分子体系自由能地貌图的变分分析及AI算法实现

杜泊船; 田圃

doi:10.7498/aps.73.20231800

摘要

精确描述复杂分子体系的自由能地貌图是理解和操控其行为, 并进一步实现分子设计制造工业化的重要基础. 刻画高维空间自由能地貌图的主要挑战是其往往在不同时空间尺度上具有多个层次, 每个层次都可能有不止一个亚稳态被相应的自由能垒分开, 且跨越路径有可能不止一条. 另外很多体系涉及非线性行为, 这使得理论解析和直接使用分子模拟都有很大困难. 针对这些挑战, 多年来研究者们发展了多种多样的增强采样方法, 但往往需要很多经验选择和操作, 从而一方面使得研究进程较为缓慢, 另一方面也让误差控制成为困难. 变分虽然在物理、统计和工程中已经被广泛应用并取得巨大成功, 但在复杂分子体系中的应用却随着神经网络的发展刚刚开始. 本文将对这些探索性工作的主要方向、进展和局限进行简要总结, 也对将来的可能发展给出展望, 希望能够激发更多对基于变分的分子体系自由能地貌图人工智能算法的关注和努力, 促进大分子药物、分子生物机器等实践应用的发展.

关键词:

Abstract

Accurate description of the free energy landscape (FES) is the basis for understanding complex molecular systems, and for further realizing molecular design, manufacture and industrialization. Major challenges include multiple metastable states, which usually are separated by high potential barriers and are not linearly separable, and may exist at multiple levels of time and spatial scales. Consequently FES is not suitable for analytical analysis and brute force simulation. To address these challenges, many enhanced sampling methods have been developed. However, utility of them usually involves many empirical choices, which hinders research advancement, and also makes error control very unimportant. Although variational calculus has been widely applied and achieved great success in physics, engineering and statistics, its application in complex molecular systems has just begun with the development of neural networks. This brief review is to summarize the background, major developments, current limitations, and prospects of applying variation in this field. It is hoped to facilitate the AI algorithm development for complex molecular systems in general, and to promote the further methodological development in this line of research in particular.

Keywords:

作者及机构信息

杜泊船¹,
田圃^1,2

1.
吉林大学生命科学学院, 长春　130012

2.
吉林大学人工智能学院, 长春　130012

通信作者: 田圃, tianpu@jlu.edu.cn

基金项目: 吉林大学“学科交叉融合创新”项目(批准号: JLUXKJC2021ZZ05)资助的课题.

Authors and contacts

Du Bo-Chuan¹,
Tian Pu^1,2

1.
School of Life Sciences, Jilin University, Changchun 130012, China

2.
School of Artificial Intelligence, Jilin University, Changchun 130012, China

Corresponding author: Tian Pu, tianpu@jlu.edu.cn

Funds: Project supported by the Interdisciplinary Integration and Innovation Project of JLU, China (Grant No. JLUXKJC2021ZZ05).

文章全文 : translate this paragraph

参考文献

[1]	Thomas C, Tampe R 2020 Annu. Rev. Biochem. 89 605 Google Scholar
[2]	Jiang F, Doudna J A 2017 Annu. Rev. Biophys. 46 505 Google Scholar
[3]	Latorraca N R, Venkatakrishnan A J, Dror R O 2017 Chem. Rev. 117 139 Google Scholar
[4]	Wei G, Xi W, Nussinov R, Ma B 2016 Chem. Rev. 116 6516 Google Scholar
[5]	Dignon G L, Best R B, Mittal J 2020 Annu. Rev. Phys. Chem. 71 53 Google Scholar
[6]	Choi J M, Holehouse A S, Pappu R V 2020 Annu. Rev. Biophys. 49 107 Google Scholar
[7]	Sponer J, Bussi G, Krepl M, et al. 2018 Chem. Rev. 118 4177 Google Scholar
[8]	Bussi G, Laio A 2020 Nat. Rev. Phys. 2 200 Google Scholar
[9]	Mobley D L, Gilson M K 2017 Annu. Rev. Biophys. 46 531 Google Scholar
[10]	Rodnina M V, Beringer M, Wintermeyer W 2007 Trends Biochem. Sci. 32 20 Google Scholar
[11]	Bernardi R C, Melo M C R, Schulten K 2015 Biochim. Biophys. Acta 1850 872 Google Scholar
[12]	Sugita Y, Okamoto Y 1999 Chem. Phys. Lett. 314 141 Google Scholar
[13]	Faraldo-Gomez J D, Roux B 2007 J. Comput. Chem. 28 1634 Google Scholar
[14]	Laio A, Parrinello M 2002 Proc. Natl. Acad. Sci. U. S. A. 99 12562 Google Scholar
[15]	Barducci A, Bussi G, Parrinello M 2008 Phys. Rev. Lett. 100 020603 Google Scholar
[16]	Maragliano L, Vanden-Eijnden E 2006 Chem. Phys. Lett. 426 168 Google Scholar
[17]	Abrams J B, Tuckerman M E 2008 J. Phys. Chem. B 112 15742 Google Scholar
[18]	Darve E, Rodriguez-Gomez D, Pohorille A 2008 J. Chem. Phys. 128 144120 Google Scholar
[19]	Torrie G M, Valleau J P 1977 J. Comput. Phys. 23 187 Google Scholar
[20]	Carter E A, Ciccotti G, Hynes J T, Kapral R 1989 Chem. Phys. Lett. 156 472 Google Scholar
[21]	Sprik M, Ciccotti G 1998 J. Chem. Phys. 109 7737 Google Scholar
[22]	Zwanzig R W 1954 J. Chem. Phys. 22 1420 Google Scholar
[23]	Kirkwood J G 1935 J. Chem. Phys. 3 300 Google Scholar
[24]	Oberhofer H, Dellago C, Geissler P L 2005 J. Phys. Chem. B 109 6902 Google Scholar
[25]	Chen M, Cuendet M A, Tuckerman M E 2012 J. Chem. Phys. 137 024102 Google Scholar
[26]	Lesage A, Lelievre T, Stoltz G, Henin J 2017 J. Phys. Chem. B 121 3676 Google Scholar
[27]	Tribello G A, Gasparotto P 2019 Front. Mol. Biosci. 6 46 Google Scholar
[28]	Comer J, Gumbart J C, Henin J, Lelievre T, Pohorille A, Chipot C 2015 J. Phys. Chem. B 119 1129 Google Scholar
[29]	Darve E, Pohorille A 2001 J. Chem. Phys. 115 9169 Google Scholar
[30]	Huber T, Torda A E, van Gunsteren W F 1994 J. Comput. Aided. Mol. Des. 8 695 Google Scholar
[31]	Wang F, Landau D P 2001 Phys. Rev. Lett. 86 2050 Google Scholar
[32]	Valsson O, Tiwary P, Parrinello M 2016 Annu. Rev. Phys. Chem. 67 159 Google Scholar
[33]	Husic B E, Pande V S 2018 J. Am. Chem. Soc. 140 2386 Google Scholar
[34]	Dellago C, Bolhuis P G, Csajka F S, Chandler D 1998 J. Chem. Phys. 108 1964 Google Scholar
[35]	Bolhuis P G, Chandler D, Dellago C, Geissler P L 2002 Annu. Rev. Phys. Chem. 53 291 Google Scholar
[36]	van Erp T S, Moroni D, Bolhuis P G 2003 J. Chem. Phys. 118 7762 Google Scholar
[37]	Moroni D, Bolhuis P G, van Erp T S 2004 J. Chem. Phys. 120 4055 Google Scholar
[38]	Hummer G 2004 J. Chem. Phys. 120 516 Google Scholar
[39]	Bolhuis P G, Swenson D W H 2021 Front. Data Comput. 4 2000237 Google Scholar
[40]	E W, Vanden-Eijnden E 2010 Annu. Rev. Phys. Chem. 61 391 Google Scholar
[41]	Sarich M, Banisch R, Hartmann C, Schütte C 2013 Entropy 16 258 Google Scholar
[42]	Cybenko G 1989 Math. Control Signal Syst. 2 303 Google Scholar
[43]	Leshno M, Lin V Y, Pinkus A, Schocken S 1993 Neural Netw. 6 861 Google Scholar
[44]	Zhou D X 2020 Appl. Comput. Harmon. Anal. 48 787 Google Scholar
[45]	Alzubaidi L, Zhang J, Humaidi A J, Al-Dujaili A, Duan Y, Al-Shamma O, Santamaria J, Fadhel M A, Al-Amidie M, Farhan L 2021 J. Big Data 8 53 Google Scholar
[46]	He K, Zhang X, Ren S, Sun J 2016 IEEE Conference on Computer Vision and Pattern Recognition Las Vegas, USA, 27–30 June, 2016 pp770–778
[47]	Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez A N, Kaiser Ł, Polosukhin I 2017 Advances in Neural Information Processing Systems Long Beach, USA, December 4–9, 2017
[48]	Ho J, Jain A, Abbeel P 2020 Advances in Neural Information Processing Systems Virtual pp6840–6851
[49]	Baydin A G, Pearlmutter B A, Radul A A, Siskind J M 2018 J. Mach. Learn. Res. 18 1 Google Scholar
[50]	Rumelhart D, Hinton G, Williams R 1986 Nature 323 533 Google Scholar
[51]	Michelucci U 2022 arXiv: 1312.6114 [stat. ML]
[52]	Kingma D P, Welling M 2019 arXiv: 1906.02691 [cs. LG]
[53]	Waterfall J J, Casey F P, Gutenkunst R N, Brown K S, Myers C R, Brouwer P W, Elser V, Sethna J P 2006 Phys. Rev. Lett. 97 150601 Google Scholar
[54]	Rumelhart D E, Hinton G E, Williams R J (Anderson J A, Rosenfeld E, ed) 1988 Neurocomputing (Vol. 1) (Cambridge: The MIT Press) pp696–700
[55]	Arfken G B, Weber H J, Harris F E 2011 Mathematical Methods for Physicists: A Comprehensive Guide (Cambridge: Academic Press
[56]	Blei D M, Kucukelbir A, McAuliffe J D 2017 J. Am. Stat. Assoc. 112 859 Google Scholar
[57]	Ganguly A, Earp S W 2021 arXiv: 2108.13083 [cs. LG]
[58]	Marquardt D W 1963 J. Soc. Ind. Appl. Math. 11 431 Google Scholar
[59]	Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L 2019 Advances in Neural Information Processing Systems pp8026–8037
[60]	Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M 2016 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16) Savannah, GA, USA, November 2–4, 2016 pp265–283
[61]	Ma Y, Yu D, Wu T, Wang H 2019 Front. Data Comput. 1 105 Google Scholar
[62]	Hadji I, Wildes R P 2018 arXiv: 1803.08834 [cs. CV]
[63]	Ghorbani M, Prasad S, Klauda J B, Brooks B R 2022 J. Chem. Phys. 156 184103 Google Scholar
[64]	Mardt A, Hempel T, Clementi C, Noe F 2022 Nat. Commun. 13 7101 Google Scholar
[65]	Perez-Hernandez G, Paul F, Giorgino T, De Fabritiis G, Noe F 2013 J. Chem. Phys. 139 015102 Google Scholar
[66]	Wu H, Noé F 2019 J. Nonlinear Sci. 30 23 Google Scholar
[67]	Mardt A, Pasquali L, Wu H, Noe F 2018 Nat. Commun. 9 5 Google Scholar
[68]	Kleiman D E, Shukla D 2023 J. Chem. Theory Comput. 19 4377 Google Scholar
[69]	Chen H, Roux B, Chipot C 2023 J. Chem. Theory Comput. 19 4414 Google Scholar
[70]	Schütte C, Fischer A, Huisinga W, Deuflhard P 1999 J. Comput. Phys. 151 146 Google Scholar
[71]	He Z, Chipot C, Roux B 2022 J. Phys. Chem. Lett. 13 9263 Google Scholar
[72]	Bonati L, Zhang Y Y, Parrinello M 2019 Proc. Natl. Acad. Sci. U. S. A. 116 17641 Google Scholar
[73]	Bittracher A, Mollenhauer M, Koltai P, Schütte C 2023 Multiscale Model. Simul. 21 449 Google Scholar
[74]	Wang Y, Ribeiro J M L, Tiwary P 2019 Nat. Commun. 10 3573 Google Scholar
[75]	Beyerle E R, Mehdi S, Tiwary P 2022 J. Phys. Chem. B 126 3950 Google Scholar
[76]	Zhang J, Lei Y K, Yang Y I, Gao Y Q 2020 J. Chem. Phys. 153 174115 Google Scholar
[77]	Kingma D P, Welling M 2013 arXiv: 1312.6114 [stat. ML]
[78]	Tiwary P, Berne B J 2016 Proc. Natl. Acad. Sci. U. S. A. 113 2839 Google Scholar
[79]	Wu H, Paul F, Wehmeyer C, Noe F 2016 Proc. Natl. Acad. Sci. U. S. A. 113 E3221 Google Scholar
[80]	Wu H, Mey A S, Rosta E, Noé F 2014 J. Chem. Phys. 141 214106 Google Scholar
[81]	Chodera J D, Swope W C, Noé F, Prinz J H, Shirts M R, Pande V S 2011 J. Chem. Phys. 134 244107 Google Scholar
[82]	Prinz J H, Chodera J D, Pande V S, Swope W C, Smith J C, Noe F 2011 J. Chem. Phys. 134 244108 Google Scholar
[83]	Rosta E, Hummer G 2015 J. Chem. Theory Comput. 11 276 Google Scholar
[84]	Mey A S, Wu H, Noé F 2014 Phys. Rev. X 4 041018 Google Scholar
[85]	Hinrichs N S, Pande V S 2007 J. Chem. Phys. 126 244101 Google Scholar
[86]	Noe F 2008 J. Chem. Phys. 128 244103 Google Scholar
[87]	Chodera J D, Noé F 2010 J. Chem. Phys. 133 265 Google Scholar
[88]	Schütt K, Kindermans P J, Sauceda Felix H E, Chmiela S, Tkatchenko A, Müller K R 2017 Advances in Neural Information Processing Systems Long Beach, ACM, USA, 2017 pp991–1001
[89]	Husic B E, Charron N E, Lemm D, Wang J, Perez A, Majewski M, Kramer A, Chen Y, Olsson S, de Fabritiis G, Noe F, Clementi C 2020 J. Chem. Phys. 153 194101 Google Scholar
[90]	Battaglia P W, Hamrick J B, Bapst V, et al. 2018 arXiv: 1806.01261 [stat. ML]
[91]	Kipf T N, Welling M 2016 arXiv: 1609.02907 [cs. LG]
[92]	Veličković P, Cucurull G, Casanova A, Romero A, Lio P, Bengio Y 2017 arXiv: 1710.10903 [stat. ML]
[93]	Ghorbani M, Prasad S, Klauda J B, Brooks B R 2022 arXiv:2201.04609 [physics.comp-ph]
[94]	Hempel T, Del Razo M J, Lee C T, Taylor B C, Amaro R E, Noe F 2021 Proc. Natl. Acad. Sci. U. S. A. 118 e2105230118 Google Scholar
[95]	Maragliano L, Fischer A, Vanden-Eijnden E, Ciccotti G 2006 J. Chem. Phys. 125 24106 Google Scholar
[96]	Pan A C, Sezer D, Roux B 2008 J. Phys. Chem. B 112 3432 Google Scholar
[97]	Weinan E, Ren W, Vanden-Eijnden E 2005 Chem. Phys. Lett. 413 242 Google Scholar
[98]	Branduardi D, Gervasio F L, Parrinello M 2007 J. Chem. Phys. 126 054103 Google Scholar
[99]	Leines G D, Ensing B 2012 Phys. Rev. Lett. 109 020601 Google Scholar
[100]	Invernizzi M, Parrinello M 2020 J. Phys. Chem. Lett. 11 2731 Google Scholar
[101]	Berezhkovskii A, Szabo A 2005 J. Chem. Phys. 122 14503 Google Scholar
[102]	Langer J S 1969 Ann. Phys. 54 258 Google Scholar
[103]	Valsson O, Parrinello M 2014 Phys. Rev. Lett. 113 090601 Google Scholar
[104]	Bilionis I, Koutsourelakis P S 2012 J. Comput. Phys. 231 3849 Google Scholar
[105]	Dempster A P, Laird N M, Rubin D B 2018 J. R. Stat. Soc. B 39 1 Google Scholar
[106]	Bonati L, Piccini G, Parrinello M 2021 Proc. Natl. Acad. Sci. U.S.A. 118 e2113533118 Google Scholar
[107]	Tishby N, Pereira F C, Bialek W 2000 arXiv: physics/0004057 [physics.data-an]
[108]	Still S 2014 Entropy 16 968 Google Scholar
[109]	Song Y, Kingma D P 2021 arXiv: 2101.03288 [cs. LG]
[110]	Arjovsky M, Chintala S, Bottou L 2017 International Conference on Machine Learning Sydney pp214–223
[111]	Huang Y P, Xia Y, Yang L, Wei J, Yang Y I, Gao Y Q 2021 Chin. J. Chem. 40 160 Google Scholar
[112]	Ribeiro J M L, Bravo P, Wang Y, Tiwary P 2018 J. Chem. Phys. 149 072301 Google Scholar
[113]	Chen M 2021 Eur. Phys. J. B 94 211 Google Scholar
[114]	Qiu Y, O'Connor M S, Xue M, Liu B, Huang X 2023 J. Chem. Theory Comput. 19 4728 Google Scholar
[115]	Monroe J I, Shen V K 2022 J. Chem. Theory Comput. 18 3622 Google Scholar
[116]	Ma A, Dinner A R 2005 J. Phys. Chem. B 109 6769 Google Scholar
[117]	Chen W, Ferguson A L 2018 J. Comput. Chem. 39 2079 Google Scholar
[118]	Chen H, Liu H, Feng H, Fu H, Cai W, Shao X, Chipot C 2022 J. Chem. Inf. Model. 62 1 Google Scholar
[119]	Wehmeyer C, Noe F 2018 J. Chem. Phys. 148 241703 Google Scholar
[120]	Williams M O, Kevrekidis I G, Rowley C W 2015 J. Nonlinear Sci. 25 1307 Google Scholar
[121]	Mezić I 2005 Nonlinear Dyn. 41 309 Google Scholar
[122]	H. Tu J, W. Rowley C, M. Luchtenburg D, L. Brunton S, Nathan Kutz J 2014 J. Comput. Dynam. 1 391 Google Scholar
[123]	Zhang J, Chen M 2018 Phys. Rev. Lett. 121 010601 Google Scholar
[124]	Rydzewski J, Valsson O 2021 J. Phys. Chem. A 125 6286 Google Scholar
[125]	Belkacemi Z, Gkeka P, Lelievre T, Stoltz G 2022 J. Chem. Theory Comput. 18 59 Google Scholar
[126]	Kikutsuji T, Mori Y, Okazaki K I, Mori T, Kim K, Matubayasi N 2022 J. Chem. Phys. 156 154108 Google Scholar
[127]	Sun L, Vandermause J, Batzner S, Xie Y, Clark D, Chen W, Kozinsky B 2022 J Chem Theory Comput 18 2341 Google Scholar
[128]	Wang Y, Lamim Ribeiro J M, Tiwary P 2020 Curr. Opin. Struct. Biol. 61 139 Google Scholar
[129]	Jung H, Covino R, Arjun A, Leitold C, Dellago C, Bolhuis P G, Hummer G 2023 Nat. Comput. Sci. 3 334 Google Scholar
[130]	Zhao L, Wang L 2023 Chin. Phys. Lett. 40 120201 Google Scholar
[131]	Wu T, He S, Liu J, Sun S, Liu K, Han Q L, Tang Y 2023 IEEE-CAA J. Automatica Sin. 10 1122 Google Scholar
[132]	Janson G, Valdes-Garcia G, Heo L, Feig M 2023 Nat. Commun. 14 774 Google Scholar
[133]	Naveed H, Ullah Khan A, Qiu S, Saqib M, Anwar S, Usman M, Akhtar N, Barnes N, Mian A 2023 arXiv: 2307.06435 [cs. CL]

施引文献

图 1 自编码器神经网络架构示意图, 蓝色部分表示编码器(encoder)函数$ f(\cdot) $, 橙色部分表示解码器(decoder)函数$ g(\cdot) $, 维度最低的绿色表示中间隐藏层(z), 对自编码器, 损失函数是输出($ {\tilde{\boldsymbol{x}}}_{i} $)与输入$ {\boldsymbol{x}}_{i} $的差别的函数(也可以加正则化项, 如参考文献[58] (5)式所示), 每一个输入数据点对应隐藏层空间的一个点

Fig. 1. Schematic representation of an auto-encoder neural network. The blue part on the left represents the encoder, the orange part on the right represents the decoder, and the middle green layer is the hidden layer (z). The loss is always a function of the difference between the input and the output vectors ($ {\boldsymbol{x}}_{i} $ and $ {\tilde{\boldsymbol{x}}}_{i} $), one may add some form of regularization when necessary (e.g. Eq. (5) in Ref. [58]).

下载: 全尺寸图片幻灯片

图 2 (a) VAMPnets构建VAMP打分((10)式)的神经网络总体架构示意图; (b)丙氨酸二肽轨迹分析实例中的典型神经网络架构, 各层神经元数目为 32-22-16-9-6, 前两层使用10%的dropout, 除最后的softmax层外, 其余各层激活函数均使用Relu^[67]

Fig. 2. (a) Schematic illustration of VAMP score construction from VAMPnets (see Eq. (10)). (b) A typical neural network architecture for analine dipeptide analysis, with the number of neurons being 32-22-16-9-6 for five layers. The first two layers utilized a 10% dropout. Relu was selected as the activation function for all layers except the last softmax layer^[67].

下载: 全尺寸图片幻灯片

表 1 复杂分子体系低维隐空间的变分方法简要总结, 表中所述集合空间问题类别是指引言中提到的三类问题

Table 1. A brief summary of variational methods for low-dimensional hidden spaces in complex molecular systems. The category of collective space problems mentioned in the table refers to the three types of problems defined in the introduction.

变分方法		主要目标	关注的集合空间问题类别	特点或主要局限
频谱分解分析	基组线性组合	给定构象子状态空间划分下求解集合变量和子态间转换速率	第1类、第2类	马尔可夫假设与线性基组局限, 需要人工划分构象空间子状态
频谱分解分析	神经网络实现	从给定轨迹中直接求解子态划分和对应转换速率	第2类	马尔可夫假设, 没有解析表示的特征函数, 需要人工调整架构测试不同聚类数量
自由能垒跨越概率时间关联函数	基组线性组合	在选定基组空间的线性组合基础上求解状态转换路径和其上的自由能垒跨越概率	第3类	基组线性组合局限, 需要定义始末态
自由能垒跨越概率时间关联函数	神经网络实现	在和给定始末态一致的神经网络函数空间求解状态转换路径和其上的自由能垒跨越概率	第3类	需要定义始末态
基于偏置势变分	基组线性组合	利用偏置势增强采样在基组线性组合空间快速求解给定集合变量方向自由能主要能量谷地	第2类	泛函受基组选择限制
基于偏置势变分	神经网络实现	利用偏置势增强采样在神经网络函数空间快速求解给定集合变量方向自由能主要能量谷地	第2类	泛函导数求解的采样需求导致偏置势(和对应自由能)的精度紧密相关, 收敛受KL散度非对称性限制
Lumpability 和 Decomposability		优化集合变量	第1类	有明确误差控制, 方差取决于隐空间维度, 两种定义的一致性要求可逆过程
信息瓶颈模型		求解信息瓶颈对应集合空间CV表示, 并利用偏置势加速自由能面采样	第2类	线性编码过程假设局限
变分自适应		结合粗粒化信息加速采样求解自由能面	第2类	总体架构较为复杂
变分自编码器		通过集合变量空间加速采样求解自由能面和聚类转化路径	第2类、第3类	特别关注隐空间

下载: 导出CSV

[1]	Thomas C, Tampe R 2020 Annu. Rev. Biochem. 89 605 Google Scholar
[2]	Jiang F, Doudna J A 2017 Annu. Rev. Biophys. 46 505 Google Scholar
[3]	Latorraca N R, Venkatakrishnan A J, Dror R O 2017 Chem. Rev. 117 139 Google Scholar
[4]	Wei G, Xi W, Nussinov R, Ma B 2016 Chem. Rev. 116 6516 Google Scholar
[5]	Dignon G L, Best R B, Mittal J 2020 Annu. Rev. Phys. Chem. 71 53 Google Scholar
[6]	Choi J M, Holehouse A S, Pappu R V 2020 Annu. Rev. Biophys. 49 107 Google Scholar
[7]	Sponer J, Bussi G, Krepl M, et al. 2018 Chem. Rev. 118 4177 Google Scholar
[8]	Bussi G, Laio A 2020 Nat. Rev. Phys. 2 200 Google Scholar
[9]	Mobley D L, Gilson M K 2017 Annu. Rev. Biophys. 46 531 Google Scholar
[10]	Rodnina M V, Beringer M, Wintermeyer W 2007 Trends Biochem. Sci. 32 20 Google Scholar
[11]	Bernardi R C, Melo M C R, Schulten K 2015 Biochim. Biophys. Acta 1850 872 Google Scholar
[12]	Sugita Y, Okamoto Y 1999 Chem. Phys. Lett. 314 141 Google Scholar
[13]	Faraldo-Gomez J D, Roux B 2007 J. Comput. Chem. 28 1634 Google Scholar
[14]	Laio A, Parrinello M 2002 Proc. Natl. Acad. Sci. U. S. A. 99 12562 Google Scholar
[15]	Barducci A, Bussi G, Parrinello M 2008 Phys. Rev. Lett. 100 020603 Google Scholar
[16]	Maragliano L, Vanden-Eijnden E 2006 Chem. Phys. Lett. 426 168 Google Scholar
[17]	Abrams J B, Tuckerman M E 2008 J. Phys. Chem. B 112 15742 Google Scholar
[18]	Darve E, Rodriguez-Gomez D, Pohorille A 2008 J. Chem. Phys. 128 144120 Google Scholar
[19]	Torrie G M, Valleau J P 1977 J. Comput. Phys. 23 187 Google Scholar
[20]	Carter E A, Ciccotti G, Hynes J T, Kapral R 1989 Chem. Phys. Lett. 156 472 Google Scholar
[21]	Sprik M, Ciccotti G 1998 J. Chem. Phys. 109 7737 Google Scholar
[22]	Zwanzig R W 1954 J. Chem. Phys. 22 1420 Google Scholar
[23]	Kirkwood J G 1935 J. Chem. Phys. 3 300 Google Scholar
[24]	Oberhofer H, Dellago C, Geissler P L 2005 J. Phys. Chem. B 109 6902 Google Scholar
[25]	Chen M, Cuendet M A, Tuckerman M E 2012 J. Chem. Phys. 137 024102 Google Scholar
[26]	Lesage A, Lelievre T, Stoltz G, Henin J 2017 J. Phys. Chem. B 121 3676 Google Scholar
[27]	Tribello G A, Gasparotto P 2019 Front. Mol. Biosci. 6 46 Google Scholar
[28]	Comer J, Gumbart J C, Henin J, Lelievre T, Pohorille A, Chipot C 2015 J. Phys. Chem. B 119 1129 Google Scholar
[29]	Darve E, Pohorille A 2001 J. Chem. Phys. 115 9169 Google Scholar
[30]	Huber T, Torda A E, van Gunsteren W F 1994 J. Comput. Aided. Mol. Des. 8 695 Google Scholar
[31]	Wang F, Landau D P 2001 Phys. Rev. Lett. 86 2050 Google Scholar
[32]	Valsson O, Tiwary P, Parrinello M 2016 Annu. Rev. Phys. Chem. 67 159 Google Scholar
[33]	Husic B E, Pande V S 2018 J. Am. Chem. Soc. 140 2386 Google Scholar
[34]	Dellago C, Bolhuis P G, Csajka F S, Chandler D 1998 J. Chem. Phys. 108 1964 Google Scholar
[35]	Bolhuis P G, Chandler D, Dellago C, Geissler P L 2002 Annu. Rev. Phys. Chem. 53 291 Google Scholar
[36]	van Erp T S, Moroni D, Bolhuis P G 2003 J. Chem. Phys. 118 7762 Google Scholar
[37]	Moroni D, Bolhuis P G, van Erp T S 2004 J. Chem. Phys. 120 4055 Google Scholar
[38]	Hummer G 2004 J. Chem. Phys. 120 516 Google Scholar
[39]	Bolhuis P G, Swenson D W H 2021 Front. Data Comput. 4 2000237 Google Scholar
[40]	E W, Vanden-Eijnden E 2010 Annu. Rev. Phys. Chem. 61 391 Google Scholar
[41]	Sarich M, Banisch R, Hartmann C, Schütte C 2013 Entropy 16 258 Google Scholar
[42]	Cybenko G 1989 Math. Control Signal Syst. 2 303 Google Scholar
[43]	Leshno M, Lin V Y, Pinkus A, Schocken S 1993 Neural Netw. 6 861 Google Scholar
[44]	Zhou D X 2020 Appl. Comput. Harmon. Anal. 48 787 Google Scholar
[45]	Alzubaidi L, Zhang J, Humaidi A J, Al-Dujaili A, Duan Y, Al-Shamma O, Santamaria J, Fadhel M A, Al-Amidie M, Farhan L 2021 J. Big Data 8 53 Google Scholar
[46]	He K, Zhang X, Ren S, Sun J 2016 IEEE Conference on Computer Vision and Pattern Recognition Las Vegas, USA, 27–30 June, 2016 pp770–778
[47]	Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez A N, Kaiser Ł, Polosukhin I 2017 Advances in Neural Information Processing Systems Long Beach, USA, December 4–9, 2017
[48]	Ho J, Jain A, Abbeel P 2020 Advances in Neural Information Processing Systems Virtual pp6840–6851
[49]	Baydin A G, Pearlmutter B A, Radul A A, Siskind J M 2018 J. Mach. Learn. Res. 18 1 Google Scholar
[50]	Rumelhart D, Hinton G, Williams R 1986 Nature 323 533 Google Scholar
[51]	Michelucci U 2022 arXiv: 1312.6114 [stat. ML]
[52]	Kingma D P, Welling M 2019 arXiv: 1906.02691 [cs. LG]
[53]	Waterfall J J, Casey F P, Gutenkunst R N, Brown K S, Myers C R, Brouwer P W, Elser V, Sethna J P 2006 Phys. Rev. Lett. 97 150601 Google Scholar
[54]	Rumelhart D E, Hinton G E, Williams R J (Anderson J A, Rosenfeld E, ed) 1988 Neurocomputing (Vol. 1) (Cambridge: The MIT Press) pp696–700
[55]	Arfken G B, Weber H J, Harris F E 2011 Mathematical Methods for Physicists: A Comprehensive Guide (Cambridge: Academic Press
[56]	Blei D M, Kucukelbir A, McAuliffe J D 2017 J. Am. Stat. Assoc. 112 859 Google Scholar
[57]	Ganguly A, Earp S W 2021 arXiv: 2108.13083 [cs. LG]
[58]	Marquardt D W 1963 J. Soc. Ind. Appl. Math. 11 431 Google Scholar
[59]	Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L 2019 Advances in Neural Information Processing Systems pp8026–8037
[60]	Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M 2016 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16) Savannah, GA, USA, November 2–4, 2016 pp265–283
[61]	Ma Y, Yu D, Wu T, Wang H 2019 Front. Data Comput. 1 105 Google Scholar
[62]	Hadji I, Wildes R P 2018 arXiv: 1803.08834 [cs. CV]
[63]	Ghorbani M, Prasad S, Klauda J B, Brooks B R 2022 J. Chem. Phys. 156 184103 Google Scholar
[64]	Mardt A, Hempel T, Clementi C, Noe F 2022 Nat. Commun. 13 7101 Google Scholar
[65]	Perez-Hernandez G, Paul F, Giorgino T, De Fabritiis G, Noe F 2013 J. Chem. Phys. 139 015102 Google Scholar
[66]	Wu H, Noé F 2019 J. Nonlinear Sci. 30 23 Google Scholar
[67]	Mardt A, Pasquali L, Wu H, Noe F 2018 Nat. Commun. 9 5 Google Scholar
[68]	Kleiman D E, Shukla D 2023 J. Chem. Theory Comput. 19 4377 Google Scholar
[69]	Chen H, Roux B, Chipot C 2023 J. Chem. Theory Comput. 19 4414 Google Scholar
[70]	Schütte C, Fischer A, Huisinga W, Deuflhard P 1999 J. Comput. Phys. 151 146 Google Scholar
[71]	He Z, Chipot C, Roux B 2022 J. Phys. Chem. Lett. 13 9263 Google Scholar
[72]	Bonati L, Zhang Y Y, Parrinello M 2019 Proc. Natl. Acad. Sci. U. S. A. 116 17641 Google Scholar
[73]	Bittracher A, Mollenhauer M, Koltai P, Schütte C 2023 Multiscale Model. Simul. 21 449 Google Scholar
[74]	Wang Y, Ribeiro J M L, Tiwary P 2019 Nat. Commun. 10 3573 Google Scholar
[75]	Beyerle E R, Mehdi S, Tiwary P 2022 J. Phys. Chem. B 126 3950 Google Scholar
[76]	Zhang J, Lei Y K, Yang Y I, Gao Y Q 2020 J. Chem. Phys. 153 174115 Google Scholar
[77]	Kingma D P, Welling M 2013 arXiv: 1312.6114 [stat. ML]
[78]	Tiwary P, Berne B J 2016 Proc. Natl. Acad. Sci. U. S. A. 113 2839 Google Scholar
[79]	Wu H, Paul F, Wehmeyer C, Noe F 2016 Proc. Natl. Acad. Sci. U. S. A. 113 E3221 Google Scholar
[80]	Wu H, Mey A S, Rosta E, Noé F 2014 J. Chem. Phys. 141 214106 Google Scholar
[81]	Chodera J D, Swope W C, Noé F, Prinz J H, Shirts M R, Pande V S 2011 J. Chem. Phys. 134 244107 Google Scholar
[82]	Prinz J H, Chodera J D, Pande V S, Swope W C, Smith J C, Noe F 2011 J. Chem. Phys. 134 244108 Google Scholar
[83]	Rosta E, Hummer G 2015 J. Chem. Theory Comput. 11 276 Google Scholar
[84]	Mey A S, Wu H, Noé F 2014 Phys. Rev. X 4 041018 Google Scholar
[85]	Hinrichs N S, Pande V S 2007 J. Chem. Phys. 126 244101 Google Scholar
[86]	Noe F 2008 J. Chem. Phys. 128 244103 Google Scholar
[87]	Chodera J D, Noé F 2010 J. Chem. Phys. 133 265 Google Scholar
[88]	Schütt K, Kindermans P J, Sauceda Felix H E, Chmiela S, Tkatchenko A, Müller K R 2017 Advances in Neural Information Processing Systems Long Beach, ACM, USA, 2017 pp991–1001
[89]	Husic B E, Charron N E, Lemm D, Wang J, Perez A, Majewski M, Kramer A, Chen Y, Olsson S, de Fabritiis G, Noe F, Clementi C 2020 J. Chem. Phys. 153 194101 Google Scholar
[90]	Battaglia P W, Hamrick J B, Bapst V, et al. 2018 arXiv: 1806.01261 [stat. ML]
[91]	Kipf T N, Welling M 2016 arXiv: 1609.02907 [cs. LG]
[92]	Veličković P, Cucurull G, Casanova A, Romero A, Lio P, Bengio Y 2017 arXiv: 1710.10903 [stat. ML]
[93]	Ghorbani M, Prasad S, Klauda J B, Brooks B R 2022 arXiv:2201.04609 [physics.comp-ph]
[94]	Hempel T, Del Razo M J, Lee C T, Taylor B C, Amaro R E, Noe F 2021 Proc. Natl. Acad. Sci. U. S. A. 118 e2105230118 Google Scholar
[95]	Maragliano L, Fischer A, Vanden-Eijnden E, Ciccotti G 2006 J. Chem. Phys. 125 24106 Google Scholar
[96]	Pan A C, Sezer D, Roux B 2008 J. Phys. Chem. B 112 3432 Google Scholar
[97]	Weinan E, Ren W, Vanden-Eijnden E 2005 Chem. Phys. Lett. 413 242 Google Scholar
[98]	Branduardi D, Gervasio F L, Parrinello M 2007 J. Chem. Phys. 126 054103 Google Scholar
[99]	Leines G D, Ensing B 2012 Phys. Rev. Lett. 109 020601 Google Scholar
[100]	Invernizzi M, Parrinello M 2020 J. Phys. Chem. Lett. 11 2731 Google Scholar
[101]	Berezhkovskii A, Szabo A 2005 J. Chem. Phys. 122 14503 Google Scholar
[102]	Langer J S 1969 Ann. Phys. 54 258 Google Scholar
[103]	Valsson O, Parrinello M 2014 Phys. Rev. Lett. 113 090601 Google Scholar
[104]	Bilionis I, Koutsourelakis P S 2012 J. Comput. Phys. 231 3849 Google Scholar
[105]	Dempster A P, Laird N M, Rubin D B 2018 J. R. Stat. Soc. B 39 1 Google Scholar
[106]	Bonati L, Piccini G, Parrinello M 2021 Proc. Natl. Acad. Sci. U.S.A. 118 e2113533118 Google Scholar
[107]	Tishby N, Pereira F C, Bialek W 2000 arXiv: physics/0004057 [physics.data-an]
[108]	Still S 2014 Entropy 16 968 Google Scholar
[109]	Song Y, Kingma D P 2021 arXiv: 2101.03288 [cs. LG]
[110]	Arjovsky M, Chintala S, Bottou L 2017 International Conference on Machine Learning Sydney pp214–223
[111]	Huang Y P, Xia Y, Yang L, Wei J, Yang Y I, Gao Y Q 2021 Chin. J. Chem. 40 160 Google Scholar
[112]	Ribeiro J M L, Bravo P, Wang Y, Tiwary P 2018 J. Chem. Phys. 149 072301 Google Scholar
[113]	Chen M 2021 Eur. Phys. J. B 94 211 Google Scholar
[114]	Qiu Y, O'Connor M S, Xue M, Liu B, Huang X 2023 J. Chem. Theory Comput. 19 4728 Google Scholar
[115]	Monroe J I, Shen V K 2022 J. Chem. Theory Comput. 18 3622 Google Scholar
[116]	Ma A, Dinner A R 2005 J. Phys. Chem. B 109 6769 Google Scholar
[117]	Chen W, Ferguson A L 2018 J. Comput. Chem. 39 2079 Google Scholar
[118]	Chen H, Liu H, Feng H, Fu H, Cai W, Shao X, Chipot C 2022 J. Chem. Inf. Model. 62 1 Google Scholar
[119]	Wehmeyer C, Noe F 2018 J. Chem. Phys. 148 241703 Google Scholar
[120]	Williams M O, Kevrekidis I G, Rowley C W 2015 J. Nonlinear Sci. 25 1307 Google Scholar
[121]	Mezić I 2005 Nonlinear Dyn. 41 309 Google Scholar
[122]	H. Tu J, W. Rowley C, M. Luchtenburg D, L. Brunton S, Nathan Kutz J 2014 J. Comput. Dynam. 1 391 Google Scholar
[123]	Zhang J, Chen M 2018 Phys. Rev. Lett. 121 010601 Google Scholar
[124]	Rydzewski J, Valsson O 2021 J. Phys. Chem. A 125 6286 Google Scholar
[125]	Belkacemi Z, Gkeka P, Lelievre T, Stoltz G 2022 J. Chem. Theory Comput. 18 59 Google Scholar
[126]	Kikutsuji T, Mori Y, Okazaki K I, Mori T, Kim K, Matubayasi N 2022 J. Chem. Phys. 156 154108 Google Scholar
[127]	Sun L, Vandermause J, Batzner S, Xie Y, Clark D, Chen W, Kozinsky B 2022 J Chem Theory Comput 18 2341 Google Scholar
[128]	Wang Y, Lamim Ribeiro J M, Tiwary P 2020 Curr. Opin. Struct. Biol. 61 139 Google Scholar
[129]	Jung H, Covino R, Arjun A, Leitold C, Dellago C, Bolhuis P G, Hummer G 2023 Nat. Comput. Sci. 3 334 Google Scholar
[130]	Zhao L, Wang L 2023 Chin. Phys. Lett. 40 120201 Google Scholar
[131]	Wu T, He S, Liu J, Sun S, Liu K, Han Q L, Tang Y 2023 IEEE-CAA J. Automatica Sin. 10 1122 Google Scholar
[132]	Janson G, Valdes-Garcia G, Heo L, Feig M 2023 Nat. Commun. 14 774 Google Scholar
[133]	Naveed H, Ullah Khan A, Qiu S, Saqib M, Anwar S, Usman M, Akhtar N, Barnes N, Mian A 2023 arXiv: 2307.06435 [cs. CL]

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