Patterns of molecular motors that guide and sort fibers by length and polarity

Beat Rupp and Francois Nedelec
Lab on a Chip, 2012,12, 4903-4910

Abstract

Molecular motors can be immobilized to transport filaments and loads that are attached to these filaments inside a nano-device. However, if motors are distributed uniformly over a flat surface, the motility is undirected, and the filaments move equally in all directions. For many applications it is important to control the direction in which the filaments move, and two strategies have been explored to achieve this: applying external forces and confining the filaments inside channels. In this article, we discuss a third strategy in which the topography of the sample remains flat, but the motors are distributed non-uniformly over the surface. Systems of filaments and patterned molecular motors were simulated using a stochastic engine that included Brownian motion and filament bending elasticity. Using an evolutionary algorithm, patterns were optimized for their capacity to precisely control the paths of the filaments. We identified patterns of motors that could either direct the filaments in a particular direction, or separate short and long filaments. These functionalities already exceed what has been achieved with confinement. The patterns are composed of one or two types of motors positioned in lines or along arcs and should be easy to manufacture. Finally, these patterns can be easily combined into larger designs, allowing one to precisely control the motion of microscopic objects inside a device.


Caption

Motors are indicated as blue or orange dots. Except for the spade rectifier and the concept length sorter, the simulation uses periodic boundary conditions, and the periodicity is indicated with white dashed lines. Fibers are displayed in the movies as white arrows. The files config.cym should be usable as input to cytosim to reproduce each situation (but stochasticity will change the results slightly).

Note: periodic boundary conditions are applied for display. Thus, if multiple fibers appear on the same frame, with the same shape and a horizontal or a vertical shift corresponding to the periodicity, they are different mirror images of the same fiber.


Gliding on uniform lawns

uniform-K

Kinesin

sketch

On a uniform lawn of plus-end directed motors, the fibers move in random directions with their minus-end leading.

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uniform-A

Dynein

sketch

With minus-end directed motors, the fibers move with their plus-end leading, and if the pattern is uniform, the fibers glide in random directions.

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uniform-KA

Kinesin + Dynein

With a mixed lawn of both plus-end and minus-end directed motors, the fibers tend to alternate between the plus- and the minus- directions (cf. Vale et al. Journal of cell biology; 1992 vol. 119 (6) pp. 1589-96 and Tao et al. Current Biology; vol. 16 (23) pp. 2293-2302).

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Topological tracks

uniform-K

Spade rectifier

This is a ratchet design from Van den Heuvel et al. Nanoletters, vol. 5 (6) pp. 1117-1122. The simulation illustrates the rectification of gliding by the boundaries (vertical walls), shaped as a 'spade'. The distribution of the motors is uniform inside the channel.

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Motifs for Guiding

pattern2A

Pattern 2A: Ratchets

For directional transport (fitness = -x), simple patterns are found after 20 generations with a fitness of 0.5 ± 0.3. The pattern is made of two motors of opposite directionality, alternating vertically. Almost no fibers are transported East. The design follows a ratchet principle: fibers heading East are diverted North or South onto segments made of the opposite motors. Thus Eastbound fibers are forced to re-adjust their path. Westbound fibers are however guided along horizontal tracks. There is one such tracks for each type of motor (Fig. 2A, arrowhead). Eventually, all fibers move West, with the plus-end (resp. minus-end) leading if they travel on a minus-end (resp. plus-end) directed motors.

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pattern2B

Pattern 2B

Humanly optimized ratchet pattern. The principle of operation remains the same, but the fitness is increased to 0.7 ± 0.3 by extending the curved section by a straight segment.

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pattern2C

Pattern 2C: Tiles

One motor is sufficient for directional transport. Here, the design is composed of arcs arranged in a tiled pattern. It works by deflecting gliding fibers consistently in one direction. A fiber transported by an arc of motor may be guided and released after reaching the end of this arc. Because on this pattern, the ends of all arcs point in the same direction, fiber are eventually directed in this direction.

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pattern2D

Pattern 2D

Modified one-motor pattern of higher fitness. All arcs point in the same direction, and their tip directed West, implicitely define the horizontal tracks along which transport occurs. The fitness is 0.77 ± 0.13.

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pattern2E

Pattern 2E: Crossing segments

The segments with plus- and minus-end directed motors are oriented at ± 30°. Because the simulation runs with periodic boundaries, this would create two set of crossing lines, but the segments terminate before the putative intersection point. The gaps left and right of this point are not equal, and this asymmetry biases the motility of fibers, taking advantage of their finite bending elasticity. Indeed, the force necessary to buckle a fiber over a length L follows Euler's formula (1/L2), and is below the motor's force for long gaps but not for short ones. Consequently, while Westbound fibers travel freely, Eastbound fibers buckle while trying to breach the large gap, leading them to a different path. Eventually, all fibers move West.

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pattern2F

Pattern 2F

An optimized pattern 2E. By properly placing the gap at both diagonal tracks, the transport efficiency reaches 0.7 ± 0.3. With two motors, this is our simplest pattern, and although it directs fibers precisely, the resulting fitness is low because fibers move with a ± 30° angle with respect to the horizontal axis. However, this allows for another interesting property: fibers moving with their minus-end (resp. plus-end) leading move North (resp. South). Thus, the pattern sorts fibers vertically depending on their orientation, and this may be very interesting functionally.

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pattern2G

Pattern 2G: Robust directional transport

Since stabilized microtubules follow a broad length distribution, we also searched for patterns that sustain directional transport irrespective of the length of the fiber. For this, we combined the final positions of a 10 µm long fiber and that of a 5 µm long one, into the fitness: f = - (x_10+x_5). The pattern exhibits robust directional transport with a fitness of 0.83 ± 0.05.

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Pattern with fast and slow motors

pattern2H

Pattern 2H : Flipper

Directional motion can be achieved with two motors of opposite directionality (pattern 2A), or at lower efficiency with one motor (pattern 2C). This pattern illustrates that directionality can also be obtained with two motors of similar directionality (eg. both moving in the direction of the plus-end), if one of them is much faster than the other. This pattern flips any Eastbound filament. The leading end of the fiber is first captured by the central spot of slow motor (blue). The fiber then pivots around this captured end, pushed by the trailing part gliding on the fast track. Eventually, the trailing end is released, and the fiber is re-injected on the horizontal fast track, but now oriented to move West.

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Motifs for length-dependent guiding

pattern3A

Pattern 3A: Length-dependent guiding

We could search for the ability to sort filaments by length by using the fitness f = x10 - x5. This rewards patterns where short fibers are transported West and long fibers East. The pattern has a sorting fitness of 0.4 ± 0.4. The direction of transport is dependent on the length of the fiber (Fig. 3B). Short fiber glide West, in a similar manner to how fibers move on a ratchet design (pattern 2A). Surprisingly, long fibers are transported in a folded configuration, belly pointing East. In this configuration, the terminal parts of the fiber are transported on segments made of opposite motors. When the speed of the minus- and plus-end directed motors are equal, this configuration is stable, and the fitness is maximal, but length-sorting capacity is retained even when the motors have different speed.

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pattern3C

Pattern 3C: Optimized length-dependent guiding

Optimized length sorter based on the principles of pattern 3A, reaching a fitness of 0.7 ± 0.1. Its transfer function exhibits a sharp transition for a length of ~7 µm (Fig. 3D).

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pattern3E

Pattern 3E: Orthogonal length-dependent guiding

With one type of motor, we did not find a pattern where short fibers would move West while long fibers would move East, but when we used the fitness f = - y10 - x5, we found that one motor was sufficient to sort fibers by length, in orthogonal directions. The pattern shown on the left sends short fibers West, and long ones South. It combines design features from pattern 2A (horizontal ratchet, black arrowhead) and from pattern 2C (South pointing tiles).

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Length sorter concept

concept-sorter

The motifs discussed so far have the ability to guide fibers in different directions depending on their length. They can be thus used to build a device that sort fibers according to their length (Fig.~4). The motif can be repeated into a periodic pattern, but in addition, translational invariance should be broken. This concept length sorter uses two strategies: the fibers are initially all in the same position, and they are trapped West and East, at the desired sorting location (by the molecules displayed in pink). There are other ways to use the motifs, depending on the design requirements. movie.mp4config.cym



Please, check our work on cytosim.org.