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Cold denaturation of RNA secondary structures with loop entropy and quenched disorder

Flavio Iannelli, Yevgeni Mamasakhlisov, and Roland R. Netz
Phys. Rev. E 101, 012502 – Published 31 January 2020
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  • INTRODUCTION
  • DISORDERED RNA
  • HOMOPOLYMER RESULTS
  • CONSTRAINED ANNEALING WITH LOOP ENTROPY
  • CONCLUSIONS
    • References

    Abstract

    The critical behavior of ribonucleic acid (RNA) secondary structures with quenched sequence randomness is studied by means of the constrained annealing method. A thermodynamic phase transition is induced by including the conformational weight of loop structures. In addition to the expected melting at high temperature, a cold-melting transition appears when the disorder strength induces competition between favorable and unfavorable base pairs. Our results suggest that the cold denaturation of RNA found experimentally might be triggered by quenched sequence disorder. We calculate hot- and cold-melting critical temperatures for competing favorable and unfavorable base-pair energies and present a folding phase diagram as a function of the loop exponent and temperature.

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    • Received 5 October 2018

    DOI:https://doi.org/10.1103/PhysRevE.101.012502

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Phase transitionsPolymer folding
    1. Physical Systems
    Disordered systemsPolymer glassesPolymer meltsPolymers
    Physics of Living SystemsPolymers & Soft MatterStatistical Physics & Thermodynamics

    Authors & Affiliations

    Flavio Iannelli*

    • Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, 12481 Berlin, Germany

    Yevgeni Mamasakhlisov

    • Department of Molecular Physics, Yerevan State University, 1 Alex Manougian Street, Yerevan 0025, Armenia

    Roland R. Netz

    • Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany

    • *iannelli.flavio@gmail.com

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    Vol. 101, Iss. 1 — January 2020

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    • Figure 1
      Figure 1

      Upper panel: Secondary structure representation with stacks and loops. Nucleotides are the green and yellow dots for U and A, respectively. The wavy lines identify the hydrogen bonds for favorable (red) and unfavorable (blue) base pairs, while the black solid lines are the nested backbone links. Thick gray denote the non-nested backbone links. In this secondary structure, a pseudoknot is formed between two loops. Lower panel: The two possible types of base pairings that are considered.

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    • Figure 2
      Figure 2

      Hierarchical recursive scheme for the partition function given by Eq. (9) without loop entropy. The subchain partition function from base i to base j+1 is the sum of the partition function from base i to base j and the partition functions of all base pairings formed between base j+1 and base k∈(i,j) [3].

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    • Figure 3
      Figure 3

      Recursion scheme for the partition function given by Eq. (12) including loops. The partition function of a RNA sequence ranging from i to j+1 with M+1 non-nested backbones (thick gray lines) is computed from the sequence ranging from i to j with M non-nested backbones by adding a base at position j+1 and considering all possible pairings with a base k∈(i,j) [26]. Each of these pairings defines a structure which has zero non-nested backbones, i.e., an arbitrary substrand that is terminated by a helix. The explicit diagrams for the latter are shown in the lower panel and are obtained by considering the sum over all non-nested backbones with associated statistical weight given by Eq. (11).

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    • Figure 4
      Figure 4

      Helicity degree for (a) ε0=−1 and (b) ε0=+1 with ε=0.5|ε0| in the absence of loop entropy with probability of U base occurrence p=0.75. Each red curve corresponds to the helicity of a single RNA sequence realization of the disorder. The quenched average (black line) is obtained from the exact computation of the partition function for 30 random sequences of length N=50. The black dashed line denotes the high-temperature limit θ∞ given by Eq. (25).

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    • Figure 5
      Figure 5

      Phase diagram of homopolymeric RNA in the T−c plane featuring the unfolded and folded phase for (a) repulsive (ε0>0) and (b) attractive (ε0<0) base-pair interaction energy. The critical lines, obtained by solving wm(c)=w=e−βε0, diverge for c=c1 for both attractive and repulsive interaction energy. For c≤2, the molecule is always folded and at c*≈2.479 the critical weight wm(c) diverges, so that for c>c* no folded phase can exist.

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    • Figure 6
      Figure 6

      Specific heat with loops in the folded phase (a),(b) with c=cRW=1.5 and (c),(d) without loops (corresponding to c=0). The quenched average (black curve) is numerically obtained from averaging over 30 random sequences (red curves) with N=50, p=0.75, and interaction energy (a),(c) ε=0.5|ε0| and (b),(d) ε=1.5|ε0|. The analytical result using the constrained annealing method is shown in blue. Here, the partition function of each sequence is computed with the loop recursive equation (12) and the constrained annealing free energy is fbca(μ̃) defined by Eq. (58). Insets: The corresponding free energies from numerics and the constrained annealing method from which the specific-heat curves in the main panels are obtained.

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    • Figure 7
      Figure 7

      Statistical weight in the constrained annealing approximation in the competitive regime (Λ=2/3) with ε=1.5|ε0| (orange) and in the noncompetitive regime (Λ=2) with ε=0.5|ε0| (violet) as a function of the temperature at p=0.75. Inset: Homopolymer weight w=e−βε0 with ε0>0 (orange) and ε0<0 (violet), for repulsive and attractive background interaction, respectively.

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    • Figure 8
      Figure 8

      Specific heat for c=cRW=1.5 and different probabilities, from light to dark, p=0.5, p=0.6, p=0.7, p=0.8, p=0.9, and p=1.0, in the constrained annealing approximation in the (a) noncompetitive regime (Λ=2) with ε=0.5|ε0| and (b) competitive regime (Λ=2/3) with ε=1.5|ε0|. Insets: Corresponding helicity degrees. All curves converge to the asymptotic value of Eq. (25) in the high-temperature molten phase.

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    • Figure 9
      Figure 9

      Fraction η− of favorable (dashed line) and η+ of unfavorable (solid line) base pairs for c=cRW=1.5 and different probabilities, from light to dark, p=0.5, p=0.6, p=0.7, p=0.8, p=0.9, and p=1.0, in the constrained annealing approximation in the (a) noncompetitive regime (Λ=2) with ε=0.5|ε0| and (b) competitive regime (Λ=2/3) with ε=1.5|ε0|.

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    • Figure 10
      Figure 10

      (a) Phase diagram in the noncompetitive regime (Λ=2) with ε=0.5|ε0| and ε0<0. The critical line that defines hot melting is defined in the range c1<c<c* [see Eqs. (37) and (38)] so that for c<c1, the RNA molecule is always folded. (b) Phase diagram in the competitive regime (Λ=2/3) with ε=1.5|ε0| and ε0<0. For c<cmin, the molecule is always folded since wm(c)<wca(T), ∀T. For cmin<c<c1, there is only hot melting, while for c>cmax, the molecule is always unfolded since wm(c)>wca[T,μ̃(T)], ∀T. In the range c1<c<cmax, both hot and cold melting take place. In both panels, the color of each curve corresponds to a different value of the probability p, from light to dark, p=0.5, p=0.6, p=0.7, p=0.8, p=0.9, and p=1.0. Inset: Critical line for p=0.75. Since 0.75>p* in the range c1<c<cmax, the double intersection between wm(c) and wca[T,μ̃(T)] introduces the additional cold-melting phase transition.

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    • Figure 11
      Figure 11

      Specific heat for the disordered model using the constrained annealing approximation with p=0.75 and ε=1.5|ε0| in (a) the competitive regime with Λ=2/3 and (b) for a homopolymer with ε0<0. In both cases, the different curves correspond to different values of the loop exponent c. In the disordered model, the two critical temperatures Tcm≈0.488|ε0|/kB (blue circle) and Thm≈2.961|ε0|/kB (red circle) correspond to the value of the loop exponent c=2.26, while for the homopolymer there is only a single melting transition at Tm≈2.605|ε0|/kB. In (c) and (d), the corresponding helicity degrees are shown.

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