pith. sign in

arxiv: 2606.25824 · v1 · pith:GKJOEJWKnew · submitted 2026-06-24 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· cs.NA· math.NA

Kinetic metric for basins of attraction of RNA secondary structures and analysis of the ultrametricity of the energy landscape

A.P. Zubarev This is my paper

Pith reviewed 2026-06-25 19:51 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechcs.NAmath.NA
keywords RNA secondary structuresenergy landscapeultrametricitykinetic metricbasins of attractiontransition ratesMahalanobis distancenucleotide order
0
0 comments X

The pith

For fixed nucleotide composition, the order of nucleotides determines the degree of nontrivial ultrametricity of the RNA energy landscape.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a method to test the hypothesis of ultrametric organization in the energy landscape of RNA secondary structures by examining transition kinetics between basins of attraction. The method builds a kinetic metric from the spectral decomposition of the symmetrized Kramers transition rate matrix combined with the Mahalanobis distance, which is not ultrametric by design. A complete computational framework is provided that includes automatic filtering of noisy eigenmodes and handling of disconnected structure graphs from stochastic sampling. Application to reference and random sequences shows that, when nucleotide composition is held fixed, the specific order of nucleotides controls the degree of nontrivial ultrametricity. This kinetic approach offers a way to probe landscape organization through observed transition behavior rather than static distance measures alone.

Core claim

The central claim is that a kinetic metric derived from the spectral decomposition of the symmetrized Kramers transition rate matrix and the Mahalanobis distance can be used to test ultrametric organization of RNA energy landscapes. When applied to sequences with fixed nucleotide composition, the analysis establishes that the degree of nontrivial ultrametricity is set by the order of the nucleotides. The framework supports this by filtering noisy modes and managing sampling-induced graph disconnections, demonstrating the result on both reference and random RNA sequences.

What carries the argument

The kinetic metric built from the spectral decomposition of the symmetrized Kramers transition rate matrix together with the Mahalanobis distance, which quantifies how far basin transitions deviate from ultrametric geometry.

If this is right

  • The method distinguishes ultrametric from non-ultrametric features in RNA folding landscapes through observed kinetics.
  • Nucleotide order, not merely composition, sets the ultrametric character of the landscape for any given base counts.
  • Automatic eigenmode filtering and disconnected-graph handling make the analysis practical on stochastically sampled structures.
  • The approach works equally on reference sequences and randomly generated ones with controlled composition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same kinetic construction might be applied to test ultrametricity in other discrete energy landscapes such as protein folding or lattice models.
  • If ultrametricity depends on sequence order, evolutionary pressures could favor or disfavor particular arrangements beyond composition alone.
  • The framework could be used to predict how changes in sequence order alter folding pathway statistics without recomputing the full landscape.

Load-bearing premise

The kinetic metric from the transition rate matrix spectral decomposition and Mahalanobis distance can be used to test and quantify the ultrametric organization hypothesis of the energy landscape.

What would settle it

Finding no systematic variation in the ultrametricity measure across sequences that differ only in nucleotide order while sharing the same composition would falsify the claim that order determines the degree of nontrivial ultrametricity.

read the original abstract

A method is proposed for testing the hypothesis of ultrametric organization of the energy landscape of RNA secondary structures, based on the analysis of transition kinetics between basins of attraction. The method relies on a kinetic metric constructed from the spectral decomposition of the symmetrized Kramers transition rate matrix and the Mahalanobis distance, which is not an ultrametric by construction. A complete computational framework is developed, including automatic filtering of noisy eigenmodes and a procedure for analyzing disconnected structure graphs arising from stochastic sampling. The method is demonstrated on a set of reference and random RNA sequences. It is shown that, for a fixed nucleotide composition, the degree of nontrivial ultrametricity of the energy landscape is determined by the order of nucleotides.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a method to test the ultrametric organization hypothesis for RNA secondary structure energy landscapes. It constructs a kinetic metric from the spectral decomposition of the symmetrized Kramers transition rate matrix combined with the Mahalanobis distance (explicitly not ultrametric by construction), develops a computational framework with automatic filtering of noisy eigenmodes and procedures for disconnected structure graphs arising from stochastic sampling, and applies the approach to reference and random RNA sequences. The central result is that, at fixed nucleotide composition, the degree of nontrivial ultrametricity is controlled by the order of nucleotides.

Significance. If the computational results and metric construction hold under scrutiny, the work supplies a non-circular, falsifiable procedure for quantifying ultrametricity in biomolecular landscapes and reports a concrete dependence on nucleotide ordering. This could inform models of RNA folding kinetics and sequence evolution. The explicit avoidance of ultrametricity by construction and the handling of stochastic sampling artifacts are methodological strengths.

minor comments (2)
  1. The description of the automatic filtering procedure for noisy eigenmodes would benefit from an explicit pseudocode or decision criterion in the methods section to allow independent reproduction.
  2. Figure captions should state the exact sequences, lengths, and sampling parameters used for the reference and random RNA examples to facilitate direct comparison with the reported ultrametricity measures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the assessment of its potential significance, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a kinetic metric via spectral decomposition of the symmetrized Kramers rate matrix combined with Mahalanobis distance, explicitly noting it is not ultrametric by construction. This metric is then used to computationally test the ultrametricity hypothesis on RNA basins for sequences with fixed composition but varying nucleotide order. The reported finding is an empirical outcome of applying the independent metric and filtering procedures, with no equations or steps reducing the ultrametricity measure to a fitted input, self-definition, or self-citation chain. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only abstract available; full details on parameters and assumptions not accessible.

axioms (1)
  • domain assumption RNA secondary structures can be grouped into basins of attraction with transition rates described by Kramers theory.
    This is the foundation for the kinetic analysis in the method.
invented entities (1)
  • Kinetic metric from spectral decomposition and Mahalanobis distance no independent evidence
    purpose: To provide a measure of ultrametricity that is independent of the ultrametric property by construction.
    This is a new construction proposed in the paper for the analysis.

pith-pipeline@v0.9.1-grok · 5665 in / 1141 out tokens · 35724 ms · 2026-06-25T19:51:52.085399+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 44 canonical work pages · 2 internal anchors

  1. [1]

    Rammal, G

    R. Rammal, G. Toulouse, and M. A. Virasoro, Ultrametricity for physicists, Rev. Mod. Phys. 58 (3), 765–788 (1986). https://doi.org/10.1103/RevModPhys.58.765 1

    work page doi:10.1103/revmodphys.58.765 1986

  2. [2]

    Hartigan, J. A. (1975). Clustering Algorithms. John Wiley & Sons. 1

    1975

  3. [3]

    Sneath, P. H. A., & Sokal, R. R. (1973). Numerical Taxonomy: The Principles and Practice of Numerical Classification. W. H. Freeman. 1, 2

    1973

  4. [4]

    Lake, J. A. (1994). Reconstructing evolutionary trees from DNA and protein sequences: par- alinear distances. Proceedings of the National Academy of Sciences, 91(4), 1455-1459. 1 31

    1994

  5. [5]

    Parisi, Infinite number of order parameters for spin-glasses, Phys

    G. Parisi, Infinite number of order parameters for spin-glasses, Phys. Rev. Lett. 43, 1754–1756 (1979). https://doi.org/10.1103/PhysRevLett.43.1754 1

    work page doi:10.1103/physrevlett.43.1754 1979

  6. [6]

    Parisi, A sequence of approximated solutions to the S-K model for spin glasses, J

    G. Parisi, A sequence of approximated solutions to the S-K model for spin glasses, J. Phys. A: Math. Gen. 13, L115–L121 (1980). https://doi.org/10.1088/0305-4470/13/4/009 1

    work page doi:10.1088/0305-4470/13/4/009 1980

  7. [7]

    M´ ezard, G

    M. M´ ezard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, Replica symmetry breaking and the nature of the spin glass phase, J. Phys. (Paris) 45, 843–854 (1984). https://doi.org/10.1051/jphys:01984004505084300 1

    work page doi:10.1051/jphys:01984004505084300 1984

  8. [8]

    M´ ezard, G

    M. M´ ezard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987). 1

    1987

  9. [9]

    Talagrand, Spin Glasses: A Challenge for Mathematicians, Berlin: Springer, 2003

    M. Talagrand, Spin Glasses: A Challenge for Mathematicians, Berlin: Springer, 2003. 1

    2003

  10. [10]

    Talagrand, Mean Field Models for Spin Glasses, Berlin: Springer, 2011

    M. Talagrand, Mean Field Models for Spin Glasses, Berlin: Springer, 2011. 1

    2011

  11. [11]

    Panchenko, The Sherrington-Kirkpatrick Model, New York: Springer, 2013

    D. Panchenko, The Sherrington-Kirkpatrick Model, New York: Springer, 2013. 1

    2013

  12. [12]

    Frauenfelder, F

    H. Frauenfelder, F. Parak, R.D.Young, Conformational substates in proteins, An- nual Review of Biophysics and Biophysical Chemistry 17 (1) (1988) 451-479. https://doi.org/10.1146/annurev.bb.17.060188.002315 1

    work page doi:10.1146/annurev.bb.17.060188.002315 1988

  13. [13]

    Frauenfelder, S.G

    H. Frauenfelder, S.G. Sligar, P.G. Wolynes, The energy landscapes and motions of proteins, Science 254 (5038) (1991) 1598-1603. https://doi.org/10.1126/science.1749933 1

    work page doi:10.1126/science.1749933 1991

  14. [14]

    Frauenfelder, B.H

    H. Frauenfelder, B.H. McMahon, Energy landscape and fluctuations in proteins, Ann. Phys., 512 (2000) 655-667. https://doi.org/10.1002/andp.200051209-1001 1

    work page doi:10.1002/andp.200051209-1001 2000

  15. [15]

    Dragovich, A

    B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, Onp-adic mathemat- ical physics,p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 1–17 (2009), https://doi.org/10.1134/S2070046609010014. 1

    work page doi:10.1134/s2070046609010014 2009

  16. [16]

    Dragovich, A

    B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, E. I. Zelenov,p-Adic math- ematical physics: the first 30 years,p-Adic Numbers, Ultrametric Analysis and Applications, 9 (2) (2017) 87–121. https://doi.org/10.1134/S2070046617020017 1

    work page doi:10.1134/s2070046617020017 2017

  17. [17]

    Dragovich, A

    B. Dragovich, A. Y. Khrennikov, S. V. Kozyrev, N. ˇZ. Miˇ si´ c,p-Adic mathematics and theoreti- cal biology, Biosystems, 199 104288 (2021). https://doi.org/10.1016/j.biosystems.2020.104288 1

    work page doi:10.1016/j.biosystems.2020.104288 2021

  18. [18]

    V. A. Avetisov, A. Kh. Bikulov, S. V. Kozyrev, and V. A. Osipov, p-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes, J. Phys. A: Math. Gen. 35, 177–189 (2002). https://doi.org/10.1088/0305-4470/35/2/305 1 32

    work page doi:10.1088/0305-4470/35/2/305 2002

  19. [19]

    V. A. Avetisov and A. Kh. Bikulov, Ultrametricity of fluctuation dynamic mobility of protein molecules, Tr. Mat. Inst. Steklova, 265 (2009) 75–81. https://doi.org/10.1134/S0081543809020060 1

    work page doi:10.1134/s0081543809020060 2009

  20. [20]

    V. A. Avetisov, A. Kh. Bikulov, A. P. Zubarev, Ultrametric random walk and dynamics of protein molecule, Tr. Mat. Inst. Steklova, 285 (2014) 3–25. https://doi.org/10.1134/S0081543814040026. 1

    work page doi:10.1134/s0081543814040026 2014

  21. [21]

    A. Kh. Bikulov, and A. P. Zubarev, Ultrametric theory of conformational dynamics of protein molecules in a functional state and the description of experiments on the kinetics of CO binding to myoglobin, Physica A: Statistical Mechanics and its Applications 583 126280-1–126280-18 (2021) . https://doi.org/10.1016/j.physa.2021.126280 1

    work page doi:10.1016/j.physa.2021.126280 2021

  22. [22]

    Khrennikov and S

    A. Khrennikov and S. V. Kozyrev, Genetic code on the dyadic plane, Physica A 381, 265–272 (2007). https://doi.org/10.1016/j.physa.2007.03.018 1

    work page doi:10.1016/j.physa.2007.03.018 2007

  23. [23]

    S. V. Kozyrev, p-Adic numbers in bioinformatics: from genetic code to PAM matrix, arXiv:0801.0213 [q-bio.GN] (2008). https://doi.org/10.48550/arXiv.0801.0213 1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.0801.0213 2008

  24. [24]

    Dragovich and A

    B. Dragovich and A. Yu. Dragovich, A p-adic model of DNA sequence and genetic code, p-Adic Numbers Ultrametric Anal. Appl. 1, 34–41 (2009). https://doi.org/10.1134/S2070046609010038 1

    work page doi:10.1134/s2070046609010038 2009

  25. [25]

    Dragovich, p-Adic structure of the genetic code, Neuroquantology 9 (4), 716–727 (2011)

    B. Dragovich, p-Adic structure of the genetic code, Neuroquantology 9 (4), 716–727 (2011). https://doi.org/10.14704/nq.2011.9.4.498 1

    work page doi:10.14704/nq.2011.9.4.498 2011

  26. [26]

    Dragovich, A

    B. Dragovich, A. Yu. Khrennikov, and N. Z. Miˇ si´ c, Ultrametrics in the genetic code and the genome, Biosystems 206, 104427 (2021). https://doi.org/10.1016/j.biosystems.2021.104427 1

    work page doi:10.1016/j.biosystems.2021.104427 2021

  27. [27]

    Brion and E

    P. Brion and E. Westhof, Hierarchy and dynamics of RNA folding, Annu. Rev. Biophys. Biomol. Struct. 26 (1), 113–137 (1997). https://doi.org/10.1146/annurev.biophys.26.1.113 1

    work page doi:10.1146/annurev.biophys.26.1.113 1997

  28. [28]

    D. E. Draper, Themes in RNA-protein recognition, J. Mol. Biol. 293 (2), 255–270 (1999). https://doi.org/10.1006/jmbi.1999.2991 1

    work page doi:10.1006/jmbi.1999.2991 1999

  29. [29]

    Tinoco, O

    I. Tinoco, O. C. Uhlenbeck, and M. D. Levine, Estimation of Secondary Structure in Ribonu- cleic Acids, Nature 230, 362–367 (1971). https://doi.org/10.1038/230362a0 1

    work page doi:10.1038/230362a0 1971

  30. [30]

    D. M. Crothers, P. E. Cole, C. W. Hilbers, R. G. Shulman, The molecular mechanism of thermal unfolding of Escherichia coli formylmethionine transfer RNA, J. Mol. Biol. 84 (1), 63–88 (1974). https://doi.org/10.1016/0022-2836(74)90560-9 1 33

    work page doi:10.1016/0022-2836(74)90560-9 1974

  31. [31]

    S. M. Freier, R. Kierzek, J. A. Jaeger, N. Sugimoto, M. H. Caruthers, T. Neilson, and D. H. Turner, Improved free-energy parameters for predictions of RNA duplex stability, Proc. Natl. Acad. Sci. USA 83 (24), 9373–9377 (1986). https://doi.org/10.1073/pnas.83.24.9373 1

    work page doi:10.1073/pnas.83.24.9373 1986

  32. [32]

    D. H. Mathews, J. Sabina, M. Zuker, and D. H. Turner, Expanded sequence dependence of thermodynamic parameters improves prediction of RNA secondary structure, J. Mol. Biol. 288 (5), 911–940 (1999). https://doi.org/10.1006/jmbi.1999.2700 1

    work page doi:10.1006/jmbi.1999.2700 1999

  33. [33]

    Zuker, On finding all suboptimal foldings of an RNA molecule, Science 244, 48–52 (1989)

    M. Zuker, On finding all suboptimal foldings of an RNA molecule, Science 244, 48–52 (1989). https://doi.org/10.1126/science.2468181 1

    work page doi:10.1126/science.2468181 1989

  34. [34]

    I. L. Hofacker, W. Fontana, P. F. Stadler, L. S. Bonhoeffer, M. Tacker, and P. Schuster, Fast folding and comparison of RNA secondary structures, Monatsh. Chem. 125, 167–188 (1994). https://doi.org/10.1007/BF00818163 1, 3, 9

    work page doi:10.1007/bf00818163 1994

  35. [35]

    Flamm, W

    C. Flamm, W. Fontana, I. L. Hofacker, and P. Schuster, RNA folding at elementary step resolution, RNA 6(3), 325–338 (2000). https://doi.org/10.1017/S1355838200992136 3, 9

    work page doi:10.1017/s1355838200992136 2000

  36. [36]

    B. L. Golden, A. R. Gooding, E. R. Podell, and T. R. Cech, A preorganized active site in the crystal structure of the Tetrahymena ribozyme, Science 282 (5387), 259–264 (1998). https://www.science.org/doi/10.1126/science.282.5387.259 1

    work page doi:10.1126/science.282.5387.259 1998

  37. [37]

    Kiss, Small nucleolar RNAs: An abundant group of noncoding RNAs with diverse cellular functions, Cell 109 (2), 145–148 (2002)

    T. Kiss, Small nucleolar RNAs: An abundant group of noncoding RNAs with diverse cellular functions, Cell 109 (2), 145–148 (2002). https://doi.org/10.1016/S0092-8674(02)00718-3 1

    work page doi:10.1016/s0092-8674(02)00718-3 2002

  38. [38]

    Hainzl, S

    T. Hainzl, S. Huang, and A. E. Sauer-Eriksson, Structure of the SRP19-RNA complex and implications for signal recognition particle assembly, Nature 417 (6890), 767–771 (2002). https://doi.org/10.1038/nature00768 1

    work page doi:10.1038/nature00768 2002

  39. [39]

    P. G. Higgs, Overlaps between RNA secondary structures, Phys. Rev. Lett. 76 (4), 704–707 (1996). https://doi.org/10.1103/PhysRevLett.76.704 1

    work page doi:10.1103/physrevlett.76.704 1996

  40. [40]

    S. R. Morgan, and P. G. Higgs, Barrier heights between ground states in a model of RNA secondary structure, Journal of Physics A: Mathematical and General, 31 (14), 3153-3170 (1998). https://doi.org/10.1088/0305-4470/31/14/005 1, 2, 2

    work page doi:10.1088/0305-4470/31/14/005 1998

  41. [41]

    A. P. Zubarev, Ultrametricity of Energy Minimum Configurations of RNA Sec- ondary Structures in the Nussinov Model, arXiv preprint arXiv:2605.26570 (2026). https://doi.org/10.48550/arXiv.2605.26570 1

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2605.26570 2026

  42. [42]

    Nussinov, G

    R. Nussinov, G. Pieczenik, J. R. Griggs, and D. J. Kleitman, Algorithms for loop matchings, SIAM J. Appl. Math. 35 (1), 68–82 (1978). http://www.jstor.org/stable/2101031 1 34

    arXiv 1978

  43. [43]

    Levandowsky, and D

    M. Levandowsky, and D. Winter, Distance between sets, Nature 234 (5323), 34–35 (1971). https://doi.org/10.1038/234034a0 1

    work page doi:10.1038/234034a0 1971

  44. [44]

    A. P. Zubarev, Degree of nontrivial ultrametricity for RNA macrostates, [Computer software], Zenodo. https://doi.org/10.5281/zenodo.20818030 1, 9

    work page doi:10.5281/zenodo.20818030

  45. [45]

    53 (D1), D20–D29 (2024)

    NCBI Resource Coordinators, Database resources of the National Center for Biotechnology In- formation, Nucleic Acids Res. 53 (D1), D20–D29 (2024). https://doi.org/10.1093/nar/gkae979 1, 8, 9

    work page doi:10.1093/nar/gkae979 2024

  46. [46]

    D. H. Turner, and D. H. Mathews, NNDB: the nearest neighbor parameter database for predicting stability of nucleic acid secondary structure, Nucleic Acids Research, 38 (Database issue), D280–D282 (2010). https://doi.org/10.1093/nar/gkp892 2

    work page doi:10.1093/nar/gkp892 2010

  47. [47]

    Lorenz, S

    R. Lorenz, S. H. Bernhart, C. H¨ oner zu Siederdissen, H. Tafer, C. Flamm, P. F. Stadler, and I. L. Hofacker, ViennaRNA Package 2.0, Algorithms for Molecular Biology, 6 (1), 26 (2011). https://doi.org/10.1186/1748-7188-6-26 2, 7

    work page doi:10.1186/1748-7188-6-26 2011

  48. [48]

    H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7 (4), 284–304 (1940). https://doi.org/10.1016/S0031-8914(40)90098-2 3

    work page doi:10.1016/s0031-8914(40)90098-2 1940

  49. [49]

    Onsager, Reciprocal relations in irreversible processes

    L. Onsager, Reciprocal relations in irreversible processes. I, Physical Review, 37 (4), 405–426 (1931). https://doi.org/10.1103/PhysRev.37.405 3

    work page doi:10.1103/physrev.37.405 1931

  50. [50]

    D. M. Zuckerman, Statistical Physics of Biomolecules: An Introduction, Boca Raton: CRC Press, 2010. 3, 9

    2010

  51. [51]

    P. C. Mahalanobis, On the generalized distance in statistics, Sankhya: The Indian Journal of Statistics, Series A (2008-), 80, S1–S7 (2018). https://www.jstor.org/stable/48723335 4

    arXiv 2008

  52. [52]

    G. H. Golub, and C. F. Van Loan, Matrix Computations, 4th ed., Baltimore: Johns Hopkins University Press, 2013. 5, 7

    2013

  53. [53]

    I. T. Jolliffe, Principal Component Analysis, 2nd ed., New York: Springer, 2002. 5

    2002

  54. [54]

    A. P. Zubarev, On stochastic generation of ultrametrics in high-dimensional Euclidean spaces,p-Adic Numbers, Ultrametric Analysis, and Applications 6 (2) (2014) 155-165. https://doi.org/10.1134/S2070046614020046 6

    work page doi:10.1134/s2070046614020046 2014

  55. [55]

    A. P. Zubarev, On the ultrametric generated by random distribution of points in Euclidean spaces of large dimensions with correlated coordinates, Journal of Classification 34 (3) (2017) 366-383. https://doi.org/10.1007/s00357-017-9236-8 6 35

    work page doi:10.1007/s00357-017-9236-8 2017

  56. [56]

    A. Kh. Bikulov, and A. P. Zubarev, A toy model of a protein prototype reveals non- trivial ultrametricity of the energy landscape, arXiv preprint arXiv:2603.13012 (2026). https://doi.org/10.48550/arXiv.2603.13012 6 36

    work page doi:10.48550/arxiv.2603.13012 2026