Weighted co-expression network construction and identification of core modules

To select core models related to AD, we performed WGCNA. The Pearson correlation coefficient was used to cluster GSE140830 samples and removed outliers. When the soft threshold was 6 (R2 = 0.86), the scale-free network was constructed (Fig. 1A, B). Then, the adjacency matrix was established and the TOM was constructed. According to the different expression types of genes, 19 co-expression modules were ultimately obtained (Fig. 1C). The study analyzed the correlation between characteristic genes and phenotype in the modules and ultimately found that three modules were associated with AD. These three modules included the magenta module (508 genes) (Cor = 0.15, P = 5.7e−4), the cyan module (266 genes) (Cor = −0.11, P = 7.6e−3), and the blue module (3081 genes) (Cor = −0.11, P = 0.01) (Fig. 1D).

Fig. 1

The results of weighted gene co-expression network analysis (WGCNA). A Analysis of the scale-free index for various soft-threshold powers (β). B Analysis of the mean connectivity for various soft-threshold powers. C WGCNA module picture, the upper part of the tree represents the initial module, while the lower part represents the final module. Different colors represent different modules, while gray represents unclassified genes. D WGCNA module and clinical phenotype correlation diagram, behavior module, column clinical phenotype

In this study, 185 apoptosis-related genes were downloaded from the Pathway database, and 44 cuproptosis-related genes were collected from the article. Venn diagram showed that there were 42 apoptosis-related overlapped genes in the intersection of the blue module and apoptosis-related genes, and 9 cuproptosis-related overlapped genes in the intersection of the blue module and cuproptosis-related genes (Fig. 2A, B). There were 6 apoptosis-related overlapped genes in the intersection of the cyan module and apoptosis-related genes, and 1 cuproptosis-related overlapped genes in the intersection of the cyan module and cuproptosis-related genes (Fig. 2C, D). There were 5 apoptosis-related overlapped genes at the intersection of the magenta module and apoptosis-related genes, but there was no cuproptosis-related overlapped gene intersection between the magenta module and cuproptosis-related genes (Fig. 2E, F). Based on the above results, the overlapped genes of blue module and apoptosis-related genes and the overlapped genes of blue module and cuproptosis-related genes were selected for further analysis.

Fig. 2

Venn diagram showing the number of AD-related module genes and apoptosis-related genes, as well as genes that overlap with cuproptosis-related genes. A,B Venn diagram of blue module genes and apoptosis genes, as well as blue module genes and cuproptosis genes. C,D Venn diagram of cyan module genes and apoptosis genes, as well as cyan module genes and cuproptosis genes. E,F Venn diagram of magenta module genes and apoptosis genes, as well as magenta module genes and cuproptosis genes

Enrichment analysis of overlapped genes

To investigate the possible pathways of the apoptosis gene set and cuproptosis gene set, we performed a function enrichment analysis. The GO analysis results (including BP, CC, and MF) of 42 apoptosis-related overlapped genes are reflected in Fig. 3A. The results showed that overlapped genes were mainly concentrated in cell death, apoptotic process, and regulation of cell death. The results of the KEGG pathway analysis indicated that overlapped genes involved in apoptosis, NOD-like receptor signaling pathway, apoptosis-multiple species, p53 signaling pathway, and MAPK signaling pathway (Fig. 3B).

Fig. 3

Enrichment analysis of overlapped genes. A Gene ontology (GO) analysis of apoptosis-related overlapped genes. B Analysis of Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway of apoptosis-related overlapped genes. C GO analysis of cuproptosis-related overlapped genes. D KEGG pathway analysis of cuproptosis-related overlapped genes

The GO analysis of 9 overlapped genes related to cuproptosis exhibited that overlapped genes were closely related to metal ion transport, copper ion transport, and cuproptosis (Fig. 3C). The KEGG pathways that overlapped genes involved in included prion diseases, HIF-1 signaling pathway, apoptosis, and MAPK signaling pathway (Fig. 3D).

PPI network construction and hub gene identification

To screen for apoptosis genes related to AD, the study used CytoHubba in Cytoscape to calculate the closeness, betweenness, and degree between various proteins and ranked them according to their correlation. The study selected the first 10 genes to cross, and finally, 8 overlapped genes were obtained, including BCL2, BIRC2, CASP1, CASP2, CASP3, CASP9, FADD, and TRADD (Fig. 4A). PPI analysis was performed on 8 apoptosis-related hub genes and 9 cuproptosis-related hub genes using the STRING database. Finally, the pivotal genes related to apoptosis and cuproptosis were visualized using Cytoscape (Fig. 4B, C).

Fig. 4

Construction of protein–protein interaction (PPI) network. A Screening of apoptosis-related overlapped genes according to the criteria. B PPI network construction of apoptosis-related hub genes. C PPI network construction of cuproptosis-related hub genes

Construction and evaluation of the nomogram model

To explore the predictive efficacy of the hub genes, we first performed multivariate logistic regression analysis. As shown in Table 1, apoptosis hub genes including TRADD, FADD, BIRC2, and CASP2 were significantly related to AD (P < 0.05), which were used for apoptosis model construction and visualized by nomogram (Fig. 5A). Calibration curve results showed that nomogram predictions showed an agreement with the actual observations (Fig. 5B). ROC curve (AUC = 0.638) and DCA results exhibited that the apoptosis model had good performance in predicting AD (Fig. 5C, D).

Table 1 Multivariate logistic regression analysis of apoptosis hub genesFig. 5

The efficiency of a model constructed from apoptosis-related hub genes in predicting AD. A The nomogram predicts the occurrence of AD, which includes apoptotic genes TRADD, FADD, BIRC2, and CASP2. B Calibration curve of the nomogram. C,D ROC curve analysis and DCA were used to evaluate the prediction performance of the apoptosis gene model. TPR, true positive rate; FPR, false positive rate; AUC, area under the curve; CI, confidence interval

Meanwhile, the multivariate logistic regression analysis results of the module genes related to cuproptosis revealed that MAP2K1, SLC31A1, and PDHB were notably associated with AD (P < 0.05) (Table 2). These three genes were used to cuproptosis model establishment and demonstrated by nomogram (Fig. 6A). Calibration curve results showed that nomogram predictions showed an agreement with the actual observations (Fig. 6B). The results of the ROC curve (AUC = 0.576) and DCA represented that the cuproptosis model had a certain predictive effect on AD (Fig. 6C, D). However, the AUCs between two groups were not statistically different (P = 0.055) (Table 3).

Table 2 Multivariate logistic regression analysis of cuproptosis hub genesFig. 6

The efficiency of a model constructed from cuproptosis-related hub genes in predicting AD. A The nomogram predicts the occurrence of AD, and the genes responsible for cuproptosis in the column chart include MAP2K1, SLC31A1, and PDHB. B Calibration curve of the nomogram. C,D ROC curve analysis and DCA of the cuproptosis model. TPR, true positive rate; FPR, false positive rate; AUC, area under the curve; CI, confidence interval

Table 3 ROC curve analysis of two prediction modelsSubgroup analysis

To further evaluate the efficacy between the apoptosis model and the cuproptosis model in predicting AD, we performed the subgroup analysis. In participants aged ≤65 or male sex, the results of the ROC curve and DCA revealed that the efficacy of the apoptosis model in predicting AD was not significantly different from that of the cuproptosis model (P > 0.05) (Fig. 7A, C, E, G; Table S1). In those with age >65 or sex female, the performance of the apoptosis model in predicting AD was higher than that of the cuproptosis model (P < 0.05) (Fig. 7B, D, F, H; Table S1).

Fig. 7

The performance of the apoptosis model and cuproptosis model in predicting AD in different subgroups. A–D ROC curve of AD predicted by the apoptosis model and cuproptosis model in each clinical subgroup. E–H The DCA of AD was predicted by the apoptosis model and cuproptosis model in each clinical subgroup. TPR, true positive rate; FPR, false positive rate; AUC, area under the curve; CI, confidence interval; A, apoptosis; C, cuproptosis

External validation of the model efficacy

To further explore the accuracy and reliability of the model, the performance of the apoptosis model and cuproptosis model in predicting AD was verified in the GSE26927 dataset. ROC curve and DCA results exhibited that the apoptosis model and cuproptosis model had certain ability in predicting AD, and there was no significant difference between the two models (Fig. 8A, B; Table 4). In the age ≥65, male and female subgroups, the apoptosis model had no significant difference in predicting AD compared with the cuproptosis model (P > 0.05) (Fig. 8D–F, H–J; Table S2). However, cuproptosis model had higher predictive ability of AD than apoptosis model in the age <65 group (P < 0.05) (Fig. 8C, G; Table S2).

Fig. 8

External validation of the apoptosis model and cuproptosis model. A,B ROC curves and DCA of apoptosis model and cuproptosis model in GSE26927. C–F The models predicted the ROC curve of AD in each clinical subgroup. G–J The models predicted the DCA of AD in each clinical subgroup. TPR, true positive rate; FPR, false positive rate; AUC, area under the curve; CI, confidence interval; A, apoptosis; C, cuproptosis

Table 4 ROC curve analysis of two prediction models in GSE26927Gene–clinical model construction

To further explore the difference between the apoptosis model and cuproptosis model in predicting AD, we combined clinical features associated with AD to construct new predictive models. TRADD, FADD, BIRC2, CASP2, and age were significantly related to AD, which was used for the construction of the apoptosis-clinical model (P < 0.05) (Table S3). MAP2K1, PDHB, PDHB, and age were significantly related to AD, which was used for the construction of the cuproptosis-clinical model (P < 0.05) (Table S4). The ROC curve and DCA indicated that the ability of the apoptosis-clinical model and cuproptosis-clinical model to predict AD was similar (P = 0.116) (Fig. 9A, B; Table S5).

Fig. 9

Comparison of apoptosis and cuproptosis models in predicting AD performance after incorporating clinical features. A Two models predict AD’s ROC curve analysis. B Two models predict the DCA of AD. TPR, true positive rate; FPR, false positive rate; AUC, the area under the curve. AUC, area under the curve; 95% CI, 95% confidence interval