Patterns of molecular motors that guide and sort fibers
Abstract
Molecular motors can be immobilized to transport filaments and any load attached to them inside a nano-device. While a single motor is able to push a filament, it is however insufficient to impose the direction in which the filament will move. Hence if many motors are distributed uniformly over a flat surface, the motility is undirected, and it occurs equally in all directions. For many applications it is important to control the direction of filament motility, and that can be done in two ways: external forces and confinement inside ratchet-like channels. In this article, we discuss a third strategy in which the motors are distributed sparsely over a flat surface according to some pattern. To identify interesting patterns, we used stochastic simulations to numerically calculate the motility of filaments. Patterns were then selected using an evolutionary algorithm, to optimize their capacity to precisely control the paths of the filaments. We found patterns that can direct the filaments in a particular direction, and other patterns that can separate short and long filaments. These functionalities already exceeds what has been achieved with confinement. The patterns are composed of one or two kind of motors positioned in line or along arcs and should be easy to manufacture. Finally, these patterns can easily be 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
Topological tracks
Motifs for Guiding
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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.
movie.mp4
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config.cym
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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.
movie.mp4
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config.cym
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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.
movie.mp4
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config.cym
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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.
movie.mp4
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config.cym
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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.
movie.mp4
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config.cym
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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.
movie.mp4
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config.cym
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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.
movie.mp4
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config.cym
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Pattern with fast and slow motors
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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.
movie.mp4
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config.cym
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Motifs for length-dependent guiding
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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.
movie.mp4
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config.cym
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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).
movie.mp4
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config.cym
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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).
movie.mp4
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config.cym
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Length sorter concept
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.mp4
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config.cym
