Necklace-cloverleaf transition in associating RNA-like diblock copolymers

We consider a $A_m{B}_n$ diblock copolymer, whose links are capable of forming local reversible bonds with each other. We assume that the resulting structure of the bonds is RNA like—i.e., topologically isomorphic to a tree. We show that, depending on the relative strengths of $A$-$A$, $A$-$B$, and $B$-$B$ contacts, such a polymer can be in one of two different states. Namely, if a self-association is preferable (i.e., $A$-$A$ and $B$-$B$ bonds are comparatively stronger than $A$-$B$ contacts), then the polymer forms a typical randomly branched cloverleaf structure with the so-called roughness exponent $\gamma =1∕2$. On the contrary, if alternating association is preferable (i.e., $A$-$B$ bonds are stronger than $A$-$A$ and $B$-$B$ contacts), then the polymer tends to form a generally linear necklace structure with $\gamma =1$. The transition between cloverleaf and necklace states is studied in detail, and it is shown that it is a second-order phase transition.

Figure 1

(a) The treelike topology of the chain in the presence of strong cooperativity. (b) The pseudoknot in the presence of strong cooperativity.

Figure 2

Schematic picture of possible thermodynamic states of RNA-like heteropolymers.

Figure 3

(a),(b) Schematic picture of allowed (a) cactuslike and prohibited (b) pseudoknot configurations of the bonds; (c),(d) Arc diagrams corresponding to configurations (a) and (b), respectively [note the intersection of arcs in (d)]; (e) The height diagram corresponding to the bond configuration (a).

Figure 4

Diagrammatic form of Dyson-type equations: (a) for Eq. (4) and (b) for Eq. (5). Thin lines correspond to free propagators $g(s,\mathbf{q})$, thick hatched lines to the homopolymer propagators $G(s,\mathbf{q}\mid v)$, and thick gray lines to the diblock copolymer propagators $G_2(s_1,s_2,\mathbf{q}\mid v_1,v_2,u)$. Finally, open and solid circles correspond to the intraspecies and interspecies bonds, respectively.

Figure 5

Sketches of typical conformations of diblock copolymers in the necklace (a) and cloverleaf (b) phases. $A$ block is represented by solid circles on the black string, $B$ block by open circles on the dark gray string. Intraspecies ($A$-$A$ and $B$-$B$) and interspecies ($A$-$B$) bonds are shown by black and light gray, respectively.

Figure 6

Free energy of the system defined by Eqs. (15, 16) as a function of $u$ (a), $v_1$ (b), and $\alpha$ (c). (a) (b) Thin lines correspond to $v_1=v_2=1$, thick lines to $v_2=2$, $v_1=0.5$, and dashed lines to $v_2=5$, $v_1=0.2$. (a) Curves corresponding to $\alpha =$ 0.1, 0.5, and 0.9 are marked by numbers $1$, $2$, and $3$, respectively. (b) Curves corresponding to $u=$ 1 (the transition point), 10, and 100 are marked by numbers $1$, $2$, and $3$, respectively. (c) Thick lines correspond to $u=2$, $v_2=1$, thin lines, both in the main figure and the inset, to $u=2$, $v_2=20$. Curves marked by 1, 2, 3, 4, and 5 correspond to $\alpha =$ 0.9, 0.7, 0.5, 0.3, and 0.1, respectively. Note that the thin lines intersect at the same point $v_1=v_2$ which, in this case, belongs to the cloverleaf phase. This is not true for the thick lines since the point $v_1=v_2$ belong to the necklace phase in this case.

Figure 7

Phase diagram of the cloverleaf-necklace phase transition with ideal loop weights. Thin, dashed, and thick lines correspond to the product $v_1{v}_2=$ 100, 10, and 1, respectively.

Figure 8

Free energy of the system with ideal loop weights as a function of $\alpha$. Thin, thick, and dashed lines correspond to $v_1=v_2=14.1$; $v_1=10$, $v_2=20$; and $v_1=4$, $v_2=50$, respectively. The curves corresponding to $u=40$, $u=80$, and $u=400$ are marked by $a$, $b$, and $c$, respectively. Note that $u=40$ (curves $a$) correspond to a point in the necklace phase just above the transition. In the cloverleaf phase [see Eq. (C2)] the free energy is linear in $\alpha$.

Figure 9

Diagrammatic structure of Eq. (26).

Figure 10

Examples of possible domains of analyticity. The curves (A7) corresponding to different values of $\alpha$ $({\alpha }_1<{\alpha }_2)$ are also shown. The touching points may (b) or may not (a) depend on $\alpha$.

Figure 11

Examples of the analyticity domains of Eq. (12) for $v_1=1,v_2=5$. Rectangular domain with borders $s_{1,2}^{*}=(1+2\sqrt{v_{1,2}}{)}^{-1}$ corresponds to values of $u$ below the necklace-cloverleaf transition (i.e., $u<u_{\mathrm{tr}}=\sqrt{5}$). The curves marked by $a$, $b$, $c$, $d$, and $e$ are the pole lines corresponding to $u=5$, 10, 20, 50, and 200, respectively.