Physical Properties Determining Self-Organization of Motors and Microtubules Thomas Surrey, François Nédélec, Stanislas Leibler, and Eric Karsenti

Supplementary Material

Methods:
Determination of the Size of Ncd/Antibody Complexes: To determine the oligomerization state of GST-Ncd with bound antiGST antibody, we performed equilibrium ultracentrifugation, gelfiltration and fluorescence correlation spectroscopy (FCS) experiments. We determined an average molecular weight for the complex of 900 kDa (10% error) by analytical ultracentrifugation (Beckmann Optima XL-A analytical ultracentrifuge, An60Ti rotor, 3500 rpm, 20 h). Gelfiltration showed that the complex exists as a polydispers solution. On a Superose 6 (Pharmacia), the most prominent peak corresponded (assuming spherical particles) to 1.2 MDa (60%), 600 kDa (20%) and 300 kDa (15%). The apparent average molecular weight was about 1.0 MDa. No free antibody was detected, some higher oligomers were visible (less than 3%). FCS measurements were in agreement with these numbers for the average molecular weight. These results indicate that the complexes consist mainly of roughly 8-12 GST-Ncd molecules complexed to about 2 antibodies.

Effect of Paclitaxel on Self-Organization: MT nucleation strongly affected self-organization. We adjusted the concentration of paclitaxel (a taxol derivative, Molecular Probes) for each batch of purified tubulin to ensure that MTs always nucleated visibly about 10 seconds after the temperature shift to 30°C (typically, 50-100 experiments could be performed with one such tubulin batch). Some tubulin batches did not require the addition of paclitaxel, and we observed the same types of structures as in the presence of paclitaxel. Preparation of Polarity Marked MT Seeds: Polarity-marked MT seeds were made using N-ethylmaleimide labeled tubulin as described (1) with two modifications. The seeds were stabilized after polymerization by taxol and were then separated from unpolymerized tubulin by repeated dilutions and filtration through 0.1 mm spin filters (Millipore, No. UFC30VVNB) at 1000 rpm in a tabletop centrifuge. The seeds were kept at room temperature and used within a few hours.

Modelling and Numerical Simulations: Model and simulations: The simulations implement a model describing the main properties of MTs and motors relevant for the experiments shown here. Starting from a uniform and random distribution, they stochastically compute the movement of all individual MTs and motors, and therefore the evolution of the system in time. A motor complex is a set of two equivalent processive entities ('single-motor') that can each bind to a MT and move along it. Each single motor is either detached, or attached to a MT, and transitions between these two states are stochastic: A single motor binds with a rate p_on = 50 s^-1 to a MT that is closer than 50 nm, and a bound motor detaches with a rate p_off = 0.5 s^-1. (The total number of single-motors allowed to bind to a single MT-rod (see below) is set to 120). A single-motor having reached the end of a MT is immobile and detaches with a rate p_off,end. The action of a complex is determined by the combined states of its two single motors. Free complexes diffuse (20 mm²/s), while complexes attached to only one MT are transported along this MT, and complexes attached to two MTs link them with a linear force F=Kdx (vectorial). A lower limit for K has been measured for kinesin in ref. 2, we adopted K=100 pN/µm. A bound single-motor moves in the direction defined by its intrinsic polarity (plus or minus end directed), at a speed V=V_max(1-F/F_max), where F is the projection of the resisting force F on the axis of the MT. In this model, a motor pulling a load along a MT never exceeds its maximum force F_max = 5 pN nor it's maximum velocity V_max= 1 mm/s see ref. 3. The unbinding of the motor is not dependent on the exerted force F (we obtained similar results with simulations where motor immediately unbind if F² > F_max²). All these values can be obtained from measured experimental data for kinesin. MTs are modeled as linear, infinitely thin and oriented objects. MTs grow from both ends, with an equal rate which is proportional to the free tubulin concentration (rates taken from ref. 4). The free tubulin concentration is assumed to be uniform across the sample and to decrease when MTs grow. Therefore, the MT growth speed decreases exponentially with time. To approximate the MT length distribution in experiments, growth is partially stochastic such that the MT length varied from 2+/-0.0 mm to 44 +/-0.8 mm during 500 seconds of simulated time. We also performed simulations with higher/lower variability and obtained similar results. To model the flexibility of the MTs, they are divided into successive straight rods of 2 mm that join at their ends. Non-aligned successive rods are subject to a torque, that tends to bring them back into alignment. This torque is linear in the deviation angle, and the constant of proportionality is obtained by dividing the MT bending modulus (20 pN mm², Ref. 5) by the length of the 2 mm-rods. For the numerical integration, the space is continuous and two-dimensional. The model reduces to a large number of rigid rods connected by linear forces exerted by motor complexes bound to two MTs and by links between adjacent rods. The movement of the rods is completely damped by the viscosity of the fluid (0.2 pN s mm^-2), and their speed is therefore proportional to the sum of the forces that applies to it. The equations of motion are integrated using a forward Euler method with a time step of 10^-4 s. All simulations were performed with an artificially high viscosity (200x more viscous than water), where a single motor is slowed down significantly by the load of dragging a MT in the fluid (6). This high viscosity is intended to replace friction effects present in the experiment.

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