Impact of noise on inverse design: the case of NMR spectra matching

Abstract

Despite its fundamental importance and widespread use for assessing reaction success in organic chemistry, deducing chemical structures from nuclear magnetic resonance (NMR) measurements has remained largely manual and time consuming. To keep up with the accelerated pace of automated synthesis in self driving laboratory settings, robust computational algorithms are needed to rapidly perform structure elucidations. We analyse the effectiveness of solving the NMR spectra matching task encountered in this inverse structure elucidation problem by systematically constraining the chemical search space, and correspondingly reducing the ambiguity of the matching task. Numerical evidence collected for the twenty most common stoichiometries in the QM9-NMR database indicate systematic trends of more permissible machine learning prediction errors in constrained search spaces. Results suggest that compounds with multiple heteroatoms are harder to characterize than others. Extending QM9 by ∼10 times more constitutional isomers with 3D structures generated by Surge, ETKDG and CREST, we used ML models of chemical shifts trained on the QM9-NMR data to test the spectra matching algorithms. Combining both ¹³C and ¹H shifts in the matching process suggests twice as permissible machine learning prediction errors than for matching based on ¹³C shifts alone. Performance curves demonstrate that reducing ambiguity and search space can decrease machine learning training data needs by orders of magnitude.

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1. Introduction

Current development times of novel molecular materials can span several decades from discovery to commercialization. In order for humanity to react to global challenges, the digitization^4–8 of molecular and materials discovery aims to accelerate the process to a few years. Long experiment times severely limit the coverage of the vastness of chemical space, making the development of self driving laboratories for autonomous robotics experimentation crucial for high throughput synthesis of novel compounds (Fig. 1a)).^9–15 To keep the pace of automated synthesis, fast and reliable characterization of reaction products through spectroscopic methods is required, an often manual, time intense and possibly error prone task. One of the most common methods to elucidate the structure of reaction products are nuclear magnetic resonance (NMR) experiments.¹⁶ Through relaxation of nuclear spins after alignment in a magnetic field, an NMR spectrum, characteristic of local atomic environments of a compound, i.e. functional groups, can be recorded. In particular, ¹H and ¹³C NMR experiments are routinely used by experimental chemists to identify the chemical structure or relevant groups just from the spectrum. For larger compounds, however, the inverse problem of mapping spectrum to structure becomes increasingly difficult, ultimately requiring NMR of additional nuclei, stronger magnets, or more advanced two-dimensional NMR experiments.^17,18

Fig. 1 Schematic workflow for autonomous chemical discovery as well as scaling of constitutional isomer space versus data availability in the QM9 (ref. 1) database. (a) After the chemical synthesis of molecular compounds, reaction products are characterized using spectroscopic methods such as nuclear magnetic resonance (NMR). The measured ¹H and ¹³C spectra are automatically processed and potential candidate structures suggested via machine learning. (b) Number of constitutional isomers for 20 stoichiometries considered.

Computer-assisted structure elucidation algorithms aim to iteratively automatize the structure identification process.^19–23 Current workflows include repeated predictions of chemical shifts for candidate structure inputs through empirical or ab initio methods.^24–26 Albeit accurate even in condensed phase through use of plane-waves²⁷ or QM/MM setup,²⁸ the cost of density functional theory (DFT) calculations severely limits the number of candidate structures that can be tested, leaving the identification of unknown reaction products out of reach for all but the smallest search spaces. Data driven machine learning models leveraging experimental or theoretical NMR databases^29–32 provide orders of magnitude of speedup over ab initio calculations, reaching 1–2 ppm ¹³C mean-absolute-error (MAE) w.r.t. experiment or theory, respectively.^31,33–38 However, while the stoichiometry of the reaction product is usually known, e.g. through prior mass spectrometry experiments, the number of possible constitutional isomers exhibits NP hard scaling in number of atoms, quickly spanning millions of valid molecular graphs already for molecules of modest size (Fig. 1b)). As such, the inverse problem of inferring the molecular structure from an NMR spectrum still poses a major challenge even for rapid solvers.

Recent machine learning approaches tackle the inverse problem using a combination of graph generation and subsequent chemical shift predictions for candidate ranking.^39–41 First explored by Jonas,³⁹ a Top-1 ranking with 57% reconstruction success-rate was achieved using deep imitation learning to predict bonds of molecular graphs. Sridharan et al.⁴¹ used online Monte Carlo tree search to build molecular graphs resulting in a similar Top-1 ranking of 57.2%. Huang et al.⁴⁰ relied on substructure predictions from which complete graphs can be constructed, reaching 67.4% top-1 accuracy by ranking substructure profiles instead of shifts. A commonality between all algorithms is the subsequent ranking of candidates using spectra matching or other heuristics. Consequently, even though the correct query compound could be detected early, similar candidates might be ranked higher, making the ranking process as critical as the candidate search itself.

In this work, we analyse the effectiveness of the NMR spectra matching task encountered in the inverse structure elucidation problem. As stagnating improvements²⁶ in chemical shift predictions due to limited public NMR data aggravate candidate rankings, results suggest that both the prediction error of machine learning models and the number of possible candidates are crucial factors for elucidation success. By systematically controlling the size of chemical search space and accuracy of chemical shifts, we find that higher error levels become permissible in constrained search spaces. Moreover, results indicate that increasing the uniqueness through including both ¹³C and ¹H shifts in the matching process, rather than relying on a single type of shift, significantly reduces ambiguity and enhances error tolerance. To evaluate the spectra matching task throughout chemical compound space, we systematically control the accuracy of 1D ¹³C and ¹H chemical shifts of the 20 most common stoichiometries in QM9-NMR^1,31 by applying distinct levels of Gaussian white noise. Note that while we focus on DFT based 1D NMR in this work, future studies could include experimental data and 2D NMR information. Comparisons amongst stoichiometries suggest that chemical spaces with increasing amounts of heteroatoms and number of constitutional isomers are harder to characterize than others. To test the spectra matching method on a large search space, we extended QM9-NMR to 56 k C₇O₂H₁₀ constitutional isomers. Controlling the chemical shift accuracy through machine learning models trained at increasing training set sizes, performance curves again indicate a trade-off between search space and accuracy. Hence, as less accurate shift predictions become useful, results show that machine learning training data needs can be reduced by multiple orders of magnitude.

2. Theory & methods

2.1. NMR spectra matching

Consider a query ¹³C or ¹H spectrum with a set of N possible candidate constitutional isomer spectra. We chose the squared euclidean distance as a metric to rank candidate spectra against the query spectrum (see ESI Fig. 3† for comparison against other metrics):

With δ being a sorted spectrum of n chemical shifts (¹³C or ¹H), q being the query, i being the i-th of N candidates, and j being the j-th chemical shift in a spectrum, respectively. To use both ¹³C and ¹H shifts simultaneously for spectra matching, a total distance can be calculated as follows:

with γ = 64 being a scaling factor determined via cross-validation (see ESI Fig. 1†) to ensure similar weighting. Final rankings are obtained by sorting all candidates by distance. The top-1 accuracy is calculated as the proportion of queries correctly ranked as the closest spectrum, respectively.

2.2. Elucidation performance curves

To analyse the spectra matching elucidation accuracy, we systematically control the number of possible candidates N and the accuracy of chemical shifts, respectively. For each constitutional isomer set, we choose 10% as queries and 90% as search pool, respectively. Next, we randomly sample N spectra from the search pool, including the query spectrum. Each sample size is drawn ten times and the top-1 accuracy averaged across all runs. To control the accuracy of chemical shifts, we apply Gaussian white noise (up to 1 or 10 σ for ¹H and ¹³C, respectively) or use the machine learning error as a function of training set size (c.f. ESI Fig. 5† for learning curves). For each N and chemical shift accuracy, results are presented as elucidation performance curves, showing the elucidation success as a function of chemical shift accuracy in terms of mean absolute deviation (MAD) for Gaussian noise (c.f.Fig. 2a and b)) or mean absolute error (MAE) for machine learning predictions (c.f.Fig. 4).

Fig. 2 Elucidation performance curves of C₇O₂H₁₀, C₅N₃OH₇, C₈OH₁₄ spectra using Gaussian noise to control chemical shift accuracy in terms of mean absolute deviation (MAD) corresponding to of the standard deviation.² (a) and (b) ¹³C and ¹H spectra matching. Individual points were obtained by calculating the percentage of queries where noisy and noise free query spectra have the lowest distance. All points have been fitted using eqn (3). Solid curves correspond to candidate numbers N_QM9 from QM9.¹ Dashed curves are an extrapolation to candidate numbers N_Surge as obtained via graph enumeration.³ The legend corresponds to both (a) and (b), respectively. (c) Spectra matching using both ¹H and ¹³C shifts. Dashed lines correspond to the accuracy required to correctly elucidate 95% of queries when only ¹H or ¹³C spectra are being used, respectively.

2.3. Chemical shift prediction

We relied on kernel ridge regression (KRR) for machine learning ¹³C and ¹H chemical shifts as presented in ref. 31 and commonly being used in learning NMR properties from quantum chemical calculations.^37,42–46 We use a Laplacian kernel and the local atomic Faber–Christensen–Huang–Lilienfeld (FCHL19 (ref. 47)) representation with a radial cutoff³¹ of 4 Å. The kernel width and regularization coefficient have been determined through 10-fold cross-validation on a subset of 10′000 chemical shifts of the training set. Note that while we relied on KRR within this work, other NMR shift estimation methods could have been used such as Hierarchically ordered spherical environment (HOSE) codes^48,49 or neural network based approaches.^50–53

2.4. Data

The QM9-NMR^1,31 dataset was used in this work, containing 130′831 small molecules up to nine heavy atoms (CONF) with chemical shieldings at the mPW1PW91/6-311+G(2d,p)-level of theory. We used the 20 most common stoichiometries (Fig. 1b)), having a minimum of 1.7 k constitutional isomers available in the dataset.

To extend the QM9-NMR C₇O₂H₁₀ constitutional isomers space, we used the systematic graph enumeration software Surge³ to generate 54′641 SMILES. 3D geometries of all SMILES have been generated using the ETKDG⁵⁴ method in RDKit. Lowest lying conformer structures were sampled using the CREST⁵⁵ algorithm, using the GFN2-xTB/GFN-FF composite method in a meta-dynamics based sampling scheme, with a final relaxation at the GFN2-xTB level. Adding all successfully generated structures to QM9, a total pool size of 56.95 k C₇O₂H₁₀ isomers was obtained.

For the training of chemical shift machine learning models, we selected C₈OH₁₂, C₈OH₁₀, C₈OH₁₄, C₇O₂H₈ and C₇O₂H₁₂ constitutional isomers, yielding a total of 143 k ¹³C and 214 k ¹H training points, respectively.

3. Results & discussion

3.1. Spectra matching accuracy with synthetic noise

To analyse the influence of noise and number of candidates on the elucidation success, we applied Gaussian noise to ¹³C and ¹H shifts of C₇O₂H₁₀, C₅N₃OH₇ and C₈OH₁₄ constitutional isomers, respectively. Fig. 2a and b) depicts a sigmoidal shaped trend of top-1 elucidation performances as a function of mean absolute deviation (MAD) corresponding to of the standard deviation² caused by applying the Gaussian noise. Note that increasing the maximum candidate pool size N_QM9 leads to an offset of the trend towards less permissible errors. A possible explanation is the correlation of the density of chemical space with increasing numbers of candidate spectra N.⁵⁶ As shift predictions need to become more accurate, limiting N through prior knowledge of the chemical space could be beneficial. Similar findings have been reported by Sridharan et al.,⁴¹ noting that brute force enumerations of chemical space lead to worse rankings than constrained graph generation. Note that while the trends in ¹³C and ¹H elucidation are similar, less error is permissible when using ¹H shifts.

To further reduce the ambiguity, we include both ¹³C and ¹H shifts into the matching problem as per eqn (2). Results suggest 50% and ∼150% more permissible ¹³C and ¹H errors when both spectra are considered in the matching process (Fig. 2c)). Similar to how chemists solve the elucidation problem, the inclusion of more distinct properties increases the uniqueness and can improve the elucidation success.

3.2. Extrapolating the search space

Due to the limited amount of constitutional isomers in databases compared to the number of possible graphs faced during inverse design (Fig. 1b)), assessing the chemical shift accuracy for successful elucidation is severely limited. As such, we extrapolate elucidation performance curves to obtain estimates about chemical shift accuracies in candidate pool sizes larger than QM9. We fit each elucidation performance curve (Fig. 2a and b)), respectively, using a smoothly broken power law function:

With x_b controlling the upper bend and offset, d changing the curvature and α changing the tilt of the function (see ESI Fig. 2†), respectively. The parameters of eqn (3) as a function of N can again be fitted using a power law function (see ESI Fig. 2†) and extrapolated to the total number of graphs N_Surge, respectively.

Results of the extrapolation (Fig. 2a and b) dashed) indicate significant differences in elucidation efficiency among stoichiometries. For instance, C₈OH₁₄ queries are potentially easier to elucidate than C₅N₃OH₇ structures. Possible reasons are the limited number of C₈OH₁₄ graphs compared to millions of C₅N₃OH₇ isomers. Moreover, the number of heteroatoms of the C₅N₃OH₇ stoichiometry might hamper the characterization when only relying on ¹³C or ¹H, respectively. Hence, to solve the inverse structure elucidation problem using experimental data of compounds larger than QM9, reducing ambiguities through including both ¹³C and ¹H shifts as well as to reduce the candidate space is critical for elucidation success.

3.3. Trends in chemical space

To analyse the elucidation efficiency throughout chemical space, we applied the Gaussian noise and extrapolation procedure to the 20 most common stoichiometries in QM9 (Fig. 1b)). Fig. 3a) shows the MAD required for 95% elucidation success as a function of N_Surge. Results suggest that less error is permissible for stoichiometries with large N_Surge and fewer carbon atoms. As such, using only ¹³C shifts might not be sufficient to fully characterize the compound. Again, similar to how chemists use multiple NMR spectra to deduct chemical structures, additional information such as ¹H shifts are beneficial to extend the information content.

Fig. 3 Trends in QM9 (ref. 1) chemical compound space to correctly elucidate queries at 95% accuracy. The mean absolute deviation (MAD) of the standard deviation.² (a) Extrapolated MAD at candidate numbers N_Surge of the 20 most common stoichiometries in QM9.¹ (b) MAD using only ¹³C spectra (¹³C_single) against ¹³C and noise-free ¹H spectra combined (¹³C_combined) at candidate numbers N_QM9 from QM9.¹

Fig. 4 Elucidation accuracy of C₇O₂H₁₀ spectra using machine learning ¹³C and ¹H shift predictions. Mean absolute error (MAE) refers to the predictive accuracy of the machine learning models, respectively. (a) and (b) ¹³C and ¹H spectra matching at increasing search pool sizes N. The inset depicts the decay of the elucidation accuracy of the best performing machine learning model at increasing levels of Gaussian noise on query spectra (MAE_Q). (c) and (d) Spectra matching accuracy when restricting the search pool to contain only known functional groups. The inset in (d) depicts the search pool size N restricted to compounds with similar functional groups as the query, respectively. (e) Spectra matching using ¹H and ¹³C shifts combined. (f) Accuracy required to reach 85% correct elucidation at increasing N when using both ¹H and ¹³C shifts combined.

In Fig. 3b), the error permissiveness of spectra matching using only ¹³C (see ESI Fig. 4† for ¹H) versus combining both ¹³C and ¹H is being compared, revealing a linear trend between both. Note that the C₇NOH₇ stoichiometry shows the smallest benefit from adding additional ¹H information. Interestingly, a hierarchy for C₇NOH_X stoichiometries of different degrees of unsaturation is visible, indicating an inverse correlation between number of hydrogens and ¹³C_single MAD (Fig. 3b) green). Similar hierarchies are also observed for other stoichiometries such as C₇O₂H_X and C₈OH_X (Fig. 3b) blue and orange). On average, the combination of ¹³C and ¹H for spectra matching increases the error permissiveness of ¹³C and ¹H by 85% and 261% (see ESI Fig. 4†), respectively.

3.4. Comparison to machine learned shift predictions

To test the elucidation performance using machine learning predictions, we trained ¹³C and ¹H KRR models at increasing training set sizes (see ESI Fig. 5† for learning curves) and predicted chemical shifts of 56 k C₇O₂H₁₀ constitutional isomers. Note that within this proof of concept application we rely on xTB-GFN2 relaxed geometries as queries, which on average are within 0.06 Å RMSD of C₇O₂H₁₀ B3LYP level of theory structures.⁵⁷ Results again show similar trends as observed with Gaussian noise (Fig. 4a and b)), however, indicate more permissive accuracy thresholds. For instance, KRR ¹³C predictions at 2 ppm MAE can identify 64% of queries rather than only 17% suggested by the Gaussian noise experiment. The difference could be explained due the systematic, non uniform nature of the QM9 (ref. 1) chemical space, influencing the shape and extrapolation of elucidation performance curves in Fig. 2. Moreover, Gaussian noise is applied to all shifts at random compared to possibly more systematic machine learning predictions. Note that the trade-off between error and N is consistent and that the exact parameters will depend on the machine learning model and the finite sampling of constitutional isomer space.

To model possible experimental noise on query spectra, we apply Gaussian noise to query spectra and evaluate the elucidation performance of the best performing machine learning model (see insets in Fig. 4a and b)). Results indicate a halving of elucidation accuracy when the query spectrum contains up to 2 ppm MAE_Q in ¹³C and 0.15 ppm MAE in ¹H error, respectively. Thus, in the presence of experimental measurement noise even higher prediction accuracies might be necessary. Combining both ¹³C and ¹H spectra for matching improves the elucidation performance up to 90% (Fig. 4e)). Again, the combination of spectra for elucidation highlights the effectiveness of reducing the ambiguity of the matching problem by including additional properties.

Investigating potential strategies to reduce the constitutional isomer search space, we constrained N based on functional groups (see ESI Table 1†). Randomly selecting functional groups present in each query, N can be reduced by 50% and 62% on average (see Fig. 4d) inset for distributions), respectively. Results in Fig. 4c and d) indicate an increase of the elucidation accuracy by 5% in ¹³C and up to 10% for ¹H, respectively, in agreement with the elucidation performance in Fig. 4a and b). Note that the knowledge of two functional groups only led to marginal improvements. However, fragmentation could be more beneficial for larger compounds than present in QM9,¹ as reported by Yao et al.⁵⁸ Using both ¹³C and ¹H shifts on the reduced search space only lead to marginal improvements of 0.5% over the results of the full search space.

3.5. Balancing search space and accuracy

We use performance curves to analyse the relationship between the elucidation performance of C₇O₂H₁₀ queries, machine learning prediction errors and candidate pool sizes N. Similar to learning curves, showing the systematic decay of out-of-sample machine learning prediction errors as a function of training data, elucidation performance curves show for a specific elucidation threshold, e.g. 90%, the machine learning prediction error as a function of pool size. Note that while learning curves of chemical shift predictions only show the predictive accuracy, e.g. in terms of MAE, the addition of elucidation performance allow a multifaceted evaluation of new spectra estimation algorithms, considering data efficiency as well as pool size. The systematic decay of performance curves (Fig. 5 red and blue) again demonstrates that constraining N with prior knowledge allows for less accurate shift predictions to be applicable. Extrapolating the ¹³C_single performance curves indicates a machine learning MAE of 0.93 ppm to correctly rank 90% of queries out of 56 k possible candidates (Fig. 5 red), 0.02 ppm lower than suggested by Gaussian noise. To reach an MAE of 0.93 ppm, four million training instances are required (Fig. 5 orange). Using both ¹³C and ¹H shifts requires two orders of magnitude less training data (Fig. 5 blue). As such, facing expensive experimental measurements and ab initio calculations, more effective inverse structure elucidation could be achieved by balancing machine learning data needs through reduced search spaces and incorporation of additional properties.

Fig. 5 Performance curves (red, blue) of the MAE permissible to correctly identify 60, 70, 80, 90% of C₇O₂H₁₀ query spectra at a given pool size N using machine learning shifts predictions, respectively. ¹³C_single (red) only uses ¹³C shifts for elucidation, whereas ¹³C_combined uses ¹³C and ¹H spectra combined, assuming a ¹H MAE of 0.15 ppm. The learning curve (orange) indicates the systematic improvement of QM9 (ref. 1) ¹³C chemical shift predictions as a function of training set size N_train using KRR with the FCHL19 (ref. 47) representation.

4. Conclusion

We have presented an analysis of the effectiveness of the NMR spectra matching task encountered in the inverse structure elucidation problem. By systematically controlling the predictive accuracy of ¹³C and ¹H chemical shifts, we found consistent trends throughout chemical compound space, suggesting that higher errors become permissible as the number of possible candidates decreases. Note that while we relied on 1D ab initio NMR data, similar analysis could be performed using 1D or 2D experimental spectra. Applications to the most common constitutional isomers in QM9 highlight that chemical spaces with many heteroatoms are harder to characterize when only relying on a single type of chemical shift. Using both ¹³C and ¹H chemical shifts increases the error permissiveness by 85% and 261% on average, respectively. Machine learning predictions for 56 k C₇O₂H₁₀ compounds showed that using both ¹³C or ¹H shifts increased elucidation success to 90% compared to only 64% and 36% when used alone, respectively. The usefulness of the analysis is expressed via performance curves, showing that training demands can be reduced by orders of magnitude compared to relying on specific shifts alone.

We believe that as the accuracy of machine learning models to distinguish spectra is limited, constrained search spaces or inclusion of more distinct properties are necessary to improve candidate rankings. Rather than solely relying on more accurate models, future approaches could deal with estimating the applicability of machine learning models to successfully elucidate unseen chemical spaces, as well as including explicit knowledge of chemical reactions, functional groups or data from mass spectrometry, infrared- or Raman spectroscopy,^59–64 respectively.

Finally, explicitly accounting for atomic similarities and chemical shift uncertainties via the DP5 probability might further increase the confidence in structure assignments.²³

Data availability

The QM9-NMR³¹ dataset is openly available at https://doi.org/10.17172/NOMAD/2021.10.16-1. The code and additional data used in this study are available at https://doi.org/10.5281/zenodo.8126379.

Conflicts of interest

The authors have no conflict of interest.

Acknowledgements

O.A.v.L. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 772834). O.A.v.L. has received support as the Ed Clark Chair of Advanced Materials and as a Canada CIFAR AI Chair. This research is part of the University of Toronto's Acceleration Consortium, which receives funding from the Canada First Research Excellence Fund (CFREF). Icons in Fig. 1 are from DBCLS, Openclipart and Simon Dürr from http://bioicons.com under CC-BY 4.0 and CC0, respectively.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3dd00132f

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