"Inchworm Filaments": Motility and Pattern Formation
““Inchworm Filaments”: Motility and Pattern Formation
Nash Rochman ∗ Department of Chemical and Biomolecular Engineering, The Johns Hopkins University
Sean X. Sun † Departments of Mechanical Engineering and Biomedical Engineering,The Johns Hopkins University, Baltimore MD 21218
Abstract
In a previous paper, we examined a class of possible conformations for helically patterned fila-ments in contact with a bonding surface. In particular, we investigated geometries where contactbetween the pattern and the surface was improved through a periodic twisting and lifting of thefilament. A consequence of this lifting is that the total length of the filament projected onto thesurface decreases after bonding. When the bonding character of the surface is actuated, this phe-nomenon can lead to both lifelike “inchworm” behavior of the filaments and ensemble movement.We illustrate, through simulation, how pattern formation may be achieved through this mechanism. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ q - b i o . S C ] D ec . INTRODUCTION In previous work[1], we explored a toy model for the Amyloid beta fibril CF-PT and aclass of conformations for that model which lead to large bonding energies when in contactwith a flat surface. Motivated by the proposition that CF-PT (cylindrical filament withperiodic thinning) is a precursor filament for PHF (paired helical filament)[2], we showedthat one such strongly bonded conformation for our model is a helix like that of PHF. Thisearlier investigation was limited to a surface with a static, uniform bonding energy. Herewe consider the possibility of a flat, nonbonding surface with a moving bonding region anddiscuss the dynamic conformational change and translation of the filament associated withmoving the region beneath it. To clarify, we do not consider the case where the surface itselfis moving but rather that the region of the surface which has the potential to bond with thefilament varies in time (due to electrical charge, chemical variation, etc.). As in the previousstudy, we consider a “close contact” approximation for the nature of the bonding betweenthe surface and the filament; that is, the bonding energy between a point on the filamentand a point on the surface is nonzero only if these two points are in contact.Our model filament consists of a cylinder with a helical bonding pattern, of period L , suchthat only the patterned region of the filament may bond with the surface. We will refer to asegment of the filament of length L where the bonding pattern is in contact with the surfaceat the beginning and end of the segment as a “monomer”. A cartoon of two monomers isshown below in Figure 1 A. To assume the helical conformation, each monomer twists toalign the bonding pattern with the surface at both ends, and bends in the center. A sketchof a monomer in the helical conformation is shown below in Figure 1 B. A consequence ofthis bending is that the monomer lifts off the surface in the center and the length of themonomer projected onto the surface decreases by some amount ∆ L Bond . This is illustratedin Figure 1 C, and a cartoon of a filament comprising three monomers assuming the helicalconformation is displayed in Figure 1 D, shown below.2 L Non-Bonded L Bonded ΔL bond Figure 1: A. Cartoon of two filament monomers. B. Sketch of a single monomer in thehelical conformation. The red lines indicate the bonding pattern and the black dashed linesthe twist of the filament. C. Illustration of the shortened projected length of the helicalconformation. D. Cartoon of a filament comprising three monomers assuming the helicalconformation.
II. SIMULATION
With the conformational change discussed above in the presence of a bonding surface, weknow that if we can actuate the surface in such a way that the bonding character changeswith time, we can control the filament shape as well. Furthermore, we can show that withthe right actuation, the filament is subject to not only a temporary shape change whilebonded, but a net displacement. Thus with continued surface actuation, filament motilityand “migration” can be achieved.This displacement stems from an asymmetric shortening and lengthening of the filamentwhen a bonding region is introduced and removed. When the filament binds to the surface,each monomer curls up, as shown above, decreasing the contact length by an amount ∆ L bond and moving the endpoints of the whole filament inwards. Similarly, when the filament3nbinds, each monomer stretches out, moving the endpoints outwards. If the bondingregion is brought in contact with and removed from the entire filament simultaneously, thismovement of the endpoints is symmetric and the filament faces no net displacement; however,if a bonding surface is present beneath one end of the filament and not the other, the motionof the bonded end can be expected to be more restricted than that of the unbonded end (dueto increased friction with the surface etc.). This leads to filament motion contrary to themotion of the bonding region. As the bonding region is introduced, the filament preferentiallyshortens from the unbonded end moving towards the oncoming bonding section. As thebonding region is removed, the filament preferentially lengthens away from its direction ofretraction. This motion qualitatively resembles that of an inchworm.When simulating motion due to surface actuation, we need to approximate how motionof bonded monomers is restricted compared to that of non-bonded monomers. In realitythis will widely vary based on both the filament and surface materials but for the purposesof constructing an illustrative simulation, we will assume the following conditions. 1) eachbound monomer confers the same restriction to motion when bound and 2) in the limitwhere only one monomer is unbound, only the unbound monomer moves. Now let N be thetotal number of monomers in the filament, n be the number of those bonded, for illustrativepurposes assume a large ∆ L Bond of L , and denote p (1) and p ( N ) as the positions of the firstand last monomers respectively. We may now consider the displacement of the monomer at p (1) in the direction of p ( N ) − p (1) . Displacement of p (1) = contractingextending only p (1) is boundonly p ( N ) is boundelse L (cid:0) − nN (cid:1) L (cid:0) nN (cid:1) L only p (1) is boundonly p ( N ) is boundelse − L (cid:0) − nN (cid:1) − L (cid:0) nN (cid:1) − L (1)Using this displacement rule, we may propagate the position of the filament as we move thebonding region. Motion of a single filament with a rightward moving, green bonding regionis shown below in Figure 2. 4 D Filament Translation
Student Version of MATLAB
Figure 2: Leftward filament migration (time progresses from left to right and top to bottom)in response to a rightward moving, green bonding section.We may note that filaments will self-order by length (see Figure 3, Sub-Figure a, below)as with each pass of the bonding region, a longer filament experiences a greater translation(due to a larger number of monomers).Filament translation in 2D can be achieved by introducing motion of the bonding regionalong two perpendicular axes. In this case, the shift of a filament caused by motion alongone axis can oppose that from the other in part or in full resulting in reduced or null motionfor filaments of the proper orientation. In the simulation displayed in Figure 3, Sub-Figureb, the bonding region moves from left to right and bottom to top. If a filament is orientedsuch that the angle its tangent axis makes with the x -axis is π or π , the motion resultantfrom the vertical bonding actuation completely cancels that from the horizontal and thefilament does not move. Orientations corresponding to angles from π to π and π to π experience a similar reduction in motion. 5 ariable Length (a) Self ordering of filaments by length: as time progresses from left to right, longer filaments movefarther. Variable Orientation (b) 2D translation (Time progresses from left to right.): the bonding region moves from left toright and bottom to top. If a filament is oriented such that the angle its tangent axis makeswith the x -axis is π or π , the motion resultant from the vertical bonding actuation completelycancels that from the lateral and the filament does not move (top left filament). To contrast, afilament oriented such that the angle its tangent axis makes with the x -axis is π or π benefitsfrom maximum translation (bottom left filament). Filaments aligned with either axis move onlyalong that axis. Figure 3: Self organization of filaments by length and 2D translation.When many filaments with different orientations are introduced to a bonding region whichmoves along two perpendicular axes, as described above, ring formation occurs. Filaments“migrate” with a direction and magnitude dependent on the angle their tangent axes makewith the x -axis. 6 ing Formation Figure 4: 2D bonding actuation applied to two-hundred filaments, thirty monomers in lengthof variable orientation.
III. DISCUSSION
We’ve shown that when in contact with a time-varying attractive surface, helically-patterned filaments may self-sort by length and orientation. When many of these filamentsare placed on such a surface, they can self-assemble into rings. We hope that this mechanismhas potential applications in the self-organization of materials. Additionally, the phenomenadiscussed in this work may have some pathological relevance in explaining the aggregationof proteins. Many diseases are characterized by protein disorganization and aggregatione.g. Amyotrophic Lateral Sclerosis[3] and other neurodegenerative pathologies[4] thoughthis mechanism is unlikely to have any physiological relevance, perhaps the idea of surface-mediated filament migration has some merit. The filament migration described above, ifobserved in a bounded region, would result in aggregation at the boundaries and stochasticsurfaces charges seen in biological systems may resemble the surface actuation describedabove.
IV. REFERENCES [1] Nash D Rochman and Sean X Sun. The twisted tauopathies: surface interactions of helicallypatterned filaments seen in alzheimer’s disease and elsewhere.
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Microscopy research and technique , 67(3-4):175–195, 2005.[3] Lewis P Rowland and Neil A Shneider. Amyotrophic lateral sclerosis.
New England Journal ofMedicine , 344(22):1688–1700, 2001.[4] Christopher A Ross and Michelle A Poirier. Protein aggregation and neurodegenerative disease.2004., 344(22):1688–1700, 2001.[4] Christopher A Ross and Michelle A Poirier. Protein aggregation and neurodegenerative disease.2004.