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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

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10.1038/s41598-024-71864-8
Article
Inferring gene regulatory networks with graph convolutional network based on causal feature reconstruction
Ji Ruirui 19077982@qq.com

12
Geng Yi 1
Quan Xin 1
1 https://ror.org/01zzmf129 grid.440733.7 0000 0000 8854 4301 School of Automation and Information Engineering, Xi ’an University of Technology, No.5, Jinhua South Road, Xi’an, 710048 Shaanxi China
2 Key Laboratory of Shaanxi Province for Complex System Control and Intelligent Information Processing, Xi’an, 710048 Shaanxi China
12 9 2024
12 9 2024
2024
14 2134213 6 2024
2 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ 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. 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. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Inferring gene regulatory networks through deep learning and causal inference methods is a crucial task in the field of computational biology and bioinformatics. This study presents a novel approach that uses a Graph Convolutional Network (GCN) guided by causal information to infer Gene Regulatory Networks (GRN). The transfer entropy and reconstruction layer are utilized to achieve causal feature reconstruction, mitigating the information loss problem caused by multiple rounds of neighbor aggregation in GCN, resulting in a causal and integrated representation of node features. Separable features are extracted from gene expression data by the Gaussian-kernel Autoencoder to improve computational efficiency. Experimental results on the DREAM5 and the mDC dataset demonstrate that our method exhibits superior performance compared to existing algorithms, as indicated by the higher values of the AUPRC metrics. Furthermore, the incorporation of causal feature reconstruction enhances the inferred GRN, rendering them more reasonable, accurate, and reliable.

Keywords

Gene regulatory network
Causal relationship
Graph convolutional network
Link prediction
Autoencoder
Subject terms

Data mining
Gene regulatory networks
Machine learning
Computational biology and bioinformatics
Systems biology
National Natural Science Foundation of China61702410 Ji Ruirui issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Gene regulatory network (GRN)1 describe the complex regulatory relationships among genes and is one of the key tools to assist researchers in analyzing and understanding biological processes at the molecular level. The advancement of high-throughput sequencing technology has resulted in the accumulation of a substantial volume of gene expression data. Mining the regulatory relationships among genes accurately based on gene expression data has become the research focus of computational biology and bioinformatics. This research is of great significance in promoting the development of biomedicine and uncovering potential biological processes.

At the outset, researchers utilized statistical-based methods2 to infer GRN. However, purely statistical methods only consider the statistical patterns existing in gene expression data3, disregarding the causal relationships among gene expression data which leads to low accuracy and no biological significance in the inference results. Therefore, researchers have begun to focus on analyzing the causal regulatory relationships between genes4. For example, Ma et al.5 proposed a nonlinear differential model based on time series data. This model achieves network inference by establishing a functional relationship between target genes and their regulatory genes. The model parameters are subsequently optimized using the Random Forest algorithm. Friedman et al.6 used Bayesian networks to establish a causal skeleton and realized network inference based on the network skeleton. Ajmal et al.7 utilized a dynamic Bayesian network to infer the network and achieve the simulation of somatic regulatory relationships between genes with multiple time lags, however, leads to a significant demand for computational resources as the number of time points increases. Olsen et al.8 proposed a method for inferring causal edges in a network by analyzing whether a variable is causally influenced by two or more variables based on a network skeleton. Feng et al.9 inferred the network skeleton using multiple-time Transfer Entropy for each pair of genes and filtered out low-confidence edges in the network using a threshold, and then retained directed edges based on the threshold through enumeration or searching. Sun et al.10,11 proposed the concept of causal entropy and an inference method based on optimal causal entropy, which incorporates time-series data to enhance the perception of causal relationships between variables.

The accuracy of network inference is improved and biologically meaningful results are generated by causal network inference methods. The causal network inference approaches consists of two steps. Firstly, the causal skeleton or the major threshold is determined by the approaches. Secondly, an enumeration or heuristic algorithm identifies additional causal edges based on the skeleton. However, these approaches suffer from two main problems: excessive computational time and resource consumption, and a lack of effective constraints during the inference process, leading to the potential generation of erroneous results.

With the advancement of deep learning, researchers have started using deep learning methods to infer GRN more effectively12. Compared to traditional causal network inference methods, deep learning methods can learn more intricate regulatory relationships from expression data with higher accuracy. Wei Liu et al.13 proposed a circRNA disease-association prediction model based on automatically selected meta-path and contrastive learning, in which GNN is used to extract node features. Li et al.14 proposed a gated convolutional recurrent network with residual learning to predict translation initiation sites. Guo et al.15 proposed a variational gated autoencoder-based feature extraction mode to potential disease-miRNA associations. Meroua et al.16 constructed deep neural networks based on known regulatory pairs for network inference. MacLean et al.17 extracted regulatory features by regulatory pairs of microarray gene expressions and constructed convolutional neural networks (CNN) for network inference. These methods only use one-to-one known regulatory pairs as labels to construct neural networks for predicting potential regulatory relationships, however, it is challenging to learn the intricate regulatory relationships of genes within the network topology. Graph Neural Networks (GNN)18 is the network model capable of processing graph data, which enables efficient representation and inference of graph structures. Therefore, the use of graph neural networks to mine more complex regulatory topology based on known gene regulatory relationships is gradually emerging as a new tool for inferring GRN. Wang et al.19 proposed a GNN link prediction method to predict regulatory relationships between genes using gene expression data as a feature matrix, and employed the network skeleton to calculate the Neighbor Aggregation of each order of genes in the network to perform inference in a semi-supervised approach, which brings in additional data requirements. Chen et al.20 proposed a graph attention network to infer latent interactions between transcription factors and target genes in GRN, however, graph-based attention mechanisms incur huge computational and memory overheads.Graph Convolutional Neural Networks (GCN)21 is a graph-based method build on GNN. By convolutional operations and hierarchical aggregation, GCN is more stable and accurate than GNN in generating neighbourhood aggregation, making it widely used in biological networks.. S. Ganeshamoorthy et al.22 used a 1D-CNN to extract key features from gene expression data. then utilized the extracted features and known regulatory pairs as input for a Graph Variational Autoencoder (GVAE) which is composed of GCN to achieve GRN inference. Mao et al.23 proposed a GCN-based interaction encoder infer GRN, by neighbor aggregation to capturing interdependencies between nodes in the network, due to which the performance of the model is impacted by the precision of neighbour aggregation. The accuracy of inferring GRN can be enhanced by using graph neural networks to generate a graph representation of known regulatory relationships through neighbor aggregation. However, the neighbor information is easily lost during aggregation and which lead to unreliable accuracy in downstream tasks. Therefore, it is the key to infer GRN by graph neural networks that ensuring the causal relationships are not overlooked during neighbor nodes aggregation, which embedded in known regulatory pairings and expression data.

Transfer Entropy (TE)24 is a kind of index that quantitatively compute the flow of information from one series to another. Currently, TE has been combined with graph model for prediction tasks. Duan et al.25 used TE to extract the causality among the time series and construct the TE graph as a priori-information to guide the forecasting task. Zhang et al.26 proposed a rutting prediction model based on multi-variate transfer entropy and GNN. TE has been used as a pre-processing before graph networks in approaches described above, inspired by which, TE can be introduced to measure the flow of neighbour information during the order-by-order node aggregation in GCN and enhance the accuracy of which. Since Transfer Entropy reflects the direction and strength of the transferred neighbour information at the same time, it indicates the essential causal relationship between neighbour aggregation of each order in GRN. Reinforcing the causal relationship in the neighbour aggregation with a method guided by transfer entropy will render the inferred GRN more reasonable and reliable.

Therefore, in this paper, the GCN based on Causal Feature Reconstruction method is firstly proposed for inferring GRN. The method employs Transfer Entropy to quantify the loss of causal information in the GCN during neighbor aggregation. Subsequently, the representations of the node features are obtained through Causal Feature Reconstruction. This method results in a more comprehensive node feature representation output from the GCN, enhancing the accuracy of the downstream link prediction task.

The main contributions of this paper are as follows: Firstly proposing a causal information-guided GCN method for inferring GRN by using Transfer Entropy to measure and enhance the neighbor aggregation. It also incorporates linear layers to complete the causal feature reconstruction of neighbor aggregation, aiming to reduce the loss of neighbor information in the GCN during the training process.

Incorporating a Gaussian kernel into an Autoencoder method to extract features from gene expression data. The Gaussian-kernel Autoencoder extracts gene expression data into significantly separable features, which is reliable and comprehensive. The method aims to enhance the subsequent computational efficiency of GCN and precision of causal reconstruction.

Validating the method on the E.coli, the S.cerevisiae, and the mDC datasets, the experimental results demonstrate that the model in this paper achieves higher network inference accuracy and credibility.

Link prediction and inference of GRN using GCN based on causal feature reconstruction

Overall framework

In this paper, the GCN based on causal feature reconstruction is investigated to infer GRN. The proposed model consists of three main components: a Gaussian-kernel Autoencoder module, a GCN based on causal feature reconstruction, and a link prediction module. The framework of the model is illustrated in Fig. 1. Firstly, the Gaussian-kernel autoencoder is used to extract the gene expression features X from the gene expression data. Subsequently, the gene expression features X and the neighbor matrix A which is obtained from the known regulatory pairs, are fed into the GCN module. The output of the GCN is reconstructed by causal features, thereafter the inference of the GRN is completed using link prediction.Fig. 1 The framework of inferring GRN by GCN based on causal feature reconstruction (where A is the neighbourhood matrix extracted from regulatory pairs, X is the gene expression feature extracted by the Gaussian-kernel Autoencoder, G is the neighbourhood matrix of the inference network, and G⊤ is the transposition of the gene expression data, A^ is the inferred network).

Extracting gene expression features

Gene expression features

Gene expression data describes the intensity of gene expression at a specific condition, with higher values indicating greater intensity. The expression intensity provides insight into the regulatory relationship to some extent. In static gene expression data, the expression values of different genes in the same sample can reflect the activated or inhibited state of genes at a certain condition. This, in turn, reflects a specific regulatory state. Additionally, the expression values of the same genes in different groups of data also vary, indicating the expression state of genes in different states of the GRN27. Therefore, extracting key effective features from gene expression data is crucial for accurately inferring the GRN.

Mirzal et al.28 used non-negative matrix decomposition to process gene expression data, providing guidelines for key gene screening. Fan et al.29 improved the accuracy of large-scale gene regulatory network inference by performing singular value decomposition on gene expression data. Ganeshamoorthy et al.22 extracted the key features of gene expression using 1D-CNN and utilized these features as inputs to the GVAE, which led to improved accuracy of the inferred network. The Gaussian kernel30 enable the data to be linearly separable, which have been used to extract separable feature in multi-omics data integration task31. The methods mentioned above are capable of extracting the required expression features. However, it is difficult for the inference algorithm to prioritise the most important features as there are small but significant differences in these features, which in turn affects the accuracy of the inference. The autoencoder has been widely used in Biodata feature extraction, such as a autoencoder-based model to classify the glioma subtype32, and a stacked autoencoder to predict the potential miRNA-disease associations33, the autoencoder is able to extract deep feature from Biodata.

Therefore, in this paper, the Gaussian kernel is incorporated into the Autoencoder, and the features are enhanced by the Gaussian kernel. This results in the original features becoming distinguishable in a separable manner. The Gaussian-kernel Autoencoder is capable of extracting deep features from expression data in both rows and columns, allowing for differentiation between them. After completing the feature extraction for gene expression, the resulting feature matrix is then used as input for the GCN. This enhances the accuracy and confidence of link prediction.

Gaussian-kernel autoencoder

The structure of the Gaussian-kernel Autoencoder is shown in Fig. 2. The Autoencoder consists of two inputs, multiple encoders and decoders, and a Gaussian kernel module. The row values of the gene expression data reflect reflect the state of specific gene under different regulatory relationships, however, it is difficult to reflect the state of the gene under different regulatory relationships. Transposing the gene expression data to invert the rows and columns, which reflecting different genes under specific regulatory relationship, thus the gene expression data and its transposition are used as inputs to the encoder to extract the regulatory state features embedded in the row and column values, which are further encoded and fused in depth by the merging layer to obtain more accurate regulatory features. Multiple merge layers share the weights in the encoder34. The merge layer consists of MLP layers which fuse and amplify the input before splitting it into output vectors. Next, a Gaussian kernel module is used to capture differential features and ultimately extract key expression features of genes.

The Gaussian-kernel as shown in Eq. (1): 1 K(x,y)=e-||x-y||22σ2

where σ is is the parameter of the Gaussian kernel.Fig. 2 The Structure of Gaussian-kernel Autoencoder (where G represents gene expression data, G⊤ represents gene expression data transposition).

The loss function measures the error between the input and output layers of the Gaussian-kernel Autoencoder, which is composed of Mean Squared Error (MSE) and Kullback-Leibler (KL) divergence. When the loss is extremely small, it indicates that the expression features accurately reflect the deep features contained in the rows and columns, as shown in Eq. (2):2 loss=∑n=1Nx1(n)-x1′(n)2∑n=1Nx2(n)-x2′(n)2-∑n=1N(KL(x1′(n)‖x1(n))+KL(x2′(n)‖x2(n)))+λ‖W‖2

where x1 and x2 denote the input features of the Autoencoder, x1′ and x2′ denote the inferred outputs, W denotes the weight and λ denotes the regularization factor.

GCN module based on causal feature reconstruction

Fig. 3 The framework of GCN link prediction based on causal feature reconstruction (where X represents the feature matrix, A represents the adjacency matrix and A^ represents the adjacency matrix of the inferred network).

The convolution layer of a GCN calculates the interactions between each node and its neighboring nodes through neighbor aggregation. It then combines this information with the input features to generate a new representation for each node. Multiple layers of neighbor aggregation enable the GCN to progressively gather neighbor aggregation of nodes of all orders. As a result, a more comprehensive feature representation is computed. It is essential that ensuring the validity and causality of the neighbor information to improve the accuracy of link prediction. For this purpose, Transfer Entropy is utilized to quantify the causality of each order of neighbor information aggregation, which allows for a more comprehensive and accurate feature representation of the nodes in the causal reconstruction module. Consequently, it enhances the causality and dependency among the neighbors of each order, ultimately improving the accuracy of GRN inference. The framework of the GCN based on causal feature reconstruction is depicted in Fig.3, which includes a GCN module and a causal feature reconstruction module.

Graph convolution layer

The graph convolution layer outputs the potential representation of the nodes through convolution operations on the gene expression features and the adjacency matrix.

The formula for aggregation in the convolution layer is shown in Eq. (3)35:3 Hk=σ(D~-12A~D~-12Hk-1Wk-1)

where Hk denotes the node feature representation at the k-th layer of the node, the adjacency matrix after Laplace normalization is denoted as D~-12A~D~-12, σ denotes the activation function sigmoid, W denotes the weight, and Hk-1 denotes the node feature representation at the k-1th layer.

The nodes obtain the neighbor aggregation of the current order through the graph convolution operation and pass it to the next convolution layer. After the network training process, the node feature representation that reflects the features of the entire graph is ultimately obtained.

Transfer entropy

During the process of order-by-order aggregation, the neighbor information of nodes is continuously updated. However, this continuous updating leads to the loss of certain original neighbor information. As a result, it is challenging for the final node feature representation to fully capture the entire graph, leading to a decrease in the accuracy of the downstream task. To tackle this issue, measuring the degree of acceptance and retention of the current node feature representation following neighbor aggregation by Transfer Entropy, which measures the causal relationship between two time series by quantitatively describing the flow of information from one time series to another.

The value of Transfer Entropy indicates the strength of the causal relationship between the two time series, as shown in Eq. (4):4 TEY→X=∑xn+1,xn,ynp(xn+1,xn,yn)logp(xn+1∣xn,yn)p(xn+1∣xn)

where X and Y denote two discrete time series, xn and yn are the discrete values of X and Y in the n-th instance, respectively, p(xn+1|xn,yn) is the joint probability of xn, xn+1, yn, and p(xn+1∣xn) is the conditional probability of xn and xn+1.

In order to measure the information loss of node-by-node aggregation, the historical information of neighbor aggregation is retained during network training. Transfer Entropy is then calculated using both the historical information and the current value. The calculated value is saved and utilized for causal feature reconstruction calculations.

Causal feature reconstruction

After calculating the Transfer Entropy value, the neighbor aggregation with lower acceptance is weighted according to Eq. (5). Then, the neighbor information of each order is fused in the causal feature reconstruction module to obtain a more comprehensive and accurate representation of the causal node features. The larger the value of the Transfer Entropy, the less neighbor information is lost during transmission, and the lower weights assigned to that information. The structure of the causal feature reconstruction module is shown in Fig. 4.Fig. 4 Structure of the causal reconstruction module.

5 Zj′=1TEi→jk+ζGCNi(A,X)

where TEi→jk denotes the Transfer Entropy of the accumulated ith-order neighbourhood information passed to the jth-order neighbourhood information at the kth training, and ζ is a small constant, taken as ζ=0.001 to avoid computational errors when TEi→jk could be zero.

The weighted neighborhood information is input into the MLP to complete the reconstruction. The Kullback-Leibler divergence is used to measure the discrepancy in the reconstruction, with the aim of minimizing the discrepancy throughout the neural network training process. Finally, the reconstruction feature Z^ is obtained, as shown in Eq. (6):6 Z^=MLP(concat(Zj′,Zj);Ω)

where concat(·) denotes the collocation operation, MLP(·;Ω) denotes the linear layer, Ω is the parameter of the linear layer, Zj is the jth-order neighbourhood aggregation, and Zi′ is the feature weighted by Transfer Entropy.

The loss function of the entire CRGCN model consists of Binary Cross Entropy (BCEloss) and two Kullback-Leibler divergence, where the BCEloss function computes the classification error between labels build by known regulatory relationships and labels predicted by the model, both of the Kullback-Leibler divergence measure the reconstructed error and the weighted error of CRGCN, which could reinforce the causal information in neighbourhood aggregation. The function is shown as following:7 loss=BCEloss-∑n=1N(KL(Z^‖Zj′)+KL(Z^‖Zj))+λ‖W‖2

where Z^ is the reconstruction causal feature, Zj′ is the feature weighted by Transfer Entropy, Zj is raw features exported from GCN, W denotes the weight and λ denotes the regularization factor.

After completing the causal feature reconstruction, the obtained reconstructed feature Zj′ is normalized to smooth the training process and prevent gradient explosion. This is done in preparation for the subsequent link prediction task.

Link prediction module

After obtaining the causal reconstruction features of the nodes through the causal feature reconstruction module, the Preferential Preferential (PA)36 is used to predict the similarity scores. These scores indicate the similarity between the current network node feature representations and the inferred network node feature representations. This method is computationally efficient, performs well in densely linked networks, and is particularly suitable for gene regulatory networks. The equation for the scores is (8):8 score=Z^i⊤Z^j

where Z^i denotes current network node feature representation and Z^j denotes inferred network node feature representation.

Using link prediction, the inferred network is obtained by analyzing node-by-node networks and calculating the probability of connectivity between nodes. This process generates the neighborhood matrix of the predicted network, which represents the inferred GRN A.

Realisation implementation

The specific steps for predicting inferred GRN using GCN based on causal feature reconstruction are shown in Algorithm 1.

Algorithm 1 Inference of gene regulatory networks

Experimental process and result analysis

Data set and evaluation indicators

The DREAM5 dataset provided by the DREAM CHALLENGES37 and the mDC networks (Mouse dendritic cell)38 are used in this paper. The specific information about the dataset is presented in Table 1. The S.cerevisiae network has more genes, fewer samples and TFs, the true-positive edges are less than the true-negative edges, which induce class imbalances. Table 1 The details of the DREAM5 dataset.

Network	Number of genes	Number of samples	TF	
E.coli	4511	805	334	
S.cerevisiae	5950	536	333	
mDC	7371	383	1480	

The implementation of gene regulatory network inference by GCN link prediction involved several steps. The hyperparameter for the Gaussian kernel function was set by several experiments. The Autoencoder hidden nodes were set to 805, 536 and 383 corresponding to the number of samples in the E.coli network, S.cerevisiae and mDC network, respectively. The Adam optimizer was used, with a learning rate of 0.001 which is chosen by experiments. Additional L2 regularization was applied during training to prevent parameter overfitting, the L2 rate is 0.001 which is chosen by experiments. After obtaining the features, they are fed into the GCN. The dataset was divided into a training set (70%) and a test set (30%). The Adam optimizer parameters is same as Gaussian-kernel Autoencoder.

In this paper, AUROC (Area Under the Receiver Operating Characteristic Curve) and AUPRC (Area Under the Precision–Recall Curve) are used as evaluation metrics for link prediction. AUROC represents the area under the curve with the axes of True Positive Rate (TPR) and False Positive Rate (FPR), while AUPRC represents the area under the curve with the axes of Precision and Recall.9 TPR=TPTP+FN

10 FPR=FPFP+TN

11 Precision=TPTP+FP

12 Recall=TPTP+FN

where TP is the number of true positives, TN is the number of true negatives, FP is the number of false positives and FN is the number of true negatives.

In order to verify the effectiveness of the method, three experiments are set up to verify the effectiveness of the feature extraction method proposed in this paper, the effectiveness of causal feature reconstruction and the effectiveness of link prediction, separately.

Experiment 1: validating the effectiveness of feature extraction methods

To evaluate the effectiveness of the proposed feature extraction method in this paper, we chose a range of input features, including the original gene expression data, the sample expression features extracted solely by the Gaussian kernel function (GKF), the features obtained through singular value decomposition (SVD)29, the features obtained through non-negative matrix factorization (NMF)28, the fusion with the features extracted by the 1DCNN method22, and the features extracted by the Gaussian-kernel Autoencoder (gAE). For the link prediction task, a two-layer GCN was selected as the network model, and the number of iterations for network training was determined by E.coli, S.cerevisiae and mDC networks, and comparative results are depicted in Figs. 5, 6, 7.Fig. 5 Comparison of results from different feature extraction methods in the E.coli network.

Fig. 6 Comparison of results from different feature extraction methods in the S.cerevisiae network.

Fig. 7 Comparison of results from different feature extraction methods in the mDC network.

Figures 5, 6,7 demonstrate that the original gene expression features resulted in the lowest AUROC and AUPRC metrics for the E.coli network, the S.cerevisiae network and the mDC network. The NMF and SVD methods achieved higher metrics by compressing and filtering of the expression data. Using the GKF, the AUROC and AUPRC metrics improved to 0.804 and 0.801 in the E.coli network, 0.801 and 0.711 in the S.cerevisiae network, 0.656 and 0.642 in the mDC network, these metrics are higher than using the original data, 1DCNN, NMF, and SVD, indicating that the separable features can improve the accuracy of inferring gene regulatory networks.

In the E.coli network, using the gAE to extract features achieved the highest AUROC and AUPRC metrics, surpassing the GKF method by approximately 3% in the AUPRC. In the S.cerevisiae network, the AUROC metric for feature extraction by the gAE was 5.8% higher than using the GKF, however, the AUPRC metrics were slightly lower due to the imbalance in the categories of this network, which has more negative edges. In the mDC network, the AUROC metric and AUPRC metric surpassing the GKF method approximately by 13% and 19%, which achieved the highest AUROC and AUPRC metrics. Therefore, the gAE is able to mine deeper, more complex features to improve prediction accuracy compared to GKF. The result shows that the gAE feature extraction method is effective in the E.coli, the S.cerevisiae and the mDC network, which provides sufficient guidance for subsequent link prediction tasks.

In order to assess the reliability of the Gaussian-kernel Autoencoder in the separable features, and the effect of the Gaussian kernel parameter σ (in eqution(1)) in separable features and prediction results, the parameter σ was taken to be 0.1, 0.5, 1, 2, and 5, tested by a two-layer GCN and a GCN based on causal feature reconstruction (CRGCN), the results of the tests are shown in Table 2. Table 2 Comparison of the Gaussian kernel parameter.

σ	Network	CRGCN	GCN	
AUROC	AUPRC	AUROC	AUPRC	
σ=0.1	E.coli	0.500	0.500	0.544	0.563	
S.cerevisiae	0.735	0.704	0.709	0.575	
mDC	0.616	0.669	0.511	0.520	
σ=0.5	E.coli	0.835	0.822	0.687	0.672	
S.cerevisiae	0.802	0.769	0.715	0.641	
mDC	0.688	0.711	0.569	0.586	
σ=1	E.coli	0.885	0.854	0.808	0.827	
S.cerevisiae	0.911	0.894	0.848	0.704	
mDC	0.870	0.884	0.742	0.766	
σ=2	E.coli	0.836	0.842	0.785	0.784	
S.cerevisiae	0.865	0.845	0.725	0.683	
mDC	0.856	0.864	0.638	0.657	
σ=5	E.coli	0.794	0.806	0.776	0.769	
S.cerevisiae	0.847	0.845	0.706	0.565	
mDC	0.856	0.864	0.638	0.657	
Bold values indicate the best metrics.

From the Table 2, it can be seen that when the parameter σ of the Gaussian kernel is taken as 1, the E.coli, the S.cerevisiae and the mDC network have the highest AUROC and AUPRC metrics.

The T-SNE method39 is used to visually analyse the original features and the separable features. Figures 8, 9, 10, 11, 12 demonstrating that the deep and separable features are extracted by the gAE. In each Fig, the blue represents the E.coli network, the green represents S.cerevisiae network, the orange represents mDC network, and in each sub-graph of the Fig, the raw features are shown on the left and the separable features are shown on the right.Fig. 8 Visualization of the network features, when σ=0.1.

Fig. 9 Visualization of the network features, when σ=0.5.

Fig. 10 Visualization of the network features, when σ=1.

Fig. 11 Visualization of the network features, when σ=2.

The parameter σ determines the distribution of the data in feature space, the larger σ, the features are leaded into more sparse space, made the features over-separated, conversely, the smaller σ, the features are leaded into more denser space, made the features unseparated. Both the larger σ and the smaller σ ineffectively extract separable features, resulting in lower AUROC and AUPRC metrics on CRGCN and GCN. The more properly separable features are extracted by Gaussian kernel when σ=1, therefore σ=1 is selected in the subsequent experiments.Fig. 12 Visualization of the network features, when σ=5.

Overall, the method of extracting gene expression data into separable expression features is effective. Additionally, using the Autoencoders to combine these two features can better preserve the underlying information of the original expression data. This allows the graph neural network to obtain a more precise and comprehensive representation of node features during the node aggregation stage, ultimately enhancing the accuracy of the subsequent link prediction task.

Experiment 2: validating the effectiveness of causal feature reconstruction

In order to validate the effectiveness of the causal feature reconstruction method, the SVD, NMF, GKF, and gAE are selected as the methods for feature extraction. A two-layer GCN and a GCN based on causal feature reconstruction (CRGCN) are used as the network models for the link prediction task, and tested on E.coli and S.cerevisiae networks.Fig. 13 Comparison of results from different network models in the E.coli network.

The former four groups in Figs. 13, 14, 15 display the results of different feature extraction methods combined with GCN in the link prediction task, the latter four groups show the results of the methods combined with CRGCN.Fig. 14 Comparison of results from different network models in the S.cerevisiae network.

Fig. 15 Comparison of results from different network models in the mDC network.

As shown in Figs. 13, 14, 15, both the AUROC and AUPRC metrics showed significantly higher values in the latter four groups compared to the former four groups. In the E.coli network, compared to the gAE-GCN method, the gAE-CRGCN method improved the AUROC metrics by 9.5% and the AUPRC metrics with 3.2%. In S.cerevisiae network the AUROC metric improved with 7.4% and the AUPRC metric improved by 26%. Similarly, in mDC network the AUROC metric improved with 17.3% and the AUPRC metric improved by 15.4%.The results illustrate that using causal feature reconstruction can lead to a deeper causal features, which in turn improves the accuracy and precision of preferential connection prediction.

Overall, causal feature reconstruction enables the GCN model to obtain a more comprehensive representation of node features by enhancing the causal relationship between neighboring nodes at each order. It is able to capture deeper details from the gene expression features, ultimately improving the accuracy of link prediction, when combined with an effective feature extraction method for gene expression data.

Experiment 3: validating the effectiveness of link prediction using GCN based on Causal feature reconstruction

To make the model more accurate, the learning rate was chosen as 0.01, 0.005, 0.001, 1e−4 and 1e−5, the results are shown in Figs. 16 and 17.Fig. 16 The AUROC with different learning rate.

Fig. 17 The AUPRC with different learning rate.

From Figs. 16 and 17, it can be seen that when the learning rate is chosen to 0.001, the model achieves the best AUROC and AUPRC, therefore, the learning rate is chosen to 0.001 in subsequent experiments.

To further validate the reliability and effectiveness of the gAE-CRGCN, 10-fold cross-validation is performed on the E.coli, the S.cerevisiae and the mDC networks, the results are shown in Figs. 18, 19, 20.Fig. 18 10-fold cross-validation on the E.coli network.

Fig. 19 10-fold cross-validation on the S.cerevisiae network.

Fig. 20 10-fold cross-validation on the mDC network.

Table 3 Comparison of Inference Results from Different Algorithms.

Methods	E.coli	S.cerevisiae	mDC	Additional data requirements	
AUROC	AUPRC	AUROC	AUPRC	AUROC	AUPRC	
SVM	0.757	0.649	0.604	0.544	0.586	0.544	NO	
RF	0.837	0.758	0.773	0.691	0.751	0.649	NO	
1DCNN-GAE	0.754	0.705	0.709	0.679	0.682	0.664	NO	
1DCNN-VGAE	0.834	0.808	0.826	0.794	0.753	0.709	NO	
GRGNN	0.903	0.805	0.880	0.797	0.868	0.737	YES	
GRDGNN	0.877	0.813	0.970	0.870	0.853	0.769	YES	
GENELink	0.845	0.809	0.863	0.860	0.830	0.854	NO	
GNNLink	0.866	0.820	0.886	0.843	0.841	0.814	NO	
gAE-CRGCN(ours)	0.885	0.854	0.911	0.894	0.870	0.884	NO	
Bold values indicate the best metrics.

In the 10-fold cross-validation experiment analysis, the E.coli, the S.cerevisiae, and the mDC networks are divided into 10 equal folds. The gAE-CRGCN is trained on 9 folds and tested on the remaining fold, the process is repeated 10 times, each time with a different fold to test, which helps to assess the performance and generalisation ability of the gAE-CRGCN. The Figs. 18, 19, 20 shown that the performers of the gAE-CRGCN model is stable within certain intervals.

Table 3 displays the AUROC and AUPRC scores for both existing methods and the methods proposed in this paper. As shown in Table 3, it can be seen that the SVM method40 demonstrates poor performance on large-scale biological networks and is unable to learn complex regulatory relationships. The RF method41 achieves better results by constructing multiple decision tree models to infer biological networks. However, the AUPRC metric for the S.cerevisiae network is only 0.691, which is lower than that of the GNN method. This indicates that it is difficult for the RF algorithm to reliably infer the class-imbalanced networks. VGAE obtains new node feature representations by sampling the distribution of node feature representations using VGAE, however, data regeneration from the latent space has KL vanishing36, resulting in a poor metric. The GRGNN method combines the network skeleton predicted by known regulatory relationships, Pearson coefficients, and the network skeleton predicted by mutual information to obtain the input neighborhood matrix, GRDGNN42 uses a multi-order neighborhood graph additionally. The GENELink20 is composed of the Graph Attention Network, which incur huge computational and memory overhead than GCN, due to its graph-based attention mechanisms. The GNNLink23 is a GCN-based interaction encoder, by capturing interdependencies between neighbors in the network to infer GRN.

The time consumption (second) of methods based on graph network is as shown in Table 4. Table 4 Time Consumption (second) of Methods Based on Graph Network.

Methods	E.coli	S.cerevisiae	mDC	
1DCNN-GAE	374.2	410.3	741.4	
1DCNN-VGAE	428.1	497.6	900.5	
GRGNN	1379.4	2012.1	6279.1	
GRDGNN	1495.4	2308.1	6563.1	
GENELink	276.0	326.4	803.9	
GNNLink	26.1	41.6	73.8	
gAE-CRGCN (ours)	117.1	140.3	326.4	

From Table 4, it can be seen that the proposed method has the second lowest running time, which means that the proposed method achieves better performance with less computational cost.

Network inference for the E.coli, the S.cerevisiae and the mDC was completed using a Gaussian-kernel Autoencoder with GCN based on causal feature reconstruction (gAE-CRGCN). The GRGNN and GRDGNN methods achieve higher AUROC metrics on E.coli and S.cerevisiae networks, by attaching extra network skeletons and obtaining input neighborhood matrices, however increasing the additional demand for data. The AUROC metric for the gAE-CRGCN method on the E.coli network was slightly lower than the GRGNN, however, the AUPRC metric was 6% higher than the GRGNN, 4.1% and 5.5% higher than GNNLink and GENELink, which are the state-of-the-art methods. Similarly, the AUROC metric of the gAE-CRGCN method on the S.cerevisiae network was slightly lower than GRDGNN, however, the AUPRC metric was 2.8% higher than GRDGNN, 6% and 4% higher than GNNLink and GENELink, the AUPRC is more valued in the GRN inference. The AUROC metric for the gAE-CRGCN method on the mDC network was 0.23% and 2% higher than GRGNN and GRDGNN, the AUPRC metrics was 8.6% and 3.5% higher than GNNLink and GENELink achieved the highest metrics. The gAE-CRGCN achieved the highest AUPRC in the three datasets, indicating that the proposed method has better prediction accuracy, due to the Causal Feature Reconstruction and Gaussian-kernel Autoencoder. The gAE-CRGCN method does not have any additional data requirements and improves the accuracy of node representations through causal reconstruction, which is capable of generating more accurate prediction results for class-imbalanced gene regulatory networks, with improved recall and precision.Fig. 21 Sub-graph of the E.coli inferred network.

Fig. 22 Sub-graph of the S.cerevisiae inferred network.

The sub-graph network are extracted from the inferred network and visualised as shown in are shown in Figs. 21, 22, 23, which intended to show the details of the inferred network sub-graphs. It can be seen the different GRN have different densities of regulatory relationships. Figure 21 shows that a number of gene regulatory relationships in E.coli are dispersed among one another. As shown in Fig. 22, a number of gene regulatory relationships in the E. coli network are dispersed among one another. As shown in Fig. 22a, some genes like YLR121C, YJR141W, YNL156C, and YGR165W have rather more regulatory relationships, and as shown in Fig. 22b gene YNL167C has the most regulatory relationships.

Overall, the gAE-CRGCN method has higher AUPRC scores, which implies the model has better precision and is more suitable for inferring the GRN. The gAE-CRGCN method enhances the node aggregation at each order, resulting in more detailed and comprehensive node feature representations. This is achieved by combining the fusion features extracted by a Gaussian-kernel Autoencoder. The enhanced node feature representations lead to higher similarity in predicting link priority connections, ultimately improving the accuracy of network inference. Experiments have confirmed that the method proposed in this paper is effective.Fig. 23 Sub-graph of the mDC inferred network.

Conclusion

GRN inference using graph GCN has become one of the advanced research. In order to improve the reliability and accuracy of GRN inference, the causal relationships of the node features gained by GCN should be tightened. In this paper, we propose a GCN for inferring GRN that is guided by causal information. The approach acquires the causal and comprehensive node representations to mitigate the loss of information during neighbor aggregation by reconstructing causal features. The Gaussian-kernel Autoencoder is proposed to extract significantly separable features from gene expression data, which improves the computational efficiency and reliability of causal feature reconstruction and the accuracy of inferring GRN. Experiments conducted on the DREAM5 dataset and the mDC dataset demonstrate that the approach proposed in this paper achieves superior prediction accuracy. Furthermore, it successfully reduce limitation of GCN in preserving the information of neighboring nodes at each order, resulting in improved accuracy. The proposed GCN link prediction method, which relies on causal feature reconstruction, enables the acquisition of node feature representations that possess causal feature. Consequently, this approach facilitates the construction of gene regulatory networks that are both reasonable and accurate, while also instilling a sense of credibility.

The proposed model can infer the gene regulatory network accurately, however, the model is impaired by class imbalance and known regulatory pairs insufficiency. In the future, we will fully exploit the priori knowledge of gene regulatory relationships to improve the performance with class imbalance, and explore integration of the multi-omics data to expand the biodata and information on regulatory relationships.

Author contributions

R.J. conceived methodology and modified the manuscript. Y.G. wrote the manuscript and implemented the software. X.Q. performed the validation and visualization. All authors have read and approved the manuscript.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 61702410.

Data availability

The Dream5 dataset can be found at https://dreamchallenges.org/closed-challenges/. The mDC dataset can be found at https://zenodo.org/records/3701939.

Competing interests

The authors declare no competing interests.

Ethical approval and consent to participate

This study did not include the use of any animals, human or otherwise, so did not require ethical approval. Informed consent was obtained from all individuals included in this study.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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