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

A three-stranded DNA with short range base pairings only is known to exhibit a classical analog of the quantum Efimov effect, viz., a three-chain bound state at the two-chain melting point where no two are bound. By using a nonperturbative renormalization-group method for a rigid duplex DNA and a flexible third strand, with base pairings and strand exchange, we show that the Efimov-DNA is associated with a limit cycle type behavior of the flow of an effective three-chain interaction. The analysis also shows that thermally generated bubbles play an essential role in producing the effect. A toy model for the flow equations shows the limit cycle in an extended three-dimensional parameter space of the two-chain coupling and a complex three-chain interaction.

Figure 1

Schematic diagram of a strand exchange and the equivalent coarse-grained three-chain interaction $g_3$. (a) A single strand [blue line marked (1)] pairs with one strand of a bubble on a duplex [brown lines marked (2)]. The short vertical (green) lines indicate base pairings, with the energy per unit length $\epsilon$. The junction weight $g_2$ is associated with each fork or the interface on a duplex. In (a) there are four interfaces as indicated by the arrows. (b) A coarse-grained version of (a) where the duplex is represented by a thick line interacting with the single line. The filled circle represents the three-chain interaction $g_3$.

Figure 2

Basic building blocks. Panel (a) represents $Z(\mathbf{k},s)$ for a Gaussian chain, (b) $Z_{\mathsf{b}}(\mathbf{k},s)$ for a two-chain bound state, (c) a Y fork representing the interface between a bound pair and two open strands. It has a weight $g_2$.

Figure 3

(a) The duplex partition function as an infinite series of bound pairs and bubbles; (b) Y fork for a duplex. (c) A three-chain interaction, $g_3$, involving a free chain and a duplex.

Figure 4

Interaction of free chain with a bound state in absence of bubbles.

Figure 5

Diagrammatic representation of the three-chain partition function. The hatched circle is the effective interaction $W$. This figure translates into an integral equation involving interactions to all order.

Figure 6

Closed elliptical trajectories in the complex $H$ plane as $\Lambda$ is varied. These are drawn for different starting values of $(H_1,H_2)$. All loops in the upper half plane have the fixed point $(1+i{s}_0)/(1-i{s}_0)$ as a focus while $(1-i{s}_0)/(1+i{s}_0)$ as a focus for the lower half plane.

Figure 7

Plot of $H$ as a function of $\Lambda$ showing zeros and divergences with ${\Lambda }_{*}=1$.

Figure 8

Schematic flow diagram in the ${\widehat{g}}_2\text{−}{\widehat{g}}_3$ plane. ${\widehat{g}}_2=0$ corresponds to the bound state (no bubbles) while ${\widehat{g}}_2={\widehat{g}}_2^{*}$ (box) is the duplex melting point. The flow along the thick vertical line through ${\widehat{g}}_2={\widehat{g}}_2^{*}$ is periodic. Along the ${\widehat{g}}_2=0$ line, there is a stable fixed point ${\widehat{g}}_3={\widehat{g}}_{3c}$ (filled disk) which represents the peeling of one polymer from the rigid bound state. A typical flow from a point away from the melting point is shown.

Figure 9

A bound state with one bubble showing variables in the Fourier-Laplace space variable obeying the k and the $s$ conservations.

Figure 10

A schematic diagram of the flow of the RG equation, Eq. (C2), showing the approach to the limit cycle. The limit cycle is an ellipse in the ${\widehat{g}}_2=0.5$ plane [blue line marked (a)]. A point with off-critical ${\widehat{g}}_2$ is shown to approach the planar limit cycle as $\Lambda \to \infty$ [red line marked (b)].