# Metastable tight knots in a worm-like polymer

Alexander Y. Grosberg and Yitzhak Rabin Department of Physics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA Department of Physics, Bar–Ilan University, Ramat-Gan 52900, Israel
February 8, 2021
###### Abstract

Based on an estimate of the knot entropy of a worm-like chain we predict that the interplay of bending energy and confinement entropy will result in a compact metastable configuration of the knot that will diffuse, without spreading, along the contour of the semi-flexible polymer until it reaches one of the chain ends. Our estimate of the size of the knot as a function of its topological invariant (ideal aspect ratio) agrees with recent experimental results of knotted dsDNA. Further experimental tests of our ideas are proposed.

While everyday experience suggests that knots are very common in long linear strings of any kind, and they get quite tight whenever a string is not carefully handled, the study of knots in polymers concentrated almost exclusively on closed loops. It is understandable in the sense that knots are not mathematically well defined for open strings. However, if the string is long enough, while the knot occupies a short fragment of it far from the ends, then distinguishing between different knots, or between knots and no knots, becomes sufficiently unambiguous.

The recent achievement in the theory of knotted loops Guitter and Orlandini (1999); Metzler et al. (2002); Marcone et al. (2005) is the idea of knot localization. As simulations show Marcone et al. (2005), when a polymer loop with a knot is placed in a good or -solvent, it typically adopts a conformation in which most of the polymer forms a long unknotted loop, while the knot gets somewhat tightened in a small part of the contour. A theoretical explanation of the knot localization phenomenon is given Metzler et al. (2002) for the so-called flat knots, confined in a thin slit between two planes. In this case, one can consider a two-dimensional network which corresponds to any configuration of the knot by identifying chain crossings as the cross-links. This mapping allows one to resort to the sophisticated theory of 2D networks with excluded volume Duplantier (1986, 1989) and conclude that the most entropically favorable network is obtained when one of the knot arcs is made long at the expense of all others, thus localizing the knot.

In the present note, we would like to use much simpler hand-waving arguments to show that, at least for a worm-like polymer, there exists a local (metastable) minimum of free energy corresponding to a tight state of the knot even when the chain itself is open (not a loop). That means that if we intentionally tie a sufficiently tight knot somewhere on a very long polymer chain, the knot will spontaneously shrink or expand to a well-defined size and then, on much longer time scales, it will diffuse along the polymer (by polymer self-reptating through the knot) until, finally, the knot is released through the chain end. We expect that this knot will diffuse along the polymer as a soliton, in the sense that its size will remain relatively stable as it diffuses over large distances. The size of such a solitary knot depends on the complexity of the knot and on the persistence length of the polymer; we estimate that the knot will tighten to a size smaller than the persistence length of the polymer.

To put forward our argument, we employ the knot entropy estimates based on the idea Grosberg et al. (1996), similar to that in reptation theory Doi and Edwards (1986) , that non-crossing constraints imposed on the chain in the knot can be self-consistently described by confining the chain in an effective tube. Physically most natural construction of such a tube corresponds to making maximally inflated or, equivalently, shortest length tube consistent with the given knot topology. This approach to estimate knot entropy was invented by us Grosberg et al. (1996). It was also used in a number of other contexts and widely popularized under the name “ideal knots” Stasiak et al. (1998). The configuration and aspect ratio, , of the “ideal” tube represent topological invariants of the knot. This is sketched in the figure 1.

Within the framework of this approach, we imagine that a tight knot, characterized by ideal aspect ratio , has been tied in a very long polymer, and that currently the degree of tightening of the knot is such that the diameter of its self-consistently confining tube is . The length of the tube is and the size of the knot in space is , such that or

 R∼p1/3D . (1)

Throughout this paper, we drop all numerical coefficients, emphasizing only the scaling aspects of our considerations. We want to estimate the free energy of the knot as a function of or . Figure 1: (Color online) A knot is tied on a polymer chain. Knot size in space is R, chain within the knot is confined to a self-consistent tube of diameter D. Length of this tube is pD, where p is the topological invariant of the knot.

Although the original estimates of knot entropy were designed for a Gaussian chain Grosberg et al. (1996), they can be readily adapted to a worm-like polymer. Consider a knot with , where is the persistence length of the polymer. In this case, the polymer is quite tightly confined in the tube and can only wiggle a little bit around its centerline. Therefore, the length of polymer within the knot is close to , and its free energy consists of bending energy and confinement entropy. We estimate the bending energy as , assuming temperature is expressed in energy units (), and assuming that the radius of curvature of the tube is about the knot size , which is natural for an “ideal” knot. We further assume that the confinement entropy is the same as that for a straight tube for which case it was computed by Odijk Odijk (2002) and turns out to be about unity per every so-called deflection length ; this results in the free energy contribution about . Thus, using also (1), we obtain the following free energy estimate:

 ΔFT∼p1/3ℓD+p(Dℓ)1/3 . (2)

where is the free energy penalty for forming the knot (taking the reference free energy to be that of the unknotted chain). Obviously, it has a minimum at

 D∗∼ℓp−1/2 ,    or   R∗∼ℓp−1/6 . (3)

Notice that the resulting optimal meets the condition , so our estimate is self-consistent in this respect.

Of course, our result applies only as long as , where is the thickness of the polymer itself. Furthermore, in the practically important case of a charged polymer, such as DNA, we should also require that tube diameter exceeds Debye screening length, . In general, we can write roughly . More complex knots get so tight that their further collapse is stopped by either the excluded volume or electrostatic repulsion. As an example, for the dsDNA, the ratio under physiological conditions Rybenkov et al. (1993), while all knots with 7 or fewer crossings on the projection have about or less Pieran’ski et al. (2001). Therefore, in practice the condition on knot complexity is not very restrictive.

Let us now try to understand the physical meaning of our result (3), because at the first glance it might seem counterintuitive. Indeed, the optimal results from the competition of two factors, each of which, as it seems, disfavors tightening! One, chain bending energy, obviously favors more loose states of the knot, or increasing . The other one, however, related to the confinement entropy, favors tube widening if the chain length in the tube is fixed. In our case, when a knot tightens, it reduces the length of the chain, , confined in the knot. In other words, the part of the chain that remains in the tube suffers more when decreases, but the other part of the chain gets completely free of restriction, and that factor wins the whole game. That means, what really tightens the knot is the entropy gain of the chain tails outside of the knot. Alternatively, we can say that the knot gets compressed by the pressure of Rouse modes (or bending phonons) of the outside chain tails. Similar force can be observed during the translocation of a long polymer through a narrow channel across a membrane; while two long polymer ends are outside the membrane, their entropic favorability results in a force stretching the polymer portion inside the channel, and, accordingly, compressing the membrane. Notice that while in this planar geometry this force increases logarithmically with the length of the polymer tails, in our case of two long polymer ends sticking out of a compact knot, the force on the knot is independent of the length of these ends.

How tight should be the knot in the first place in order for our mechanism to take over and to bring the knot to its metastable size ? In other words, how wide is the basin of attraction of our metastable free energy minimum? We argue that the knot should be initially tightened to the state in which tube diameter is smaller or about chain persistence length . Indeed, when knot tube is wider, , the chain inside the tube is roughly Gaussian, with blobs of size . Each blob contains about of polymer contour length, and the entire tube contains polymer length . At the same time, confinement entropy is about unity per blob, which results in overall confinement entropy of about , independent of . Thus, our entropic knot tightening effect does not work if the tube is wider than persistence length, and it comes into play only when the knot is prepared in a compact enough state such that . To achieve this, the knot should be initially prepared by the pulling the ends with force , where .

Let us discuss now the dynamics aspect of the situation. Imagine once again that a sufficiently tight knot was initially tied in the polymer, similar to how it was done in the experimental work Bao et al. (2003) (see also Bao et al. (2005)) with DNA. Suppose now that the chain is released and is free to move. We predict then that the knot tightens spontaneously within a time which is roughly independent of the total chain length . After that, the knot will diffuse along the chain in pretty much the same way as it was observed experimentally Bao et al. (2003, 2005) and numerically Vologodskii (2006) for the stretched chain (even though in the experiment of ref. Bao et al. (2003) the chain ends were held at fixed separation throughout the diffusion process). As regards the knot diffusion coefficient along the chain, it was shown in the work Bao et al. (2003) that it can be quite accurately expressed in terms of the friction coefficient, , of a polymer with length and diameter , moving in the viscous solvent inside a tube of diameter : . The diffusion coefficient is then determined by Einstein’s relation as , and the full relaxation time of knot diffusion to the chain end is about . In these formulae, is assumed to be the bulk viscosity of the solvent. Figure 2: (Color online) Knot size R∗ against the stretching force f. Force is indicated in piconewtons and knot size in nanometers, both under the assumptions that ℓ=50 nm, and that there are no any numerical coefficients to worry about.

The above description of a soliton-like knot diffusing as a whole along the chain holds as long as diffusion time is shorter than the time of thermally-activated loosening of the knot. Indeed, we argued that the knot gets to its metastable size only when it is compact enough to begin with, such that and, therefore, a free energy barrier exists at (knots with will loosen up spontaneously). Using formula (2), we can estimate the barrier height as . As one could have expected, this barrier is very high for complex knots, but even for , one gets a barrier of about which should be sufficient to keep a knot locked while diffusing over a distance of few hundred persistence lengths, typical of DNA manipulation experiments. For smaller values of , the barrier may be too small to stabilize the tight knot and without the applied stretching it will spread due to thermal fluctuations.

As a corollary to our result, let us mention the following rather unexpected prediction. Let us take the polymer with the knot tightened according to our mechanism, and now let us gently pull the chain ends by a weak force. The applied stretching will suppress transverse fluctuations of the open chain ends. But these fluctuations are exactly the reason why the knot was tightened in the first place. Therefore, when these fluctuations are suppressed by the applied force, the knot swells - instead of further tightening which one could have naively expected. Of course, at the larger forces the loosening of the knot stops and normal tightening takes over.

Let us support this physical argument by a little calculation. If the chain of contour length is stretched by a weak force , it represents the succession of Pincus blobs, each involving the number of persistence lengths such that ; the stretching free energy is about per blob, or (see, e.g., deGennes (1979) for further details). In our case, only the chain outside of the knot is subject to this stretching effect, which yields the overall free energy

 ΔFT∼p1/3ℓD+p(Dℓ)1/3+L−pDℓ(ℓfT)2 , (4)

subject to optimization with respect to (see Eq. (6) below); here, the term is large, but can be dropped as independent of . Formula (4) remains valid as long as Pincus blob is larger than persistence length .

If we stretch the outside chain even further, beyond the Gaussian regime, it crosses-over to the so-called Marco-Siggia or Odijk regime in which stretching free energy is about per Odijk deflection length. In this regime,

 ΔFT∼p1/3ℓD+p(Dℓ)1/3+L−pDℓ(ℓfT)1/2 ; (5)

once again, this has to be optimized with respect to .

Notice that in both equations (4) and (5) we have neglected the effect of force on the chain part inside of the knot. This is justified as long as the amount of transverse fluctuations of the outside chain, , which is either the blob size in Pincus regime or the tube diameter in Odijk regime, remains smaller than the optimized tube diameter inside the knot, . At still larger forces, the chain inside the knot is just as stretched as outside. Therefore, we can summarize all results of optimization in the following way:

 R∗∼⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ℓp−1/6+ℓp−1/2(fℓT)2forfℓT<1ℓp−1/6+ℓp−1/2(fℓT)1/2for1

These results are sketched in figure 2. In accordance with our qualitative argument, the application of weak force loosens the tight knot instead of tightening it further!

In practice, for the case of dsDNA, since and , and taking , we get that both cross-overs and are within an experimentally feasible force range (about a few tenth and about a few piconewtons, respectively).

Speaking about experimental tests of our theory, we should of course consider first the experiment by Quake et al Bao et al. (2003, 2005) in which the knots were tied on DNA by the use of optical tweezers and then the knot motion along DNA was observed. Our theory indicates that the optimal tube diameter in the knot, , does depend in a certain way on the knot complexity, on . In the experiment, authors were unable to observe directly the knot and measure its size , but they were able to detect the amount of fluorescence coming from the knot region in excess of the fluorescence from linear DNA. That means, in our notations, that authors measured quantity , because is the length of DNA within the knot (proportional to the amount of fluorescence), while is the length of DNA that would have been in that place if there were no knot. Given the normalization condition (1), one could write . Authors of the work Bao et al. (2003), assuming a’priori that is independent of , plotted against , fitted the data to the linear function, and interpreted its slope as the tube diameter . By contrast, our theory predicts that does depend on knot complexity , such that . This is illustrated in the figure 3, where both the linear fit and our theory results are plotted along with the data points reproduced from the work Bao et al. (2003). At present, it is impossible to decide which theory is a better fit to the data. Figure 3: (Color online) The excess DNA length in the knot, pD∗−R∗ against p−p1/3. The meaning of these variables is explained in the text. Data points are taken from the work Bao et al. (2003, 2005). Linear fit corresponds to the assumption that D∗ does not depend on the knot type, on p. Curved line corresponds to our theory.

To further test our predictions, one should prepare the knot and then consider what happens to it if the force is subsequently switched off or significantly reduced. Another test of our theory would be to see how the knot diffusion time, determined by the friction coefficient , depends on the knot type through . Yet another possibility is to use granular macroscopic chain experiments, as described in Ben-Naim et al. (2001). In the movie available on the site

http://cnls.lanl.gov/~ebn/research/pics/chain.mov


the knot does move along the chain like a soliton, it does not increase in size while moving, in accord with our theory. It would be interesting to make a detailed statistics on this subject, although we should also put in question the applicability of our arguments based on worm-like chain model to the granular chain examined in Ben-Naim et al. (2001).

To conclude, we have shown that a sufficiently complex knot in a worm-like polymer will attain a well-defined compact size which is smaller that the persistence length. This configuration is metastable and the knot will diffuse along the chain as a soliton, whether the ends of the chain are subjected to external force or are free to move. Notice that since the results depend crucially on the elastic properties of the polymer on length scales comparable to and smaller than the persistence length, the effects described in this work are quite sensitive to the choice of the polymer model and can not be captured by locally stiff (e.g., freely jointed or freely rotating) chain models. Likewise, solitary knots will not form in Gaussian chains for which the persistence length and the thickness of the chain coincide. Such knots are expected to form in semi-flexible polymers such as double stranded DNA for which (a) separation of length scales exists, and (b) elastic behavior on length scales smaller than was demonstrated in single molecule experiments Smith et al. (1992); Strick et al. (2000).

###### Acknowledgements.
We would like to acknowledge useful discussion with Eli Ben-Naim. We thank Xiaoyan Robert Bao for providing us with the data points used above in Figure 3. The work of AG was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR-0212302. This research was supported in part by a grant from the US-Israel Binational Science Foundation.

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