Edited by Leo P. Kadanoff, college of Chicago, Chicago, IL, and also approved July 30, 2007 (received for evaluation December 21, 2006)



Fig. 4.

Properties the the circulation of observed knot types. (A) number of unique knots it was observed (per trial) vs. String length. The heat is a fit come a an easy sigmoidal function N = N 0/(1 + (L/L 0)b), through N 0 = 0.16, L 0 = 5 ft, and b = −2.6. (B) mean minimum crossing number vs. Cable length. The line is a fit to a straightforward exponential duty P = P 0(1 − exp(−bL)), v P 0 = 5.6 and b = 0.54. (C) portion of total possible types observed vs. Minimum crossing number (points), contrasted with the total variety of types possible (bars).

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Discussion

Although our experiment involve only mechanical activity of a one-dimensional object and occupation that a finite number of well defined topological states, the complexity introduced by knot development raises a extensive question: Can any kind of theoretical framework, next to impractical brute-force calculation under Newton"s laws, guess the development of knots in ours experiment?

Many computational studies have examined knotting of arbitrarily walks. Although the conformations of ours confined string are not simply random walks (being much more ordered), some similarities were observed. Specifics computational studies uncover that the probability 1 − P that not creating a knot decreases exponentially with random walk length (13, 14). In our experiments through the medium-stiffness string, we discover the very same trend because that lengths varying from L = 0.46 to 1.5 m, yet P approached a value of 13, 20), P was uncovered to boost with boosting confinement, and this impact has to be proposed to explain the high probability of knotting the DNA confined in specific viruses (6). However, this trend is in contrast to the observed in our experiment. Our movies disclose that in our case, boosting confinement the a stiff string in a box reasons increased wedging the the string versus the walls of the box, i beg your pardon reduces the tumbling activity that facilitates knotting. Interestingly, a similar effect has likewise been proposed come restrict the probability that knotting that the umbilical cord of fetuses as result of confinement in the amniotic sac (21).

Calculations on numerical random walks likewise find the the probability of occurrence of any certain knot decreases greatly with that complexity, as measured by the minimum cross number (16). We discover that such habits holds fairly strikingly in our experiment too (Fig. 5 A). This finding says that, although ours string conformations space not random walks, random movements do play critical role.


Fig. 5.

Dependence the the probability of knotting on procedures of node complexity. (A) natural log the P K plotted matches theoretically calculate knot energy (25). (B) herbal log of the probability P K of forming a specific knot plotted vs. Minimum cross number c(K). Each worth was normalized by the probability P 0 of forming the unknot. The filled one are outcomes with string lengths L > 1.5 m and the open up circles space with L 1 knot, which especially did no follow the all at once trend, were plotted as triangles.


Another measure of knot intricacy is “knot energy.” to investigate even if it is optimal spatial forms exist because that knots, mathematicians have connected energy attributes with knotted curves and also sought minimizers (22–24). A class of features studied in detail was inverse-power potentials, mimicking loops v uniform charge density. A regularized potential ≈1/r 2 was uncovered to be useful as the energy can be make scale-invariant and invariant under Möbius transformations. Freedman, He, and also Wang (24) proved the existence of minimizers for such features and set certain upper bounds on possible knot energies. Kusner and also Sullivan (25) provided a gradient descent algorithm to numerically calculation minimum power states for plenty of different knots and showed the they can distinguish various knots having the very same minimum cross number. Although our string mirrors no far-reaching static fee (see Materials and Methods), its flexural rigidity would penalize facility knot development in a qualitatively similar manner as the Möbius knot power (23). In fact, we observe a strong correlation (an approximately exponential decrease) of the probability P K of creating a details knot with the minimum energies calculated in ref. 25 (Fig. 5 B), back the 51 knot deviated significantly from the trend.

Comparison with Previous Studies.

Several previous studies have investigated knots in agitated ball-chains. Ben-Naim et al. (8) tied straightforward 31 knots in the chains and also studied their unknotting top top a vibrating plate. They found that the knot survival probability complied with a global scaling role independent of the chain length, and also that the dynamics could be modeled by 3 random walks connecting via excluded volume in one spatial dimension.

Belmonte et al. (7) observed spontaneous knotting and also unknotting of a propelled hanging ball-chain. Miscellaneous knots were formed, however only 31 and also 41 knots were specifically identified. It was discovered that back 41 is an ext complex, that occurred more frequently than 31. Additional studies verified that the 31 node (and other “torus knots”; e.g., 51 71, 91, 111) slips more easily turn off the bottom the the hanging chain (26). These experiments show that unknotting deserve to have a strong influence on the probability of obtaining a certain knot after a addressed agitation time and may aid to define our monitoring of a lower probability because that the 51 knot family member to the trend in Fig. 5 B (although we keep in mind that 31 developed with greater probability than 41 in ours experiment).

Hickford et al. (9) freshly examined the knotting and also unknotting dynamics the a ball-chain top top a vibrating plate. The chain was quick enough the almost all of the knots were simple 31 knots and the tying and also untying events might be detect by video image analysis. They discovered that the knotting rate was elevation of chain length but that the unknotting rate enhanced rapidly through length. It was displayed that the probability P of detect a node after a details time relied on the balance in between tying and also untying kinetics. Although our experimental geometry is different, ours measured dependence of P on size (Fig. 2) is quite similar to that observed through Hickford et al., saying that a similar mechanism might apply. In our study, however, the wire is much longer, lot more facility knots space formed, and also we focus on characterizing the relative probabilities of formation of different knots.

Simplified version for knot Formation.

Because the segments of a solid string can not pass through each other, the ethics of topology dictate the knots deserve to only nucleate at the ends of the string. Around speaking, the string end must map a path that coincides to a details knot topology in stimulate for that knot come form. This procedure has been straight visualized for straightforward 31 knots in the studies of vibrated ball-chains (9). In principle, knots may kind independently at both ends of the string, yet principles that knot theory dictate that this would result in the development of “nonprime” knots (3). Because that example, if a different 31 node is developed at each end of a string, they can be slid together at the facility of the string yet cannot merge to type a single prime knot. That the majority of the observed knots to be prime argues that knotting mostly occurs at one end of the cable in our experiment. Therefore, in occurring our model, we minimal our attention to the dynamics in ~ one end and ignored the other end.

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The photos and movies of our tumbled string display that string stiffness and also confinement in the box promote a conformation consisting (at least partly) of concentric coils having a diameter on the order of package size. Based on this observation, us propose a minimal, simplified design for knot formation, as depicted schematically in Fig. 6. Us assume the multiple parallel strands lied in the vicinity that the wire end and that knots form when the end segment weaves under and also over adjacent segments. Interestingly, our model corresponds very closely to the mathematical depiction of knots in a “braid diagram,” and also the weaving corresponds to “braid moves,” which provides additional insights (3). The relationship in between a braid diagram and a knot is established by the assumed connectivity of the group of line segments, as suggested by the dashed present in the figure. One may overlook the regional motions of these sections that the string due to the fact that they cannot adjust the topology. In our straightforward model, us assume the the finish segment renders random weaves, with a 50% possibility of moving up vs. Down and also a 50% opportunity of relocating under vs. End an surrounding segment. This model permits for both knotting and also unknotting to occur.