Random knotting

In principle, the most obvious way to obtain a molecular knot relies on the direct random knotting occurring in a difunctionalized single molecular strand before its cyclization (Figure 5). Despite the simplicity of the concept, this approach has, to our knowledge, never been attempted because of the low probability of a chain tying a knot before its ends A and B find each other and connect. 

Deruchi, T., Tsurusaki, K., 1994. A statistical study of random knotting using the Vassiliev invariants. J. Knot Theory and Its Ramifications 3 321-353. 

Fig. 4.27 Restoring force of two different diamond topology interpenetrating networks, with and without random knots, as indicated for Ns 12. (From Ref. 13).

Fig. 4.28 Conformation plot of a stretched network with random knots. In the nonlinear regime there are a few strongly stretched paths, which are shown dark and thick. The less the stretching, the finer the bonds are shown. (From Ref. 13).

If every calibration point (with the exception of replicates that can be treated by averaging their response values) is treated as a separate knot, two different situations can be distinguished. In case of very precisely defined response values, y., obtained in practice by a high number of replicates in presence of small random errors, it is possible to use interpolating splines. Presumbly, the more frequent case envisaged will be the one, where relatively few data points whose random errors are not negligible and/or that are not highly replicated span the concentration (or mass) domain. 

Gillberg, C., Melander, H., von Knotting, A., Janols, L., Thernlund, G., Heggel, B., Edievall-Walin, L., Gustafsson, P., and Kopp, S. (1997) Long-term central stimulant treatment of children with attention-deficit hyperactivity disorder. A randomized double-blind placebo-controlled trial. Arch Gen Psychiatry 54 857—864. 

As seen above, the randomness of the Mobius strip approach and numerous difficult steps in Schill s directed approach were highly limiting factors in a trefoil knot synthesis. Both these major obstacles might be circumvented by the use of an unambiguous templated synthesis procedure. 

Besides hairpin turns and broader U-tums, a protein chain may turn out and fold back to reenter a P sheet from the opposite side. Such crossover connections, which are necessarily quite long, often contain helices. Like turns, crossover connections have a handedness and are nearly always right-handed (Fig. 2-25).117/219 Most proteins also contain poorly organized loops on their surfaces. Despite their random appearance, these loops may be critical for the functioning of a protein.220 In spite of the complexity of the folding patterns, peptide chains are rarely found to be knotted.221... 

In particular, the most obvious topological question, concerning the probability of knotting during random closure of polymer chain, cannot be answered using the Gauss invariant. 

The dependence p( N) of the chain self-knotting probability p is determined as a function of chain length N by the random chain closure [29-31, 34]. In a recent work [36], the simulation procedure was extended up to the chains of order N 2000, where for trivial knot formation probability the exponential asymptotics of the type... 

It is easy to understand that the trivial knot formation probability under random linear chain closure, q(a) = 1 — p(a), can be defined by the relation ... 

The synthetic chemist can see beauty in an approach to a chemical object, be it in the form of the perceived elegance, efficiency, directness, or combination of approaches. Covalent bond forming, coordination chemistry and purely non-covalent possibilities are all important of course, but each one on its own is weak. Take the statistical approach to link formation A flexible linear molecule in random motion is not likely to form a knot, and it is even less likely that we could separate and characterise it even if it did. There is no direction in the reactivity. Covalent bonds can be used to hold fragments together, and then those bonds used to orient the fragments are removed to leave the linked molecule. With today s control over molecular conformation and covalent bond making and breaking, this has to be considered a viable approach. But, those approaches based on coordination chemistry and non-covalent bonds are more direct and efficient for the moment. 

A new type of covariate screening method is to use partially linear mixed effects models (Bonate, 2005). Briefly, the time component in a structural model is modeled using a penalized spline basis function with knots at usually equally spaced time intervals. Under this approach, the knots are treated as random effects and linear mixed effects models can be used to find the optimal smoothing parameter. Further, covariates can be introduced into the model to improve the goodness of fit. The LRT between a full and reduced model with and without the covariate of interest can be used to test for the inclusion of a covariate in a model. The advantage of this method is that the exact structural model (i.e., a 1-compartment or 2-compartment model with absorption) does not have to be determined and it is fast and efficient at covariate identification. 

Almost 20 years after theoretical estimations of the probability of DNA knotting were first published (Vologodskii et al., 1974 [76] Frank-Kamenetskii et al., 1975 [21]), quantitative experimental data have been reported (Rybenkov et al., 1993 [61] Shaw and Wang, 1993 [65]) which fully agreed with the theory. In these experiments, the equilibrium fraction of knotted DNA molecules at various ionic conditions was quantitatively measured while molecules carrying "cohesive" ends randomly closed, in the absence of any proteins. Comparing the fraction with theoretical predictions of Klenin et al. (1988) [36],... 

And how about experiments on polymer knots The most important and, luckily, also the easiest subject of such experiments is double helical DNA. One nice experiment can de done using DNA with sticky ends - a long double helix with each chain extending at one end by 15 or so unpaired nucleotides beyond the counterpart chain. If the sequences of these extending pieces are complementary to each other, they will stick upon first collision due to the random fluctuations of the double helical coil. Can we then determine the topology of the product ... 

Knots can of course be addressed also for toy lattice proteins discussed in section 10.10. The shortest cubic lattice knot has 24 monomers (Figure C11.7 a), but it is not space filling. The shortest space filling (open ended) lattice polymer to have knot is 36-mer. Its conformation with a trefoil knot is shown in the Figure C11.7. By the way, if the sequence of monomer species in it is properly selected (see Section 10.6), it folds in virtuo, of course) quite successfully, and not much slower that the corresponding chain without a knot — which opens even wider the question as to why real proteins are statistically less likely to have knots than random. 

Furthermore, power laws and imderlying fractal properties are also seen if one looks at the evolution from the point of view of protein conformations, not sequences. Here, we should mention that protein conformation appear to have been somehow selected. One aspect of it we already discussed in Section 11.6 conformations with knots seem to be under-represented compared at random. But quite apart from that, as C. Chotia of Cambridge University in England pointed out, only relatively few conformations, not more than several thousand, are featured in the proteins (his paper had an interesting title Proteins. One thousand families for the molecular... 

Is there anything similar in the physics of biopolymers, any general laws that are not affected by the random choices There certainly are They control the formation of knots in DNA (see Section 2.6), the hydrophobic-hydrophilic separation of a globular protein (Section 5.7), and many other properties most of these laws may still be unknown. 

Micheletti, C., Marenduzzo, D., Orlandini, E., and Sumners, D. W., 2006. Knotting of random ring polymers in confined spaces, /. Chem. Phys., 124 064903. 

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