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Physical Properties Determining Self-Organization of Motors and Microtubules

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Science  11 May 2001:
Vol. 292, Issue 5519, pp. 1167-1171
DOI: 10.1126/science.1059758

Abstract

In eukaryotic cells, microtubules and their associated motor proteins can be organized into various large-scale patterns. Using a simplified experimental system combined with computer simulations, we examined how the concentrations and kinetic parameters of the motors contribute to their collective behavior. We observed self-organization of generic steady-state structures such as asters, vortices, and a network of interconnected poles. We identified parameter combinations that determine the generation of each of these structures. In general, this approach may become useful for correlating the morphogenetic phenomena taking place in a biological system with the biophysical characteristics of its constituents.

A central question in biology concerns the origin of complex macroscopic structures. Two fundamentally different mechanisms can account for the generation of large-scale structures from random mixtures of small molecules. One mechanism is self-assembly near thermodynamic equilibrium (1, 2). A very different mechanism is self-organization in energy-dissipating systems. Although they do not reach thermodynamic equilibrium, these systems can reach steady states; kinetic parameters can influence or determine the final structures (3, 4). In eukaryotic cells, organization of the intracellular architecture is largely determined by the collective behavior of the ensemble of proteins that constitute the cytoskeleton (5, 6). A remarkable property of the cytoskeleton resides in the versatility of all patterns that can be produced. Indeed, similar sets of components are found to be organized into very different assemblies, depending on the cell type and cell cycle stage. Motor proteins and filaments play an important role in determining the final structure (7–9). How their concentrations and the combination of plus- and minus-end motors contribute to morphogenetic processes is not understood. Moreover, other parameters like the speed of different motors, their processivity, or the time they spend bound at microtubule (MT) ends may strongly influence the patterns generated at steady state.

Here, we analyzed the contribution of multiple parameters to the formation of steady-state patterns generated by MTs and soluble motor complexes. Plus-end motors and MTs can self-organize and create structures such as vortices and asters when these motors are multimeric (10). The structures emerge within 10 to 20 min after the addition of kinesin complexes to purified tubulin in the presence of adenosine 5′-triphosphate (ATP) and guanosine 5′-triphosphate (GTP) (Fig. 1). As an example of a multimeric minus-end motor, we prepared a complex of glutathione-S-transferase–nonclaret disjunctional (GST-Ncd) fusion proteins (11) bound to antibodies to GST (anti-GST) and compared directly the effect of the oligomeric kinesin and Ncd constructs on MT organization (Fig. 2A) (12). At low motor concentrations, MTs remained randomly oriented. When the concentration was increased, kinesin formed MT vortices at intermediate concentration and asters at higher concentration. Vortices and asters appear to be true steady-state structures (13). At no time did we observe the transformation of a vortex into an aster. In contrast to kinesin, Ncd did not form vortices at any of the concentrations tested. As soon as the Ncd concentration was high enough to organize MTs, asters formed. Asters organized by Ncd or kinesin looked similar, but the MT orientation was different. In Ncd-organized asters, the minus ends of the MTs pointed toward the center (as revealed by the orientation of added MT seeds), whereas in kinesin-organized asters, the plus ends were concentrated in the center (14). In both kinds of asters, the motors accumulated in the aster center, as revealed by the use of fluorescently labeled motor complexes.

Figure 1

Kinetics of aster formation by multimeric kinesin complexes as observed in vitro by dark-field microscopy or as reproduced in numerical computer simulations. Experimental conditions are as in Fig. 2, and simulation parameters as in Fig. 3.

Figure 2

MT patterns organized in vitro by the action of multimeric motor complexes. (A) One species of motor complex: Variation of the concentration of multimeric kinesin complexes (a to c) and of multimeric Ncd complexes (d to f). MTs are visualized by dark-field microscopy. The concentrations of the monomeric motors are indicated. The kinesin (monomer)/streptavidin (tetramer) ratio was maintained at 16, and the GST-Ncd (monomer)/anti-GST ratio at 5.3. Tubulin concentrations were 26 μM in kinesin experiments (a to c) and 23 μM in Ncd experiments (d to f). MT orientation in Ncd asters is shown with tetramethylrhodamine-labeled polarity-marked MT seeds (g: darkfield, h: fluorescence). The bright part of the seeds indicates the minus end. Motor localization is shown in asters organized by tetramethylrhodamine-labeled Ncd com- plexes (i: dark-field, j: fluorescence). (B) Simultaneous action of multimeric kinesin and multimeric Ncd: Variation of the motor/tubulin ratio (without changing the Ncd/kinesin ratio) (a to c). The concentrations of kinesin, Ncd, and tubulin were 1.2, 4.0, and 28 μM (a); 1.5, 4.9, and 28 μM (b); and 1.7, 5.6, and 26 μM (c), respectively. Variation of the kinesin/Ncd concentration ratio (d to f). Kinesin and Ncd concentrations were 1.2 and 5.6 μM (d), 1.7 and 5.6 μM (e), and 2.0 and 4.6 μM (f), respectively. The tubulin concentration was 28 μM. The localization of kinesin in MT networks is shown with the use of fluorescein-labeled kinesin (g: dark-field, h: fluorescence, i: overlay; conditions as in c). Images were taken 10 to 20 min (A) or 25 to 30 min (B) after the start of MT polymerization. For methods, see (12, 28).

We then investigated whether structures with new topologies could form by mixing the two motor complexes of opposite polarity (Fig. 2B). We started with low motor concentrations corresponding to the regime in which kinesin alone would form MT vortices and Ncd alone would form small MT asters. This resulted in the assembly of a pattern containing Ncd asters and kinesin vortices. Increasing the total motor/tubulin ratio without changing the motor/motor ratio led to local alignment of MTs and, at higher ratios, to a network of poles connected by aligned MTs. The network formed when the kinesin/Ncd ratio was about 0.3. If this ratio was decreased or increased by about 30%, one of the motors won the competition, resulting in the generation of either Ncd or kinesin asters.

Using labeled kinesin complexes, we found that kinesin localized to every second “pole” of the network (Fig. 2B). Thus, the MTs were uniformly oriented in the structures, with one pole made of minus ends and the other of plus ends. Plus- and minus-end poles alternated throughout the network, and the MT ends touched each other in a minus-minus or plus-plus manner. The network contained no antiparallel MT overlaps.

To explore how the physical properties of motors and microtubules determine the generation of asters, vortices, and networks, we performed numerical computer simulations (15–17). We modeled the motor complexes as two identical motors, which could be bound to one or two MTs or diffuse freely in solution (Fig. 3A). Motors bind and unbind stochastically from MTs and move along them in a force-dependent manner. Upon reaching the MT end, motors detach with a finite off-rate p off,end. Motor complexes bound to two MTs exert a force on the MTs, causing them to move. This is the ultimate reason for self-organization in this system. MTs are in a state of growth, but approach a finite length owing to subunit depletion. Because of the stochastic nature of the model, structural details are not fully determined by the initial conditions of a simulation—e.g., the exact number of MTs per self-organized aster can vary. But in most cases, the characteristic features of the generated steady-state patterns arise in a deterministic manner, just as in the experiments. We can compare meaningfully the predictions of the theoretical model with experimental results because of the minimal composition of our experimental system. The present biophysical characterization of its components provides most of the model parameters.

Figure 3

Numerical computer simulations. (A) Schematic representation of the model used for the simulations. Unless otherwise stated, simulations were performed with kinesin-like parameters: binding rate p on = 50 s−1, unbinding rate p off = 0.5 s−1, maximum speedV max = 1 μm s−1, maximum force F max = 5 pN, diffusion coefficientD = 20 μm2 s−1. For other parameters and the details of the model and the numerical simulations, see (16, 17) and supplementary material (15). (B) Effect of the motor density on the intensity of aster organization by one kind of motor complex. Asters generated by simulations in a periodic box of 50 μm by 50 μm with an average MT density of 0.05 μm−2 and average motor densities M of 0.05, 0.5, and 2.0 μm−2 (125 MTs and 125, 1250, and 5000 motors, respectively). The unbinding rate from MT ends was p off,end = 2.5 s−1. (C) Variation of kinetic parameters. The total average number of MTs in the three largest asters characterizes the intensity of aster organization and is plotted as a function of the motor density. The simulated time was 500 s. Parameter sets [V (s−1), P off(s−1), and P off,end(s−1)] are as follows: (○—○) 1, 0.5, 2.5; (□) 1, 1, 2.5; (×) 1, 5, 2.5; (○···○) 0.2, 0.5, 2.5; (▵) 0.2, 0.5, 0.5; (+) 0.2, 0.5, 0.05.

We first examined the effect of motor concentration on aster formation by performing simulations, using the kinetic parameter set for kinesin at various motor densities [because simulations are two-dimensional (2D), 3D concentrations are replaced by 2D densities]. Although at very low density no ordered structures emerged, asters formed at higher motor densities (Fig. 3B). Increasing the motor density led to an increase in the incorporation of MTs into the generated asters, i.e., to an increase in the “intensity of aster formation.” Decreasing the processivity (v/p off )—i.e., the average distance a motor moves along a MT before detaching—reduced the intensity of aster formation at all motor densities (Fig. 3C) (18). An increase in motor density could—within limits—compensate for a decreased processivity. Kinetic parameters could also compensate for changes in other kinetic parameters. This explains why motor constructs with different kinetic properties, like kinesin and Ncd, can generate similar asters in our in vitro system. Changing the motor density, speed (v), and unbinding rate (p off) did not induce a transition from aster to vortex, i.e., did not affect the topology of the self-organized structures.

In addition to these quantitative changes, some parameter variations also caused qualitative transitions between structures of different topologies. Increasing p off,end to 70 s−1 or more (i.e., decreasing the average motor residence time at MT ends to 15 ms or less in the background of kinesin-like parameters) induced the formation of vortices (Fig. 4A). We did not observe a vortex-to-aster transition in simulations by changing the value of another parameter and keeping p off,end constant [for a comparison of this result with experimental results, see (19)]. Thus, the residence time at MT ends plays a crucial role with regard to the capacity of motors to focus MTs to a pole (because motor links between MT ends and other MTs can eventually bring MT ends together, if these links are stable). The residence time has not yet been determined for any motor protein. However, in our experimental system,p off,end appears to be higher for the kinesin construct than for the Ncd construct (18), because kinesin, in contrast to Ncd, is capable of forming vortices.

Figure 4

Numerical computer simulations of different MT patterns generated by one kind of motor complex and two kinds of motor complexes with opposite directionalities (represented by blue and green colors). (A) The aster-to-vortex transition in the presence of one species of motor complex is governed by the probability that motors will fall off from MT ends. Examples are shown for two end-unbinding probabilities and two motor densities. Values for the low and high end-unbinding probability (p off,end) are 2.5 and 70 s−1, respectively, and for the low and high motor density (M) are 0.03 and 0.5 μm−2, respectively. (B) Two motors with opposite directionality (represented by blue and green colors) but otherwise identical parameters (e.g., highM 1 = M 2, lowp off,end,1 =p off,end,2) form a network of poles connected by aligned MTs. (C) Patterns generated by mixtures of two species of motors with opposite directionalities and identical, low off-rates from MT ends (low p off,end,1 =p off,end,2) (top) or with one low and one high off-rate from MT ends (p off,end,1p off,end,2) (bottom). Both cases are shown for low and high motor densities (M 1= M 2). The size of the periodic box is 100 μm by 100 μm, of which 60% are shown. The simulated time is 1500 s [or 500 s for the simulation with p off,end = 2.5 s−1and M = 0.5 μm−2 in (A), because these asters form more rapidly than the other structures]. For all other parameters, see supplementary material (15).

We performed simulations of 1:1 mixtures of two motors with identical kinesin-like properties but with opposite directionalities. At high motor densities, when each of the individual motors alone formed asters, a mixture of two such motors generated an irregular network of connected poles (Fig. 4B) similar to the one observed in the experiment (Fig. 2B). Decreasing the motor density from 0.5 to 0.03 μm−2 in such a mixture caused the generation of isolated asters of both polarities (Fig. 4C). We then studied a 1:1 mixture of an aster-forming motor complex (long residence time at MT ends) and a vortex-forming motor complex (short residence time at MT ends) with opposite directionality of movement. At low motor densities, mixtures of asters and vortices did form, whereas at higher motor densities asters of one polarity were formed by one of the motors, while the other one induced the formation of curved MT bundles between these asters (Fig. 4C). Networks of connected poles did not form [for a comparison of these simulation results with experimental results, see (19)]. This demonstrates that the formation of these networks is also sensitive to the residence time of the motors at MT ends (i.e., to p off,end). We also simulated mixtures of motors with different kinetic parameters and found that they would form networks at a given density, as long as both motors could, on their own at this density, organize all MTs into asters.

The MT structures classically found in cells are asters, bundles, spindles, or more complex architectures found in protozoa (20). From the results of our simulations, we predict that the motor that forms asters in vivo, dynein (21), should form processive oligomers that have a long residence time at MT ends, emphasizing the importance of motor-MT interactions at MT ends (22, 23). The other structures observed in this study, vortices and networks, do not seem to have counterparts in the living world. Yet, in the networks formed by two motors, the two oppositely oriented sets of parallel aligned MTs are reminiscent of the two half-spindles in a bipolar mitotic spindle. There are important differences, however. In the network, both MTs and motors are entirely sorted. In the mitotic spindle midzone, MTs form stable antiparallel overlaps in which motors are present (24–26). It will be interesting to determine which properties are responsible for the stabilization of such antiparallel MT overlaps.

In exploring the generic steady-state patterns that could emerge from mixtures of MTs and one or two oligomeric motors of opposite directionality, we have found a limited number of patterns: radial MT structures, either asters or vortices, or networks of poles connected by aligned MTs. Using computer simulations, we found that changes in the value of many parameters did not affect the topology of the pattern, whereas changes in other parameter values did. Those parameters are potential key targets for regulation. Many complex biological structures are also collective out-of-equilibrium assemblies. In the past, they have been described mainly by attributing qualitative “functions” to some of their constituent molecules. Here, we have used kinetic parameters describing the properties and interactions of the molecules to deduce the structures produced by the ensemble.

  • * These authors contributed equally to this work.

  • Present address: Departments of Physics and Molecular Biology, Princeton University, Princeton, NJ 08544, USA.

  • To whom correspondence should be addressed. E-mail: karsenti{at}embl-heidelberg.de

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