1
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Yang G, Jiang S, Luo Y, Wang S, Jiang J. Cross-Modal Prediction of Spectral and Structural Descriptors via a Pretrained Model Enhanced with Chemical Insights. J Phys Chem Lett 2024; 15:8766-8772. [PMID: 39163398 DOI: 10.1021/acs.jpclett.4c02129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/22/2024]
Abstract
Proposing and utilizing machine learning descriptors for chemical property prediction and material screening have become a cutting-edge field in artificial intelligence-enabled chemical research. However, a single descriptor typically captures only partial features of a chemical object, resulting in an information deficiency and limiting generalizability. Obtaining a comprehensive set of descriptors is essential but challenging, especially when accessing some microlevel structural and electronic features due to technological limitations. Herein, we exploit multimodal chemical descriptors to construct an encoder-decoder machine learning framework that enables the cross-modal prediction of spectral and structural descriptors. By pretraining the model to endow it with chemical insights, the multimodal data fusion is implemented in a descriptor-encoded hidden layer. The model's capabilities are validated in the system of CO/NO adsorption on Au/Ag surfaces, demonstrating successful reciprocal prediction of infrared spectra, Raman spectra, and internal coordinates. This work provides a proof-of-concept for the feasibility of cross-modal predictions between different chemical features and will significantly reduce the machine learning model's dependence on complete physicochemical parameters and improve its multitarget prediction capabilities.
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Affiliation(s)
- Guokun Yang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
| | - Shuang Jiang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
| | - Yi Luo
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
| | - Song Wang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
| | - Jun Jiang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
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2
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Wu F, Huang Y, Yang G, Ye S, Mukamel S, Jiang J. Unraveling dynamic protein structures by two-dimensional infrared spectra with a pretrained machine learning model. Proc Natl Acad Sci U S A 2024; 121:e2409257121. [PMID: 38917009 PMCID: PMC11228460 DOI: 10.1073/pnas.2409257121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2024] [Accepted: 05/28/2024] [Indexed: 06/27/2024] Open
Abstract
Dynamic protein structures are crucial for deciphering their diverse biological functions. Two-dimensional infrared (2DIR) spectroscopy stands as an ideal tool for tracing rapid conformational evolutions in proteins. However, linking spectral characteristics to dynamic structures poses a formidable challenge. Here, we present a pretrained machine learning model based on 2DIR spectra analysis. This model has learned signal features from approximately 204,300 spectra to establish a "spectrum-structure" correlation, thereby tracing the dynamic conformations of proteins. It excels in accurately predicting the dynamic content changes of various secondary structures and demonstrates universal transferability on real folding trajectories spanning timescales from microseconds to milliseconds. Beyond exceptional predictive performance, the model offers attention-based spectral explanations of dynamic conformational changes. Our 2DIR-based pretrained model is anticipated to provide unique insights into the dynamic structural information of proteins in their native environments.
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Affiliation(s)
- Fan Wu
- Key Laboratory of Precision and Intelligent Chemistry, Hefei National Research Center for Physical Sciences at the Microscale, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei230026, Anhui, China
| | - Yan Huang
- Key Laboratory of Precision and Intelligent Chemistry, Hefei National Research Center for Physical Sciences at the Microscale, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei230026, Anhui, China
| | - Guokun Yang
- Key Laboratory of Precision and Intelligent Chemistry, Hefei National Research Center for Physical Sciences at the Microscale, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei230026, Anhui, China
| | - Sheng Ye
- Anhui Provincial Engineering Research Center for Unmanned System and Intelligent Technology, School of Artificial Intelligence, Anhui University, Hefei230601, Anhui, China
| | - Shaul Mukamel
- Department of Chemistry and of Physics & Astronomy, University of California, Irvine, CA92697
| | - Jun Jiang
- Key Laboratory of Precision and Intelligent Chemistry, Hefei National Research Center for Physical Sciences at the Microscale, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei230026, Anhui, China
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3
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Chikkanayakanahalli Mukunda D, Rodrigues J, Chandra S, Mazumder N, Vitkin A, Kishore Mahato K. Protein classification by autofluorescence spectral shape analysis using machine learning. Talanta 2024; 267:125167. [PMID: 37714041 DOI: 10.1016/j.talanta.2023.125167] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2023] [Revised: 08/23/2023] [Accepted: 09/04/2023] [Indexed: 09/17/2023]
Abstract
Depending on the relative numbers and spatial arrangement of Tryptophan (Trp; W) and Tyrosine (Tyr; Y) residues, different proteins produce distinct autofluorescence (AF) spectral shapes when excited at ∼280 nm. Yet, considering the vast number and heterogeneous forms in nature, visual analysis and precise identification of proteins based on their AF spectra is challenging and further compounded in cases when different proteins produce substantially similar AF spectral shapes. There is, thus, a serious need to develop a methodology to address this problem. The current study proposes a practical technology to quickly identify proteins using machine learning (ML) algorithms based on their AF spectra. Specifically, AF spectra of fifteen different standard proteins of varying origin with distinct structural and Trp/Tyr compositions were recorded; based on the spectral features selected by the Minimum-Redundancy-Maximum-Relevance (mRMR) algorithm, a multiclass Support Vector Machine (SVM) learning model with Radial Basis Function (RBF), Polynomial, and Linear kernels classified the proteins with high accuracy of 99.06%, 99.03%, and 98.29% respectively. Since protein identification is the key to understand biological functions and disease diagnosis, the proposed methodology could offer a viable alternative to and improve the existing protein identification techniques.
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Affiliation(s)
| | - Jackson Rodrigues
- Department of Biophysics, Manipal School of Life Sciences, Manipal Academy of Higher Education, Manipal, 576104, Karnataka, India
| | - Subhash Chandra
- Department of Biophysics, Manipal School of Life Sciences, Manipal Academy of Higher Education, Manipal, 576104, Karnataka, India
| | - Nirmal Mazumder
- Department of Biophysics, Manipal School of Life Sciences, Manipal Academy of Higher Education, Manipal, 576104, Karnataka, India
| | - Alex Vitkin
- Department of Medical Biophysics, University of Toronto, Toronto, Ontario, M5G 1L7, Canada
| | - Krishna Kishore Mahato
- Department of Biophysics, Manipal School of Life Sciences, Manipal Academy of Higher Education, Manipal, 576104, Karnataka, India.
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4
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Zhao Y, Li H, Shan J, Zhang Z, Li X, Shi JQ, Jiao Y, Li H. Machine Learning Confirms the Formation Mechanism of a Single-Atom Catalyst via Infrared Spectroscopic Analysis. J Phys Chem Lett 2023:11058-11062. [PMID: 38048178 DOI: 10.1021/acs.jpclett.3c02896] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/06/2023]
Abstract
Single-atom catalysts (SACs) offer significant potential across various applications, yet our understanding of their formation mechanism remains limited. Notably, the pyrolysis of zeolitic imidazolate frameworks (ZIFs) stands as a pivotal avenue for SAC synthesis, of which the mechanism can be assessed through infrared (IR) spectroscopy. However, the prevailing analysis techniques still rely on manual interpretation. Here, we report a machine learning (ML)-driven analysis of the IR spectroscopy to unravel the pyrolysis process of Pt-doped ZIF-67 to synthesize Pt-Co3O4 SAC. Demonstrating a total Pearson correlation exceeding 0.7 with experimental data, the algorithm provides correlation coefficients for the selected structures, thereby confirming crucial structural changes with time and temperature, including the decomposition of ZIF and formation of Pt-O bonds. These findings reveal and confirm the formation mechanism of SACs. As demonstrated, the integration of ML algorithms, theoretical simulations, and experimental spectral analysis introduces an approach to deciphering experimental characterization data, implying its potential for broader adoption.
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Affiliation(s)
- Yanzhang Zhao
- School of Chemical Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
| | - Huan Li
- School of Chemical Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
| | - Jieqiong Shan
- School of Chemical Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
- Department of Chemistry, City University of Hong Kong, Kowloon 999077, Hong Kong Special Administrative Region of the People's Republic of China
| | - Zhen Zhang
- Australian Institute for Machine Learning, The University of Adelaide, Adelaide, South Australia 5000, Australia
| | - Xinyu Li
- Australian Institute for Machine Learning, The University of Adelaide, Adelaide, South Australia 5000, Australia
| | - Javen Qinfeng Shi
- Australian Institute for Machine Learning, The University of Adelaide, Adelaide, South Australia 5000, Australia
| | - Yan Jiao
- School of Chemical Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
| | - Haobo Li
- School of Chemical Engineering, The University of Adelaide, Adelaide, South Australia 5005, Australia
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5
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Yang T, Zhou D, Ye S, Li X, Li H, Feng Y, Jiang Z, Yang L, Ye K, Shen Y, Jiang S, Feng S, Zhang G, Huang Y, Wang S, Jiang J. Catalytic Structure Design by AI Generating with Spectroscopic Descriptors. J Am Chem Soc 2023. [PMID: 38019281 DOI: 10.1021/jacs.3c09299] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2023]
Abstract
Generative artificial intelligence has depicted a beautiful blueprint for on-demand design in chemical research. However, the few successful chemical generations have only been able to implement a few special property values because most chemical descriptors are mathematically discrete or discontinuously adjustable. Herein, we use spectroscopic descriptors with machine learning to establish a quantitative spectral structure-property relationship for adsorbed molecules on metal monatomic catalysts. Besides catalytic properties such as adsorption energy and charge transfer, the complete spatial relative coordinates of the adsorbed molecule were successfully inverted. The spectroscopic descriptors and prediction models are generalized, allowing them to be transferred to several different systems. Due to the continuous tunability of the spectroscopic descriptors, the design of catalytic structures with continuous adsorption states generated by AI in the catalytic process has been achieved. This work paves the way for using spectroscopy to enable real-time monitoring of the catalytic process and continuous customization of catalytic performance, which will lead to profound changes in catalytic research.
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Affiliation(s)
- Tongtong Yang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
- Institute of Intelligent Innovation, Henan Academy of Sciences, Zhengzhou, Henan 451162, P. R. China
| | - Donglai Zhou
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Sheng Ye
- School of Artificial Intelligence, Anhui University, Hefei, Anhui 230601, China
| | - Xiyu Li
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
| | - Huirong Li
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yi Feng
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zifan Jiang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Li Yang
- Institutes of Physical Science and Information Technology, Anhui University, Hefei, Anhui 230601, China
| | - Ke Ye
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yixi Shen
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Shuang Jiang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Shuo Feng
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Guozhen Zhang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yan Huang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Song Wang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jun Jiang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
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6
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Guo S, Jiang J, Ren H, Wang S. Fusion of Multiple Spectra for Investigating Chemical Bonding Properties via Machine Learning. J Phys Chem Lett 2023; 14:7461-7468. [PMID: 37579021 DOI: 10.1021/acs.jpclett.3c01709] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/16/2023]
Abstract
Chemical bonding properties are crucial to understanding the chemical behavior of molecules. Spectroscopy is a versatile technical tool to study various microscopic properties, but its interpretation suffers from human biases and the loss of high-dimensional information. Here, we present a machine learning approach to predict diverse bonding properties, including the bond dissociation energy, bond length, and α-C connectivity of hydroxyls in organic molecules, by fusing multiple spectra with different physical mechanisms. Combining nuclear magnetic resonance and vibrational spectroscopy exhibits higher prediction accuracy than what they did separately. On the hold-out test data set, the models achieve a mean absolute error of 1.243 kcal/mol and 1.041 × 10-4 Å for BDE and bond length and an accuracy of 95.09% for hydroxyl α-C connectivity. Our models demonstrate strong extrapolation capabilities when they are transferred to different molecules, external electric fields, and solvation environments. These end-to-end models pave the way to investigating chemical bonding properties by using spectroscopic observables.
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Affiliation(s)
- Sibei Guo
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Jun Jiang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
- Hefei National Laboratory, University of Science and Technology of China, Hefei, Anhui 230088, China
| | - Hao Ren
- School of Materials Science and Engineering, China University of Petroleum (East China), Qingdao, Shandong 266580, China
| | - Song Wang
- Key Laboratory of Precision and Intelligent Chemistry, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, Anhui 230026, China
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7
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Du W, Yang X, Wu D, Ma F, Zhang B, Bao C, Huo Y, Jiang J, Chen X, Wang Y. Fusing 2D and 3D molecular graphs as unambiguous molecular descriptors for conformational and chiral stereoisomers. Brief Bioinform 2023; 24:bbac560. [PMID: 36528804 PMCID: PMC9851338 DOI: 10.1093/bib/bbac560] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2022] [Revised: 10/28/2022] [Accepted: 11/15/2022] [Indexed: 12/23/2022] Open
Abstract
The rapid progress of machine learning (ML) in predicting molecular properties enables high-precision predictions being routinely achieved. However, many ML models, such as conventional molecular graph, cannot differentiate stereoisomers of certain types, particularly conformational and chiral ones that share the same bonding connectivity but differ in spatial arrangement. Here, we designed a hybrid molecular graph network, Chemical Feature Fusion Network (CFFN), to address the issue by integrating planar and stereo information of molecules in an interweaved fashion. The three-dimensional (3D, i.e., stereo) modality guarantees precision and completeness by providing unabridged information, while the two-dimensional (2D, i.e., planar) modality brings in chemical intuitions as prior knowledge for guidance. The zipper-like arrangement of 2D and 3D information processing promotes cooperativity between them, and their synergy is the key to our model's success. Experiments on various molecules or conformational datasets including a special newly created chiral molecule dataset comprised of various configurations and conformations demonstrate the superior performance of CFFN. The advantage of CFFN is even more significant in datasets made of small samples. Ablation experiments confirm that fusing 2D and 3D molecular graphs as unambiguous molecular descriptors can not only effectively distinguish molecules and their conformations, but also achieve more accurate and robust prediction of quantum chemical properties.
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Affiliation(s)
- Wenjie Du
- School of Software Engineering, University of Science and Technology of China, Hefei 230026, China
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China
| | - Xiaoting Yang
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China
- School of Computer Science and Technology, University of Science and Technology of China, Hefei 230026, China
| | - Di Wu
- School of Software Engineering, University of Science and Technology of China, Hefei 230026, China
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China
| | - FenFen Ma
- Suzhou Laboratory , Suzhou 215123, Jiangsu, China
| | - Baicheng Zhang
- School of Chemistry and Materials Science, University of Science and Technology of China, Hefei 230026, China
| | - Chaochao Bao
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China
| | - Yaoyuan Huo
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China
| | - Jun Jiang
- School of Chemistry and Materials Science, University of Science and Technology of China, Hefei 230026, China
| | - Xin Chen
- Suzhou Laboratory , Suzhou 215123, Jiangsu, China
| | - Yang Wang
- School of Software Engineering, University of Science and Technology of China, Hefei 230026, China
- Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou, Jiangsu 215123, China
- School of Computer Science and Technology, University of Science and Technology of China, Hefei 230026, China
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8
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An T, Wen J, Dong Z, Zhang Y, Zhang J, Qin F, Wang Y, Zhao X. Plasmonic Biosensors with Nanostructure for Healthcare Monitoring and Diseases Diagnosis. SENSORS (BASEL, SWITZERLAND) 2022; 23:445. [PMID: 36617043 PMCID: PMC9824517 DOI: 10.3390/s23010445] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/14/2022] [Revised: 12/20/2022] [Accepted: 12/20/2022] [Indexed: 06/17/2023]
Abstract
Nanophotonics has been widely utilized in enhanced molecularspectroscopy or mediated chemical reaction, which has major applications in the field of enhancing sensing and enables opportunities in developing healthcare monitoring. This review presents an updated overview of the recent exciting advances of plasmonic biosensors in the healthcare area. Manufacturing, enhancements and applications of plasmonic biosensors are discussed, with particular focus on nanolisted main preparation methods of various nanostructures, such as chemical synthesis, lithography, nanosphere lithography, nanoimprint lithography, etc., and describing their respective advances and challenges from practical applications of plasmon biosensors. Based on these sensing structures, different types of plasmonic biosensors are summarized regarding detecting cancer biomarkers, body fluid, temperature, gas and COVID-19. Last, the existing challenges and prospects of plasmonic biosensors combined with machine learning, mega data analysis and prediction are surveyed.
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Affiliation(s)
- Tongge An
- College of Materials and Environmental Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Jiahong Wen
- The College of Electronics and Information, Hangzhou Dianzi University, Hangzhou 310018, China
- Shangyu Institute of Science and Engineering, Hangzhou Dianzi University, Shaoxing 312000, China
| | - Zhichao Dong
- College of Materials and Environmental Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Yongjun Zhang
- College of Materials and Environmental Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Jian Zhang
- College of Materials and Environmental Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Faxiang Qin
- Institute for Composites Science Innovation, School of Materials Science and Engineering, Zhejiang University, Hangzhou 310027, China
| | - Yaxin Wang
- College of Materials and Environmental Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Xiaoyu Zhao
- College of Materials and Environmental Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
- Zhejiang Laboratory, Hangzhou 311100, China
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9
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Parker KA, Schultz JD, Singh N, Wasielewski MR, Beratan DN. Mapping Simulated Two-Dimensional Spectra to Molecular Models Using Machine Learning. J Phys Chem Lett 2022; 13:7454-7461. [PMID: 35930790 DOI: 10.1021/acs.jpclett.2c01913] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Two-dimensional (2D) spectroscopy encodes molecular properties and dynamics into expansive spectral data sets. Translating these data into meaningful chemical insights is challenging because of the many ways chemical properties can influence the spectra. To address the task of extracting chemical information from 2D spectroscopy, we study the capacity of simple feedforward neural networks (NNs) to map simulated 2D electronic spectra to underlying physical Hamiltonians. We examined hundreds of simulated 2D spectra corresponding to monomers and dimers with varied Franck-Condon active vibrations and monomer-monomer electronic couplings. We find the NNs are able to correctly characterize most Hamiltonian parameters in this study with an accuracy above 90%. Our results demonstrate that NNs can aid in interpreting 2D spectra, leading from spectroscopic features to underlying effective Hamiltonians.
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Affiliation(s)
- Kelsey A Parker
- Department of Chemistry, Duke University, Durham, North Carolina 27708, United States
| | - Jonathan D Schultz
- Department of Chemistry and Institute for Sustainability and Energy, Northwestern University, Evanston, Illinois 60208-3113, United States
| | - Niven Singh
- Program in Computational Biology and Bioinformatics, Center for Genomics and Computational Biology, Duke University School of Medicine, Durham, North Carolina 27710, United States
| | - Michael R Wasielewski
- Department of Chemistry and Institute for Sustainability and Energy, Northwestern University, Evanston, Illinois 60208-3113, United States
| | - David N Beratan
- Department of Chemistry, Duke University, Durham, North Carolina 27708, United States
- Department of Biochemistry, Duke University, Durham, North Carolina 27710, United States
- Department of Physics, Duke University, Durham, North Carolina 27708, United States
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