Estimates of lateral and longitudinal bond energies within the microtubule lattice

Thermal fluctuations of grafted microtubules provide evidence of a length-dependent persistence length

Microtubules are hollow cylindrical structures that constitute one of the three major classes of cytoskeletal filaments. On the mesoscopic length scale of a cell, their material properties are characterized by a single stiffness parameter, the persistence length l(p). Its value, in general, depends on the microscopic interactions between the constituent tubulin dimers and the architecture of the microtubule. Here, we use single-particle tracking methods combined with a fluctuation analysis to systematically study the dependence of l(p) on the total filament length L. Microtubules are grafted to a substrate with one end free to fluctuate in three dimensions. A fluorescent bead is attached proximally to the free tip and is used to record the thermal fluctuations of the microtubule's end. The position distribution functions obtained with this assay allow the precise measurement of l(p) for microtubules of different contour length L. Upon varying L between 2.6 and 47.5 mum, we find a systematic increase of l(p) from 110 to 5,035 mum. At the same time we verify that, for a given filament length, the persistence length is constant over the filament within the experimental accuracy. We interpret this length dependence as a consequence of a nonnegligible shear deflection determined by subnanometer relative displacement of adjacent protofilaments. Our results may shine new light on the function of microtubules as sophisticated nanometer-sized molecular machines and give a unified explanation of seemingly uncorrelated spreading of microtubules' stiffness previously reported in literature.

Insights into the mechanism of microtubule stabilization by Taxol

The antitumor drug Taxol stabilizes microtubules and reduces their dynamicity, promoting mitotic arrest and cell death. Upon assembly of the alpha/beta-tubulin heterodimer, GTP bound to beta-tubulin is hydrolyzed to GDP reaching a steady-state equilibrium between free tubulin dimers and microtubules. The binding of Taxol to beta-tubulin in the polymer results in cold-stable microtubules at the expense of tubulin dimers, even in the absence of exogenous GTP. However, there is little biochemical insight into the mechanism(s) by which Taxol stabilizes microtubules. Here, we analyze the structural changes occurring in both beta- and alpha-tubulin upon microtubule stabilization by Taxol. Hydrogen/deuterium exchange (HDX) coupled to liquid chromatography-electrospray ionization MS demonstrated a marked reduction in deuterium incorporation in both beta-and alpha-tubulin when Taxol was present. Decreased local HDX in peptic peptides was mapped on the tubulin structure and revealed both expected and new dimer-dimer interactions. The increased rigidity in Taxol microtubules was distinct from and complementary to that due to GTP-induced polymerization. The Taxol-induced changes in tubulin conformation act against microtubule depolymerization in a precise directional way. These results demonstrate that HDX coupled to liquid chromatography-electrospray ionization MS can be effectively used to study conformational effects induced by small ligands on microtubules. The present study also opens avenues for locating drug and protein binding sites and for deciphering the mechanisms by which their interactions alter the conformation of microtubules and tubulin dimers.

Elastic response, buckling, and instability of microtubules under radial indentation

We tested the mechanical properties of single microtubules by lateral indentation with the tip of an atomic force microscope. Indentations up to approximately 3.6 nm, i.e., 15% of the microtubule diameter, resulted in an approximately linear elastic response, and indentations were reversible without hysteresis. At an indentation force of around 0.3 nN we observed an instability corresponding to an approximately 1-nm indentation step in the taxol-stabilized microtubules, which could be due to partial or complete rupture of a relatively small number of lateral or axial tubulin-tubulin bonds. These indentations were reversible with hysteresis when the tip was retracted and no trace of damage was observed in subsequent high-resolution images. Higher forces caused substantial damage to the microtubules, which either led to depolymerization or, occasionally, to slowly reannealing holes in the microtubule wall. We modeled the experimental results using finite-element methods and find that the simple assumption of a homogeneous isotropic material, albeit structured with the characteristic protofilament corrugations, is sufficient to explain the linear elastic response of microtubules.

Compartment volume influences microtubule dynamic instability: a model study

Microtubules (MTs) are cytoskeletal polymers that exhibit dynamic instability, the random alternation between growth and shrinkage. MT dynamic instability plays an essential role in cell development, division, and motility. To investigate dynamic instability, simulation models have been widely used. However, conditions under which the concentration of free tubulin fluctuates as a result of growing or shrinking MTs have not been studied before. Such conditions can arise, for example, in small compartments, such as neuronal growth cones. Here we investigate by means of computational modeling how concentration fluctuations caused by growing and shrinking MTs affect dynamic instability. We show that these fluctuations shorten MT growth and shrinkage times and change their distributions from exponential to non-exponential, gamma-like. Gamma-like distributions of MT growth and shrinkage times, which allow optimal stochastic searching by MTs, have been observed in various cell types and are believed to require structural changes in the MT during growth or shrinkage. Our results, however, show that these distributions can already arise as a result of fluctuations in the concentration of free tubulin due to growing and shrinking MTs. Such fluctuations are possible not only in small compartments but also when tubulin diffusion is slow or when many MTs (de)polymerize synchronously. Volume and all other factors that influence these fluctuations can affect MT dynamic instability and, consequently, the processes that depend on it, such as neuronal growth cone behavior and cell motility in general.

Polymerization dynamics of double-stranded biopolymers: chemical kinetic approach

The polymerization dynamics of double-stranded polymers, such as actin filaments, is investigated theoretically using simple chemical kinetic models that explicitly take into account some microscopic details of the polymer structure and the lateral interactions between the protofilaments. By considering all possible molecular configurations, the exact analytical expressions for the growth velocity and dispersion for two-stranded polymers are obtained in the case of the growing at only one end, and for the growth from both polymer ends. Exact theoretical calculations are compared with the predictions of approximate multilayer models that consider only a finite number of the most relevant polymer configurations. Our theoretical approach is applied to analyze the experimental data on the growth and fluctuations dynamics of individual single actin filaments.

Nonequilibrium self-assembly of linear fibers: microscopic treatment of growth, decay, catastrophe and rescue

Many of the large structures of cells are constructed from fibers. These fibers self-assemble from individual proteins in a far-from-equilibrium fashion. Nonequilibrium self-assembly results in a highly dynamic process at the subcellular level that can be regulated and tuned to carry out many of the biological functions of the cell: growth, division and locomotion. We construct and analyze a nonequilibrium model of the dynamic end of a biological fiber that possesses site-resolved resolution. We solve for the steady states of this nonequilibrium system using a variational method. The results are compared to exact numerical solutions for systems with modest size. Using an effective reaction coordinate, we construct an effective potential from the steady-state distribution. The stochastic transitions of the system can be analyzed in this representation. We then apply this method to model microtubule systems. Predictions for macroscopic catastrophe, rescue and dynamic instability in the steady states are analyzed. We find that the length of the cap of the microtubule is small. The relations between the catastrophe/rescue rate and the growth rate are also discussed.

Monte Carlo simulations of rigid biopolymer growth processes

Rigid biopolymers, such as actin filaments, microtubules, and intermediate filaments, are vital components of the cytoskeleton and the cellular environment. Understanding biopolymer growth dynamics is essential for the description of the mechanisms and principles of cellular functions. These biopolymers are composed of N parallel protofilaments which are aligned with arbitrary but fixed relative displacements, thus giving rise to complex end structures. We have investigated rigid biopolymer growth processes by Monte Carlo simulations by taking into account the effects of such "end" properties and lateral interactions. Our simulations reproduce analytical results for the case of N = 2, which is biologically relevant for actin filaments. For the case of N = 13, which applies to microtubules, the simulations produced results qualitatively similar to the N = 2 case. The simulation results indicate that polymerization events are evenly distributed among the N protofilaments, which imply that both end-structure effects and lateral interactions are significant. The effect of different splittings in activation energy has been investigated for the case of N = 2. The effects of activation energy coefficients on the specific polymerization and depolymerization processes were found to be unsubstantial. By expanding the model, we have also obtained a force-velocity relationship of microtubules as observed in experiments. In addition, a range of lateral free-energy parameters was found that yields growth velocities in accordance with experimental observations and previous simulation estimates for the case of N = 13.

Microtubules: a ring for the depolymerization motor

Newly discovered rings around microtubules, assembled from the Dam1 protein complex, may provide the dynamic linkage at microtubule ends for force generation coupled to microtubule depolymerization and polymerization.

Altered microtubule dynamics by expression of modified alpha-tubulin protein causes right-handed helical growth in transgenic Arabidopsis plants

The proper organization of cortical microtubule arrays is essential for anisotropic growth in plants but how distinct array patterns are formed is not understood. Here, we report a relationship between microtubule dynamics and array organization using transgenic plants expressing modified tubulins. When green fluorescent protein (GFP) or a hemaglutinin epitope tag was fused to the N-terminus of tubulins and expressed in Arabidopsis plants, these tubulins were incorporated into microtubules along with endogenous tubulins. Plants expressing the modified beta-tubulins were phenotypically normal and possessed transversely oriented cortical arrays in the epidermal cells of the root elongation zone; however, the expression of modified alpha-tubulins caused right-handed helical growth, increased trichome branching, and a shallow left-handed (S-form) helical array organization. In cells expressing the modified alpha-tubulins, microtubule dynamicity was suppressed and polymerization was promoted, and GFP-EB1 (End Binding 1) labeled larger regions of the microtubule end more frequently, when compared with control cells. We propose that the N-terminal appendage introduced into alpha-tubulin inhibits GTP hydrolysis, thus producing polymerization-prone microtubules with an extended GTP cap. Consistent with this interpretation, plants expressing an alpha-tubulin mutated in the GTPase-activating domain exhibited similar microtubule properties, with regard to dynamics and the localization of GFP-EB1, and showed right-handed helical growth.

Mechanochemical model of microtubule structure and self-assembly kinetics

Microtubule self-assembly is largely governed by the chemical kinetics and thermodynamics of tubulin-tubulin interactions. An important aspect of microtubule assembly is that hydrolysis of the beta-tubulin-associated GTP promotes protofilament curling. Protofilament curling presumably drives the transition from tip structures associated with growth (sheetlike projections and blunt ends) to those associated with shortening (rams' horns and frayed ends), and transitions between these structures have been proposed to be important for growth-shortening transitions. However, previous models for microtubule dynamic instability have not considered such structures or mechanics explicitly. Here we present a three-dimensional model that explicitly incorporates mechanical stress and strain within the microtubule lattice. First, we found that the model recapitulates three-dimensional tip structures and rates of assembly and disassembly for microtubules grown under standard conditions, and we propose that taxol may stabilize microtubule growth by reducing flexural rigidity. Second, in contrast to recent suggestions, it was determined that sheetlike tips are more likely to undergo catastrophe than blunt tips. Third, partial uncapping of the tubulin-GTP cap provides a possible mechanism for microtubule pause events. Finally, simulations of the binding and structural effects of XMAP215 produced the experimentally observed growth and shortening rates, and tip structure.

Mechanical stress induced mechanism of microtubule catastrophes

Microtubules assembled in vitro from pure tubulin can switch occasionally from growing to shrinking states or resume assembly, an unusual behavior termed "dynamic instability of microtubule growth". Its origin remains unclear and several models have been proposed, including occasional switching of the microtubules into energetically unfavorable configurations during assembly. In this study, we have asked whether the excess energy accumulated in these configurations would be of sufficient magnitude to destabilize the capping region that must exist at the end of growing microtubules. For this purpose, we have analyzed the frequency distribution of microtubules assembled in vitro from pure tubulin, and modeled the different mechanical constraints accumulated in their wall. We find that the maximal excess energy that the microtubule lattice can store is in the order of 11 kBT per dimer. Configurations that require distortions up to approximately 20 kBT are allowed at the expense of a slight conformational change, and larger distortions are not observed. Modeling of the different elastic deformations suggests that the excess energy is essentially induced by protofilament skewing, microtubule radial curvature change and inter-subunit shearing, distortions that must destabilize further the tubulin subunits interactions. These results are consistent with the hypothesis that unfavorable closure events may trigger the catastrophes observed at low tubulin concentration in vitro. In addition, we propose a novel type of representation that describes the stability of microtubule assembly systems, and which might be of considerable interest to study the effects of stabilizing and destabilizing factors on microtubule structure and dynamics.

Tension-dependent regulation of microtubule dynamics at kinetochores can explain metaphase congression in yeast

During metaphase in budding yeast mitosis, sister kinetochores are tethered to opposite poles and separated, stretching their intervening chromatin, by singly attached kinetochore microtubules (kMTs). Kinetochore movements are coupled to single microtubule plus-end polymerization/depolymerization at kinetochore attachment sites. Here, we use computer modeling to test possible mechanisms controlling chromosome alignment during yeast metaphase by simulating experiments that determine the 1) mean positions of kinetochore Cse4-GFP, 2) extent of oscillation of kinetochores during metaphase as measured by fluorescence recovery after photobleaching (FRAP) of kinetochore Cse4-GFP, 3) dynamics of kMTs as measured by FRAP of GFP-tubulin, and 4) mean positions of unreplicated chromosome kinetochores that lack pulling forces from a sister kinetochore. We rule out a number of possible models and find the best fit between theory and experiment when it is assumed that kinetochores sense both a spatial gradient that suppresses kMT catastrophe near the poles and attachment site tension that promotes kMT rescue at higher amounts of chromatin stretch.

Elastic vibrations in seamless microtubules

Parameters characterizing elastic properties of microtubules, measured in several recent experiments, reflect an anisotropic character. We describe the microscopic dynamical properties of microtubules using a discrete model based on an appropriate lattice of dimers. Adopting a harmonic approximation for the dimer-dimer interactions and estimating the lattice elastic constants, we make predictions regarding vibration dispersion relations and vibration propagation velocities. Vibration frequencies and velocities are expressed as functions of the elastic constants and of the geometrical characteristics of the microtubules. We show that vibrations which propagate along the protofilament do so significantly faster than those along the helix.

Anisotropic elastic properties of microtubules

We review and model the experimental parameters which characterize elastic properties of microtubules. Three macroscopic estimates are made of the anisotropic elastic moduli, accounting for the molecular forces between tubulin dimers: for a longitudinal compression of a microtubule, for a lateral force and for a shearing force. These estimates reflect the anisotropies in these parameters observed in several recent experiments.

A molecular-mechanical model of the microtubule

Dynamic instability of MTs is thought to be regulated by biochemical transformations within tubulin dimers that are coupled to the hydrolysis of bound GTP. Structural studies of nucleotide-bound tubulin dimers have recently provided a concrete basis for understanding how these transformations may contribute to MT dynamic instability. To analyze these ideas, we have developed a molecular-mechanical model in which structural and biochemical properties of tubulin are used to predict the shape and stability of MTs. From simple and explicit features of tubulin, we define bond energy relationships and explore the impact of their variations on integral MT properties. This modeling provides quantitative predictions about the GTP cap. It specifies important mechanical features underlying MT instability and shows that this property does not require GTP-hydrolysis to alter the strength of tubulin-tubulin bonds. The MT plus end is stabilized by at least two layers of GTP-tubulin subunits, whereas the minus end requires at least one; this and other differences between the ends are explained by asymmetric force balances. Overall, this model provides a new link between the biophysical characteristics of tubulin and the physiological behavior of MTs. It will also be useful in building a more complete description of MT dynamics and mechanics.

The physical basis of microtubule structure and stability

Microtubules are cylindrical polymers found in every eukaryotic cell. They have a unique helical structure that has implications at both the cellular level, in terms of the functions they perform, and at the multicellular level, such as determining the left-right symmetry in plants. Through the combination of an atomically detailed model for a microtubule and large-scale computational techniques for computing electrostatic interactions, we are able to explain the observed microtubule structure. On the basis of the lateral interactions between protofilaments, we have determined that B lattice is the most favorable configuration. Further, we find that these lateral bonds are significantly weaker than the longitudinal bonds along protofilaments. This explains observations of microtubule disassembly and may serve as another step toward understanding the basis for dynamic instability.