Formation of base pairs between the nucleotides of a ribonucleic acid (RNA) sequence gives rise to a complex and often highly branched RNA structure. While numerous studies have demonstrated the functional importance of the high degree of RNA branching—for instance, for its spatial compactness or interaction with other biological macromolecules—RNA branching topology remains largely unexplored. Here, we use the theory of randomly branching polymers to explore the scaling properties of RNAs by mapping their secondary structures onto planar tree graphs. Focusing on random RNA sequences of varying lengths, we determine the two scaling exponents related to their topology of branching. Our results indicate that ensembles of RNA secondary structures are characterized by annealed random branching and scale similarly to self-avoiding trees in three dimensions. We further show that the obtained scaling exponents are robust upon changes in nucleotide composition, tree topology, and folding energy parameters. Finally, in order to apply the theory of branching polymers to biological RNAs, whose length cannot be arbitrarily varied, we demonstrate how both scaling exponents can be obtained from distributions of the related topological quantities of individual RNA molecules with fixed length. In this way, we establish a framework to study the branching properties of RNA and compare them to other known classes of branched polymers. By understanding the scaling properties of RNA related to its branching structure, we aim to improve our understanding of the underlying principles and open up the possibility to design RNA sequences with desired topological properties.
REFERENCES
1.
A. S.
Kulkarni
and G.
Beaucage
, “Quantification of branching in disordered materials
,” J. Polym. Sci., Part B: Polym. Phys.
44
, 1395
–1405
(2006
).2.
B. I.
Voit
and A.
Lederer
, “Hyperbranched and highly branched polymer architectures—synthetic strategies and major characterization aspects
,” Chem. Rev.
109
, 5924
–5973
(2009
).3.
J. R.
van der Maarel
, Introduction to Biopolymer Physics
(World Scientific Publishing Company
, 2007
).4.
A. B.
Cook
and S.
Perrier
, “Branched and dendritic polymer architectures: Functional nanomaterials for therapeutic delivery
,” Adv. Funct. Mater.
30
, 1901001
(2020
).5.
J.
Wiedemann
, J.
Kaczor
, M.
Milostan
, T.
Zok
, J.
Blazewicz
, M.
Szachniuk
, and M.
Antczak
, “RNAloops: A database of RNA multiloops
,” Bioinformatics
38
, 4200
(2022
).6.
M. A.
Boerneke
, J. E.
Ehrhardt
, and K. M.
Weeks
, “Physical and functional analysis of viral RNA genomes by SHAPE
,” Annu. Rev. Virol.
6
, 93
(2019
).7.
H.
Wan
, R. L.
Adams
, B. D.
Lindenbach
, and A. M.
Pyle
, “The in vivo and in vitro architecture of the hepatitis C virus RNA genome uncovers functional RNA secondary and tertiary structures
,” J. Virol.
96
, e0194621
(2022
).8.
A. M.
Yoffe
, P.
Prinsen
, A.
Gopal
, C. M.
Knobler
, W. M.
Gelbart
, and A.
Ben-Shaul
, “Predicting the sizes of large RNA molecules
,” Proc. Natl. Acad. Sci. U. S. A.
105
, 16153
–16158
(2008
).9.
L.
Tubiana
, A. L.
Božič
, C.
Micheletti
, and R.
Podgornik
, “Synonymous mutations reduce genome compactness in icosahedral ssRNA viruses
,” Biophys. J.
108
, 194
–202
(2015
).10.
S. W.
Singaram
, R. F.
Garmann
, C. M.
Knobler
, W. M.
Gelbart
, and A.
Ben-Shaul
, “Role of RNA branchedness in the competition for viral capsid proteins
,” J. Phys. Chem. B
119
, 13991
–14002
(2015
).11.
R. F.
Garmann
, M.
Comas-Garcia
, C. M.
Knobler
, and W. M.
Gelbart
, “Physical principles in the self-assembly of a simple spherical virus
,” Acc. Chem. Res.
49
, 48
–55
(2016
).12.
C.
Beren
, L. L.
Dreesens
, K. N.
Liu
, C. M.
Knobler
, and W. M.
Gelbart
, “The effect of RNA secondary structure on the self-assembly of viral capsids
,” Biophys. J.
113
, 339
–347
(2017
).13.
A. L.
Božič
, C.
Micheletti
, R.
Podgornik
, and L.
Tubiana
, “Compactness of viral genomes: Effect of disperse and localized random mutations
,” J. Phys.: Condens. Matter
30
, 084006
(2018
).14.
L.
Marichal
, L.
Gargowitsch
, R. L.
Rubim
, C.
Sizun
, K.
Kra
, S.
Bressanelli
, Y.
Dong
, S.
Panahandeh
, R.
Zandi
, and G.
Tresset
, “Relationships between RNA topology and nucleocapsid structure in a model icosahedral virus
,” Biophys. J.
120
, 3925
–3936
(2021
).15.
R.
Zandi
and P.
van der Schoot
, “Size regulation of ss-RNA viruses
,” Biophys. J.
96
, 9
–20
(2009
).16.
P.
van der Schoot
and R.
Zandi
, “Impact of the topology of viral RNAs on their encapsulation by virus coat proteins
,” J. Biol. Phys.
39
, 289
–299
(2013
).17.
J.
Wagner
, G.
Erdemci-Tandogan
, and R.
Zandi
, “Adsorption of annealed branched polymers on curved surfaces
,” J. Phys.: Condens. Matter
27
, 495101
(2015
).18.
G.
Erdemci-Tandogan
, J.
Wagner
, P.
van der Schoot
, R.
Podgornik
, and R.
Zandi
, “RNA topology remolds electrostatic stabilization of viruses
,” Phys. Rev. E
89
, 032707
(2014
).19.
G.
Erdemci-Tandogan
, J.
Wagner
, P.
van der Schoot
, R.
Podgornik
, and R.
Zandi
, “Effects of RNA branching on the electrostatic stabilization of viruses
,” Phys. Rev. E
94
, 022408
(2016
).20.
A.
Gopal
, D. E.
Egecioglu
, A. M.
Yoffe
, A.
Ben-Shaul
, A. L. N.
Rao
, C. M.
Knobler
, and W. M.
Gelbart
, “Viral RNAs are unusually compact
,” PLoS One
9
, e105875
(2014
).21.
R. F.
Garmann
, A.
Gopal
, S. S.
Athavale
, C. M.
Knobler
, W. M.
Gelbart
, and S. C.
Harvey
, “Visualizing the global secondary structure of a viral RNA genome with cryo-electron microscopy
,” RNA
21
, 877
–886
(2015
).22.
H. H.
Gan
, S.
Pasquali
, and T.
Schlick
, “Exploring the repertoire of RNA secondary motifs using graph theory; implications for RNA design
,” Nucleic Acids Res.
31
, 2926
–2943
(2003
).23.
T.
Schlick
, “Adventures with RNA graphs
,” Methods
143
, 16
–33
(2018
).24.
D.
Vaupotič
, A.
Rosa
, R.
Podgornik
, L.
Tubiana
, and A.
Božič
, “Viral RNA as a branched polymer
,” arXiv:2212.00829 [physics.bio-ph] (2022
).25.
A.
Borodavka
, S. W.
Singaram
, P. G.
Stockley
, W. M.
Gelbart
, A.
Ben-Shaul
, and R.
Tuma
, “Sizes of long RNA molecules are determined by the branching patterns of their secondary structures
,” Biophys. J.
111
, 2077
–2085
(2016
).26.
P. J.
Flory
, Principles of Polymer Chemistry
(Cornell University Press
, Ithaca, NY
, 1953
).27.
S. M.
Bhattacharjee
, A.
Giacometti
, and A.
Maritan
, “Flory theory for polymers
,” J. Phys.: Condens. Matter
25
, 503101
(2013
).28.
R.
Everaers
, A. Y.
Grosberg
, M.
Rubinstein
, and A.
Rosa
, “Flory theory of randomly branched polymers
,” Soft Matter
13
, 1223
–1234
(2017
).29.
M.
Rubinstein
and R. H.
Colby
, Polymer Physics
(Oxford University Press
, New York
, 2003
).30.
Z.-G.
Wang
, “50th anniversary perspective: Polymer conformation—A pedagogical review
,” Macromolecules
50
, 9073
–9114
(2017
).31.
L. T.
Fang
, W. M.
Gelbart
, and A.
Ben-Shaul
, “The size of RNA as an ideal branched polymer
,” J. Chem. Phys.
135
, 155105
(2011
).32.
H. A.
Kramers
, “The behavior of macromolecules in inhomogeneous flow
,” J. Chem. Phys.
14
, 415
–424
(1946
).33.
T. C.
Lubensky
and J.
Isaacson
, “Statistics of lattice animals and dilute branched polymers
,” Phys. Rev. A
20
, 2130
–2146
(1979
).34.
M.
Daoud
and J. F.
Joanny
, “Conformation of branched polymers
,” J. Phys.
42
, 1359
–1371
(1981
).35.
E. J.
Janse van Rensburg
and N.
Madras
, “A nonlocal Monte Carlo algorithm for lattice trees
,” J. Phys. A: Math. Gen.
25
, 303
(1992
).36.
A. Y.
Grosberg
and R.
Bruinsma
, “Confining annealed branched polymers inside spherical capsids
,” J. Biol. Phys.
44
, 133
–145
(2018
).37.
J.
Kelly
, A. Y.
Grosberg
, and R.
Bruinsma
, “Sequence dependence of viral RNA encapsidation
,” J. Phys. Chem. B
120
, 6038
–6050
(2016
).38.
P. G.
Higgs
, “RNA secondary structure: A comparison of real and random sequences
,” J. Phys.
3
, 43
–59
(1993
).39.
R.
Bundschuh
and T.
Hwa
, “Statistical mechanics of secondary structures formed by random RNA sequences
,” Phys. Rev. E
65
, 031903
(2002
).40.
E. A.
Schultes
, A.
Spasic
, U.
Mohanty
, and D. P.
Bartel
, “Compact and ordered collapse of randomly generated RNA sequences
,” Nat. Struct. Mol. Biol.
12
, 1130
–1136
(2005
).41.
P.
Clote
, F.
Ferré
, E.
Kranakis
, and D.
Krizanc
, “Structural RNA has lower folding energy than random RNA of the same dinucleotide frequency
,” RNA
11
, 578
–591
(2005
).42.
F.
Chizzolini
, L. F. M.
Passalacqua
, M.
Oumais
, A. I.
Dingilian
, J. W.
Szostak
, and A.
Lupták
, “Large phenotypic enhancement of structured random RNA pools
,” J. Am. Chem. Soc.
142
, 1941
–1951
(2019
).43.
A.
Rosa
and R.
Everaers
, “Computer simulations of randomly branching polymers: Annealed versus quenched branching structures
,” J. Phys. A: Math. Theor.
49
, 345001
(2016
).44.
A.
Rosa
and R.
Everaers
, “Computer simulations of melts of randomly branching polymers
,” J. Chem. Phys.
145
, 164906
(2016
).45.
A.
Rosa
and R.
Everaers
, “Beyond Flory theory: Distribution functions for interacting lattice trees
,” Phys. Rev. E
95
, 012117
(2017
).46.
In the original work of Rosa and Everaers,45 the exponents θ and t and the related numerical constants C and K bear the subscript ℓ to distinguish them from analogous quantities appearing in other distribution functions. Here, to lighten up the notation, we drop this subscript.
47.
C. T.
Woods
, L.
Lackey
, B.
Williams
, N. V.
Dokholyan
, D.
Gotz
, and A.
Laederach
, “Comparative visualization of the RNA suboptimal conformational ensemble in vivo
,” Biophys. J.
113
, 290
–301
(2017
).48.
R.
Lorenz
, S. H.
Bernhart
, C.
Höner zu Siederdissen
, H.
Tafer
, C.
Flamm
, P. F.
Stadler
, and I. L.
Hofacker
, “ViennaRNA package 2.0
,” Algorithms Mol. Biol.
6
, 26
(2011
).49.
S. W.
Singaram
, A.
Gopal
, and A.
Ben-Shaul
, “A Prüfer-sequence based algorithm for calculating the size of ideal randomly branched polymers
,” J. Phys. Chem. B
120
, 6231
–6237
(2016
).50.
S.
Poznanović
, C.
Wood
, M.
Cloer
, and C.
Heitsch
, “Improving RNA branching predictions: Advances and limitations
,” Genes
12
, 469
(2021
).51.
M.
Ward
, A.
Datta
, M.
Wise
, and D. H.
Mathews
, “Advanced multi-loop algorithms for RNA secondary structure prediction reveal that the simplest model is best
,” Nucleic Acids Res.
45
, 8541
–8550
(2017
).52.
S.
Poznanović
, F.
Barrera-Cruz
, A.
Kirkpatrick
, M.
Ielusic
, and C.
Heitsch
, “The challenge of RNA branching prediction: A parametric analysis of multiloop initiation under thermodynamic optimization
,” J. Struct. Biol.
210
, 107475
(2020
).53.
A.
Clauset
, C. R.
Shalizi
, and M. E. J.
Newman
, “Power-law distributions in empirical data
,” SIAM Rev.
51
, 661
–703
(2009
).54.
D. H.
Turner
and D. H.
Mathews
, “NNDB: The nearest neighbor parameter database for predicting stability of nucleic acid secondary structure
,” Nucleic Acids Res.
38
, D280
–D282
(2010
).55.
D. H.
Mathews
and D. H.
Turner
, “Prediction of RNA secondary structure by free energy minimization
,” Curr. Opin. Struct. Biol.
16
, 270
–278
(2006
).56.
M.
Andronescu
, A.
Condon
, H. H.
Hoos
, D. H.
Mathews
, and K. P.
Murphy
, “Computational approaches for RNA energy parameter estimation
,” RNA
16
, 2304
–2318
(2010
).57.
W. B.
Langdon
, J.
Petke
, and R.
Lorenz
, “Evolving better RNAfold structure prediction
,” in European Conference on Genetic Programming
(Springer
, 2018
), pp. 220
–236
.58.
M.
Tacker
, P. F.
Stadler
, E. G.
Bornberg-Bauer
, I. L.
Hofacker
, and P.
Schuster
, “Algorithm independent properties of RNA secondary structure predictions
,” Eur. Biophys. J.
25
, 115
–130
(1996
).59.
A. Y.
Grosberg
, “Annealed lattice animal model and Flory theory for the melt of non-concatenated rings: Towards the physics of crumpling
,” Soft Matter
10
, 560
–565
(2014
).60.
G.
Parisi
and N.
Sourlas
, “Critical behavior of branched polymers and the Lee-Yang edge singularity
,” Phys. Rev. Lett.
46
, 871
–874
(1981
).61.
According to Kramers’ theorem,29,32,34 each bond contributes ≃⟨b2⟩ to the mean-square radius of gyration. In terms of the notation of this paper, ⟨b2⟩ can be defined as the ratio between the mean-square end-to-end distance between tree paths with average ladder distance ⟨ALD⟩ and the last one, i.e.,where b is the mean bond length as in Eq. (17) and νpath = ν/ρ28 is the scaling exponent for the spatial structure of tree linear paths. For noninteracting trees with ν = 1/4 and ρ = 1/2,28 νpath = 1/2 and ⟨b2⟩ ≃ b2 does not depend on N. Conversely, for interacting trees, we do have ⟨ALD⟩/b ∼ Nρ [Eq. (2)] and the above equation then implies that ⟨b2⟩ ≃ b2N2ν−ρ.
62.
C.
Hyeon
, R. I.
Dima
, and D.
Thirumalai
, “Size, shape, and flexibility of RNA structures
,” J. Chem. Phys.
125
, 194905
(2006
).63.
Z.-H.
Guo
, L.
Yuan
, Y.-L.
Tan
, B.-G.
Zhang
, and Y.-Z.
Shi
, “RNAStat: An integrated tool for statistical analysis of RNA 3D structures
,” Front. Bioinform.
1
, 809082
(2022
).64.
A. M.
Gutin
, A. Y.
Grosberg
, and E. I.
Shakhnovich
, “Polymers with annealed and quenched branchings belong to different universality classes
,” Macromolecules
26
, 1293
–1295
(1993
).65.
S.
Jain
, A.
Laederach
, S. B.
Ramos
, and T.
Schlick
, “A pipeline for computational design of novel RNA-like topologies
,” Nucleic Acids Res.
46
, 7040
–7051
(2018
).66.
S.
Jain
, Y.
Tao
, and T.
Schlick
, “Inverse folding with RNA-As-Graphs produces a large pool of candidate sequences with target topologies
,” J. Struct. Biol.
209
, 107438
(2020
).67.
L.
Rolband
, D.
Beasock
, Y.
Wang
, Y.-G.
Shu
, J. D.
Dinman
, T.
Schlick
, Y.
Zhou
, J. S.
Kieft
, S.-J.
Chen
, G.
Bussi
et al, “Biomotors, viral assembly, and RNA nanobiotechnology: Current achievements and future directions
,” Comput. Struct. Biotechnol. J.
20
, 6120
(2022
).68.
J. M.
Herrero
, T.
Stahl
, S.
Erbar
, K.
Maxeiner
, A.
Schlegel
, T.
Bacic
, L.
Cavalcanti
, M.
Schroer
, D.
Svergun
, U.
Sahin
et al, “Ultra-compacted single self-amplifying RNA molecules as quintessential vaccines
,” Res. Square
(published online 2022).69.
D.
Vaupotič
, A.
Rosa
, L.
Tubiana
, and Anže
Božič
(2023). “Scaling properties of RNA as a randomly branching polymer
,” V. 1, Materials Cloud Archive, Dataset. https://doi.org/10.24435/materialscloud:js-fx© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
