AbstractIncreasing the binding affinity of an antibody to its target antigen is a crucial task in antibody therapeutics development. This paper presents a pretrainable geometric graph neural network, GearBind, and explores its potential in in silico affinity maturation. Leveraging multi-relational graph construction, multi-level geometric message passing and contrastive pretraining on mass-scale, unlabeled protein structural data, GearBind outperforms previous state-of-the-art approaches on SKEMPI and an independent test set.

摘要提高抗体与其靶抗原的结合亲和力是抗体治疗学发展的关键任务。本文提出了一种可预训练的几何图神经网络GearBind，并探讨了其在计算机亲和力成熟中的潜力。利用多关系图构建，多级几何信息传递和大规模对比预训练，未标记的蛋白质结构数据，GearBind优于以前在SKEMPI和独立测试集上的最新方法。

A powerful ensemble model based on GearBind is then derived and used to successfully enhance the binding of two antibodies with distinct formats and target antigens. ELISA EC50 values of the designed antibody mutants are decreased by up to 17 fold, and KD values by up to 6.1 fold. These promising results underscore the utility of geometric deep learning and effective pretraining in macromolecule interaction modeling tasks..

然后推导出基于GearBind的强大集成模型，并用于成功增强具有不同形式和靶抗原的两种抗体的结合。。这些有希望的结果强调了几何深度学习和有效预训练在大分子相互作用建模任务中的实用性。。

IntroductionAntibody plays a crucial role in the human immune system and serves as a powerful diagnostic and therapeutic tool, due to its ability to bind selectively and specifically to target antigens with high affinity. In vivo, antibodies go through affinity maturation, where the target-binding affinity gradually increases as a result of somatic hypermutation and clonal selection1.

引言抗体在人类免疫系统中起着至关重要的作用，由于其能够以高亲和力选择性和特异性结合靶抗原，因此它是一种强大的诊断和治疗工具。在体内，抗体经历亲和力成熟，其中靶标结合亲和力由于体细胞超突变和克隆选择而逐渐增加1。

When a new antigen surfaces, therapeutic antibody leads repurposed from known antibodies or screened from a natural or de novo designed library often require in vitro affinity maturation to enhance their binding affinity to a desired, usually sub-nanomolar, level.Wet lab experimental methods for in vitro antibody affinity maturation usually involve constructing mutant libraries and screening with display technology2,3,4,5.

当新抗原表面时，从已知抗体中重新利用或从天然或从头设计的文库中筛选的治疗性抗体导致通常需要体外亲和力成熟以将其结合亲和力增强至所需的（通常为亚纳摩尔）水平。体外抗体亲和力成熟的湿实验室实验方法通常涉及构建突变体文库和用展示技术2,3,4,5进行筛选。

These methods, while significantly improved during the past few years, are still labor-intensive and costly in general, taking 2–3 months or more to complete the process. Let’s consider the combinatorial search space of possible mutations. There are usually 50–60 residues on the complementarity-determining region (CDR) of an antibody, which are hypervariable in vivo and contribute to the majority of the binding free energy ΔGbind6.

这些方法虽然在过去几年中得到了显着改进，但总体上仍然是劳动密集型的，成本高昂，需要2-3个月或更长时间才能完成。让我们考虑可能突变的组合搜索空间。抗体的互补决定区（CDR）上通常有50-60个残基，这些残基在体内是高变的，并贡献了大部分结合自由能ΔGbind6。

Previous works show that multiple point mutations are often needed for successful affinity maturation7,8. Performing experiments on all combinations of over a thousand possible point mutations in antibody CDR regions (60 residues × 19 residues per residue) is difficult if not prohibitive. Therefore, a fast and accurate computational method for narrowing down the search space is much desired.Nevertheless, it is nontrivial for computational affinity maturation methods to balance speed and accuracy.

以前的工作表明，成功的亲和力成熟通常需要多个点突变7,8。如果不是禁止的话，对抗体CDR区域中超过1000个可能的点突变的所有组合（每个残基60个残基×19个残基）进行实验是困难的。因此，迫切需要一种快速准确的计算方法来缩小搜索空间。然而，计算亲和力成熟方法平衡速度和准确性是不平凡的。

Molecular dynamics methods based on empirical f.

基于经验f的分子动力学方法。

bind predictorThe GearBind framework is designed to extract geometric representations from wild-type and mutant structures via multi-level and multi-relational message passing to predict the binding free energy change ΔΔGbind. GearBind leverages information within a protein complex at three different levels with complementary insights, namely, atom-level information holding precise spatial and chemical characteristics, edge-level information capturing angular relationships, and residue-level information highlighting broader context within the protein complex.

绑定预测器GearBind框架旨在通过多层次和多关系信息传递从野生型和突变型结构中提取几何表示，以预测结合自由能变化ΔΔGbind。GearBind利用蛋白质复合物中三个不同层次的信息，具有互补的见解，即原子级信息具有精确的空间和化学特征，边缘级信息捕获角度关系，残基级信息突出蛋白质复合物中更广泛的背景。

Merging these distinct yet interconnected tiers of information allows for a more holistic view of protein complexes, potentially enhancing model capabilities.More formally, when a protein complex structure is input to GearBind, a multi-relational interface atom graph is first constructed to model the detailed interactions within the complex.

合并这些不同但相互关联的信息层可以更全面地查看蛋白质复合物，从而潜在地增强模型功能。更正式地说，当蛋白质复合物结构输入到GearBind时，首先构建多关系界面原子图来模拟复合物内的详细相互作用。

The relations defined cover both sequential proximity (for atoms on the same chain) and spatial proximity (which includes k-nearest-neighbor and within-r-radius relations). Atom-level representations are obtained by applying a geometric relational graph neural network (GearNet22) on the interface graph.

定义的关系包括顺序接近（对于同一链上的原子）和空间接近（包括k近邻和r半径内的关系）。通过在界面图上应用几何关系图神经网络（GearNet22）获得原子级表示。

On top of that, a line graph is constructed by treating each edge in the atom graph as a line node, connecting adjacent line nodes, and encoding the angular information as line edge features. Edge-level interactions are then captured by performing message passing on the line graph, similar to a sparse version of AlphaFold’s triangle attention20.

在此基础上，通过将原子图中的每条边视为线节点，连接相邻的线节点，并将角度信息编码为线边缘特征来构造线图。然后通过在线图上执行消息传递来捕获边缘级交互，类似于AlphaFold三角形注意的稀疏版本20。

Finally, after aggregating atom and edge representations for each residue, a geometric graph attention layer is applied to pass messages between residues. This multi-level message-passing scheme injects multi-granularity structural infor.

最后，在聚合每个残基的原子和边缘表示后，应用几何图注意层在残基之间传递消息。这种多级消息传递方案注入了多粒度的结构信息。

(1)

(1)

$${{{{\bf{e}}}}}_{(j,i,r)}^{(l)}\leftarrow {{{\rm{EdgeMP}}}}\left({{{{\bf{e}}}}}_{(j,i,r)}^{(l-1)}\right),$$

$${{{{\bf{e}}}}}}}{（j，i，r）}}}{（l）}\左箭头{{{\rm{EdgeMP}}}}\左（{{{{\bf{e}}}}}{（j，i，r）}^{（l-1）}\右）$$

(2)

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$${{{{\bf{a}}}}}_{i}^{(l)}\leftarrow {{{{\bf{a}}}}}_{i}^{(l)}+{{{\rm{AGGR}}}}\left({{{{\bf{e}}}}}_{(j,i,r)}^{(l)}\right),$$

$${{{{\bf{a}}}}}}}{{i}}}}{（l）}\leftarrow{{{{\bf{a}}}}}}}}}{i}}{（l）}+{{{\rm{AGGR}}}}}\左（{{\bf{e}}}}}}左（{（j，i，r）}^{（l）}\右）$$

(3)

(3)

$${{{{\bf{a}}}}}_{{{{\rm{C}}}}\alpha (i)}^{(l)}\leftarrow {{{{\bf{a}}}}}_{{{{\rm{C}}}}\alpha (i)}^{(l)}+{{{\rm{ResAttn}}}}\left({{{{\bf{a}}}}}_{{{{\rm{C}}}}\alpha (i)}^{(l)}\right).$$

$${{{{\bf{a}}}}}}}}{{{{\rm{C}}}}}\alpha（i）}^{（l）}\leftarrow{{{{\bf{a}}}}}}}{{{\rm{C}}}}}\alpha（i）}{（l）}+{{\rm{ResAttn}}}}}\left（{{{{\bf{a}}}}}}{{{{\rm{C}}}}}}\alpha（i）}^{（l）}\右）$$

(4)

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First, we perform atom-level message passing (AtomMP) on the atom graph. Then, a line graph is constructed for the message passing between edges (EdgeMP) so as to learn effective representations between atom pairs. The edge representations are used to update atom representations via an aggregation function (AGGR).

首先，我们在原子图上执行原子级消息传递（AtomMP）。然后，为边之间的消息传递（EdgeMP）构造线图，以学习原子对之间的有效表示。边缘表示用于通过聚合函数（AGGR）更新原子表示。

Finally, we take the representations \({{{{\bf{a}}}}}_{{{{\rm{C}}}}\alpha (i)}^{(l)}\) of the alpha carbon as residue representation and perform a residue-level attention mechanism (ResAttn), which can be seen as a special kind of message passing on a fully connected graph. In the following paragraphs, we will discuss these components in detail.Atom-level message passingFollowing GearNet22, we use a relational graph neural network (RGCN)38 to pass messages between atoms.

最后，我们将α碳的表示形式“（{{{{\bf{a}}}}}}{{{\rm{C}}}}\alpha（i）}^{（l）}”）作为残基表示，并执行残基级注意机制（ResAttn），这可以看作是在完全连通图上传递的一种特殊消息。在以下段落中，我们将详细讨论这些组件。原子级消息传递根据GearNet22，我们使用关系图神经网络（RGCN）38在原子之间传递消息。

In a message-passing step, each node aggregates messages from its neighbors to update its own representation. The message is computed as the output of a relation (edge type)-specific linear layer when applied to the neighbor representation. Formally, the message-passing step is defined as:$${{{\rm{AtomMP}}}}\left({{{{\bf{a}}}}}_{i}^{(l-1)}\right)={{{{\bf{a}}}}}_{i}^{(l-1)}+\sigma \left({{{\rm{BN}}}}\left({\sum}_{r\in {{{\mathcal{R}}}}}{{{{\bf{W}}}}}_{r}^{(a)}{\sum}_{(j,i,r)\in {{{\mathcal{E}}}}}{{{{\bf{a}}}}}_{j}^{(l-1)}\right)\right),$$where BN( ⋅ ) denotes batch norm and σ( ⋅ ) is the ReLU activation function.Edge-level message passing and aggregationModeling sequential proximity or spatial distance alone is not enough for capturing the complex protein–protein interactions (PPI) contributing to the binding.

在消息传递步骤中，每个节点聚合来自其邻居的消息以更新其自己的表示。当应用于邻居表示时，消息被计算为特定于关系（边类型）的线性层的输出。形式上，消息传递步骤定义为：$$${{{{\rm{atomp}}}}}\ left（{{{\bf{a}}}}}ui}^{（l-1）}\ right）={{{{\bf{a}}}}ui}^{（l-1）}+\ sigma \ left（{{\rm{BN}}}}}\ left（{\ sum}}}在{{{\mathcal{r}}}}}}{{{\bf{W}}}}}}}}}}}}}}}{{（a）}}}{\sum}}{（j，i，r）}}}{{{\mathcal{E}}}}}}}}}}}{{\bf{a}}}}}}}}}}}{j}^{（l-1）}}\右）}}中，$$BN（⋅）表示批次范数，σ（⋅）表示ReLU激活函数。边缘级消息传递和聚合仅对序列接近度或空间距离进行建模不足以捕获有助于结合的复杂蛋白质-蛋白质相互作用（PPI）。

Multiple works have demonstrated the benefits of incorporating angular information using edge-level message passing20,22,39. Here we construct a line graph40, i.e., a rela.

多项工作已经证明了使用边缘级消息传递结合角度信息的好处20,22,39。在这里，我们构建了一个线图40，即rela。

(5)

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where h(wt) and h(mt) denote the representations for wild-type and mutant complexes and \(\widetilde{\Delta \Delta {G}_{{{{\rm{bind}}}}}}\) is the predicted ΔΔGbind from our GearBind model.Modeling energy landscape of proteins via noise contrastive estimationAs paired binding free energy change data is of relatively small size, it would be beneficial to pretrain GearBind with massive protein structural data.

其中h（wt）和h（mt）表示野生型和突变型复合物的表示{G}_{{{{\rm{bind}}}}}是我们的GearBind模型预测的ΔΔGbind。通过噪声对比估计模拟蛋白质的能量分布由于配对结合自由能变化数据的规模相对较小，因此有利于用大量蛋白质结构数据预训练GearBind。

The high-level idea of our pretraining method is to model the distribution of native protein structures, which helps identify harmful mutations yielding unnatural structures. Denoting a protein structure as x, its distribution can be modeled with Boltzmann distribution as:$$p({{{\bf{x}}}};{{{\boldsymbol{\theta }}}})=\frac{\exp (-E({{{\bf{x}}}};{{{\boldsymbol{\theta }}}}))}{A({{{\boldsymbol{\theta }}}})},\, A({{{\boldsymbol{\theta }}}})=\int\exp (-E({{{\bf{x}}}};{{{\boldsymbol{\theta }}}}))d{{{\bf{x}}}},$$.

我们的预训练方法的高级思想是模拟天然蛋白质结构的分布，这有助于识别产生非天然结构的有害突变。将蛋白质结构表示为x，其分布可以用玻尔兹曼分布建模为：$$p（{{{\bf{x}}}；{{{\boldsymbol{\theta}}}）=\frac{\exp（-E（{{\bf{x}}；{{\boldsymbol{\theta}}}））}}{a（{{{\boldsymbol{\theta}}}}}}，，\，a（{{{\boldsymbol{\theta}}}}）=\int\exp（-E（{{\bf{x}}；{{\boldsymbol{\theta}}}}））d{{\bf{x}}}}，$$。

(6)

(6)

where θ denotes learnable parameters in our encoder, E(x; θ) denotes the energy function for the protein x and A(θ) is the partition function to normalize the distribution. The energy function is predicted by applying a linear layer on the GearBind representations h(x) of protein x:$$E({{{\bf{x}}}};{{{\boldsymbol{\theta }}}})={{{\rm{Linear}}}}({{{\bf{h}}}}({{{\bf{x}}}})).$$.

其中θ表示我们编码器中的可学习参数，E（x；θ）表示蛋白质x的能量函数，A（θ）是归一化分布的配分函数。通过在蛋白质x的齿轮结合表示h（x）上应用线性层来预测能量函数：$$E（{{{\bf{x}}；{{{\boldsymbol{\theta}}}）={{{\rm{linear}}（{{\bf{h}}}}（{{\bf{x}}}}}））。$$。

(7)

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Given the observed dataset {x1, . . . , xT} from PDB, our objective is to maximize the probability of these samples:$${{{\rm{maximize}}}}\,\,\frac{1}{2T}{\sum}_{t}\log p({{{{\bf{x}}}}}_{t};{{{\boldsymbol{\theta }}}}).$$

给定PDB中观察到的数据集{x1，…，xT}，我们的目标是最大化这些样本的概率：$${{{\rm{maximize}}}，\，\frac{1}{2T}{\sum}}\ut}\log p（{{{{\bf{x}}}}}}ut}；{{{\boldsymbol{\theta}}}}）$$

(8)

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However, direct optimization of this objective is intractable, since calculating the partition function requires integration over the whole protein structure space. To address this issue, we adopt a popular method for learning energy-based models called noise contrastive estimation24.

然而，这个目标的直接优化是困难的，因为计算分配函数需要在整个蛋白质结构空间上进行积分。为了解决这个问题，我们采用了一种流行的方法来学习基于能量的模型，称为噪声对比估计24。

For each observed structure xt, we sample a negative structure yt and then the problem can be transformed to a binary classification task, i.e., whether a sample is observed in the dataset or not.$${{{\rm{minimize}}}}\,\,\frac{1}{2T}{\sum}_{t}\log \left[\sigma (E({{{{\bf{x}}}}}_{t};{{{\boldsymbol{\theta }}}})-E({{{{\bf{y}}}}}_{t};{{{\boldsymbol{\theta }}}}))\right],$$.

对于每个观察到的结构xt，我们对负结构yt进行采样，然后可以将问题转换为二进制分类任务，即是否在数据集中观察到样本$$。

(9)

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where σ( ⋅ ) denotes the sigmoid function for calculating the probability for a sample xt belonging to the positive class. We could see that the above training objective tries to push down the energy of the positive examples (i.e., the observed structures) while pushing up the energy of the negative samples (i.e., the mutant structures.For negative sampling, we perform random single-point mutations on the corresponding positive samples and then generate its conformation by keeping the backbone unchanged and sampling side-chain torsional angles at the mutation site from a backbone-dependent rotamer library25.

其中σ（⋅）表示用于计算属于正类的样本xt的概率的S形函数。我们可以看到，上述训练目标试图降低正面样本（即观察到的结构）的能量，同时提高负面样本（即突变结构）的能量。对于负面采样，我们对相应的正面样本进行随机单点突变，然后通过保持骨架不变并从骨架依赖的旋转异构体库中采样突变位点的侧链扭转角来产生其构象25。

Besides, to further enhance the model’s capability to distinguish structural noises, we randomly choose 30% residues to randomly rotate torsional angles when generating negative samples.After pretraining on the CATH database, we finetune the GearBind encoder on downstream tasks for prediction to avoid overfitting.Cross-validation on SKEMPIDuring cross-validation, a model is trained and tested five times, each time using a different subset as the test set and the remaining four subsets as the training set.

此外，为了进一步增强模型识别结构噪声的能力，我们在产生负样本时随机选择30%的残基来随机旋转扭转角。在对CATH数据库进行预训练后，我们在下游任务上微调GearBind编码器以进行预测，以避免过度拟合。SKEMPI上的交叉验证在交叉验证过程中，对模型进行五次训练和测试，每次使用不同的子集作为测试集，其余四个子集作为训练集。

Results are calculated for each test set, and their mean and standard error of mean are reported as the final cross-validation performance. During the process of cross-validation, each individual data point is incorporated into the test set precisely once. This ensures that a comprehensive “test result table” is compiled, which includes predictive values for each data point when it is part of the test set.

计算每个测试集的结果，并将其平均值和平均值的标准误差报告为最终的交叉验证性能。在交叉验证过程中，每个单独的数据点被精确地合并到测试集中一次。这可以确保编译一个全面的“测试结果表”，其中包括每个数据点作为测试集的一部分时的预测值。

Subsequent performance analyses are done by splitting this table by various criteria and evaluate performance on each subset.After cross-validation on SKEMPI, we obtain five sets of model parameters. During inference, we use the mean of the predic.

随后的性能分析是通过按各种标准拆分此表并评估每个子集的性能来完成的。在SKEMPI上进行交叉验证后，我们获得了五组模型参数。在推理过程中，我们使用预测的平均值。

bind contribution analysisProtein structure analysis is conducted by python scripts. The antibody–antigen complex structure after mutation was obtained from Rosetta Flex-ddG relaxation10. The relaxed protein structure can provide more accurate side-chain conformations, which are critical for accurate contact and conformational analysis.

。突变后的抗体-抗原复合物结构是从Rosetta Flex ddG松弛10获得的。松弛的蛋白质结构可以提供更准确的侧链构象，这对于准确的接触和构象分析至关重要。

The improved accuracy of such analyses enables a deeper understanding of the underlying binding mechanisms and can facilitate the identification of key characteristics involved in protein–protein interactions. The contribution scores are derived by using Integrated Gradients (IG)43, a model-agnostic attribution method, on GearBind to obtain residue-level interpretation following22.

此类分析的准确性提高，可以更深入地了解潜在的结合机制，并有助于识别蛋白质间相互作用中涉及的关键特征。贡献分数是通过在GearBind上使用积分梯度（IG）43（一种模型不可知归因方法）得出的，以获得22之后的残留水平解释。

All protein structure figures are created with PyMOL v3.0.Molecular dynamics simulationFor antibody mutation structural analysis, we conducted molecular dynamics simulations of the wild-type and mutant antibody–antigen complex. Initial structures were taken from the Rosetta Flex-ddG relaxed structures used by GearBind.

所有蛋白质结构图均使用PyMOL v3.0创建。分子动力学模拟对于抗体突变结构分析，我们对野生型和突变型抗体-抗原复合物进行了分子动力学模拟。初始结构取自GearBind使用的Rosetta Flex ddG松弛结构。

The LEaP module in the AMBER 22 suite was used for building starting structures and adding ions and solvent for the simulation44. The protonation states of the molecules were kept at the default settings as assigned by LEaP during the initial structure preparation. All systems were simulated with the ff19SB protein force field45 and solvated in boxes of water with the OPC346 solvent model.

AMBER 22套件中的LEaP模块用于构建起始结构，并为模拟添加离子和溶剂44。。所有系统均使用ff19SB蛋白质力场45进行模拟，并使用OPC346溶剂模型在水中溶解。

Simulated systems were solvated using a 10 Angstrom solvent box. All bonds involving hydrogen atoms were constrained with the SHAKE algorithm47. The particle mesh Ewald (PME) algorithm was used to calculate long-range electrostatic interactions48. Initial structures were relaxed with maximum 10,000 steps of minimization before convergence, then subjected to he.

使用10埃溶剂箱将模拟系统溶剂化。所有涉及氢原子的键都受到SHAKE算法的约束47。粒子网格Ewald（PME）算法用于计算远程静电相互作用48。初始结构在收敛之前以最大10000个最小化步骤放松，然后进行he。

Data availability

数据可用性

The raw SKEMPI database can be accessed at https://life.bsc.es/pid/skempi2. The CATH database can be accessed via https://www.cathdb.info/. The raw HER2 binders data can be accessed via https://github.com/AbSciBio/unlocking-de-novo-antibody-design/blob/main/spr-controls.csv. Source data are provided with this paper..

可以访问原始SKEMPI数据库https://life.bsc.es/pid/skempi2.可以通过以下方式访问CATH数据库https://www.cathdb.info/.https://github.com/AbSciBio/unlocking-de-novo-antibody-design/blob/main/spr-controls.csv.本文提供了源数据。。

Code availability

代码可用性

The GearBind inference code, the trained model checkpoints and the dataset preprocessing scripts are available via https://github.com/DeepGraphLearning/GearBind under the Apache 2.0 License. They can also be accessed via Zenodo51.

GearBind推理代码，经过训练的模型检查点和数据集预处理脚本可通过https://github.com/DeepGraphLearning/GearBind在Apache 2.0许可下。它们也可以通过Zenodo51访问。

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PubMed Google ScholarTianlei YingView作者出版物您也可以在

PubMed Google ScholarJian TangView author publicationsYou can also search for this author in

PubMed Google ScholarJian TangView作者出版物您也可以在

PubMed Google ScholarContributionsJ.T. conceptualized the study and supervised the project. T.Y. and Y.W. co-supervised the project. H.C. investigated related work, co-led model development, processed the datasets and in silico results, and led manuscript writing. Z.Z. investigated related work, and led model development.

PubMed谷歌学术贡献。T、 将研究概念化并监督该项目。T、 Y.和Y.W.共同监督了该项目。H、 C.调查相关工作，共同领导模型开发，处理数据集和计算机结果，并领导手稿撰写。Z、 Z.调查相关工作，并领导模型开发。

B.Z. led the analysis of the structures and MD trajectories of the proposed mutants. M.W. was in charge of the in vitro experiments and results analysis with the help of Q.L. and Y.Z. All participated in manuscript writing.Corresponding authorsCorrespondence to.

B、 Z.领导了对所提出突变体的结构和MD轨迹的分析。M、 W.在Q.L.和Y.Z.的帮助下负责体外实验和结果分析。所有人都参与了手稿撰写。通讯作者通讯。

Yanling Wu, Tianlei Ying or Jian Tang.Ethics declarations

吴燕玲，田磊英或健堂。道德宣言

Competing interests

相互竞争的利益

The authors declare no competing interests.

作者声明没有利益冲突。

Peer review

同行评审

Peer review information

同行评审信息

Nature Communications thanks Ge Liu, Pietro Sormanni and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Nature Communications感谢Ge Liu，Pietro Sormanni和另一位匿名审稿人对这项工作的同行评审做出的贡献。可以获得同行评审文件。

Additional informationPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Supplementary informationSupplementary InformationReporting SummaryPeer Review FileDescription of Additional Supplementary FilesSupplementary SoftwareSource dataSource DataRights and permissions.

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Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material.

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You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

根据本许可证，您无权共享源自本文或其部分的改编材料。本文中的图像或其他第三方材料包含在文章的知识共享许可证中，除非该材料的信用额度中另有说明。如果材料未包含在文章的知识共享许可中，并且您的预期用途不受法律法规的许可或超出许可用途，则您需要直接获得版权所有者的许可。

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Reprints and permissionsAbout this articleCite this articleCai, H., Zhang, Z., Wang, M. et al. Pretrainable geometric graph neural network for antibody affinity maturation.

转载和许可本文引用本文Cai，H.，Zhang，Z.，Wang，M。等人。用于抗体亲和力成熟的可训练几何图神经网络。

Nat Commun 15, 7785 (2024). https://doi.org/10.1038/s41467-024-51563-8Download citationReceived: 13 September 2023Accepted: 13 August 2024Published: 06 September 2024DOI: https://doi.org/10.1038/s41467-024-51563-8Share this articleAnyone you share the following link with will be able to read this content:Get shareable linkSorry, a shareable link is not currently available for this article.Copy to clipboard.

《国家公社》157785（2024）。https://doi.org/10.1038/s41467-024-51563-8Download引文接收日期：2023年9月13日接收日期：2024年8月13日发布日期：2024年9月6日OI：https://doi.org/10.1038/s41467-024-51563-8Share本文与您共享以下链接的任何人都可以阅读此内容：获取可共享链接对不起，本文目前没有可共享的链接。复制到剪贴板。

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