-hole symmetry characteristic to the BDI symmetry class stems from the distinct behaviour of the translational and rotational degrees of freedom under inversion. This and other generic properties led us to the remarkable conclusion that, by adjusting the gyration radius of the bodies, one can always simultaneously open a gap in the phonon spectrum, lock-in all the characteristic symmetries and generate a non-trivial topological invariant. The particle-hole symmetry occurs around a finite frequency, and hence we can witness a dynamical topological Majorana edge mode. Contrasting a floppy mode occurring at zero frequency, a dynamical edge mode can absorb and store mechanical energy, potentially opening new applications of topological mechanics.

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.

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.

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.

Mechanochemical modeling of dynamic microtubule growth involving sheet-to-tube transition

Microtubule dynamics is largely influenced by nucleotide hydrolysis and the resultant tubulin configuration changes. The GTP cap model has been proposed to interpret the stabilizing mechanisms of microtubule growth from the view of hydrolysis effects. Besides, the growth of a microtubule involves the closure of a curved sheet at its growing end. The curvature conversion from the longitudinal direction to the circumferential direction also helps to stabilize the successive growth, and the curved sheet is referred to as the conformational cap. However, there still lacks theoretical investigation on the mechanical–chemical coupling growth process of microtubules. In this paper, we study the growth mechanisms of microtubules by using a coarse-grained molecular method. First, the closure process involving a sheet-to-tube transition is simulated. The results verify the stabilizing effect of the sheet structure and predict that the minimum conformational cap length that can stabilize the growth is two dimers. Then, we show that the conformational cap and the GTP cap can function independently and harmoniously, signifying the pivotal role of mechanical factors. Furthermore, based on our theoretical results, we describe a Tetris-like growth style of microtubules: the stochastic tubulin assembly is regulated by energy and harmonized with the seam zipping such that the sheet keeps a practically constant length during growth.

Dynamic instability of a growing adsorbed polymorphic filament

The intermittent transition between slow growth and rapid shrinkage in polymeric assemblies is termed "dynamic instability", a feature observed in a variety of biochemically distinct assemblies including microtubules, actin, and their bacterial analogs. The existence of this labile phase of a polymer has many functional consequences in cytoskeletal dynamics, and its repeated appearance suggests that it is relatively easy to evolve. Here, we consider the minimal ingredients for the existence of dynamic instability by considering a single polymorphic filament that grows by binding to a substrate, undergoes a conformation change, and may unbind as a consequence of the residual strains induced by this change. We identify two parameters that control the phase space of possibilities for the filament: a structural mechanical parameter that characterizes the ratio of the bond strengths along the filament to those with the substrate (or equivalently the ratio of longitudinal to lateral interactions in an assembly), and a kinetic parameter that characterizes the ratio of timescales for growth and conformation change. In the deterministic limit, these parameters serve to demarcate a region of uninterrupted growth from that of collapse. However, in the presence of disorder in either the structural or the kinetic parameter the growth and collapse phases can coexist where the filament can grow slowly, shrink rapidly, and transition between these phases, thus exhibiting dynamic instability. We exhibit the window for the existence of dynamic instability in a phase diagram that allows us to quantify the evolvability of this labile phase.

Structural dynamics of an actin spring

Actin-based motility in cells is usually associated with either polymerization/depolymerization in the presence of cross-linkers or contractility in the presence of myosin motors. Here, we focus on a third distinct mechanism involving actin in motility, seen in the dynamics of an active actin spring that powers the acrosomal reaction of the horseshoe crab (Limulus polyphemus) sperm. During this process, a 60-μm bent and twisted bundle of cross-linked actin uncoils and becomes straight in a few seconds in the presence of Ca(2+). This straightening, which occurs at a constant velocity, allows the acrosome to forcefully penetrate the egg. Synthesizing ultrastructural information with the kinetics, energetics, and imaging of calcium binding allows us to construct a dynamical theory for this mechanochemical engine consistent with our experimental observations. It also illuminates the general mechanism by which energy may be stored in conformational changes and released cooperatively in ordered macromolecular assemblies.

Topological phonon modes and their role in dynamic instability of microtubules

Microtubules (MTs) are self-assembled hollow protein tubes playing important functions in live cells. Their building block is a protein called tubulin, which self-assembles in a particulate 2 dimensional lattice. We study the vibrational modes of this lattice and find Dirac points in the phonon spectrum. We discuss a splitting of the Dirac points that leads to phonon bands with nonzero Chern numbers, signaling the existence of topological vibrational modes localized at MTs edges, which we indeed observe after explicit calculations. Since these modes are robust against the large changes occurring at the edges during the dynamic cycle of the MTs, we can build a simple mechanical model to illustrate how they would participate in this phenomenon.

Regulation of microtubule dynamics by inhibition of the tubulin deacetylase HDAC6

We studied the role of a class II histone deacetylase, HDAC6, known to function as a potent alpha-tubulin deacetylase, in the regulation of microtubule dynamics. Treatment of cells with the class I and II histone deacetylase inhibitor TSA, as well as the selective HDAC6 inhibitor tubacin, increased microtubule acetylation and significantly reduced velocities of microtubule growth and shrinkage. siRNA-mediated knockdown of HDAC6 also increased microtubule acetylation but, surprisingly, had no effect on microtubule growth velocity. At the same time, HDAC6 knockdown abolished the effect of tubacin on microtubule growth, demonstrating that tubacin influences microtubule dynamics via specific inhibition of HDAC6. Thus, the physical presence of HDAC6 with impaired catalytic activity, rather than tubulin acetylation per se, is the factor responsible for the alteration of microtubule growth velocity in HDAC6 inhibitor-treated cells. In support of this notion, HDAC6 mutants bearing inactivating point mutations in either of the two catalytic domains mimicked the effect of HDAC6 inhibitors on microtubule growth velocity. In addition, HDAC6 was found to be physically associated with the microtubule end-tracking protein EB1 and a dynactin core component, Arp1, both of which accumulate at the tips of growing microtubules. We hypothesize that inhibition of HDAC6 catalytic activity may affect microtubule dynamics by promoting the interaction of HDAC6 with tubulin and/or with other microtubule regulatory proteins.

Mal3, the Schizosaccharomyces pombe homolog of EB1, changes the microtubule lattice

In vitro studies of pure tubulin have suggested that tubulin heterodimers in cells assemble into B-lattice microtubules, where the 8-nm dimers in adjacent protofilaments are staggered by 0.9 nm. This arrangement requires the tube to close by forming a seam with an A-lattice, in which the protofilaments are staggered by 4.9 nm. Here we show that Mal3, an EB1 family tip-tracking protein, drives tubulin to assemble in vitro into exclusively 13-protofilament microtubules with a high proportion of A-lattice protofilament contacts. We present a three-dimensional cryo-EM reconstruction of a purely A-lattice microtubule decorated with Mal3, in which Mal3 occupies the groove between protofilaments and associates closely with one tubulin monomer. We propose that Mal3 promotes assembly by binding to freshly formed tubulin polymer and particularly favors any with A-lattice arrangement. These results reopen the question of microtubule structure in cells.

Intrinsic bending and structural rearrangement of tubulin dimer: molecular dynamics simulations and coarse-grained analysis

Microtubules are long polymers of alphabeta-tubulin heterodimers. They undergo a process known as dynamic instability, in which the ends of a microtubule switch stochastically between phases of slow growth and rapid shrinkage. The molecular mechanisms inducing the depolymerization of microtubules were attributed to the hydrolysis of the guanosine triphosphate (GTP) nucleotide bound to the beta-tubulin. The hydrolysis of GTP is thought to cause microtubule instability by promoting outward curving of the protofilaments constituting the microtubule lattice. The bending of protofilaments is associated with the structural transformation of a tubulin dimer from straight to curved conformations. However, the nature of intrinsic bending of the dimer remains elusive. This study uses molecular dynamics (MD) simulations and coarse-grained analysis to reveal the intrinsic bending, as well as the local structural rearrangements, of the unassembled tubulin dimer as the dimer relaxes from its lattice-constrained, straight conformation of a zinc-induced tubulin sheet. The effect of the nucleotide state on dimer-bending is investigated by the introduction of gamma-phosphate into the beta-tubulin to form GTP-bound tubulin. In agreement with recent experimental studies that proposed nucleotide-independent curved conformations, both guanosine diphosphate (GDP)-bound and GTP-bound tubulin dimers were found to have curved conformations, but with a tendency toward smaller bending in the GTP-tubulin than in the GDP-tubulin. The perturbation induced through the introduction of gamma-phosphate is posited to play a role in straightening the intradimer bending. The local structural rearrangements of GDP-tubulin because of the bending mode of motion of the dimer reveal that one of the three functional domains, the intermediate domain, exhibits significantly lower bending deformation compared with the others, signifying a dynamic connection to the functionally defined domains.

EB1 regulates microtubule dynamics and tubulin sheet closure in vitro

End binding 1 (EB1) is a plus-end-tracking protein (+TIP) that localizes to microtubule plus ends where it modulates their dynamics and interactions with intracellular organelles. Although the regulating activity of EB1 on microtubule dynamics has been studied in cells and purified systems, the molecular mechanisms involved in its specific activity are still unclear. Here, we describe how EB1 regulates the dynamics and structure of microtubules assembled from pure tubulin. We found that EB1 stimulates spontaneous nucleation and growth of microtubules, and promotes both catastrophes (transitions from growth to shrinkage) and rescues (reverse events). Electron cryomicroscopy showed that EB1 induces the initial formation of tubulin sheets, which rapidly close into the common 13-protofilament-microtubule architecture. Our results suggest that EB1 favours the lateral association of free tubulin at microtubule-sheet edges, thereby stimulating nucleation, sheet growth and closure. The reduction of sheet length at microtubule growing-ends together with the elimination of stressed microtubule lattices may account for catastrophes. Conversely, occasional binding of EB1 to the microtubule lattice may induce rescues.

Structures and functions of yeast kinetochore complexes

The kinetochore is a key cell division organelle that enables high-fidelity transmission of genetic information by coupling chromosomes to spindle microtubules during mitosis and meiosis. Despite its cytological description more than a century ago, remarkably little information is available on kinetochore function at a molecular level. Recently, important advances elucidating the overall organization of kinetochores, as well as information about the structures and molecular mechanisms of kinetochore function, have been achieved through a detailed analysis of the kinetochores of the budding yeast Saccharomyces cerevisiae. Here we review the current understanding of kinetochore function in budding yeast and draw comparisons to recent findings in other organisms.

Drosophila as a genetic and cellular model for studies on axonal growth

One of the most fascinating processes during nervous system development is the establishment of stereotypic neuronal networks. An essential step in this process is the outgrowth and precise navigation (pathfinding) of axons and dendrites towards their synaptic partner cells. This phenomenon was first described more than a century ago and, over the past decades, increasing insights have been gained into the cellular and molecular mechanisms regulating neuronal growth and navigation. Progress in this area has been greatly assisted by the use of simple and genetically tractable invertebrate model systems, such as the fruit fly Drosophila melanogaster. This review is dedicated to Drosophila as a genetic and cellular model to study axonal growth and demonstrates how it can and has been used for this research. We describe the various cellular systems of Drosophila used for such studies, insights into axonal growth cones and their cytoskeletal dynamics, and summarise identified molecular signalling pathways required for growth cone navigation, with particular focus on pathfinding decisions in the ventral nerve cord of Drosophila embryos. These Drosophila-specific aspects are viewed in the general context of our current knowledge about neuronal growth.

Mitosis: riding the protofilament curl

More than 50 years ago, microtubule depolymerization was proposed as the force responsible for chromosome movement. New studies measure the force produced by depolymerization and show that protein ring complexes can couple depolymerization to movement. These results have implications for anaphase chromosome motility and mitotic evolution.

Structural intermediates in microtubule assembly and disassembly: how and why?

Microtubules are cytoskeletal polymers made of repeating alphabeta-tubulin heterodimers that play essential roles in all eukaryotic cells. The complex dynamic behavior of microtubules, which is ultimately due to the tubulin subunit structure and its intrinsic GTPase activity, is key to the functions of these ubiquitous polymers. Microtubule assembly and disassembly do not take place by simple helical growth and shrinkage via individual subunits, but rather involve transient polymer intermediates, distinct from the microtubule, without parallel in other biological self-assembly systems. The discovery of these intermediates a decade ago has been followed recently by quantitative descriptions of their structure and their relationship to nucleotide state.

The Dam1 kinetochore ring complex moves processively on depolymerizing microtubule ends

Chromosomes interact through their kinetochores with microtubule plus ends and they are segregated to the spindle poles as the kinetochore microtubules shorten during anaphase A of mitosis. The molecular natures and identities of coupling proteins that allow microtubule depolymerization to pull chromosomes to poles during anaphase have long remained elusive. In budding yeast, the ten-protein Dam1 complex is a critical microtubule-binding component of the kinetochore that oligomerizes into a 50-nm ring around a microtubule in vitro. Here we show, with the use of a real-time, two-colour fluorescence microscopy assay, that the ring complex moves processively for several micrometres at the ends of depolymerizing microtubules without detaching from the lattice. Electron microscopic analysis of 'end-on views' revealed a 16-fold symmetry of the kinetochore rings. This out-of-register arrangement with respect to the 13-fold microtubule symmetry is consistent with a sliding mechanism based on an electrostatically coupled ring-microtubule interface. The Dam1 ring complex is a molecular device that can translate the force generated by microtubule depolymerization into movement along the lattice to facilitate chromosome segregation.