{"paper_id":"29edade2-3514-4992-b1fb-d834ebf79dc3","body_text":"NMR as a Video-(game): Constructing Super-Resolution Cross-peak \nTrajectories in Protein Spectroscopy  \nChng Jing Hang1, Yevheniia Kuznietsova 1, Mikhail Fillipov2 and Konstantin \nPervushin1  \n \n1 School of Biological Sciences, Nanyang Technological University, 60 Nanyang \nDrive, Singapore 637551, Singapore \n \n2 School of Computing, National University of Singapore, Computing 1, 13 \nComputing Drive, Singapore 117417, Singapore \n \n \nAbstract \nHigh-resolution multidimensional NMR spectroscopy of proteins remains limited \nby long acquisition times, sensitivity constraints, and severe peak overlap, \nparticularly for larger systems. Conventional 3D and higher-dimensional \nexperiments trade experimental efficiency for resolution, while post-acquisition \nanalysis often becomes the dominant bottleneck. Here, we present a new \nframework that redefines both how NMR experiments are constructed and how \nthey are executed and analyzed, by treating an AI agent-controllable series of 2D \nspectra as a spatiotemporal dataset analogous to a video. Our approach is \nbased on temperature-dependent series of reduced-dimensionality 2D HSQC \nand novel RDL-TROSY experiments, in which each 2D [\n1H,15N] cross-peak is \ncontrollably shifted and split in proportion to the 13C chemical shift of the J-\ncoupled carbons. We propose treating a variable-temperature (VT) series as a \npseudo-temporal video sequence in which each cross-peak traces a physically \nmotivated trajectory through frequency space. The proportionality coefficient (\nα ) \nof this reduced-dimensionality encoding is systematically and programmatically \nvaried together with the temperature providing full control for constructing optimal \ncross-peak trajectories. As a result, individual resonances follow predictable, \nspectral acquisition time-controllable trajectories in the 2D spectral plane across \nthe series, which can be executed by an autonomous AI agent directly interacting \nwith the NMR GUI layer. Each spectrum represents a single “frame,” while \ntemperature and RD controls serves as the temporal dimension. We describe \ntwo complementary super-resolution strategies: a cross-peak model-independent  \napproach based on the deep-learning video super-resolution that leverages \ntemporal redundancy to sharpen per-frame peak shapes, and a model-based \napproach that derives the exact mathematical form of the peak trajectories and \nuses it to design acquisition schedules that render individual peak paths \nmaximally distinct and amenable for algorithmic deconvolution. As a result, we \nobtained full backbone resonance assignment in the wide temperature range \n(279–315 K) with one degree Kelvin resolution in a test protein in an automatic \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nmanner in the time frame typically required for collection of a single 3D NMR \ndataset.    \n \nKeywords: RD-HSQC; variable-temperature nuclear magnetic resonance (NMR); \nspatiotemporal super-resolution; peak trajectory; spectral-width modulation; \nresonance assignment; ubiquitin; agentic optimization. \nIntroduction \nBackbone resonance assignment is the prerequisite for all quantitative protein \nnuclear magnetic resonance (NMR) spectroscopy (1, 2), and its primary obstacle \nis spectral overlap. Conventional solutions add indirect dimensions, spreading \npeaks across additional frequency axes that form orthogonal spectral dimensions \nafter Fourier transformation, at the cost of linearly increased acquisition time, \nwhich becomes prohibitive for experiments requiring large numbers of indirect \ntime-domain sampling points (3). Variable-temperature (VT) NMR offers an \nindependent additional axis. Backbone amide \n1H, 15N, and 13C/g2961  chemical shifts \nall respond to temperature in a residue-specific manner (3–5), reflecting local \nhydrogen bond geometry and backbone dynamics. The temperature coefficient \n∂δ /g2892/g2898 /∂ T  is linear over a wide temperature range (269–369 K) (5) and takes \nvalues spanning roughly −16 to +2 ppb K/g2879/g2869  for amide protons, with buried \nresidues showing attenuated sensitivity (5). Since this linearity is well established, \nwe acquire 40 spectra across a temperature range falling within this window, at 1 \nK intervals. This generates a pseudo-temporal sequence in which each \ntemperature step corresponds to a time point, and each cross-peak traces a \ndistinct, predictable path through frequency space, encoding the differential \nchange in chemical shift drift between successive spectra. Two peaks \noverlapping at one temperature are often separated at another temperature or \nreduced-dimensionality (RD) settings; the question is how to exploit this \ninformation systematically across the full series of acquired spectra rather than \ndiscard it through frame-by-frame analysis. \nThe conceptual core of this paper is the bridging of VT-NMR to a computer vision \nframework, in which a non-spectroscopic variable (temperature) as well as signal \nacquisition time-controlled variables are exploited to resolve physics motivated \nspectroscopic observables such as chemical shifts or selected NOEs. The CV \nfield has developed extensively, with deep learning-based methods now \nachieving robust detection and tracking of objects across successive frames, \nexemplified by YOLO (7) for detection and DeepSORT (8) for multi-object \ntracking. Each spectrum is a 2D frame; each point of the spectral intensity grid \ncan be viewed as a pixel; the temperature and RD axis is a pseudo-temporal axis; \nand cross-peak drift is a motion field. This bridge has direct operational \nconsequences: resolving transiently occluded peaks becomes a problem of \npropagating information across the temporal dimension, for which optical flow, \nrecurrent feature propagation, and super-resolution are directly applicable. \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nOur approach differs from that of Shchukina et al. (9), who sample temperature \nand indirect evolution times jointly within a single acquisition, necessitating \ncorrection for signal non-stationarity arising from temperature drift during \nacquisition. In contrast, we acquire a fully stationary RD-HSQC or RDL-TROSY \nspectrum at each discrete temperature step; each frame is recorded \nindependently and is therefore free from non-stationarity artefacts. The resulting \nchallenge lies in linking and exploiting this discrete series of stationary spectra. \nTrainor, K et al. addressed a related problem by implementing peak tracking \nacross temperatures to determine temperature coefficients, showing that the \nchemical shifts of \n1H and 15N nuclei exhibit an approximately linear dependence \non temperature. In our work, we extend this concept by exploiting the \ntemperature-dependent chemical shift dynamics of three types of spins within a \nsingle spectral series, introducing a temperature-based encoding that captures \npseudo-temporal correlations between spectra. \n \nResults \nThe experiment of choice is the reduced-dimensionality heteronuclear single \nquantum coherence (RD-HSQC) spectroscopy (10, 11). In addition, we \nintroduced novel Reduced Dimensionality L-TROSY (RDL-TROSY) experiment \nmaximizing achievable spectral resolution in all spectral dimensions. In this \nexperiment, the \n13C/g2961  evolution time is linearly co-incremented with the 15N \nevolution time during the indirect dimension, so that the apparent 15N chemical \nshift is a linear combination of the true 15N and 13C/g2961  offsets (10). In a \nconventional 1H–15N HSQC, two residues with similar 1H and 15N shifts overlap \nregardless of their 13C/g2961  differences. The RD-HSQC encodes the 13C/g2961  frequency \noffset into the detected 15N dimension via a constant-time evolution element with \na mixing coefficient β7g3ω,ωS W H /g2898 /SWH /g2887 , producing two symmetrically displaced \ncross-peaks per residue—the S/g2878  and S/g2879  components—at apparent 15N positions: \nη /g4666 /g2921 /g4667 ,/g3399 7gω666n 7gω66x 7g3ω,ωΔ ν /g2898\n/g4666 /g2921 /g4667 7gω666n 7gω66x 7g3399β 7gω666 n 7gω66x 7g 3,yΔν /g2887\n/g4666 /g2921 /g4667 7gω666 n 7gω66x # 7gω6661 7gω66x  \n \nwhere β7gω666n7gω66x 7g3ω,ω SWH /g2898 7gω666n7gω66x/SWH /g2887 7gω666n7gω66x  is the frame-dependent mixing ratio (ratio of \n15N to 13C/g2961  spectral widths, both in Hz), k  indexes residues, and n  indexes \ntemperature frames. The original two heteronuclear coordinates are recovered \nexactly by sum-and-difference inversion: \nΔν /g2898\n/g4666 /g2921 /g4667 7gω666 n 7gω66x 7g3ω,ω 1\n2 7gω6x|η /g4666 /g2921 /g4667 ,/g2878 7gω666 n 7gω66x 7g339xη /g4666 /g2921 /g4667 ,/g2879 7gω666 n 7gω66x7gω6x3# 7gω666 2 7gω66x  \n \nΔν /g2887\n/g4666 /g2921 /g4667 7gω666n 7gω66x 7g3ω,ω η /g4666 /g2921 /g4667 ,/g2878 7gω666 n 7gω66x 7g339yη /g4666 /g2921 /g4667 ,/g2879 7gω666 n 7gω66x\n2β 7gω666 n 7gω66x # 7gω666 3 7gω66x  \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\n \nThe encoding is lossless when β7g3ω,a0 . By modulating SWH /g2887 7gω666n7gω66x  across the \ntemperature series and thereby varying β7gω666n7gω66x , residues with different /g2869/g2871 C/g2961  offsets \nare displaced by different amounts at each frame, converting latent 13C diversity \ninto a time-varying differential displacement that renders otherwise parallel \ntrajectories divergent and individually trackable. Note that modulating SWH /g2898 7gω666n7gω66x  \nachieves the same effect; in practice SWH /g2887 7gω666n7gω66x  is the more convenient variable \nbecause the 13C/g2961  window is narrower and more easily stepped without aliasing. \nTwo complementary resolution strategies follow from this reframing. The cross-\npeak model-independent strategy adopts video super-resolution (12, 13) to \npropagate spectral features bidirectionally across temperature steps without \nassumptions about the specific cross-peak point spread function (PSF). The \ncross-peak model-based strategy uses the exact master equation for RD-HSQC \ntrajectories to design acquisition schedules that maximize trajectory separability. \nTogether they operate at the two levels where the video reframing has traction: \nthe reconstruction of per-frame peak profiles, and the design of the frame \nsequence itself. \n \nMaster equation for RD-HSQC trajectories \nLet the chemical shift offsets from carrier for residue \nk  at temperature step n  be \nΔν /g2898\n/g4666/g2921/g4667 7gω666n7gω66x , Δν /g2887\n/g4666/g2921/g4667 7gω666n7gω66x , and Δν /g2892\n/g4666/g2921/g4667 7gω666n7gω66x  for 15N, 13C/g2961 , and 1H/g2898  respectively. The \ntemperature dependence of backbone amide shifts is approximately linear over \nmoderate ranges (4): \nΔν /g2908\n/g4666 /g2921 /g4667 7gω666 n 7gω66x 7g3ω,ωΔ ν /g2908\n/g4666 /g2921 /g4667\n7gω666 0 7gω66x 7g339x n7g 3,yΔT7g 3,yv /g2908\n/g4666 /g2921 /g4667 # 7gω666 4 7gω66x  \n \nwhere ΔT  is the temperature increment per step and v /g2908\n/g4666/g2921/g4667  is the per-residue drift \nvelocity (Hz K/g2879/g2869 ) for nucleus X . All three nuclei share a common carrier drift \ncomponent that displaces the entire peak cloud coherently: \nΔν /g2908\n/g4666 /g2921 /g4667 7gω666 n 7gω66x 7g3ω,ωΔ ν /g2908\n/g4666 /g2921 /g4667\n7gω666 n 7gω66x 7g339xF /g2868,/g2908 7gω666 n 7gω66x # 7gω666 5 7gω66x  \n \nwhere F /g2868,/g2908 7gω666n7gω66x  is a shared, frame-dependent carrier correction (arising, for \nexample, from deuterium lock drift or thermal expansion of the probe). \nThe complete observed peak positions in the 2D RD-HSQC spectrum are then \n \n                                     \n7g x9,7gω666n7gω66x 7g3ω,ω 7gω6x,7g|x||7g xyy 7gω666 0 7gω66x 7g339xn 7g 3,y Δ T 7g 3,y 7g x967g339y7g xy, 7gω666 n 7gω66x7gω6x 7g xxa7gω666n7gω66x /g2883                              (6) \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\n \nwhere 7g|x||7g xyy7gω66607gω66x 7g ωyy 7g 33x /g2897/g3400/g2871  is the matrix of initial offset coordinates for all M  residues \n(columns ordered as Δν /g2898 , Δν /g2887 , Δν /g2892 ), 7g x967g ωyy7g 33x /g2897/g3400/g2871  is the per-residue drift velocity \nmatrix, 7g xy,7gω666n7gω66x  is a rank-1 broadcast matrix encoding shared carrier drift, and: \n \n7g xxa 7gω666 n 7gω66x 7g3ω,ω7g3ω3x\n17g339x β 7gω666 n 7gω66x 0\n17g339y β 7gω666 n 7gω66x 0\n001\n7g3ωω # 7gω666 7 7gω66x  \n \nEach row-pair of 7g x9,7gω666n7gω66x  gives the observed 7gω666η /g2878 ,η /g2879 ,Δ ν /g2892 7gω66x coordinates of one \nresidue at step n . Because 7g xy,7gω666n7gω66x  is rank-1 and maps identically to all residues, it \ncancels exactly in all pairwise peak differences and has no effect on inter-residue \nseparability; its sole role is to keep the peak cloud centered within the spectral \nwindow. The drift velocity matrix 7g x96 , drawing per-residue coefficients from \ndistributions well-characterised for soluble proteins (4), is the fundamental source \nof trajectory distinctiveness. The RD projection matrix \n7g xxa7gω666n7gω66x  translates 13C/g2961  offset \ndiversity into differential separation in the mixed 15N dimension. \nComplementary S/g2878  and S/g2879  spectral planes as independent channels \nThe spin-state-selective editing of the RD-HSQC yields two spectral planes, S/g2878  \nand S/g2879 , each containing one cross-peak per residue. The construction of planes \nis achieved by acquiring x and y quadrature components of the evolving \n13C/g2961  signal, S /g2919/g2924  and S /g2911/g2924/g2930/g2919 , then computing S /g3399 7g3ω,ωS /g2919/g2924 7g3399S /g2911/g2924/g2930/g2919  (10, 11). The resulting \nS/g2878  and S/g2879  spectra function as two independent video sequences whose motion \nfields are coupled by Eq. (1). For a correct S/g2878 /S/g2879  pair, the sum η /g4666/g2921/g4667,/g2878 7gω666n7gω66x 7g339x\nη /g4666/g2921/g4667,/g2879 7gω666n7gω66x 7g3ω,ω 2Δν /g2898\n/g4666/g2921/g4667 7gω666n7gω66x  must vary smoothly with temperature, providing a stringent \ncross-validation constraint: spurious pairings produce erratic sums, correct \npairings produce smooth 15N trajectories. Because S/g2878  and S/g2879  peaks are spatially \nseparated by |Δ /g2902/g2888\n/g4666/g2921/g4667 | 7g3ω,ω 2β|Δν /g2887\n/g4666/g2921/g4667 | , peaks occluded in one channel may be resolved \nin the other. The S/g2878 /S/g2879  splitting magnitude is directly proportional to the 13C/g2961  \noffset from the 13C carrier. Residues near the carrier (Δν /g2887 7g3ω,60 ) produce nearly \ncoincident S/g2878 /S/g2879  pairs and therefore contribute little discriminating information \nfrom the RD dimension; the acquisition schedule should set the carrier to \nmaximise the spread of |Δν /g2887\n/g4666/g2921/g4667 |  values across all residues.  \nFigure 1 shows overlay of 40 spectral planes of S/g2878  and S/g2879  RD-HSQC measured \nwith 76 amino acid test protein (Ubiquitin). Placing the 13C/g2961  carrier at 52 ppm \n(near the centre of the C/g2961  chemical shift range for amino acids) achieves \nadequate coverage for most residues in the protein. The ¹³C spectral width \nSWH_C(n) and the coupled ¹/i2 N spectral width SWH_N(n) were modulated as \nlinear functions of the frame index n across the 40-step temperature ramp, \nsubject to the reciprocal-sum constraint 1/SWH_N(n) + 1/SWH_C(n) = C_const \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\n(Eq. 8), which ensures that β (n) evolves in a controlled, monotonic fashion. This \nframe-by-frame modulation of β (n) actively steers the trajectory of each S⁺ /S⁻  \npeak pair in the mixed ¹/i2 N dimension: residues with larger |Δν ⁽ /i2⁾ _C| are \ndisplaced more strongly at each frame, converting latent ¹³Cα  chemical shift \ndiversity into a time-varying differential displacement that renders otherwise \nparallel trajectories divergent and individually trackable across the series. \nStreak length and orientation in each panel encode the magnitude and direction \nof each residue's net drift across the 40 K ramp. Compact, nearly circular spots \nindicate residues whose amide and ¹³Cα  chemical shifts are weakly temperature-\nsensitive, consistent with deeply buried, strongly hydrogen-bonded backbone \npositions. Elongated streaks indicate residues whose amide shifts are strongly \nmodulated by temperature, consistent with solvent-exposed or weakly hydrogen-\nbonded positions where backbone dynamics are sensitive to thermal perturbation. \nPeak motion with temperature originates primarily from changes in hydrogen \nbonding geometry at each backbone amide site, which modulates the ¹H and \n¹\n/i2 N chemical shifts at residue-specific rates governed by the drift velocity matrix \nV (Eq. 4 of the master equation). \nThe observed net drift direction incorporates a superposition of two contributions. \nThe intrinsic contribution reflects genuine temperature-dependent changes in \nhydrogen bonding geometry, which typically drive backbone amide ¹H and ¹/i2 N \nshifts upfield as temperature increases. Superimposed on this is a deuterium lock \nartifact: as the HDO resonance shifts upfield with increasing temperature, the \nspectrometer's field-frequency lock compensates by adjusting B₀  upward to \nreturn the deuterium lock signal to its reference position, inadvertently shifting all \nobserved resonances downfield. The apparent peak velocities visible in the \noverlaid trajectories therefore reflect a superposition of intrinsic upfield drift from \nhydrogen bond weakening and a lock-induced downfield displacement common \nto all residues — a systematic effect that must be corrected before per-residue \ntemperature coefficients can be interpreted in terms of hydrogen bond geometry. \nThis controlled trajectory design, achieved through spectral-width modulation, \ntransforms the variable-temperature series from a collection of discrete, \nindependently analyzed observations into a continuous, physically trackable \nmotion field amenable to video-processing-based analysis.  \nSpectral-width modulation as a design variable \nGiven the master equation, the acquisition schedule—the sequence of \nSWH /g2887 7gω666n7gω66x  \nand 13C/g2961  carrier offset ω /g2887 /g32097gω666n7gω66x  values across temperature steps—becomes a \ndesign variable. Modulating SWH /g2887 7gω666n7gω66x  varies the mixing coefficient β7gω666n7gω66x 7g3ω,ω\nSWH /g2898 7gω666n7gω66x/SWH /g2887 7gω666n7gω66x  and thereby the 13C/g2961 -driven S/g2878 /S/g2879  splitting at each frame. \nWe parameterise SWH /g2887 7gω666n7gω66x  as a low-order polynomial in frame index: \nSWH /g2887 7gω666n7gω66x 7g3ω,ω 1\n2 a /g2903/g2907/g2887 7g 3,yn /g2870 7g339xv /g2903/g2907/g2887 7g 3,y n7g339xS W H /g2887,/g2868  \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nsubject to a minimum guard SWH /g2887 7gω666n7gω66x 7g3ω , SWH /g2887,/g2923/g2919/g2924  to avoid aliasing of 13C/g2961  \nresonances. Because β7gω666n7gω66x 7g3ω,ω SWH /g2898 /SWH /g2887 7gω666n7gω66x  and SWH /g2898  is held fixed, Eq. (8) \nuniquely determines the β7gω666n7gω66x  trajectory. An equivalent formulation holds if SWH /g2898  \nis modulated instead; both descriptions are physically interchangeable. Only \nmodulation of β7gω666n7gω66x , and thereby the relative displacement of S/g2878  and S/g2879  peaks, \ngenuinely alters pairwise separability in Hz-space; carrier drift, being rank-1, \ncannot change pairwise separability between residues. Varying β  converts the \nstatic /g2869/g2871 C-driven splitting into a time-varying differential displacement that grows \nor contracts frame by frame, with the largest effect on residues with the largest \n|Δν /g2887\n/g4666/g2921/g4667 | . \nCross-peak model-independent video super-resolution. \nEach temperature step shifts peaks by a small amount in frequency space. \nAcross the series, the same peak is observed at many slightly different positions, \nproviding more localization information than any single frame. We adapt the \nBasicVSR architecture (12, 13) to propagate spectral features bidirectionally \nacross temperature steps—from lower temperatures forward and from higher \ntemperatures backward—so that each frame is informed by resolved \nobservations from both directions. Alignment is guided by the cross-peak \nintensity distribution, which plays the role of the optical flow field in natural video. \nThe analogy holds because peak positions vary slowly and approximately linearly \nwith temperature, producing a smooth, well-conditioned flow field amenable to \nstandard optical-flow estimation. The approach makes no assumptions about drift \nlinearity, residue-specific rates, or trajectory continuity, and is therefore robust to \npathological cases such as conformational exchange broadening or cis/trans \nproline isomerization where a physical model would produce discontinuous or \nmultistate trajectories.  \nBasicVSR++ (14) offers enhanced propagation and was therefore used. Figure 2 \nillustrates the improvement: reconstructed contours sharpen peak profiles \nrelative to raw contours, with no systematic displacement of centroids. The \nprocessing pipeline applied to produce the blue (reconstructed) spectra operates \nin two sequential stages. In the first stage, the raw RD-HSQC frames are loaded \ninto a spatio-temporal contrast-based spectral reconstruction (STSR) module. \nThis step does not increase the spatial (spectral) resolution beyond the digital \nresolution ceiling set by the acquisition spectral width and number of time-domain \npoints; rather, it stabilizes intensity across the temperature sequence by applying \nmultiplicative 95th-percentile scaling with additive mean alignment to remove \nframe-to-frame gain variation, suppresses noise through temporal averaging of \ncorrelated spectral features, and eliminates baseline fluctuations, ensuring that \neach frame entering the super-resolution stage has a consistent, well-conditioned \nintensity profile. In the second stage, the normalised frame sequence is \nprocessed by a BasicVSR video super-resolution model trained on the Vimeo-\n90K dataset — a corpus of high-resolution natural video with smooth inter-frame \nmotion. BasicVSR propagates spectral features bidirectionally across \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\ntemperature steps (from lower temperatures forward and from higher \ntemperatures backward through the series), aligning and fusing information from \nresolved frames to sharpen per-frame peak profiles at frames where peaks are \nclose or overlapping. The output has four times the spatial resolution of the input \nframe sequence. \nThe region of interest selected for display contains two cross-peaks whose \ntrajectories converge with decreasing temperature (Fig. 2). From panel (a) to (f), \nthe groove — the intensity minimum between the two peaks in both the direct (¹H) \nand indirect (RD ¹\n/i2 N) dimensions — progressively narrows as the peaks \napproach one another. In the raw spectra (red), by panel (f) the two peaks have \nmerged into a single, unresolved contour, consistent with their separation falling \nbelow the digital resolution set by the acquisition parameters. In the \nreconstructed spectra (blue), the two peaks remain individually distinguishable as \nseparate contours in panel (f), with a preserved intensity minimum between them, \ndemonstrating that the temporal information pooled across the temperature \nseries by the BasicVSR propagation step recovers per-frame resolution that is \nlost in single-frame analysis of the raw data. This resolution recovery carries no \npeak displacement: the centroids of well-resolved blue peaks are coincident with \ntheir red counterparts in the panels where both are resolved, confirming that the \nreconstruction introduces no systematic bias into downstream peak position \nestimates. The pipeline thus constitutes an effective preprocessing method for \ncross-peak tracking across variable-temperature RD-HSQC series, improving the \nfidelity of trajectory recovery in regions of the spectral window where resonances \ntransiently overlap. \nThe cross-peak model-independent and model-based strategies are \ncomplementary in depth. A peak consistently occluded throughout the series \ncannot be recovered by model-independent methods alone; trajectory \nconstruction is indispensable in that case. Conversely, agnostic smoothing \nprovides a quality floor that benefits all downstream stages regardless of \nacquisition quality. xxx \nPeak Tracking and Frequency Recovery \nPer-frame peak detections—obtained by watershed segmentation (15) of the 2D \nintensity surface—are linked into continuous trajectories by a constant-velocity \nKalman filter (16) with state vector \n7gω666x, y, v /g2934 ,v /g2935 7gω66x, where x  and y  denote the 1H and \napparent 15N frequencies respectively. The constant-acceleration prior is \nappropriate because backbone amide group spins shift temperature coefficients \nare approximately constant over moderate temperature ranges (4), producing \ntrajectories that are nearly linear or slightly curved in frame index. A higher-order \ndynamical model would generate systematic prediction errors in the initial frames \nbefore the velocity state is well estimated, causing data-association failures \nprecisely where they are most costly. The filter predicts each peak’s next position \nand updates upon detection; bidirectional initialization reduces track \nfragmentation at series endpoints. \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nThe process noise covariance 7g x9  and measurement noise covariance 7g x9|  are tuned \nfrom the empirical distribution of per-step peak displacements in a held-out \ncalibration subset of the ubiquitin dataset. For each accepted detection, the \ninnovation (residual between predicted and measured position) is retained as a \ndiagnostic: large, persistent innovations flag residues with non-linear temperature \ndependence, which are candidates for higher-order modelling or flagging as \nexchange-broadened. \nGraph-based S\n/g2878 /S/g2879  pairing and frequency recovery \nTrajectory fragments from the S/g2878  and S/g2879  channels are first consolidated using a \nunion-find data structure (17, 18) to group frame-level detections that belong to \nthe same continuous trajectory. Pairing across channels exploits the constraint \nfrom Eq. (1): candidate S/g2878 /S/g2879  pairs are scored by the smoothness of their sum \ntrajectory η /g4666/g2921/g4667,/g2878 7gω666n7gω66x 7g339x η /g4666/g2921/g4667,/g2879 7gω666n7gω66x , which should follow a smooth, near-linear \ntemperature dependence consistent with a pure 15N chemical shift trajectory. \nPairings satisfying this smoothness criterion above a confidence threshold are \nretained as high-confidence assignments and used to bootstrap recovery of \nthree-nucleus chemical shift trajectories via Eqs. (2) and (3). The recovery is \nlossless when β7g3ω,a0 . \nIn the reported series (Fig. 1) the 13C spectral width SWH /g2887 7gω666n7gω66x  and 15N spectral \nwidth SWH /g2898 7gω666n7gω66x  were modulated as functions of the frame index n  across the 40-\nstep temperature ramp (279–318 K, 1 K increments). Modulation was subject to \nthe reciprocal-sum constraint 1/SWH /g2898 7gω666n7gω66x 7g339x 1/SWH /g2887 7gω666n7gω66x 7g3ω,ω C /g2913/g2925/g2924/g2929/g2930  (Eq. 8). This \nconstraint maintained consistent digital resolution while enabling monotonic \nvariation of the mixing coefficient β7gω666n7gω66x  throughout the series. All expected \ntrajectories were successfully recovered from the dataset. Figure 6 reports \nreconstructed cross-peak trajectories encoding the detailed temperature \ndependencies of all backbone resonances in the test protein.  \n \nConstruction of optimal VT RD-HSQC Peak Trajectories \nTo study optimal construction of peak trajectories in RD-HSQC experiments, we \nsimulated the evolution of cross-peak positions in temperature-resolved /g2869 H–/g2869/g2873 N \nHSQC spectra of proteins with variable controls. Simulations were performed for \ntemperature values ranging from 279 K to 315 K with a step of 1 K. At each \ntemperature point the spectral coordinates of all peaks were calculated using \nbackbone chemical shifts obtained from BMRB data. In these simulations each \nresidue generates a pair of peaks whose coordinates evolve as temperature and \ncontrols change. Figure 5 shows the trajectories of these peaks through a \nsequence of temperature reprsented as moving points in a two–dimensional \nspectral plane expressed in frequency units (Hz). The detailed description of the \nsimulations including temperature effect and trajectory controls is described in \nthe Supplementary Information (section 1).     \n \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nWe formalize trajectory quality as a scalar scoring function over the space of \nacquisition schedules. Three criteria contribute: (i) repulsion—mean nearest-\nneighbour separation across the series, computed in the 2D observed spectral \nplane 7gω666η /g3399 ,Δ ν /g2892 7gω66x; (ii) step displacement—frame-to-frame displacement variance, \npenalising schedules that produce large sudden jumps that would break tracker \ncontinuity; and (iii) boundary compliance—fraction of trajectory points within the \nspectral window, penalising configurations where peaks alias outside the \nacquired region and (iv) trajectories collision resolution. These criteria are \ncombined into a single differentiable objective evaluated on simulated trajectories \ngenerated from Eq. (6) with chemical shifts statistics drawn from protein \nsequence databases (4). \nThis formalization permits principled optimization of the trajectories controls and, \ncritically, enables agentic control of the design loop: the scoring function defines \nwhat constitutes a good acquisition schedule, and an autonomous agent \ntraverses the low-dimensional parameter space (\na /g2903/g2907/g2887 , v /g2903/g2907/g2887 , SWH /g2887,/g2868 , and ω /g2887 /g3209) to \nmaximise it. Example optimised trajectories are shown in Figure 5, with penalty \ncomponents evaluated throughout the temperature sweep. \n \nNovel RDL-TROSY experiment.  \nWe developed a reduced-dimensionality longitudinal relaxation optimized \nexperiment, denoted RDL-TROSY (Fig. 3), to maximize spectral resolution and \ncontrollability of cross-peak trajectories in temperature-resolved 2D [¹H,¹\n/i2 N] \ncorrelation spectra. In this sequence, excitation is restricted to amide protons or \nwater and aliphatic protons using band-selective ¹H pulses, enhancing \nlongitudinal recovery of \n1HN polarization. The key design feature is that the entire \nperiod during which ¹/i2 N magnetization resides in the transverse plane is utilized \nfor chemical shift encoding, thereby achieving maximal effective evolution time \nand, consequently, the highest attainable resolution in the indirectly detected \ndimension. Reduced-dimensionality encoding is implemented through controlled \ntransfer between ¹\n/i2 N and ¹³Cα  spins, producing the characteristic S(+) and S(–) \ncomponents. Importantly, the sequence incorporates weak selective RF \nirradiation on the ¹³C channel during the ¹/i2 N→¹³ C transfer period, or \nsimultaneously on both ¹³C and ¹/i2 N channels, enabling targeted attenuation of \nspecific cross-peaks. This feature allows dynamic suppression of selected \nresonances at defined time points, providing an experimental handle to resolve \ntrajectory collisions and cross-peak occlusions in a temperature series.  \nIn contrast to conventional RD-HSQC implementations derived from folding the \n¹³C\nα  dimension of a 3D HNCA experiment into the ¹/i2 N dimension, the RDL-\nTROSY sequence is specifically optimized for longitudinal relaxation and \nmaximal encoding efficiency. Standard RD-HSQC inherits limitations from the \nparent 3D experiment, including fragmented evolution periods and reduced \neffective evolution time for ¹\n/i2 N, which constrain achievable resolution and lead \nto less controllable trajectory geometry. Furthermore, conventional approaches \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nlack mechanisms for selective, time-resolved attenuation of individual peaks, \nmaking them vulnerable to peak overlap during trajectory evolution. By \ncombining full-length ¹/i2 N evolution, TROSY-based relaxation optimization, and \nprogrammable weak selective irradiation, RDL-TROSY enables both improved \nspectral resolution and active manipulation of cross-peak trajectories. This \nenhanced control is essential for avoiding trajectory collisions and for enabling \ndownstream computational strategies, such as reinforcement learning–guided \noptimization of acquisition parameters. \n \nSelective control of the trajectories at the collision point. \nThe ability of the RDL-TROSY experiment to selectively attenuate individual \ncross-peaks in the S\n⁺  and S-  spectra is demonstrated in Figure 4. Panel (A) \nshows an overlay of two S⁺  spectra acquired with near zero (red) and maximal \n(black) weak RF irradiation applied on the 13C channel at a carrier position \ncentered at 60 ppm, corresponding to the Cα  chemical shift of Ile31. Under these \nconditions, a single targeted cross-peak is efficiently and selectively suppressed, \nwhile the remainder of the spectrum remains largely unaffected. This highlights \nthe high residue specificity of the weak RF irradiation scheme, which exploits the \nnarrow-band excitation profile to address individual resonances without \nperturbing neighboring peaks. The absence of noticeable distortions in \nsurrounding signals confirms that the applied RF field of 28 Hz is sufficiently \nweak to avoid global perturbation while still achieving complete attenuation of the \nselected resonance. \nFigure 4B further quantifies this effect by showing one-dimensional slices through \nthe suppressed peak and two neighboring reference peaks across a series of \nincreasing RF irradiation strengths (0–143 Hz). The intensity of the targeted peak \ndecreases smoothly and monotonically with increasing RF power, ultimately \nreaching near-complete suppression, whereas the flanking peaks remain largely \nunchanged across the entire power range. This continuous and controllable \nattenuation demonstrates that weak RF irradiation can be used as a finely \ntunable parameter to modulate peak intensities in a predictable manner. The \nattenuation effect of the weak rf-irradiation was extensively numerically modeled \n(Supplementary material, section 4) confirming experimentally observed \nattenuation levels (Fig. 3S). Interestingly, our simulations show that to almost \ncomplete and selective peak intensity attenuation can be achieved with a \ncombined weak \n15N and 13C irradiation of only 5 to-10 Hz in the field strength (Fig. \n4S) creating a very precise selection of the targeted trajectory.    \nImportantly, this introduces a new experimental control dimension for trajectory \noptimization: selective, time-resolved masking of specific resonances. In the \ncontext of AI-driven trajectory construction, such weak RF interventions can be \nformulated as discrete, low-cost actions that temporarily suppress colliding peaks, \nenabling the agent to resolve trajectory overlaps without altering the underlying \nspectral encoding. This capability provides a direct experimental counterpart to \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nalgorithmic “masking” operations and significantly enhances the feasibility of \nagentic control over peak trajectories in complex spectra. \nDiscussion \nIn this work, we introduce a conceptual and experimental framework that \nredefines variable-temperature NMR as a spatiotemporal problem in which cross-\npeaks evolve along a controllable trajectory axis. Rather than treating \ntemperature as a perturbation to be minimized or corrected, we exploit it as a \nstructured, physically meaningful variable that encodes additional information into \nthe spectral domain. In combination with reduced-dimensionality encoding, this \ntransforms a series of 2D spectra into a trajectory-resolved dataset in which \notherwise overlapping resonances can be separated through their differential \nmotion. In contrast to conventional multidimensional NMR, where additional \nspectral resolution is obtained by extending Fourier dimensions at the cost of \nacquisition time, our approach redistributes dimensionality into a controllable \nexternal variable, enabling efficient extraction of multi-nuclear chemical shift \ninformation within a comparable experimental time. \nThis framework provides both an experimental and computational advantage. On \nthe experimental side, modulation of the RD mixing parameter and spectral \nwidths allows active steering of cross-peak trajectories, converting latent ¹³C \nchemical shift diversity into time-dependent separability. The RDL-TROSY \nexperiment further maximizes the achievable resolution by fully utilizing the ¹\n/i2 N \ntransverse evolution period and introduces a novel control dimension through \nweak selective RF irradiation. This selective attenuation acts as a precise, \nresidue-specific intervention that resolves trajectory collisions without perturbing \nthe global spectral encoding. On the computational side, the trajectory \nformulation enables direct application of computer vision methodologies, \nincluding video super-resolution and multi-object tracking, which leverage \ntemporal redundancy to recover information lost in individual frames. Together, \nthese elements establish a unified framework in which acquisition design and \ndata analysis are intrinsically coupled. \nA key implication of this work is that spectroscopic dimensionality need not be \nrestricted to traditional frequency axes. External variables such as temperature, \npH, ionic strength, or ligand concentration can be incorporated as controlled \ntrajectory-generating dimensions, provided that their effects on chemical shifts \nare sufficiently systematic. This generalization suggests a broader paradigm of \n“trajectory-encoded spectroscopy,” in which the experiment is designed to \nproduce maximally informative motion of resonances rather than static \nseparation in high-dimensional frequency space. Within this paradigm, the \ninformation content arises not only from instantaneous peak positions but also \nfrom their evolution, effectively introducing derivative constraints that improve \nidentifiability of overlapping signals. \nThe formulation of trajectory quality as a differentiable objective function further \nenables automated optimization of acquisition parameters. In this study, we \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\ndemonstrate that global controls such as carrier frequency and spectral width can \nbe tuned to maximize peak separability while maintaining smooth, trackable \nmotion. Importantly, the addition of selective RF attenuation introduces a discrete \ncontrol mechanism that can be used to resolve transient peak overlaps. This \nnaturally lends itself to a reinforcement learning formulation, in which an agent \niteratively adjusts acquisition parameters and selective interventions to maximize \na global reward function. Unlike conventional automation, which executes \npredefined protocols, such an agentic framework directly influences the physical \nexperiment in real time, enabling adaptive and protein-specific optimization of the \nacquisition process. \nSeveral limitations should be noted. The approach relies on approximately linear \ntemperature dependence of chemical shifts and requires that the protein remains \nstructurally stable over the sampled temperature range. Strong conformational \nexchange, unfolding transitions, or non-linear drift may complicate trajectory \nmodeling and tracking. In addition, the requirement for dense sampling across \ntemperature introduces a trade-off between temporal resolution and total \nacquisition time, although this is partially mitigated by the use of efficient 2D \nexperiments. Finally, while the current implementation focuses on backbone \nassignment, extension to side-chain correlations or distance restraints will require \nfurther development of both pulse sequences and trajectory models. \nOverall, the present work establishes a bridge between NMR spectroscopy, \ntrajectory-based signal encoding, and modern computer vision methodologies. \nBy integrating experimental control, physical modeling, and data-driven analysis \nwithin a unified framework, it opens a pathway toward adaptive, agent-guided \nNMR experiments in which acquisition and interpretation are co-optimized. This \nperspective suggests that future developments in NMR may increasingly rely not \nonly on improved hardware or pulse sequences, but also on intelligent control \nstrategies that actively shape the information content of the experiment. \nConclusions \nWe have described a framework that reconceptualizes the VT-NMR experiment \nas a video processing problem and demonstrated concrete consequences of that \nreconceptualization for cross-peak resolution and residue assignment. The \ntemperature axis, conventionally treated as a complication, is recast as a \npseudo-temporal axis along which inter-frame pooling resolves peaks \nirresolvable in any individual spectrum. The RD-HSQC is the natural vehicle: its \nexplicit \n13C/g2961  encoding provides spectral, temporal, and topological discrimination \nsimultaneously, and its exact mathematical form provides the basis for principled \nacquisition design. \nThe agnostic and model-based strategies are complementary: the former \nimproves per-frame quality without physical assumptions; the latter converts \nlatent 13C/g2961  diversity into engineered inter-trajectory separation through spectral-\nwidth modulation. A formal scoring function, a selective intensity-control pulse \nsequence element, and the agentic design loop together define a path toward \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nfully automated acquisition optimisation. The framework is protein-independent; \nits inputs are backbone chemical shift databases and the target primary \nsequence, and its output is an optimised acquisition schedule calibrated to the \nspecific assignment problem at hand. \nMaterials and Methods \nHuman ubiquitin 15N/13C uniformly labeled was purchased from Sigma-Aldrich \n(MilliporeSigma). NMR spectra were measured using Bruker Avance III NMR \nspectrometer operating at 700 MHz magnetic field strength. Specgra were \nreferenced relative to the internal DSSThe RD-HSQC was acquired quadrature \nselective 13C/g2961  editing, yielding separate S/g2878  and S/g2879  free induction decays (FIDs) \nby acquiring the in-phase (S /g2919/g2924 ) and antiphase (S /g2911/g2924/g2930/g2919 ) spectra in an interleaved \nfashion (10, 11). The temperature ramp spanned 279–318 K in 1 K increments \n(40 steps); one complete RD-HSQC was acquired per step following 1 min \nthermal equilibration. Temperature calibration was performed using the standard \nmethanol sample (2). \nRaw Bruker TopSpin free induction decays (FIDs) were apodised (cosine-bell in \nt /g2869 , exponential in t /g2870 ), zero-filled to twice the acquisition size in both dimensions, \nFourier transformed, and phased. S/g2878  and S/g2879  subspectra were computed by \naddition and subtraction of the S /g2919/g2924  and S /g2911/g2924/g2930/g2919  spectra. Both channels were \nassembled into video sequences at 4 fps with 1:1 array-to-pixel grayscale \nencoding and normalised by multiplicative 95th-percentile scaling with additive \nmean alignment. Inter-frame consistency was assessed by the structural \nsimilarity index measure (SSIM) (20). Agnostic super-resolution, per-frame peak \ndetection, Kalman tracking, graph-based classification, and frequency-coordinate \nrecovery were implemented in Python 3.10 with graphics processing unit (GPU) \nacceleration (CUDA 11.8). \n \nFigure 1. Variable-temperature RD-HSQC spectra of human ubiquitin (279–\n318 K) with linearly modulated spectral widths. Forty contour plots are \noverlaid for each subspectrum panel, with temperature encoded by a continuous \ncolor gradient from blue (279 K, frame 1) to red (318 K, frame 40). (a) S\n/i2  and (b) \nS/i2  spectral plane overlays, each containing one cross-peak per residue. The S/i2  \nand S/i2  components are complementary spin-state-selective channels produced \nby the echo-antiecho editing of the RD-HSQC pulse sequence: at each \ntemperature step, the S/i2 /S/i2  peak pair for residue k is split symmetrically about \nthe position of the conventional ¹/i2 N–¹H HSQC cross-peak, with splitting \nmagnitude proportional to the residue's ¹³Cα  rotating-frame offset according to \n|Δ/i2/i2/i2 _RD(n)| = 2β (n)|Δν/i2/i2/i2 _C(n)| (Eq. 26), where β (n) = \nSWH_N(n)/SWH_C(n) is the frame-dependent mixing coefficient. Paired S/i2 /S/i2  \ntrajectories exhibit mirror-image motion in the mixed ¹/i2 N (RD) dimension, and \ntheir sum and difference recover the original per-residue ¹/i2 N and ¹³Cα  chemical \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nshift trajectories via the closed-form inversion Eqs. 28–30, confirming that the RD \nencoding is lossless when β  ≠  0. \n \n \nFigure 2. Peak model-independent video super-resolution reconstruction of \nRD-HSQC spectra of human ubiquitin. Panels (a)–(f) each show a cropped \nregion of interest from the 2D RD-HSQC spectral window as a two-spectrum \noverlay: the raw spectrum (red contours), as processed directly in the Bruker \nTopSpin v4.05 following standard phase correction and baseline correction, is \noverlaid with the reconstructed spectrum (blue contours) produced by the full \nspatio-temporal processing pipeline. The six panels sample a descending \ntemperature sequence from (a) 300 K to (f) 294 K in 1 K steps, so that the \nprogression from panel (a) to (f) corresponds to cooling and the associated \nupfield drift of backbone amide resonances brings a pair of initially resolved \ncross-peaks progressively closer together and into partial overlap. \n \n \nFigure 3: Experimental scheme for the Reduced-Dimensionality \nLongitudinal 1H relaxation optimized 2D [1H,15N]-TROSY (RDL-TROSY). The \nradio frequency pulses on 1H, 15N, 13C/g3080 , 13CO are applied at 7g|,33 /g3009 , 7g|,33 /g3015 , 7g|,33 /g3004 /g3328, 7g|,33 /g3004/g3016  of \n4.7, 118, 55, 174 ppm, respectively. Narrow and wide black bars indicate \nnonselective 90° and 180° pulses, respectively. The \n1H shaped pulses are: 7g|,3y /g2869 -/g2872 , \n1H/g3015  5.5–10.0 ppm band-selective 1.5 ms excitation E-Burp2 pulses with the \nphases 7gω66y7g yx6 7gω669 , 7gω66y7g yxx7gω669 , 7gω66y 7g yxx ,7g yxx ,7g yxx ,7g yxx ,7g339y 7g yxx ,7g339y 7g yxx ,7g339y 7g yxx ,7g339y 7g yxx 7gω669  and 7gω66y7g yx67gω669 , respectively. The shapes of \n7g|,3y /g2870,/g2872  are time reversed of 7g|,3y /g2869,/g2871 . The 7g|,3y /g2873  and 7g|,3y /g2874  are 1H/g3015  5.5–10.0 ppm and 5.5–1.0 \nppm band-selective 1.8 ms refocusing Re-Burp pulses with the phase 7gω66y7g yx67gω669 . The \n13C shaped pulses are 1.3 ms Gaussian cascade Q3 (dark shapes) and Q5 (light \nshapes) with the phases 7g|,3y /g2869/g2868 7g3ω,ω 7gω66y7g339y7g yx67gω66687gω66x, 7g yx67gω66687gω66x7gω669 , 7g|,3y /g2869/g2869 7g3ω,ω 7gω66y7g339y7g yx67gω66687gω66x, 7g yx67gω66687gω66x7gω669  and \n7gω66y7g339y7g yxx7gω66687gω66x, 7gω66687gω66x7gω669  for the reduced-dimensionality in-phase and antiphase spectra, 7g yωa /g3036/g3041  \nand 7g yωa /g3028/g3041/g3047/g3036 , respectively. The spectra 7g yωa7gω6667g339x7gω66x  and 7g yωa7gω6667g339y7gω66x  are reconstructed by adding \nand subtracting 7g yωa /g3036/g3041  and 7g yωa /g3028/g3041/g3047/g3036 , respectively. The phases 7g|,3y /g2868,/g2875 -/g2877  are \n7gω66y 7g yx6 ,7g339y 7g yx6 ,7g339y 7g yxx ,7g yxx ,7g yx6 ,7g339y 7g yx6 ,7g yxx ,7g339y 7g yxx 7gω669 , 7gω66y7g yxx, 7g339y7g yxx, 7g339y7g yx6 , 7g yx6 7gω669 , 7gω66y7g yxx7gω669 , 7gω66y7g yx67gω66647gω66x, 7g339y7g yx67gω66647gω66x7gω669 , respectively. The \necho-anti-echo transverse relaxation optimised spectroscopy (TROSY) phase \ndiscrimination pattern is applied to 7g|,3y /g2873  and 7g|,3y /g2868  for each 7g yωa /g3036/g3041  and 7g yωa /g3028/g3041/g3047/g3036 . The delays \nare 7g yx|7g3ω,ω2 . 7  ms and 7g yω67g3ω,ω3 0  ms. The pulsed field gradients (PFGs) are G1: 80 \nG/cm; G2: 95 G/cm; G3: 70 G/cm. The colored shapes represent narrow band-\nselective phase-modulated radio-frequency irradiation with the radio-frequency \n(RF) field strength of 7g|,| /g2869,/g3015  and 7g|,| /g2869,/g3004  with the corresponding offsets defined in the \ntext. To generate cross-peak trajectories, the 2D RDL 7g yωa 7gω6667g339x7gω66x  and 7g yωa 7gω6667g339y7gω66x  planes are \nmeasured with the temperature range (279–315 K) and variable settings of the \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\n13C carrier frequency 7g|,33 /g3004 /g3328, and the corresponding spectral width 7g yωa7g yω97g y3ω /g3015  and 7g yωa7g yω97g y3ω /g3004 . \nThe dashed box in the middle indicates the part of the pulse sequence used in \nthe numeric simulations of the propagation of the density operator of two 7g y36-\ncoupled spins 13C and 15N under the effects of the static Hamiltonian and the \nphase modulated weak RF irradiation \n \nFigure 4: Selective and tunable cross-peak attenuation in RD-HSQC. (a) \nOverlay of two S/g2878  spectra acquired with maximum (black) and minimum (red) RF \nirradiation at a targeted 13C offset. Only one peak (arrowed) is completely \nsuppressed at using weak 13C radio-frequency irradiation 7g|,| /g2869,/g3004  = 28 Hz centered \nat 60 ppm (CA chemical shift of Ile31), demonstrating residue-specific selectivity. \n(b) 1D slices through the suppressed peak (center) and two reference peaks \n(flanking) across the power series 7g|,| /g2869,/g3004  = \n{0,/i2 3.5,/i2 5,/i2 7,/i2 11,/i2 16,/i2 22,/i2 28,/i2 40,/i2 56,/i2 80,/i2 113,/i2 143} Hz (colored from \nblue to yellow in the corresponding order), showing progressive attenuation of \nthe target peak intensity as power increases. \n \nFigure 5: RD-HSQC peak trajectories generated using near-optimal control \nparameters. Each colored curve represents the temperature-dependent \ntrajectory of a backbone amide resonance of ubiquitin (76 residues) across the \nsimulated temperature range (279–315 K). The RD encoding produces two \nspectral planes (top and bottom panels), corresponding to the \n7g yωa 7gω6667g339x7gω66x  and 7g yωa 7gω6667g339y7gω66x  \ncomponents arising from folding of the 13C/g3080  chemical shift into the indirect \ndimension. The trajectories are actively reshaped through global experimental \ncontrols acting on the \n13C carrier frequency (7g|,33 /g3004 /g3328) and the 13C spectral width \n(7g yωa7g yω97g y3ω /g3004 ), whose rates and accelerations are dynamically updated with \ntemperature while maintaining the reciprocal-sum constraint 1/7g yωa7g yω97g y3ω /g3015 7gω6667g y66 7gω66x 7g339x\n1/7g yωa7g yω97g y3ω /g3004 7gω6667g y667gω66x 7g3ω,ω 7g y|9 /g3030/g3042/g3041/g3046/g3047  (Eq. 8). The control parameters used in this example were \n7g|,33 /g3004 /g33287g3ω,ω 57.65  ppm with rate 0.39 ppm/K and acceleration 0.0187 ppm/K2, and \n7g yωa7g yω97g y3ω /g3004 7g3ω,ω 2617.8  Hz with rate 7g339y196.8  Hz/K and acceleration 9.86 Hz/K2. Insets \nshow the evolution of representative penalty components used to evaluate \ntrajectory optimality during the temperature sweep, including spread, step-size, \nboundary, and collision terms. These penalty functions quantify peak separation, \nsmoothness of motion, distance from spectral boundaries, and avoidance of peak \noverlaps, respectively, and together define the objective function used for \ntrajectory optimization. \n \nFigure 6: Automated S+/S− trajectory pairing via iterative RMSD \nbootstrapping. Each colored trajectory represents a matched S+/S− trajectory \npair, with paired trajectory sharing the same color. Grey trajectories in the \nbackground indicate unmatched candidate islands. Pairing was performed by \niterative Hungarian assignment with a progressively relaxed RMSD threshold \n(starting from 0, incremented by 0.001 per round over 20,000 iterations). The \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint \n\nRMSD serves as the sole matching criterion, exploiting the physical constraint \nthat the ¹H chemical shift drift (Δν _H) is identical for the S+ and S− peaks of the \nsame residue, producing coincident horizontal trajectories across the \ntemperature series. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nAcknowledgements \nWe thank Prof Vladislav Orekhov for fruitful early discussions and his advice for \nthe project.  \nReferences \n1.  Wüthrich, K. (1986) NMR of Proteins and Nucleic Acids, Wiley-\nInterscience, New York \n2.  Cavanagh, J., Fairbrother, W. J., Palmer, A. G. I., Rance, M., and Skelton, \nN. J. 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