Renormalization Group Limit Cycle for Three-Stranded DNA

Abstract

We show that there exists an Efimov-like three strand DNA bound state at the duplex melting point and it is described by a renormalization group limit cycle. A nonperturbative RG is used to obtain this result in a model involving short range pairing only. Our results suggest that Efimov physics can be tested in polymeric systems.PACS numbers: 87.14.gk, 64.60.ae, 87.15.Zg Consider a three-particle quantum system with pairwise short-range potential. Apart from the occurrence of the usual three-body bound state, a very special phenomenon occurs at the critical two-body zero-energy state. An infinite number of three-body bound states appear though the corresponding potential is not appropriate to bind any two of them; the removal of any one of them destroys the bound state. This phenomenon, valid for any short-range interaction, is known as the Efimov effect. The size of the three-body bound states, or Efimov trimers, is large compared to the potential range, and so it is a purely quantum effect [1]. Although it was predicted in the context of nuclear physics [2,3], it has now been detected in cold atoms [4].An ideal DNA consisting of two Gaussian polymers interacting with native base pairing undergoes a critical melting transition where the two strands get detached. Maji et.al. recently showed that if, to a double stranded DNA at its melting point, a third strand is added, the three together would form a bound state instead of remaining critical [5]. The existence of a triplex has further been verified by real space renormalization group (RG) and transfer matrix calculations [5,6]. That this is an Efimov-like effect can be seen by the imaginary time transformation of the quantum problem in the path integral formulation. The paths in quantum mechanics are identified as Gaussian polymers and the equal time interaction maps onto the native base pairing. Such a bound state of a triple-stranded DNA is called an Efimov DNA.In both cases, the special effect is due to a long-range attraction generated by critical fluctuations at the transition point. For the DNA case, the large fluctuations in the bubble sizes at the melting point allow a third strand to form bound segments with the other two. The power law behavior of the size of a polymer is essential to induce a 1/R 2 interaction between any two chains [5].Universal aspects of polymers are well understood in the RG approach [7]. A single chain and many chain solutions are described by length scale dependent running parameters, which, with increasing length scales, are expected to reach certain fixed points. The purpose of this paper is to show that the triple chain bound state at the duplex melting point is of a different type. This "fewchain problem" is actually described by a renormalization group "limit cycle" [2,8]. The appearance of a limit cycle invokes log periodicity in the corresponding three-body coupling in the polymer problem. So they break the continuous scale invariance around the two-body fixed point imposing a discrete scaling symmetry, the hallmark of t...

Thermodynamic relations for DNA phase transitions

“…The dependence of melting on the nature of the space has also been studied via the choice of different fractal lattices [40][41][42][43][44], showing the possible variations in the melting transition. The mapping of the DNA melting problem to a quantum problem revealed the connection between the bubble entropy of DNA and the quantum transition [45], and to the Efimov-physics [46]. Biological applications have also been considered, especially the motion of the interface or the Y-fork [47,48].…”

Observation of the Efimov state of the helium trimer

“…Intriguingly, this three-body behavior does not depend on the details of the underlying two-body interactions. This makes the Efimov effect a universal phenomenon, with important applications in particle, nuclear (3,4), atomic (4), condensed matter (5) and biological physics (6). Figure 1 summarizes two facets of Efimov's prediction, namely the energy spectrum and the structure of an Efimov state.…”

Efimov-like phase of a three-stranded DNA and the renormalization-group limit cycle

“…This extension, which goes beyond Ref. [10], shows that the fluctuation-induced Efimov DNA is not just a specialty of the melting point but it also exists over a region where the duplex should have been unbound. Although the fractal-like lattices of Ref.…”