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Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations

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Pith's one-line read This paper proposes a deformation-first geometric framework in which ordered protein deformation histories are represented by a noncommutative quaternionic transport algebra, so the order of local perturbations is recorded even when endpoin

desk verdict A clear and honest formal scaffold for order-sensitive protein deformation; the transport-algebra core is sound, but the spectral-response layer is conditional on an unproven hypothesis and the helix example is only schematic. read the letter →

arxiv 2607.29101 v1 pith:NGSADHVP submitted 2026-07-31 q-bio.BM

Xiaoting Chen , Chon-Fai Kam , Yu Li , David Medina-Ortiz , Cedric Damour , Jean Pierre Chabriat , Alain Miranville , Miloud Bessafi
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Frederic Cadet
This is my paper · ORCID
classification q-bio.BM MSC 46L8758J5281Q1053C8092C40
keywords proteindeformationquaternionicframetransportorderedalgebranoncommutativegeometrymemoryspectralresponseDiracoperatorpath-dependent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that two protein deformation histories can be genuinely different even when their final conformations look the same, and that this difference can be captured geometrically. Its proposal is to lift local backbone frames to quaternions, encode infinitesimal rotation as Ω(ℓ) = 2q(ℓ)⁻¹∂_ℓq(ℓ), and organize ordered deformation paths into a noncommutative transport algebra in which multiplication records that deformation A followed by B need not equal B followed by A. A 'commutative shadow' map projects out exactly this order-sensitive memory, leaving endpoint-level observables. On an idealized α-helix, the paper shows that two pitch-and-bend histories can have near-identical RMSD and end-to-end distance while producing a nonzero order-memory signal. If correct, this gives a formal foundation for history-sensitive descriptors of allostery, mutation-order effects, conformational switching, and path-dependent response.

What carries the argument

The carrying object is the ordered transport algebra A_tr(Σ), generated by admissible quaternionic transport paths on the physical locus Σ inside a complex deformation space P_C, together with its commutative shadow E: A_tr(Σ) → C_c^∞(Σ). The algebra's multiplication records ordered concatenation of deformations; its kernel M_ord = ker E is the order-memory sector that endpoint-level descriptors lose. The infinitesimal input is the quaternionic frame transport Ω(ℓ) = 2q(ℓ)⁻¹∂_ℓq(ℓ), which lifts local backbone rotations to SU(2). The spectral layer is realized through a global first-order Dirac-type operator on a twisted spinor bundle, whose tangent-groupoid localization yields intrinsic loca

What would settle it

Take a real protein trajectory or all-atom simulation, identify two localized perturbations A and B, and compute the ordered transport products U_AB and U_BA together with endpoint descriptors; if swapping the order of two local backbone rotations always yields the same quaternionic transport product whenever the endpoint conformations coincide, the central distinction collapses. Concretely, a single realistic pair of perturbations with a trivial group commutator C_AB = U_A U_B U_A⁻¹ U_B⁻¹ would falsify the claim that order-dependent transport memory is generically present.

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Extended reading notes

Core claim

The central claim is that endpoint proximity and transport-history equivalence are distinct notions: two deformation histories may have almost identical final conformations yet retain different ordered quaternionic transport. The construction lifts each backbone frame to a unit quaternion q(ℓ), uses Ω(ℓ) = 2q(ℓ)⁻¹∂_ℓq(ℓ) as the infinitesimal rotational transport, and concatenates admissible deformation paths into an ordered transport algebra A_tr(Σ) with a unitary cocycle twist. The algebra is noncommutative in a way that records order; its kernel under the commutative shadow E is the order-memory sector M_ord. From this transport layer the paper builds a global Dirac-type operator D and, un

Load-bearing premise

The whole spectral-response layer rests on Hypothesis 1 (Section 7.3): the local spectral germ defined on the physical deformations must extend holomorphically to a neighborhood of that locus in the complexified deformation space, with a uniform short-time heat expansion—if that extension fails or is not uniform, the renormalized density and mixed response form are undefined.

Editorial extensions

If this is right

  • Endpoint similarity and deformation-history equivalence are distinct: two histories can have nearly identical RMSD and end-to-end distance while differing in ordered quaternionic transport.
  • The order-memory sector is invisible to endpoint-level observables and is suppressed by the commutative shadow, whereas the spectral-response sector survives the collapse.
  • The framework provides a formal foundation for history-sensitive descriptors of allosteric switching, mutation-order effects, conformational switching, and epistatic rearrangements.
  • Trajectories, NMR ensembles, and generative model outputs can in principle be converted into quaternionic transport fields from which order-memory and spectral-response signatures can be extracted.
  • The minimal helical realization shows that a nonzero order-memory signal can coexist with near-zero endpoint discrepancy, so path-dependent effects are not captured by endpoint geometry alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the framework is correct, a practical descriptor could be built by discretizing backbone frames into quaternionic increments and comparing ordered products; the cumulative log-ratio Δ_ord in the paper's Eq. (4) is a natural candidate for a computable history discrepancy.
  • The noncommutative transport structure suggests that generative models trained to match endpoint distributions may systematically discard functional information; conditioning on transport-history signatures could improve mutation-effect and allosteric prediction.
  • The spectral-response layer, being defined on a complexified deformation space, hints that response potentials may connect to holonomy or geometric-phase effects along closed conformational cycles; testing whether closed deformation loops produce nonzero holonomy would link the formalism to existing path-dependence ideas.
  • Because the paper does not construct an empirical realization map from real protein data to Σ, a testable extension is to build that map from molecular-dynamics trajectories and measure whether Δ_ord correlates with known order-dependent biological outcomes, such as mutational epistasis.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a 'deformation-first' framework for protein structures in which ordered deformation histories, not just endpoint conformations, are represented. Local backbone frames are lifted to quaternionic variables, with infinitesimal rotation encoded by Ω(ℓ)=2q(ℓ)^{-1}∂ℓq(ℓ). Ordered concatenation of admissible paths generates a noncommutative transport algebra A_tr(Σ) whose noncommutativity records the order of local perturbations. A commutative shadow E: A_tr(Σ)→C_c^∞(Σ) is introduced, with kernel M_ord = ker E called the 'order-memory sector.' From this transport structure, the paper constructs a Dirac-type operator D on a spinor bundle twisted by a Hermitian line bundle, and then a spectral-response layer consisting of intrinsic local spectral germs L_ξ(D), a renormalized local spectral density ρ_ren_χ, and a mixed response form G_χ = i ∂∂̄ ρ_ren_χ. The central claim is that endpoint proximity and transport-history equivalence are distinct: two deformation histories can have similar endpoints while retaining different ordered transport memory, and the spectral-response sector remains visible after the commutative shadow collapses the order sector. A minimal realization on an idealized α-helix compares a pitch perturbation A_α and a bending perturbation B_β applied in opposite orders, computes a nonzero order-memory signal Δ_ord, and sketches a pulled-back spectral-response density. The paper is explicitly positioned as a theoretical foundation and pre-algorithmic

Significance. If the construction were fully realized, the framework would provide a novel formal language for history-dependent protein descriptors, potentially relevant to allostery, mutation-order effects, conformational memory, and generative-model outputs. The formal transport layer is carefully built: the canonical geometric cocycle from an integral curvature form (Proposition 1), the smooth transport representation (Proposition 2), and the essential self-adjointness of the Dirac operator (Theorem 3) are standard but correctly stated, and the appendix sketches proofs. The helical example, while minimal, concretely illustrates that two noncommuting local perturbations can produce a nonzero commutator signal. However, the paper's strongest advertised contribution—the spectral-response geometry—depends entirely on an externally imposed holomorphic-extension hypothesis (Hypothesis 1) whose validity is not established for any protein-derived locus. Consequently, the significance of the full package is conditional on a future construction or proof for a concrete Σ. The separation into order-memory and response-memory sectors is partly definitional, and the empirical content of the framework rema

major comments (3)
  1. [Section 7.3, Hypothesis 1] The entire spectral-response layer—the renormalized density ρ_ren_χ (Definition 16), the mixed response form G_χ (Eq. (3)), and the local response potentials (Definition 17)—is defined only under Hypothesis 1. This hypothesis asserts that the intrinsic local spectral germ extends holomorphically to a neighborhood U⊂P_C of Σ and admits a uniform short-time heat-kernel expansion. No construction, proof, or concrete criterion is given for any protein-derived Σ. Appendix A.1's realization map E_data is only a sketch, and Section 8.3's helical realization merely pulls back ρ_ren_χ abstractly to ζ_γ(α,β); Figure 4E is explicitly labeled a 'schematic diagnostic,' and no Dirac operator, heat kernel, or asymptotic expansion is computed for the helix. If the holomorphic extension or the uniform heat expansion fails—for example, for a generic embedded submanifold Σ⊂P_C—then ρ_ren_χ and G_χ are unde
  2. [Section 8.2–8.3, helical realization] The helical realization is presented as the minimal demonstration of the paper's main point, but its quantitative content is limited. The perturbations A_α and B_β depend on unspecified bump functions f and g and amplitudes α and β; the curves in Figures 3–5 are described as 'schematic'; and the only concrete computed quantity is Δ_ord = δ_ord(N−1), the norm of the group commutator in Eq. (5). This quantity is positive by construction whenever the two transport increments do not commute. The example therefore shows that the framework can detect noncommutation, but it does not provide a parameter-free or data-driven test of the claim that endpoint proximity can coexist with distinct ordered transport histories, because no specific α, β, f, g, or endpoint geometry (beyond the stated RMSD 0.250 and end-distance difference 0.004, which are given without derivation) is specified. To make the
  3. [Section 9, Proposition 7] The collapse of the order-memory sector under the commutative shadow is definitional: M_ord is defined as ker E in Definition 5, so Proposition 7's assertion E|M_ord = 0 is immediate. Similarly, the nonzero order-memory signal in the helical example is the norm of a commutator, which is nonzero by construction for noncommuting unitary factors. The substantive claim is not that the order sector collapses—that is built into the definition—but that the response-memory sector survives and remains observable. That survival depends on the spectral layer and, through it, on Hypothesis 1 (Section 7.3), which is unsupported. The paper should separate more sharply what is introduced by definition, what requires proof, and what remains a hypothesis. As written, the distinction between 'order memory' and 'response memory' risks appearing tautological rather than as an empirical or mathematical disco
minor comments (5)
  1. [Table 1 caption] The table caption contains a typo: 'T able 1 Correspondence...' should read 'Table 1: Correspondence...'.
  2. [Definitions 6 and 10] The canonical metric-Clifford datum fixes a spin^c structure on Σ and a spinor bundle S, but the existence or choice of this structure is not discussed. Since not every manifold admits a spin or spin^c structure without topological conditions, the authors should state the necessary assumption explicitly or construct the spin^c structure from the data.
  3. [Section 8.2, composition convention] The notation γ_AB = B_β ∘ A_α is used to mean that A_α is applied first and B_β second, which is the opposite of the usual left-to-right reading of 'AB.' This convention is stated in the text, but in Eq. (4) the subscripts U_AB and U_BA are easy to misread. A short renaming (e.g., γ_{A→B} and γ_{B→A}) would improve clarity.
  4. [Appendix C.1, proof of Proposition 1] The proof invokes a 'geodesic triangulation of a filling surface' for an arbitrary admissible pair, but the existence of such a filling with triangles subordinate to the fixed good cover is not established. The integrality argument is sound once this geometric input is granted; the construction should be stated as part of the definition of admissible paths or proved explicitly.
  5. [Section 2.5 vs. Section 4] Section 2.5 formalizes histories as symbolic lists γ=(d_1,...,d_n), while Section 4 defines histories as paths γ:[0,1]→Σ. The relationship between discrete events and continuous paths is not made precise. The authors should clarify whether the path formulation is intended to be a refinement of the event-list formulation or a separate representation.

Circularity Check

2 steps flagged · score 6.0 of 10

Order-memory separation is definitional: M_ord=kerE makes Prop. 7 collapse tautological, and Δ_ord is a commutator norm; spectral layer rests on unproved Hypothesis 1.

  1. self definitional [Definition 5 (Section 4.2) and Proposition 7 (Section 9)]
    "Definition 5: 'A commutative shadow of Atr(Σ) is a linear map E : Atr(Σ) → C∞c(Σ) ... Its kernel Mord := kerE is called the order-memory sector.' Proposition 7: 'Then E|Mord = 0. Its image is the commutative coefficient algebra C∞c(Σ), so the cocycle-twisted ordered transport structure has no nontrivial commutative shadow.'"

    The order-memory sector is introduced as the kernel of E, and the response-memory sector is defined (Definition 18) as spectral data on which E is not even defined. Proposition 7's statement that E vanishes on Mord and that the ordered sector 'disappears' under the shadow is therefore exactly the definition of Mord, not a derived collapse. The claimed separation of order-memory from response-memory is built into the definitions rather than obtained from protein-geometric input or any external principle.

  2. self definitional [Section 8.2, Eqs. (4)–(5) and following text]
    "Eqs. (4)–(5): 'The order-sensitive comparison is then δord(m) = || log(UAB(m)−1UBA(m))||, ∆ord = δord(N−1). ... Equivalently, one may use the group commutator CAB = UAUBU−1A U−1B, (5) which is trivial only when the two elementary transports commute in the realized sector. Thus, ∆ord > 0 is a direct transport-memory signature.'"

    The order-memory signal is defined as the logarithm of the ratio of the two ordered products, or equivalently as the norm of their group commutator. Hence any two histories whose transport operators fail to commute produce a positive ∆ord by construction, independently of any biological or protein-specific content. The helical demonstration — nearby endpoints yet nonzero memory — follows from choosing a pitch rotation and a bend rotation about different axes in SU(2); it is an instance of noncommutativity imported from the rotation group, not a prediction derived from protein deformation data.

full rationale

The headline separation of memory sectors reduces to definitions: Mord := kerE makes Proposition 7 a tautology, and the order-memory signal is defined as a commutator norm, so the nonzero result in the helix is guaranteed by choosing noncommuting rotations. No load-bearing self-citation occurs: the only cited work with an overlapping author (Lu et al. 2025) supports a general remark about generative models and is not used to justify the construction. The spectral-response layer is not circular but is conditional: Section 7.3's Hypothesis 1 (holomorphic extension of the local germ and uniform short-time heat-kernel expansion) is asserted without proof or construction for any protein-derived locus; Appendix A.1's realization map Edata is only a sketch, and Fig. 4E is labeled a schematic diagnostic. The paper itself concedes in Section 10 that the helical realization 'is intentionally minimal' and does not provide quantitative validation. These limitations lower the stakes but do not remove the definitional circularity of the sector separation, so the score is 6 rather than higher.

Assumptions & free parameters 2 free parameters · 6 assumptions · 4 invented entities

The central construction rests on a set of strong geometric existence assumptions (totally real embedded Sigma, spin^c structure, holomorphic extension of the spectral germ) and a deferred map from real protein data to the abstract locus. No empirical input is used; the only numerical values are schematic. The invented mathematical objects carry no independent falsifiable evidence, so all load-bearing weight is on internal consistency and future realization.

free parameters (2)
  • Normalization data (base point xi_0, small-time cutoff chi) = not fitted; chosen by hand
    Definition 16 fixes xi_0 in Sigma and chi in C_c^inf([0,infinity)) with chi=1 near 0; rho_ren_chi and G_chi depend on this choice, and no rule for selecting them from protein data is given.
  • Helix perturbation parameters (amplitudes alpha, beta; bump functions f, g) = not specified numerically
    Section 8.2 defines A_alpha and B_beta through alpha f(z) and beta g(z); the reported endpoint similarity (RMSD 0.250, end-distance diff 0.004) and nonzero Delta_ord depend on these unspecified choices; no robustness scan is shown.
assumptions (6)
  • domain assumption Sigma is a connected, oriented, complete real-analytic Riemannian manifold, embedded as a totally real submanifold of a complex manifold P_C, with closed integral curvature Theta (1/2pi [Theta] in H^2(Sigma; Z)).
    Definition 6 imposes this structure; no construction of such an embedding from actual protein backbone data is provided.
  • domain assumption A spin^c structure on Sigma and a Hermitian line bundle L with unitary connection nabla^L exist with the stated curvature, and V = S tensor L tensor |Lambda|^{1/2} is the relevant spinor module.
    Definitions 9-10 and Theorem 3 use this to build the Dirac operator; the existence of a suitable spin^c structure on protein-derived loci is assumed, not verified.
  • domain assumption Deformation histories of proteins are representable as equivalence classes of piecewise C^1 paths in the admissible class P_ad(Sigma), stable under reversal and concatenation, and real trajectories map into Sigma.
    Definitions 3-4 and Section 2.5 make deformations primary; Appendix A.1 defers the empirical realization map E_data: X_data -> Sigma, so the connection to real trajectories is not established.
  • ad hoc to paper Hypothesis 1: the intrinsic local spectral germ extends holomorphically to a neighborhood U of Sigma in P_C, with local spectral operators and a uniform short-time heat-kernel expansion.
    Section 7.3 introduces this as an imposed hypothesis; the entire renormalized spectral density and mixed response form rest on it.
  • standard math The complete Riemannian manifold theorem for essential self-adjointness of Dirac-type operators is valid for the constructed V and connection.
    Theorem 3 proof in Appendix C.3 invokes the standard completeness theorem; this is a standard background result.
  • domain assumption For the helix example, ideal-helix parameters (theta ~ 100 deg, h ~ 1.5 A, R ~ 2.3 A) and the Frenet frame convention give Omega_0 = tau i + kappa k, and localized pitch/bend deformations are modeled by smooth bump functions.
    Section 8.1-8.2; the example uses idealized geometry and unspecified bump deformations, not atomistic MD.
invented entities (4)
  • Physical quaternionic locus Sigma subset P_C (complex deformation space)
    purpose: Ambient space for admissible deformation paths and spectral response
    Postulated as a smooth complete totally real submanifold; no explicit realization from protein data is given (Definitions 2 and 6, Appendix A.1).
  • Ordered transport algebra A_tr(Sigma) with generators U_gamma and cocycle sigma
    purpose: Algebraic record of ordered deformation histories
    Abstractly defined in Definition 4; the only concrete realization is the helix commutator, which is just SU(2) rotation composition.
  • Commutative shadow E and order-memory sector M_ord = ker E
    purpose: Separate order information from endpoint-level observables
    M_ord is defined as the kernel of E, so Proposition 7's collapse is by construction rather than an empirical finding.
  • Intrinsic local spectral germ L_xi(D), renormalized density rho_ren_chi, mixed response form G_chi
    purpose: Local response observables derived from the Dirac operator
    Defined under Hypothesis 1 and normalization data (chi, xi_0); no explicit computation from protein-like data is given, and Panel E is schematic.

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Pith. "Pith review of Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations." pith.science (2026). https://pith.science/paper/NGSADHVP

@misc{pith2026260729101,
  author       = {Pith},
  title        = {Pith review of: Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGSADHVP}},
  note         = {Machine review of arXiv:2607.29101}
}
abstract

Protein function may depend on both endpoint conformations and the ordered deformation histories by which they are reached. This distinction is relevant to allostery, conformational switching, mutation-induced rearrangements, and epistatic effects, where different perturbation sequences may produce similar structures while retaining distinct internal transport histories. Current state- or endpoint-centered representations may not preserve this order-sensitive information. We therefore provide a foundation for descriptors of protein deformation trajectories that distinguish ordered histories even when endpoint conformations are similar. Such descriptors could support analyses of allosteric switching, mutation-order effects, conformational memory, and path-dependent response in molecular-dynamics trajectories, NMR ensembles, structural families, and outputs of geometric generative models. We propose a deformation-first geometric framework based on quaternionic frame transport along the protein backbone. Local backbone frames are lifted to quaternionic variables, with infinitesimal rotation encoded by \(\Omega(\ell)=2\,q(\ell)^{-1}\partial_\ell q(\ell)\). Ordered concatenation of admissible deformation paths generates a noncommutative transport algebra, recording that deformation A followed by B need not be equivalent to B followed by A. From this ordered transport layer, we construct a spectral-response layer comprising a global Dirac-type operator, local spectral germs, a renormalized spectral density, and a mixed response form. A minimal realization on an idealized \(\alpha\)-helix shows how localized pitch and bending perturbations can yield similar endpoint descriptors while preserving a nonzero order-memory signal. The framework separates an order-sensitive transport-memory sector, lost under a commutative shadow, from a spectral-response sector that remains visible.

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Works this paper leans on

40 extracted references · 13 canonical work pages

  1. [1]

    Chemical reviews , volume=

    Symmetry, rigidity, and allosteric signaling: from monomeric proteins to molecular machines , author=. Chemical reviews , volume=. 2019 , publisher=

    2019

  2. [2]

    Nature Reviews Chemistry , volume=

    Loop dynamics and the evolution of enzyme activity , author=. Nature Reviews Chemistry , volume=. 2023 , publisher=

    2023

  3. [3]

    Computers in Biology and Medicine , volume=

    A review on description dynamics and conformational changes of proteins using combination of principal component analysis and molecular dynamics simulation , author=. Computers in Biology and Medicine , volume=. 2024 , publisher=

    2024

  4. [4]

    Bioinformatics , pages=

    Protein structure prediction , author=. Bioinformatics , pages=. 2022 , publisher=

    2022

  5. [5]

    and Adler, J

    Abramson, J. and Adler, J. and Dunger, J. and others. Accurate structure prediction of biomolecular interactions with AlphaFold 3. Nature. 2024. doi:10.1038/s41586-024-07487-w

    doi:10.1038/s41586-024-07487-w 2024

  6. [6]

    and Wardetzky, M

    Bergou, M. and Wardetzky, M. and Robinson, S. and Audoly, B. and Grinspun, E. Discrete elastic rods. ACM SIGGRAPH 2008 Papers. 2008. doi:10.1145/1399504.1360662

    arXiv 2008

  7. [7]

    Bishop, R. L. There is more than one way to frame a curve. Am. Math. Mon. 1975. doi:10.1080/00029890.1975.11993807

    arXiv 1975

  8. [8]

    Changeux, J. P. and Edelstein, S. J. Allosteric mechanisms of signal transduction. Science. 2005. doi:10.1126/science.1108595

    work page doi:10.1126/science.1108595 2005

Show all 40 references
  1. [9]

    Noncommutative geometry

    Connes, A. Noncommutative geometry. 1994

    1994

  2. [10]

    and Moscovici, H

    Connes, A. and Moscovici, H. The local index formula in noncommutative geometry. Geom. Funct. Anal. 1995. doi:10.1007/BF01895667

    1995 doi

  3. [11]

    and Anishchenko, I

    Dauparas, J. and Anishchenko, I. and Bennett, N. R. and others. Robust deep learning-based protein sequence design using ProteinMPNN. Science. 2022. doi:10.1126/science.add2187

    2022 doi

  4. [12]

    and Schafer, N

    Davtyan, A. and Schafer, N. P. and Zheng, W. and others. AWSEM-MD : protein structure prediction using coarse-grained physical potentials and bioinformatically based local structure biasing. J. Phys. Chem. B. 2012. doi:10.1021/jp212541y

    2012 doi

  5. [13]

    Fuchs, F. B. and Worrall, D. E. and Fischer, V. and others. SE(3) -Transformers: 3D roto-translation equivariant attention networks. Advances in Neural Information Processing Systems. 2020

    2020

  6. [14]

    Gracia-Bond \'i a, J. M. and V \'a rilly, J. C. and Figueroa, H. Elements of noncommutative geometry. 2001

    2001

  7. [15]

    Hanson, A. J. and Thakur, S. Quaternion maps of global protein structure. J. Mol. Graph. Model. 2012. doi:10.1016/j.jmgm.2012.06.004

    2012 doi

  8. [16]

    Hanson, A. J. Quaternion protein frame maps. Visualizing More Quaternions. 2024. doi:10.1016/B978-0-32-399202-2.00027-7

    2024 doi

  9. [17]

    and Kern, D

    Henzler-Wildman, K. and Kern, D. Dynamic personalities of proteins. Nature. 2007. doi:10.1038/nature06522

    2007 doi

  10. [18]

    and Vuckovic, J

    Huguet, G. and Vuckovic, J. and Fatras, K. and others. Sequence-augmented SE(3) -flow matching for conditional protein generation. Advances in Neural Information Processing Systems. 2024. doi:10.52202/079017-1039

    2024 doi

  11. [19]

    and Eismann, S

    Jing, B. and Eismann, S. and Suriana, P. and others. Learning from protein structure with geometric vector perceptrons. Proceedings of the International Conference on Learning Representations. 2021

    2021

  12. [20]

    and Evans, R

    Jumper, J. and Evans, R. and Pritzel, A. and others. Highly accurate protein structure prediction with AlphaFold. Nature. 2021. doi:10.1038/s41586-021-03819-2

    2021 doi

  13. [21]

    and Sander, C

    Kabsch, W. and Sander, C. Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded and geometrical features. Biopolymers. 1983. doi:10.1002/bip.360221211

    1983 doi

  14. [22]

    An introduction to noncommutative spaces and their geometry

    Landi, G. An introduction to noncommutative spaces and their geometry. 1997. doi:10.1007/3-540-14949-X

    1997 doi

  15. [23]

    and AlQuraishi, M

    Lin, Y. and AlQuraishi, M. Generating novel, designable, and diverse protein structures by equivariantly diffusing oriented residue clouds. Proceedings of the 40th International Conference on Machine Learning. 2023

    2023

  16. [24]

    and Chen, X

    Lu, J. and Chen, X. and Lu, S. Z. and others. Structure language models for protein conformation generation. Proceedings of the International Conference on Learning Representations. 2025

    2025

  17. [25]

    Motlagh, H. N. and Wrabl, J. O. and Li, J. and others. The ensemble nature of allostery. Nature. 2014. doi:10.1038/nature13001

    2014 doi

  18. [26]

    and Corey, R

    Pauling, L. and Corey, R. B. and Branson, H. R. The structure of proteins: two hydrogen-bonded helical configurations of the polypeptide chain. Proc. Natl. Acad. Sci. USA. 1951. doi:10.1073/pnas.37.4.205

    1951 doi

  19. [27]

    Satorras, V. G. and Hoogeboom, E. and Welling, M. E(n) equivariant graph neural networks. Proceedings of the 38th International Conference on Machine Learning. 2021

    2021

  20. [28]

    Animating rotation with quaternion curves

    Shoemake, K. Animating rotation with quaternion curves. ACM SIGGRAPH Comput. Graph. 1985. doi:10.1145/325165.325242

    1985

  21. [29]

    Starr, T. N. and Thornton, J. W. Epistasis in protein evolution. Protein Sci. 2016. doi:10.1002/pro.2897

    2016 doi

  22. [30]

    van Suijlekom, W. D. Noncommutative geometry and particle physics. 2015. doi:10.1007/978-94-017-9162-5

    2015 doi

  23. [31]

    and Wang, L

    Wang, Y. and Wang, L. and Shen, Y. and others. Protein conformation generation via force-guided SE(3) diffusion models. Proceedings of the 41st International Conference on Machine Learning. 2024

    2024

  24. [32]

    Watson, J. L. and Juergens, D. and Bennett, N. R. and others. De novo design of protein structure and function with RFdiffusion. Nature. 2023. doi:10.1038/s41586-023-06415-8

    2023 doi

  25. [33]

    and Trippe, B

    Yim, J. and Trippe, B. L. and De Bortoli, V. and others. SE(3) diffusion model with application to protein backbone generation. Proceedings of the 40th International Conference on Machine Learning. 2023

    2023

  26. [34]

    and St \"a rk, H

    Jing, B. and St \"a rk, H. and Jaakkola, T. and Berger, B. Generative modeling of molecular dynamics trajectories. Advances in Neural Information Processing Systems. 2024. doi:10.52202/079017-1282

    2024 doi

  27. [35]

    and Xu, M

    Han, J. and Xu, M. and Lou, A. and Ye, H. and Ermon, S. Geometric trajectory diffusion models. Advances in Neural Information Processing Systems. 2024. doi:10.52202/079017-0806

    2024 doi

  28. [36]

    and Winther, O

    Schreiner, M. and Winther, O. and Olsson, S. Implicit transfer operator learning: multiple time-resolution models for molecular dynamics. Advances in Neural Information Processing Systems. 2023. doi:10.52202/075280-1582

    2023 doi

  29. [37]

    and Foong, A

    Klein, L. and Foong, A. Y. K. and Fjelde, T. E. and Mlodozeniec, B. and Brockschmidt, M. and Nowozin, S. and No \'e , F. and Tomioka, R. Timewarp : transferable acceleration of molecular dynamics by learning time-coarsened dynamics. Advances in Neural Information Processing Sy...

    2023 doi

  30. [38]

    Non-commutative differential geometry

    Connes, A. Non-commutative differential geometry. Publications Math \'e matiques de l'IH \'E S. 1985. doi:10.1007/BF02698807

    1985 doi

  31. [39]

    and Gupta, J

    Ruhe, D. and Gupta, J. K. and De Keninck, S. and Welling, M. and Brandstetter, J. Geometric Clifford algebra networks. Proceedings of the 40th International Conference on Machine Learning. 2023

    2023

  32. [40]

    Hanson, A. J. and Ma, H. Visualizing flow with quaternion frames. Proceedings of Visualization '94. 1994

    1994

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.