Self-organization of microtubules and motors.

F.J. Nédélec 1,2, T. Surrey 1, A.C. Maggs 2 and S. Leibler 1,#

1 Departments of Molecular Biology and Physics, Princeton University, Princeton, N.J. 08544, U.S.A.

2 Laboratoire de Physico-Chimie Théorique, E.S.P.C.I., 10 rue Vauquelin, 75231 Paris, Cédex 05, France.

Cellular structures are established and maintained through a dynamic interplay between assembly phenomena and regulatory processes. From this perspective, self-organization of molecular components provides a variety of possible spatial structures; the regulatory machinery constantly chooses the most appropriate to express a given cellular functionc1. Here, we study the extent and the characteristics of self-organization using microtubules and molecular motors2 as a model system. These components are known to participate in the formation of many cellular structures, such as the dynamic asters found in mitotic and meiotic spindles3, 4. We show that a simple system consisting solely of multi-headed motor constructs, built with kinesin5, and stabilized microtubules is capable of organizing into asters. Dynamic asters can also be obtained starting from a homogeneous solution of tubulin and motors. By varying the relative concentrations of the components, we obtain a collection of self-organized structures. We study this process in a constrained geometry of micro-fabricated glass chambers6 and demonstrate that the same final structure can be reached through different assembly pathways.

Mitotic spindle formation is an intensively studied prototype of cell morphogenesis phenomena. In the classical scenario1, a bipolar spindle forms when mitotic chromosomes capture and selectively stabilize dynamic microtubules nucleated by two centrosomes. This assembly process can tolerate numerous variations in intermediate configurations, including different chromosome positions and geometrical and mechanical perturbations7. Spindles can also form in the absence of centrosomes, as observed in meiotic cell divisions, and, more surprisingly, in in vitro experiments in Xenopus lævis egg extracts devoid of centrosomes8, 9. In the latter case, microtubules grown in the vicinity of chromatin are organized into a bipolar spindle by molecular motors. Beyond their immediate impact on the study of mitosis, these results are important for two reasons. Firstly, they suggest that microtubules and motors alone are capable of self-organizing into quite evolved spatial structures. Secondly, from a more general perspective, they also demonstrate that formation of a cellular structure can take place through various, distinct assembly pathways 8, 9. It is easy to imagine that, as in the case of metabolic or signal transduction pathways, a given route of assembly could be selected depending on cell type, interactions with the environment or state of growth.

In order to explore the issues of self-organization of microtubules and motors, and of the existence of different assembly pathways, we have constructed an extremly simplified system with only a few purified components. In addition to tubulin, this system includes artificial molecular constructs of several kinesin heads associated through biotin-streptavidin links (see Material and Methods).We have chosen kinesin because of its extensive molecular characterization10-12. The motor constructs, which we will often refer to as kinesins or motors , are able to attach to neighboring microtubules, and then, in the presence of ATP, to move towards the microtubule plus ends13. In this way they form dynamic crosslinks between microtubules (Figure 1a).

Figure 1b shows self-organization in a system of microtubules stabilized with taxol14 and mixed with motors. The sample is sandwiched between slide and coverslip in a quasi-two dimensional geometry. The starting configuration is an isotropic gel consisting of microtubules crosslinked randomly by motors (not shown). Asters of microtubules form within a few minutes as kinesins accumulate in their centers. (Similar asters were previously observed to form in a system of kinesin-containing chromaffin vesicles mixed with microtubules15; as well as in Xenopus lævis egg extracts with the addition of taxol16 or DMSO17.) Naturally, the size of asters is determined by the length distribution of stabilized microtubules. When two neighbouring asters overlap sufficiently, they can merge18. The aster formation can be reproduced in numerical simulations (Figure 1c), in which microtubules are treated as flexible, polar rods, and kinesin-like motors are characterized by a linear force-velocity curve11, 12, high processivity11, 12 and a tendency to remain attached to microtubule plus ends19 (see Materials and Methods). The simulations demonstrate that these properties are sufficient to explain the experimentally observed behaviour. The use of taxol permits a control of microtubule's length and number, however, it is important to note that stabilizing microtubules with taxol is not necessary for aster formation; asters also form if microtubules are allowed to assemble and disassemble through the dynamic instability20. All experiments presented below are performed in the absence of taxol.

In order to study the influence of geometrical constraints on self-organization, we have encapsulated solutions of tubulin and motors in sealed micro-fabricated glass chambers6 of various shapes, with typical lateral dimensions of 100 microns, and a thickness of 5 microns (Figure 2). In a cylindrical geometry, an initially homogeneous solution first transforms into a symmetric aster centered in the chamber. As microtubules grow further and begin to buckle, the center of the aster becomes unstable and a vortex structure forms. Vortices can be simultaneously observed in hundreds of chambers etched in a single coverslip and filled with the same initial solution; we saw in equal number structures with right-handed and left-handed vorticity. Since vortices form from asters, the microtubules have their plus ends oriented towards the center of the vortex; small objects such as short microtubules or small colloidal beads with motors attached to them, circle around the core of the vortex18. This circulatory motion is reminiscent of the microtubule and motor dependent movement of cytoplasm observed in the development of Drosophila melanogaster oocytes21. Vortices can also be reproduced in numerical simulation by incorporating geometrical confinement and the dynamic assembly of microtubules (Figure 2b). This simulation exhibits an assembly pathway very similar to that in the experiments, in which an initially symmetric aster is destabilized by the growth and buckling of microtubules.

A natural question now emerges: is the assembly through a symmetric aster the only possible route leading to the final vortex structure To answer it, we have enclosed the same solution of proteins in a chamber of the same size but with the topology of a torus. We have been inspired here by experiments of micro-surgery performed on melanophores22, in which the geometry of the cell was artificially changed. Obviously, the formation of symmetric asters is now prevented by the geometry of the chamber. Nevertheless, the same final steady-state, a vortex, can be reached (Figure 2c). This demonstrates that this simple system can find alternative assembly pathways.

Further experiments have shown that the precise shape of the containers is often unimportant for the choice of final assembled structures. For instance, we have regularly observed formation of circular vortices in square chambers18, However, in a chamber of the same torus-like shape but of a larger lateral size, a different final structure is observed (Figure 2d).

A surprising variety of larger-scale patterns can be formed in an unconfined geometry by further self-organization of the previously described structural elements, asters and vortices. The final patterns depend on the initial concentrations of molecular components: a 2-fold variation in the concentration of kinesin, included in the same solution of assembling microtubules, results in distinctly different patterns (Figure 3). At low motor concentrations, a lattice of vortices forms; at slightly higher kinesin concentrations, one observes lattices of asters. As discussed above, overlapping asters tend to fuse - this process may determine the final distance between the asters. For higher motor concentrations, the fusion process increases in efficiency and the average dimensions of the asters becomes larger. Finally, at even higher motor concentrations, a distinct state is achieved in which the microtubules form bundles. This process is sensitive to the microtubule nucleation rate, and depends on the salt conditions.

Self-organization observed in the present experiments can be compared to pattern formation encountered in simple physical systems23. Here a variety of structures has been formed far from thermal equilibrium from a homogeneous solution of proteins and from a gel of fibers, connected through non-covalent dynamic crosslinks. The energy dissipated in these structures is injected into the system through ATP and GTP hydrolysis24. From this point of view, the situation is reminiscent of patterns formed in chemical reactions. The use of protein components, however, opens a possibility of additional control at the molecular level. From a physicist's perspective, a complete characterization of the phenomena described here should include an evaluation of all the relevant time and length scales, and the determination of the out-of-equilibrium phase diagrams.

These simplified experiments show that the basic structural vocabulary used by the cell is extremely rich: with two basic components, and simple local rules of interaction, one obtains a large variety of assembled structures. By extending the present system as to include other interacting components, such as nucleation centers for microtubules or different motors, it will be possible to explore the conditions and the components needed for the formation of other structures. Another direction of study would be to introduce some elements of regulation. One could then search for rules underlying the choice of different words from this large vocabulary of self-organized structures.

Material and Methods.

Proteins: Recombinant kinesin K401-bio, consisting of 401 amino-acids of the N-terminal motor domain of the heavy chain of D. melanogaster kinesin linked to the BCCP sequence of E.coli, was expressed in E.coli and purified25. These recombinant kinesins form biotinylated active dimers 25, which can then be joined by adding streptavidin26 (Molecular Probes). A solution of motor constructs was obtained by mixing streptavidin and purified K401-bio, giving final concentrations of 50 µg/ml and ~150 µg/ml, respectively. Tubulin was purified6, the stock solution was at ~15.3 mg/ml in M2B buffer (80mM Pipes+KOH to pH 6.8, 2mM MgCl2, 1mM EGTA). Assay with stabilized microtubules: To polymerize microtubules, a solution containing 1.25 mg/ml tubulin, 1mM GTP, 1% DMSO in M2B was kept at 37°C for 15 min., taxol was then added to a final concentration of 20µM. Polymerized microtubules were kept at room temperature. To start self-assembly, polymerized microtubules, motor constructs, and a ATP solution were mixed to give final concentrations of 0.27 mg/ml tubulin, 50µg/ml K401-bio, 17 µg/ml Streptavidin, 1.4mM ATP, 0.6 mM GTP, 7.7mM MgCl2, 4.3 mM KCl, 1mM EGTA, 27 µg/ml a-casein (Sigma), 0.3% DMSO, 20µM taxol, 80mM PIPES/KOH to pH 6.8. This mixture was immediately examined by microscopy. Assay with non-stabilized, dynamic microtubules: A mixture of purified (unpolymerized) tubulin, motor constructs and nucleotides was now used. Apart from an increased tubulin concentration of 5.1 mg/ml and absence of taxol, all other concentrations were kept as in the assay with stabilized microtubules, except as further specified. Polymerization of microtubules was started by heating the sample to 37°C on the microscope. Glass cleaning and coating: VWR-brand 24x60 mm N°1 coverslips were cleaned by three rounds of sonication in a hot solution (~80°C) of 10% of detergent VWR-Extran 1000 in deionized water, followed by 5 rounds of sonication in pure, hot, deionized water. Slides were then immersed in pure ethanol and air-dried. Glass cleaned this way was hydrophilic. To prevent kinesin and microtubules from sticking to the surface, slides had then to be coated. They were dipped into a solution of 0.1% agarose in deionized water and were allowed to dry. This layer was further coated by dipping the glass first into a solution of 0.2% bovine serum albumin in M2B filtered at 0.2µm and then 4 times into deionized water. Coated slides were kept humid and were used within a few hours. Fabrication of microchambers: as in 6. Glass with microfabricated chambers was cleaned and coated as usual. Sealing of the microchambers was achieved by applying a steady ~20kg/cm2 pressure for 3 minutes on the pit-containing coverslip and the slide. Microscopy and imaging: a Zeiss Axiovert 135 TV with an Olympus 100X iris objective, an Olympus 20X objective and a Zeiss dark-field ultra condenser was used. Both condenser and objective were heated to warm the immersion oil and the sample to 37°C by circulating warm water through cooper-tubing wrapped around them. A CCD camera (Paultek inc.) was used to record on S-VHS video tape, no video processing was necessary to observe microtubules. A Nikon camera was also mounted on the microscope to take slides with Kodak Ektachrome P1600 film. Simulations: simulations will be described in detail elsewhere 18. Essentially, microtubules were represented by short rigid rods of 6 µm connected with flexible links; they could grow and shrink according to a simple model of dynamic instability27 The simulations were two-dimensional; the excluded volume interactions between microtubules were neglected so that microtubules can cross one another. This corresponds well to experiments performed in thin chambers or between closely spaced glass surfaces, in which microtubules are nearly parallel to the plane of the sample. Motors formed force generating links between two microtubules. The movement of the microtubules was then calculated at each time-step by assuming a completely damped viscous regime. At each time-step, motors moved along the microtubules with a speed which is a function of the force they exert. Motors also were assumed to stay attached with a higher probability at the plus ends of microtubules19. Unbound motors could bind with small probability when two microtubules cross each others. Almost all of the relevant parameters are known from previous experiments: microtubules dynamic instability parameters: polymerisation speed 2 µm/min. 28, microtubule rigidity: persistence length 5 mm. 29, motor maximum force 5 pN and force-speed curve: linear with maximum speed 0.8 µm/s 10, 11, microtubule viscous drag and diffusion coefficients: calculated assuming microtubules are perfect cylinders moving in water30. Typical simulation covered 10 min. of real time using standard molecular dynamic methods with a time step of 10-4s. On a Silicon Graphic Indy (R4000) the simulation took ~3 hours.

Acknowledgments.

We would like to thank Edgar Young and Jeff Gelles for the generous gift of kinesin plasmids and Jill Johnson for the gift of taxol. We have appreciated the help of Tim Holy in the preparation of glass chambers. We have also benefited from numerous discussions with Tim Holy, Michael Elowitz, Etienne Wolf, Eric Karsenti, Tim Mitchison, Joe Howard and Steven Block. This work has been supported by grants from the N.I.H., the N.S.F and the H.F.S.P. Additional support through a fellowship from the Deutsche Forschungsgemeinschaft to T.S. and from the French Government to F.J.N. is also acknowledged.

Figure captions:

Figure 1. Self-organization of taxol-stabilized microtubules and kinesin constructs into asters. (a) Schematic view of a kinesin-streptavidin construct moving simultaneously along two microtubules. The kinesin constructs can be viewed as force-generating, mobile crosslinks. (b) Formation of an aster as observed by dark-field microscopy: polymerized and taxol-stabilized microtubules were mixed with motor constructs and ATP. The sample is shown after ~1 min. (center) and ~2 min. (right). The bright spot in the center of the aster is caused by light scattering from accumulated microtubules and motors. Kinesin K401-bio moves microtubules with typical speeds up to ~0.8 µm/sec in motility assays (data not shown). We have varied the relative amount of streptavidin to kinesin and have found that the maximum activity in motility assays is observed at a stochiometric ratio of 4 K401-bio dimers per streptavidin, this ratio was used for all experiments. No aster is formed without streptavidin, or with too much streptavidin. Note that in cells and cell free extracts, where the asters are organized by minus-end directed motors, such as dyneins16, 17, the asters polarity is opposite. (c) Time sequence (0, 1.25 min. and 4 min.) of a numerical simulation showing aster formation. 150 non dynamic microtubules with an exponential length distribution of average length 41µm.

Figure 2. Self-organization in the constrained geometry of micro-fabricated chambers etched in glass. (a) Formation of a vortex as observed by dark-field microscopy: a homogeneous solution of tubulin, motor constructs, ATP and GTP was sealed in a chamber with a diameter of 90µm and a depth of 5 µm. Microtubule polymerization has been initiated by warming the sample to 37°C. Left: the sample after ~0.5 min., microtubules nucleate uniformly in the sample. Middle: after ~1.5 min. an aster forms in the center of the chamber. Right: after ~3 min. and more one observes a steady-state structure: a vortex with clockwise polarity. (b) Two-dimensional numerical simulation of the same process: 100 dynamic microtubules in a round chamber of diameter of 90µm; their mean length was initially smaller than the chamber radius, but increased above 45µm in the state shown (~5 min.). (c) Vortex structure in a torus-shaped chamber of the same diameter as the previous ones, but including a central un-etched region. The final structure shown here is similar to the one in (a), but with opposite vorticity. Both vorticities are observed with equal probability. (d) Steady-state structure in a torus-shaped chamber of a bigger size (240µm). Note that the three vortices have the same core-size as those in (a) and (c).

Figure 3. Different large-scale patterns formed through self-organization of microtubules and motors. Initially uniform mixtures of proteins have been heated to 37°C, patterns resulting after 7 minutes are shown at equal magnification. The samples differ by kinesin concentrations: (a) A lattice of asters and vortices obtained at ~25 µg/ml of kinesin concentration. (b) An irregular lattice of asters obtained at ~37.5 µg/ml of kinesin. (c) A state in which microtubules form bundles obtained at ~50µg/ml of kinesin (shown in the insert in higher magnification). (d) A lattice of vortices, a different structure obtained with lower concentrations of kinesin (<15µg/ml).

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