Variational analysis and AI algorithm implementation of free energy landscapes of molecular system

Du Bo-Chuan; Tian Pu

doi:10.7498/aps.73.20231800

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:

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).

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References

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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

Figure 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]).

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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]

Figure 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].

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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

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[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
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[27]	Tribello G A, Gasparotto P 2019 Front. Mol. Biosci. 6 46 Google Scholar
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