NMR as a Video-(game): Constructing Super-Resolution Cross-peak Trajectories in Protein Spectroscopy

High-resolution multidimensional NMR spectroscopy of proteins remains limited by long acquisition times, sensitivity constraints, and severe peak overlap, particularly for larger systems. Conventional 3D and higher-dimensional experiments trade experimental efficiency for resolution, while post-acquisition analysis often becomes the dominant bottleneck. Here, we present a new framework that redefines both how NMR experiments are constructed and how they are executed and analyzed, by treating an AI agent-controllable series of 2D spectra as a spatiotemporal dataset analogous to a video. Our approach is based on temperature-dependent series of reduced-dimensionality 2D HSQC and novel RDL-TROSY experiments, in which each 2D [ 1H,15N] cross-peak is controllably shifted and split in proportion to the 13C chemical shift of the J- coupled carbons. We propose treating a variable-temperature (VT) series as a pseudo-temporal video sequence in which each cross-peak traces a physically motivated trajectory through frequency space. The proportionality coefficient ( α ) of this reduced-dimensionality encoding is systematically and programmatically varied together with the temperature providing full control for constructing optimal cross-peak trajectories. As a result, individual resonances follow predictable, spectral acquisition time-controllable trajectories in the 2D spectral plane across the series, which can be executed by an autonomous AI agent directly interacting with the NMR GUI layer. Each spectrum represents a single “frame,” while temperature and RD controls serves as the temporal dimension. We describe two complementary super-resolution strategies: a cross-peak model-independent approach based on the deep-learning video super-resolution that leverages temporal redundancy to sharpen per-frame peak shapes, and a model-based approach that derives the exact mathematical form of the peak trajectories and uses it to design acquisition schedules that render individual peak paths maximally distinct and amenable for algorithmic deconvolution. As a result, we obtained full backbone resonance assignment in the wide temperature range (279–315 K) with one degree Kelvin resolution in a test protein in an automatic .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint manner in the time frame typically required for collection of a single 3D NMR dataset.

RD-HSQC; variable-temperature nuclear magnetic resonance (NMR); spatiotemporal super-resolution; peak trajectory; spectral-width modulation; resonance assignment; ubiquitin; agentic optimization.

Backbone resonance assignment is the prerequisite for all quantitative protein nuclear magnetic resonance (NMR) spectroscopy (1, 2), and its primary obstacle is spectral overlap. Conventional solutions add indirect dimensions, spreading peaks across additional frequency axes that form orthogonal spectral dimensions after Fourier transformation, at the cost of linearly increased acquisition time, which becomes prohibitive for experiments requiring large numbers of indirect time-domain sampling points (3). Variable-temperature (VT) NMR offers an independent additional axis. Backbone amide 1H, 15N, and 13C/g2961 chemical shifts all respond to temperature in a residue-specific manner (3–5), reflecting local hydrogen bond geometry and backbone dynamics. The temperature coefficient ∂δ /g2892/g2898 /∂ T is linear over a wide temperature range (269–369 K) (5) and takes values spanning roughly −16 to +2 ppb K/g2879/g2869 for amide protons, with buried residues showing attenuated sensitivity (5). Since this linearity is well established, we acquire 40 spectra across a temperature range falling within this window, at 1 K intervals. This generates a pseudo-temporal sequence in which each temperature step corresponds to a time point, and each cross-peak traces a distinct, predictable path through frequency space, encoding the differential change in chemical shift drift between successive spectra. Two peaks overlapping at one temperature are often separated at another temperature or reduced-dimensionality (RD) settings; the question is how to exploit this information systematically across the full series of acquired spectra rather than discard it through frame-by-frame analysis. The conceptual core of this paper is the bridging of VT-NMR to a computer vision framework, in which a non-spectroscopic variable (temperature) as well as signal acquisition time-controlled variables are exploited to resolve physics motivated spectroscopic observables such as chemical shifts or selected NOEs. The CV field has developed extensively, with deep learning-based methods now achieving robust detection and tracking of objects across successive frames, exemplified by YOLO (7) for detection and DeepSORT (8) for multi-object tracking. Each spectrum is a 2D frame; each point of the spectral intensity grid can be viewed as a pixel; the temperature and RD axis is a pseudo-temporal axis; and cross-peak drift is a motion field. This bridge has direct operational consequences: resolving transiently occluded peaks becomes a problem of propagating information across the temporal dimension, for which optical flow, recurrent feature propagation, and super-resolution are directly applicable. .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint Our approach differs from that of Shchukina et al. (9), who sample temperature and indirect evolution times jointly within a single acquisition, necessitating correction for signal non-stationarity arising from temperature drift during acquisition. In contrast, we acquire a fully stationary RD-HSQC or RDL-TROSY spectrum at each discrete temperature step; each frame is recorded independently and is therefore free from non-stationarity artefacts. The resulting challenge lies in linking and exploiting this discrete series of stationary spectra. Trainor, K et al. addressed a related problem by implementing peak tracking across temperatures to determine temperature coefficients, showing that the chemical shifts of 1H and 15N nuclei exhibit an approximately linear dependence on temperature. In our work, we extend this concept by exploiting the temperature-dependent chemical shift dynamics of three types of spins within a single spectral series, introducing a temperature-based encoding that captures pseudo-temporal correlations between spectra.

The experiment of choice is the reduced-dimensionality heteronuclear single quantum coherence (RD-HSQC) spectroscopy (10, 11). In addition, we introduced novel Reduced Dimensionality L-TROSY (RDL-TROSY) experiment maximizing achievable spectral resolution in all spectral dimensions. In this experiment, the 13C/g2961 evolution time is linearly co-incremented with the 15N evolution time during the indirect dimension, so that the apparent 15N chemical shift is a linear combination of the true 15N and 13C/g2961 offsets (10). In a conventional 1H–15N HSQC, two residues with similar 1H and 15N shifts overlap regardless of their 13C/g2961 differences. The RD-HSQC encodes the 13C/g2961 frequency offset into the detected 15N dimension via a constant-time evolution element with a mixing coefficient β7g3ω,ωS W H /g2898 /SWH /g2887 , producing two symmetrically displaced cross-peaks per residue—the S/g2878 and S/g2879 components—at apparent 15N positions: η /g4666 /g2921 /g4667 ,/g3399 7gω666n 7gω66x 7g3ω,ωΔ ν /g2898 /g4666 /g2921 /g4667 7gω666n 7gω66x 7g3399β 7gω666 n 7gω66x 7g 3,yΔν /g2887 /g4666 /g2921 /g4667 7gω666 n 7gω66x # 7gω6661 7gω66x where β7gω666n7gω66x 7g3ω,ω SWH /g2898 7gω666n7gω66x/SWH /g2887 7gω666n7gω66x is the frame-dependent mixing ratio (ratio of 15N to 13C/g2961 spectral widths, both in Hz), k indexes residues, and n indexes temperature frames. The original two heteronuclear coordinates are recovered exactly by sum-and-difference inversion: Δν /g2898 /g4666 /g2921 /g4667 7gω666 n 7gω66x 7g3ω,ω 1 2 7gω6x|η /g4666 /g2921 /g4667 ,/g2878 7gω666 n 7gω66x 7g339xη /g4666 /g2921 /g4667 ,/g2879 7gω666 n 7gω66x7gω6x3# 7gω666 2 7gω66x Δν /g2887 /g4666 /g2921 /g4667 7gω666n 7gω66x 7g3ω,ω η /g4666 /g2921 /g4667 ,/g2878 7gω666 n 7gω66x 7g339yη /g4666 /g2921 /g4667 ,/g2879 7gω666 n 7gω66x 2β 7gω666 n 7gω66x # 7gω666 3 7gω66x .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint The encoding is lossless when β7g3ω,a0 . By modulating SWH /g2887 7gω666n7gω66x across the temperature series and thereby varying β7gω666n7gω66x , residues with different /g2869/g2871 C/g2961 offsets are displaced by different amounts at each frame, converting latent 13C diversity into a time-varying differential displacement that renders otherwise parallel trajectories divergent and individually trackable. Note that modulating SWH /g2898 7gω666n7gω66x achieves the same effect; in practice SWH /g2887 7gω666n7gω66x is the more convenient variable because the 13C/g2961 window is narrower and more easily stepped without aliasing. Two complementary resolution strategies follow from this reframing. The cross- peak model-independent strategy adopts video super-resolution (12, 13) to propagate spectral features bidirectionally across temperature steps without assumptions about the specific cross-peak point spread function (PSF). The cross-peak model-based strategy uses the exact master equation for RD-HSQC trajectories to design acquisition schedules that maximize trajectory separability. Together they operate at the two levels where the video reframing has traction: the reconstruction of per-frame peak profiles, and the design of the frame sequence itself. Master equation for RD-HSQC trajectories Let the chemical shift offsets from carrier for residue k at temperature step n be Δν /g2898 /g4666/g2921/g4667 7gω666n7gω66x , Δν /g2887 /g4666/g2921/g4667 7gω666n7gω66x , and Δν /g2892 /g4666/g2921/g4667 7gω666n7gω66x for 15N, 13C/g2961 , and 1H/g2898 respectively. The temperature dependence of backbone amide shifts is approximately linear over moderate ranges (4): Δν /g2908 /g4666 /g2921 /g4667 7gω666 n 7gω66x 7g3ω,ωΔ ν /g2908 /g4666 /g2921 /g4667 7gω666 0 7gω66x 7g339x n7g 3,yΔT7g 3,yv /g2908 /g4666 /g2921 /g4667 # 7gω666 4 7gω66x where ΔT is the temperature increment per step and v /g2908 /g4666/g2921/g4667 is the per-residue drift velocity (Hz K/g2879/g2869 ) for nucleus X . All three nuclei share a common carrier drift component that displaces the entire peak cloud coherently: Δν /g2908 /g4666 /g2921 /g4667 7gω666 n 7gω66x 7g3ω,ωΔ ν /g2908 /g4666 /g2921 /g4667 7gω666 n 7gω66x 7g339xF /g2868,/g2908 7gω666 n 7gω66x # 7gω666 5 7gω66x where F /g2868,/g2908 7gω666n7gω66x is a shared, frame-dependent carrier correction (arising, for example, from deuterium lock drift or thermal expansion of the probe). The complete observed peak positions in the 2D RD-HSQC spectrum are then 7g 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) .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint where 7g|x||7g xyy7gω66607gω66x 7g ωyy 7g 33x /g2897/g3400/g2871 is the matrix of initial offset coordinates for all M residues (columns ordered as Δν /g2898 , Δν /g2887 , Δν /g2892 ), 7g x967g ωyy7g 33x /g2897/g3400/g2871 is the per-residue drift velocity matrix, 7g xy,7gω666n7gω66x is a rank-1 broadcast matrix encoding shared carrier drift, and: 7g xxa 7gω666 n 7gω66x 7g3ω,ω7g3ω3x 17g339x β 7gω666 n 7gω66x 0 17g339y β 7gω666 n 7gω66x 0 001 7g3ωω # 7gω666 7 7gω66x Each row-pair of 7g x9,7gω666n7gω66x gives the observed 7gω666η /g2878 ,η /g2879 ,Δ ν /g2892 7gω66x coordinates of one residue at step n . Because 7g xy,7gω666n7gω66x is rank-1 and maps identically to all residues, it cancels exactly in all pairwise peak differences and has no effect on inter-residue separability; its sole role is to keep the peak cloud centered within the spectral window. The drift velocity matrix 7g x96 , drawing per-residue coefficients from distributions well-characterised for soluble proteins (4), is the fundamental source of trajectory distinctiveness. The RD projection matrix 7g xxa7gω666n7gω66x translates 13C/g2961 offset diversity into differential separation in the mixed 15N dimension. Complementary S/g2878 and S/g2879 spectral planes as independent channels The spin-state-selective editing of the RD-HSQC yields two spectral planes, S/g2878 and S/g2879 , each containing one cross-peak per residue. The construction of planes is achieved by acquiring x and y quadrature components of the evolving 13C/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 S/g2878 and S/g2879 spectra function as two independent video sequences whose motion fields are coupled by Eq. (1). For a correct S/g2878 /S/g2879 pair, the sum η /g4666/g2921/g4667,/g2878 7gω666n7gω66x 7g339x η /g4666/g2921/g4667,/g2879 7gω666n7gω66x 7g3ω,ω 2Δν /g2898 /g4666/g2921/g4667 7gω666n7gω66x must vary smoothly with temperature, providing a stringent cross-validation constraint: spurious pairings produce erratic sums, correct pairings produce smooth 15N trajectories. Because S/g2878 and S/g2879 peaks are spatially separated by |Δ /g2902/g2888 /g4666/g2921/g4667 | 7g3ω,ω 2β|Δν /g2887 /g4666/g2921/g4667 | , peaks occluded in one channel may be resolved in the other. The S/g2878 /S/g2879 splitting magnitude is directly proportional to the 13C/g2961 offset from the 13C carrier. Residues near the carrier (Δν /g2887 7g3ω,60 ) produce nearly coincident S/g2878 /S/g2879 pairs and therefore contribute little discriminating information from the RD dimension; the acquisition schedule should set the carrier to maximise the spread of |Δν /g2887 /g4666/g2921/g4667 | values across all residues. Figure 1 shows overlay of 40 spectral planes of S/g2878 and S/g2879 RD-HSQC measured with 76 amino acid test protein (Ubiquitin). Placing the 13C/g2961 carrier at 52 ppm (near the centre of the C/g2961 chemical shift range for amino acids) achieves adequate coverage for most residues in the protein. The ¹³C spectral width SWH_C(n) and the coupled ¹/i2 N spectral width SWH_N(n) were modulated as linear functions of the frame index n across the 40-step temperature ramp, subject to the reciprocal-sum constraint 1/SWH_N(n) + 1/SWH_C(n) = C_const .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint (Eq. 8), which ensures that β (n) evolves in a controlled, monotonic fashion. This frame-by-frame modulation of β (n) actively steers the trajectory of each S⁺ /S⁻ peak pair in the mixed ¹/i2 N dimension: residues with larger |Δν ⁽ /i2⁾ _C| are displaced more strongly at each frame, converting latent ¹³Cα chemical shift diversity into a time-varying differential displacement that renders otherwise parallel trajectories divergent and individually trackable across the series. Streak length and orientation in each panel encode the magnitude and direction of each residue's net drift across the 40 K ramp. Compact, nearly circular spots indicate residues whose amide and ¹³Cα chemical shifts are weakly temperature- sensitive, consistent with deeply buried, strongly hydrogen-bonded backbone positions. Elongated streaks indicate residues whose amide shifts are strongly modulated by temperature, consistent with solvent-exposed or weakly hydrogen- bonded positions where backbone dynamics are sensitive to thermal perturbation. Peak motion with temperature originates primarily from changes in hydrogen bonding geometry at each backbone amide site, which modulates the ¹H and ¹ /i2 N chemical shifts at residue-specific rates governed by the drift velocity matrix V (Eq. 4 of the master equation). The observed net drift direction incorporates a superposition of two contributions. The intrinsic contribution reflects genuine temperature-dependent changes in hydrogen bonding geometry, which typically drive backbone amide ¹H and ¹/i2 N shifts upfield as temperature increases. Superimposed on this is a deuterium lock artifact: as the HDO resonance shifts upfield with increasing temperature, the spectrometer's field-frequency lock compensates by adjusting B₀ upward to return the deuterium lock signal to its reference position, inadvertently shifting all observed resonances downfield. The apparent peak velocities visible in the overlaid trajectories therefore reflect a superposition of intrinsic upfield drift from hydrogen bond weakening and a lock-induced downfield displacement common to all residues — a systematic effect that must be corrected before per-residue temperature coefficients can be interpreted in terms of hydrogen bond geometry. This controlled trajectory design, achieved through spectral-width modulation, transforms the variable-temperature series from a collection of discrete, independently analyzed observations into a continuous, physically trackable motion field amenable to video-processing-based analysis. Spectral-width modulation as a design variable Given the master equation, the acquisition schedule—the sequence of SWH /g2887 7gω666n7gω66x and 13C/g2961 carrier offset ω /g2887 /g32097gω666n7gω66x values across temperature steps—becomes a design variable. Modulating SWH /g2887 7gω666n7gω66x varies the mixing coefficient β7gω666n7gω66x 7g3ω,ω SWH /g2898 7gω666n7gω66x/SWH /g2887 7gω666n7gω66x and thereby the 13C/g2961 -driven S/g2878 /S/g2879 splitting at each frame. We parameterise SWH /g2887 7gω666n7gω66x as a low-order polynomial in frame index: SWH /g2887 7gω666n7gω66x 7g3ω,ω 1 2 a /g2903/g2907/g2887 7g 3,yn /g2870 7g339xv /g2903/g2907/g2887 7g 3,y n7g339xS W H /g2887,/g2868 .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint subject to a minimum guard SWH /g2887 7gω666n7gω66x 7g3ω , SWH /g2887,/g2923/g2919/g2924 to avoid aliasing of 13C/g2961 resonances. Because β7gω666n7gω66x 7g3ω,ω SWH /g2898 /SWH /g2887 7gω666n7gω66x and SWH /g2898 is held fixed, Eq. (8) uniquely determines the β7gω666n7gω66x trajectory. An equivalent formulation holds if SWH /g2898 is modulated instead; both descriptions are physically interchangeable. Only modulation of β7gω666n7gω66x , and thereby the relative displacement of S/g2878 and S/g2879 peaks, genuinely alters pairwise separability in Hz-space; carrier drift, being rank-1, cannot change pairwise separability between residues. Varying β converts the static /g2869/g2871 C-driven splitting into a time-varying differential displacement that grows or contracts frame by frame, with the largest effect on residues with the largest |Δν /g2887 /g4666/g2921/g4667 | . Cross-peak model-independent video super-resolution. Each temperature step shifts peaks by a small amount in frequency space. Across the series, the same peak is observed at many slightly different positions, providing more localization information than any single frame. We adapt the BasicVSR architecture (12, 13) to propagate spectral features bidirectionally across temperature steps—from lower temperatures forward and from higher temperatures backward—so that each frame is informed by resolved observations from both directions. Alignment is guided by the cross-peak intensity distribution, which plays the role of the optical flow field in natural video. The analogy holds because peak positions vary slowly and approximately linearly with temperature, producing a smooth, well-conditioned flow field amenable to standard optical-flow estimation. The approach makes no assumptions about drift linearity, residue-specific rates, or trajectory continuity, and is therefore robust to pathological cases such as conformational exchange broadening or cis/trans proline isomerization where a physical model would produce discontinuous or multistate trajectories. BasicVSR++ (14) offers enhanced propagation and was therefore used. Figure 2 illustrates the improvement: reconstructed contours sharpen peak profiles relative to raw contours, with no systematic displacement of centroids. The processing pipeline applied to produce the blue (reconstructed) spectra operates in two sequential stages. In the first stage, the raw RD-HSQC frames are loaded into a spatio-temporal contrast-based spectral reconstruction (STSR) module. This step does not increase the spatial (spectral) resolution beyond the digital resolution ceiling set by the acquisition spectral width and number of time-domain points; rather, it stabilizes intensity across the temperature sequence by applying multiplicative 95th-percentile scaling with additive mean alignment to remove frame-to-frame gain variation, suppresses noise through temporal averaging of correlated spectral features, and eliminates baseline fluctuations, ensuring that each frame entering the super-resolution stage has a consistent, well-conditioned intensity profile. In the second stage, the normalised frame sequence is processed by a BasicVSR video super-resolution model trained on the Vimeo- 90K dataset — a corpus of high-resolution natural video with smooth inter-frame motion. BasicVSR propagates spectral features bidirectionally across .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint temperature steps (from lower temperatures forward and from higher temperatures backward through the series), aligning and fusing information from resolved frames to sharpen per-frame peak profiles at frames where peaks are close or overlapping. The output has four times the spatial resolution of the input frame sequence. The region of interest selected for display contains two cross-peaks whose trajectories converge with decreasing temperature (Fig. 2). From panel (a) to (f), the groove — the intensity minimum between the two peaks in both the direct (¹H) and indirect (RD ¹ /i2 N) dimensions — progressively narrows as the peaks approach one another. In the raw spectra (red), by panel (f) the two peaks have merged into a single, unresolved contour, consistent with their separation falling below the digital resolution set by the acquisition parameters. In the reconstructed spectra (blue), the two peaks remain individually distinguishable as separate contours in panel (f), with a preserved intensity minimum between them, demonstrating that the temporal information pooled across the temperature series by the BasicVSR propagation step recovers per-frame resolution that is lost in single-frame analysis of the raw data. This resolution recovery carries no peak displacement: the centroids of well-resolved blue peaks are coincident with their red counterparts in the panels where both are resolved, confirming that the reconstruction introduces no systematic bias into downstream peak position estimates. The pipeline thus constitutes an effective preprocessing method for cross-peak tracking across variable-temperature RD-HSQC series, improving the fidelity of trajectory recovery in regions of the spectral window where resonances transiently overlap. The cross-peak model-independent and model-based strategies are complementary in depth. A peak consistently occluded throughout the series cannot be recovered by model-independent methods alone; trajectory construction is indispensable in that case. Conversely, agnostic smoothing provides a quality floor that benefits all downstream stages regardless of acquisition quality. xxx Peak Tracking and Frequency Recovery Per-frame peak detections—obtained by watershed segmentation (15) of the 2D intensity surface—are linked into continuous trajectories by a constant-velocity Kalman filter (16) with state vector 7gω666x, y, v /g2934 ,v /g2935 7gω66x, where x and y denote the 1H and apparent 15N frequencies respectively. The constant-acceleration prior is appropriate because backbone amide group spins shift temperature coefficients are approximately constant over moderate temperature ranges (4), producing trajectories that are nearly linear or slightly curved in frame index. A higher-order dynamical model would generate systematic prediction errors in the initial frames before the velocity state is well estimated, causing data-association failures precisely where they are most costly. The filter predicts each peak’s next position and updates upon detection; bidirectional initialization reduces track fragmentation at series endpoints. .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint The process noise covariance 7g x9 and measurement noise covariance 7g x9| are tuned from the empirical distribution of per-step peak displacements in a held-out calibration subset of the ubiquitin dataset. For each accepted detection, the innovation (residual between predicted and measured position) is retained as a diagnostic: large, persistent innovations flag residues with non-linear temperature dependence, which are candidates for higher-order modelling or flagging as exchange-broadened. Graph-based S /g2878 /S/g2879 pairing and frequency recovery Trajectory fragments from the S/g2878 and S/g2879 channels are first consolidated using a union-find data structure (17, 18) to group frame-level detections that belong to the same continuous trajectory. Pairing across channels exploits the constraint from Eq. (1): candidate S/g2878 /S/g2879 pairs are scored by the smoothness of their sum trajectory η /g4666/g2921/g4667,/g2878 7gω666n7gω66x 7g339x η /g4666/g2921/g4667,/g2879 7gω666n7gω66x , which should follow a smooth, near-linear temperature dependence consistent with a pure 15N chemical shift trajectory. Pairings satisfying this smoothness criterion above a confidence threshold are retained as high-confidence assignments and used to bootstrap recovery of three-nucleus chemical shift trajectories via Eqs. (2) and (3). The recovery is lossless when β7g3ω,a0 . In the reported series (Fig. 1) the 13C spectral width SWH /g2887 7gω666n7gω66x and 15N spectral width SWH /g2898 7gω666n7gω66x were modulated as functions of the frame index n across the 40- step temperature ramp (279–318 K, 1 K increments). Modulation was subject to the reciprocal-sum constraint 1/SWH /g2898 7gω666n7gω66x 7g339x 1/SWH /g2887 7gω666n7gω66x 7g3ω,ω C /g2913/g2925/g2924/g2929/g2930 (Eq. 8). This constraint maintained consistent digital resolution while enabling monotonic variation of the mixing coefficient β7gω666n7gω66x throughout the series. All expected trajectories were successfully recovered from the dataset. Figure 6 reports reconstructed cross-peak trajectories encoding the detailed temperature dependencies of all backbone resonances in the test protein. Construction of optimal VT RD-HSQC Peak Trajectories To study optimal construction of peak trajectories in RD-HSQC experiments, we simulated the evolution of cross-peak positions in temperature-resolved /g2869 H–/g2869/g2873 N HSQC spectra of proteins with variable controls. Simulations were performed for temperature values ranging from 279 K to 315 K with a step of 1 K. At each temperature point the spectral coordinates of all peaks were calculated using backbone chemical shifts obtained from BMRB data. In these simulations each residue generates a pair of peaks whose coordinates evolve as temperature and controls change. Figure 5 shows the trajectories of these peaks through a sequence of temperature reprsented as moving points in a two–dimensional spectral plane expressed in frequency units (Hz). The detailed description of the simulations including temperature effect and trajectory controls is described in the Supplementary Information (section 1). .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint We formalize trajectory quality as a scalar scoring function over the space of acquisition schedules. Three criteria contribute: (i) repulsion—mean nearest- neighbour separation across the series, computed in the 2D observed spectral plane 7gω666η /g3399 ,Δ ν /g2892 7gω66x; (ii) step displacement—frame-to-frame displacement variance, penalising schedules that produce large sudden jumps that would break tracker continuity; and (iii) boundary compliance—fraction of trajectory points within the spectral window, penalising configurations where peaks alias outside the acquired region and (iv) trajectories collision resolution. These criteria are combined into a single differentiable objective evaluated on simulated trajectories generated from Eq. (6) with chemical shifts statistics drawn from protein sequence databases (4). This formalization permits principled optimization of the trajectories controls and, critically, enables agentic control of the design loop: the scoring function defines what constitutes a good acquisition schedule, and an autonomous agent traverses the low-dimensional parameter space ( a /g2903/g2907/g2887 , v /g2903/g2907/g2887 , SWH /g2887,/g2868 , and ω /g2887 /g3209) to maximise it. Example optimised trajectories are shown in Figure 5, with penalty components evaluated throughout the temperature sweep. Novel RDL-TROSY experiment. We developed a reduced-dimensionality longitudinal relaxation optimized experiment, denoted RDL-TROSY (Fig. 3), to maximize spectral resolution and controllability of cross-peak trajectories in temperature-resolved 2D [¹H,¹ /i2 N] correlation spectra. In this sequence, excitation is restricted to amide protons or water and aliphatic protons using band-selective ¹H pulses, enhancing longitudinal recovery of 1HN polarization. The key design feature is that the entire period during which ¹/i2 N magnetization resides in the transverse plane is utilized for chemical shift encoding, thereby achieving maximal effective evolution time and, consequently, the highest attainable resolution in the indirectly detected dimension. Reduced-dimensionality encoding is implemented through controlled transfer between ¹ /i2 N and ¹³Cα spins, producing the characteristic S(+) and S(–) components. Importantly, the sequence incorporates weak selective RF irradiation on the ¹³C channel during the ¹/i2 N→¹³ C transfer period, or simultaneously on both ¹³C and ¹/i2 N channels, enabling targeted attenuation of specific cross-peaks. This feature allows dynamic suppression of selected resonances at defined time points, providing an experimental handle to resolve trajectory collisions and cross-peak occlusions in a temperature series. In contrast to conventional RD-HSQC implementations derived from folding the ¹³C α dimension of a 3D HNCA experiment into the ¹/i2 N dimension, the RDL- TROSY sequence is specifically optimized for longitudinal relaxation and maximal encoding efficiency. Standard RD-HSQC inherits limitations from the parent 3D experiment, including fragmented evolution periods and reduced effective evolution time for ¹ /i2 N, which constrain achievable resolution and lead to less controllable trajectory geometry. Furthermore, conventional approaches .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint lack mechanisms for selective, time-resolved attenuation of individual peaks, making them vulnerable to peak overlap during trajectory evolution. By combining full-length ¹/i2 N evolution, TROSY-based relaxation optimization, and programmable weak selective irradiation, RDL-TROSY enables both improved spectral resolution and active manipulation of cross-peak trajectories. This enhanced control is essential for avoiding trajectory collisions and for enabling downstream computational strategies, such as reinforcement learning–guided optimization of acquisition parameters. Selective control of the trajectories at the collision point. The ability of the RDL-TROSY experiment to selectively attenuate individual cross-peaks in the S ⁺ and S- spectra is demonstrated in Figure 4. Panel (A) shows an overlay of two S⁺ spectra acquired with near zero (red) and maximal (black) weak RF irradiation applied on the 13C channel at a carrier position centered at 60 ppm, corresponding to the Cα chemical shift of Ile31. Under these conditions, a single targeted cross-peak is efficiently and selectively suppressed, while the remainder of the spectrum remains largely unaffected. This highlights the high residue specificity of the weak RF irradiation scheme, which exploits the narrow-band excitation profile to address individual resonances without perturbing neighboring peaks. The absence of noticeable distortions in surrounding signals confirms that the applied RF field of 28 Hz is sufficiently weak to avoid global perturbation while still achieving complete attenuation of the selected resonance. Figure 4B further quantifies this effect by showing one-dimensional slices through the suppressed peak and two neighboring reference peaks across a series of increasing RF irradiation strengths (0–143 Hz). The intensity of the targeted peak decreases smoothly and monotonically with increasing RF power, ultimately reaching near-complete suppression, whereas the flanking peaks remain largely unchanged across the entire power range. This continuous and controllable attenuation demonstrates that weak RF irradiation can be used as a finely tunable parameter to modulate peak intensities in a predictable manner. The attenuation effect of the weak rf-irradiation was extensively numerically modeled (Supplementary material, section 4) confirming experimentally observed attenuation levels (Fig. 3S). Interestingly, our simulations show that to almost complete and selective peak intensity attenuation can be achieved with a combined weak 15N and 13C irradiation of only 5 to-10 Hz in the field strength (Fig. 4S) creating a very precise selection of the targeted trajectory. Importantly, this introduces a new experimental control dimension for trajectory optimization: selective, time-resolved masking of specific resonances. In the context of AI-driven trajectory construction, such weak RF interventions can be formulated as discrete, low-cost actions that temporarily suppress colliding peaks, enabling the agent to resolve trajectory overlaps without altering the underlying spectral encoding. This capability provides a direct experimental counterpart to .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint algorithmic “masking” operations and significantly enhances the feasibility of agentic control over peak trajectories in complex spectra.

In this work, we introduce a conceptual and experimental framework that redefines variable-temperature NMR as a spatiotemporal problem in which cross- peaks evolve along a controllable trajectory axis. Rather than treating temperature as a perturbation to be minimized or corrected, we exploit it as a structured, physically meaningful variable that encodes additional information into the spectral domain. In combination with reduced-dimensionality encoding, this transforms a series of 2D spectra into a trajectory-resolved dataset in which otherwise overlapping resonances can be separated through their differential motion. In contrast to conventional multidimensional NMR, where additional spectral resolution is obtained by extending Fourier dimensions at the cost of acquisition time, our approach redistributes dimensionality into a controllable external variable, enabling efficient extraction of multi-nuclear chemical shift information within a comparable experimental time. This framework provides both an experimental and computational advantage. On the experimental side, modulation of the RD mixing parameter and spectral widths allows active steering of cross-peak trajectories, converting latent ¹³C chemical shift diversity into time-dependent separability. The RDL-TROSY experiment further maximizes the achievable resolution by fully utilizing the ¹ /i2 N transverse evolution period and introduces a novel control dimension through weak selective RF irradiation. This selective attenuation acts as a precise, residue-specific intervention that resolves trajectory collisions without perturbing the global spectral encoding. On the computational side, the trajectory formulation enables direct application of computer vision methodologies, including video super-resolution and multi-object tracking, which leverage temporal redundancy to recover information lost in individual frames. Together, these elements establish a unified framework in which acquisition design and data analysis are intrinsically coupled. A key implication of this work is that spectroscopic dimensionality need not be restricted to traditional frequency axes. External variables such as temperature, pH, ionic strength, or ligand concentration can be incorporated as controlled trajectory-generating dimensions, provided that their effects on chemical shifts are sufficiently systematic. This generalization suggests a broader paradigm of “trajectory-encoded spectroscopy,” in which the experiment is designed to produce maximally informative motion of resonances rather than static separation in high-dimensional frequency space. Within this paradigm, the information content arises not only from instantaneous peak positions but also from their evolution, effectively introducing derivative constraints that improve identifiability of overlapping signals. The formulation of trajectory quality as a differentiable objective function further enables automated optimization of acquisition parameters. In this study, we .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint demonstrate that global controls such as carrier frequency and spectral width can be tuned to maximize peak separability while maintaining smooth, trackable motion. Importantly, the addition of selective RF attenuation introduces a discrete control mechanism that can be used to resolve transient peak overlaps. This naturally lends itself to a reinforcement learning formulation, in which an agent iteratively adjusts acquisition parameters and selective interventions to maximize a global reward function. Unlike conventional automation, which executes predefined protocols, such an agentic framework directly influences the physical experiment in real time, enabling adaptive and protein-specific optimization of the acquisition process. Several limitations should be noted. The approach relies on approximately linear temperature dependence of chemical shifts and requires that the protein remains structurally stable over the sampled temperature range. Strong conformational exchange, unfolding transitions, or non-linear drift may complicate trajectory modeling and tracking. In addition, the requirement for dense sampling across temperature introduces a trade-off between temporal resolution and total acquisition time, although this is partially mitigated by the use of efficient 2D experiments. Finally, while the current implementation focuses on backbone assignment, extension to side-chain correlations or distance restraints will require further development of both pulse sequences and trajectory models. Overall, the present work establishes a bridge between NMR spectroscopy, trajectory-based signal encoding, and modern computer vision methodologies. By integrating experimental control, physical modeling, and data-driven analysis within a unified framework, it opens a pathway toward adaptive, agent-guided NMR experiments in which acquisition and interpretation are co-optimized. This perspective suggests that future developments in NMR may increasingly rely not only on improved hardware or pulse sequences, but also on intelligent control strategies that actively shape the information content of the experiment.

We have described a framework that reconceptualizes the VT-NMR experiment as a video processing problem and demonstrated concrete consequences of that reconceptualization for cross-peak resolution and residue assignment. The temperature axis, conventionally treated as a complication, is recast as a pseudo-temporal axis along which inter-frame pooling resolves peaks irresolvable in any individual spectrum. The RD-HSQC is the natural vehicle: its explicit 13C/g2961 encoding provides spectral, temporal, and topological discrimination simultaneously, and its exact mathematical form provides the basis for principled acquisition design. The agnostic and model-based strategies are complementary: the former improves per-frame quality without physical assumptions; the latter converts latent 13C/g2961 diversity into engineered inter-trajectory separation through spectral- width modulation. A formal scoring function, a selective intensity-control pulse sequence element, and the agentic design loop together define a path toward .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint fully automated acquisition optimisation. The framework is protein-independent; its inputs are backbone chemical shift databases and the target primary sequence, and its output is an optimised acquisition schedule calibrated to the specific assignment problem at hand.

Human ubiquitin 15N/13C uniformly labeled was purchased from Sigma-Aldrich (MilliporeSigma). NMR spectra were measured using Bruker Avance III NMR spectrometer operating at 700 MHz magnetic field strength. Specgra were referenced relative to the internal DSSThe RD-HSQC was acquired quadrature selective 13C/g2961 editing, yielding separate S/g2878 and S/g2879 free induction decays (FIDs) by acquiring the in-phase (S /g2919/g2924 ) and antiphase (S /g2911/g2924/g2930/g2919 ) spectra in an interleaved fashion (10, 11). The temperature ramp spanned 279–318 K in 1 K increments (40 steps); one complete RD-HSQC was acquired per step following 1 min thermal equilibration. Temperature calibration was performed using the standard methanol sample (2). Raw Bruker TopSpin free induction decays (FIDs) were apodised (cosine-bell in t /g2869 , exponential in t /g2870 ), zero-filled to twice the acquisition size in both dimensions, Fourier transformed, and phased. S/g2878 and S/g2879 subspectra were computed by addition and subtraction of the S /g2919/g2924 and S /g2911/g2924/g2930/g2919 spectra. Both channels were assembled into video sequences at 4 fps with 1:1 array-to-pixel grayscale encoding and normalised by multiplicative 95th-percentile scaling with additive mean alignment. Inter-frame consistency was assessed by the structural similarity index measure (SSIM) (20). Agnostic super-resolution, per-frame peak detection, Kalman tracking, graph-based classification, and frequency-coordinate recovery were implemented in Python 3.10 with graphics processing unit (GPU) acceleration (CUDA 11.8). Figure 1. Variable-temperature RD-HSQC spectra of human ubiquitin (279– 318 K) with linearly modulated spectral widths. Forty contour plots are overlaid for each subspectrum panel, with temperature encoded by a continuous color gradient from blue (279 K, frame 1) to red (318 K, frame 40). (a) S /i2 and (b) S/i2 spectral plane overlays, each containing one cross-peak per residue. The S/i2 and S/i2 components are complementary spin-state-selective channels produced by the echo-antiecho editing of the RD-HSQC pulse sequence: at each temperature step, the S/i2 /S/i2 peak pair for residue k is split symmetrically about the position of the conventional ¹/i2 N–¹H HSQC cross-peak, with splitting magnitude proportional to the residue's ¹³Cα rotating-frame offset according to |Δ/i2/i2/i2 _RD(n)| = 2β (n)|Δν/i2/i2/i2 _C(n)| (Eq. 26), where β (n) = SWH_N(n)/SWH_C(n) is the frame-dependent mixing coefficient. Paired S/i2 /S/i2 trajectories exhibit mirror-image motion in the mixed ¹/i2 N (RD) dimension, and their sum and difference recover the original per-residue ¹/i2 N and ¹³Cα chemical .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint shift trajectories via the closed-form inversion Eqs. 28–30, confirming that the RD encoding is lossless when β ≠ 0. Figure 2. Peak model-independent video super-resolution reconstruction of RD-HSQC spectra of human ubiquitin. Panels (a)–(f) each show a cropped region of interest from the 2D RD-HSQC spectral window as a two-spectrum overlay: the raw spectrum (red contours), as processed directly in the Bruker TopSpin v4.05 following standard phase correction and baseline correction, is overlaid with the reconstructed spectrum (blue contours) produced by the full spatio-temporal processing pipeline. The six panels sample a descending temperature sequence from (a) 300 K to (f) 294 K in 1 K steps, so that the progression from panel (a) to (f) corresponds to cooling and the associated upfield drift of backbone amide resonances brings a pair of initially resolved cross-peaks progressively closer together and into partial overlap. Figure 3: Experimental scheme for the Reduced-Dimensionality Longitudinal 1H relaxation optimized 2D [1H,15N]-TROSY (RDL-TROSY). The radio 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 4.7, 118, 55, 174 ppm, respectively. Narrow and wide black bars indicate nonselective 90° and 180° pulses, respectively. The 1H shaped pulses are: 7g|,3y /g2869 -/g2872 , 1H/g3015 5.5–10.0 ppm band-selective 1.5 ms excitation E-Burp2 pulses with the phases 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 7g|,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 ppm band-selective 1.8 ms refocusing Re-Burp pulses with the phase 7gω66y7g yx67gω669 . The 13C shaped pulses are 1.3 ms Gaussian cascade Q3 (dark shapes) and Q5 (light shapes) with the phases 7g|,3y /g2869/g2868 7g3ω,ω 7gω66y7g339y7g yx67gω66687gω66x, 7g yx67gω66687gω66x7gω669 , 7g|,3y /g2869/g2869 7g3ω,ω 7gω66y7g339y7g yx67gω66687gω66x, 7g yx67gω66687gω66x7gω669 and 7gω66y7g339y7g yxx7gω66687gω66x, 7gω66687gω66x7gω669 for the reduced-dimensionality in-phase and antiphase spectra, 7g yωa /g3036/g3041 and 7g yωa /g3028/g3041/g3047/g3036 , respectively. The spectra 7g yωa7gω6667g339x7gω66x and 7g yωa7gω6667g339y7gω66x are reconstructed by adding and subtracting 7g yωa /g3036/g3041 and 7g yωa /g3028/g3041/g3047/g3036 , respectively. The phases 7g|,3y /g2868,/g2875 -/g2877 are 7gω66y 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 echo-anti-echo transverse relaxation optimised spectroscopy (TROSY) phase discrimination 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 are 7g yx|7g3ω,ω2 . 7 ms and 7g yω67g3ω,ω3 0 ms. The pulsed field gradients (PFGs) are G1: 80 G/cm; G2: 95 G/cm; G3: 70 G/cm. The colored shapes represent narrow band- selective phase-modulated radio-frequency irradiation with the radio-frequency (RF) field strength of 7g|,| /g2869,/g3015 and 7g|,| /g2869,/g3004 with the corresponding offsets defined in the text. To generate cross-peak trajectories, the 2D RDL 7g yωa 7gω6667g339x7gω66x and 7g yωa 7gω6667g339y7gω66x planes are measured with the temperature range (279–315 K) and variable settings of the .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint 13C carrier frequency 7g|,33 /g3004 /g3328, and the corresponding spectral width 7g yωa7g yω97g y3ω /g3015 and 7g yωa7g yω97g y3ω /g3004 . The dashed box in the middle indicates the part of the pulse sequence used in the numeric simulations of the propagation of the density operator of two 7g y36- coupled spins 13C and 15N under the effects of the static Hamiltonian and the phase modulated weak RF irradiation Figure 4: Selective and tunable cross-peak attenuation in RD-HSQC. (a) Overlay of two S/g2878 spectra acquired with maximum (black) and minimum (red) RF irradiation at a targeted 13C offset. Only one peak (arrowed) is completely suppressed at using weak 13C radio-frequency irradiation 7g|,| /g2869,/g3004 = 28 Hz centered at 60 ppm (CA chemical shift of Ile31), demonstrating residue-specific selectivity. (b) 1D slices through the suppressed peak (center) and two reference peaks (flanking) across the power series 7g|,| /g2869,/g3004 = {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 blue to yellow in the corresponding order), showing progressive attenuation of the target peak intensity as power increases. Figure 5: RD-HSQC peak trajectories generated using near-optimal control parameters. Each colored curve represents the temperature-dependent trajectory of a backbone amide resonance of ubiquitin (76 residues) across the simulated temperature range (279–315 K). The RD encoding produces two spectral planes (top and bottom panels), corresponding to the 7g yωa 7gω6667g339x7gω66x and 7g yωa 7gω6667g339y7gω66x components arising from folding of the 13C/g3080 chemical shift into the indirect dimension. The trajectories are actively reshaped through global experimental controls acting on the 13C carrier frequency (7g|,33 /g3004 /g3328) and the 13C spectral width (7g yωa7g yω97g y3ω /g3004 ), whose rates and accelerations are dynamically updated with temperature while maintaining the reciprocal-sum constraint 1/7g yωa7g yω97g y3ω /g3015 7gω6667g y66 7gω66x 7g339x 1/7g 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 7g|,33 /g3004 /g33287g3ω,ω 57.65 ppm with rate 0.39 ppm/K and acceleration 0.0187 ppm/K2, and 7g yωa7g yω97g y3ω /g3004 7g3ω,ω 2617.8 Hz with rate 7g339y196.8 Hz/K and acceleration 9.86 Hz/K2. Insets show the evolution of representative penalty components used to evaluate trajectory optimality during the temperature sweep, including spread, step-size, boundary, and collision terms. These penalty functions quantify peak separation, smoothness of motion, distance from spectral boundaries, and avoidance of peak overlaps, respectively, and together define the objective function used for trajectory optimization. Figure 6: Automated S+/S− trajectory pairing via iterative RMSD bootstrapping. Each colored trajectory represents a matched S+/S− trajectory pair, with paired trajectory sharing the same color. Grey trajectories in the

indicate unmatched candidate islands. Pairing was performed by iterative Hungarian assignment with a progressively relaxed RMSD threshold (starting from 0, incremented by 0.001 per round over 20,000 iterations). The .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint RMSD serves as the sole matching criterion, exploiting the physical constraint that the ¹H chemical shift drift (Δν _H) is identical for the S+ and S− peaks of the same residue, producing coincident horizontal trajectories across the temperature series.

We thank Prof Vladislav Orekhov for fruitful early discussions and his advice for the project.

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It is made The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint .CC-BY 4.0 International licenseavailable under a (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 The copyright holder for this preprintthis version posted March 23, 2026. ; https://doi.org/10.64898/2026.03.19.712888doi: bioRxiv preprint