The RW algorithm is a method that is simple to operate but not easy to fall into a local minimum. We constructed a gene interaction network by known AMD-related genes and a string database. Then, we implemented RW on the gene interaction network.

f(x) is a multivariate function with n variables; x = (x₁,x₂,…,x_n) is an n dimension vector.

Step 1: Given the initial iteration point x, λ is the first walking step length, and ϵ is the control accuracy (ϵ is a very small positive number, used to control the end of the algorithm).

Step 2: Given the number of iterations control N, k is the current iteration number; set k = 1.

Step 3: When k < N, randomly generate an n-dimensional vector between (−1, 1). u = (u₁,u₂,…,u_n), (−1 < u_i < 1, i = 1, 2,…, n), and standardize it to get u′.

Let x1 = x + λu′ to complete the first step of walking.

Step 4: Calculate the value of the function, if f(x1) < f(x), which is a better point than the initial value, then reset k to 1, change x1 to x, and go back to step 2; otherwise, k = k + 1. Go back to step 3.

Step 5: If no better value can be found for N consecutive times, it is considered that the optimal solution is within the N-dimensional sphere with the current optimal solution as the center and the current step as the radius (if it is three-dimensional, it just happens to be in the space sphere). At this point, if λ < ϵ, the algorithm ends; otherwise, let λ = λ2, go back to step 1, and start a new round of walking.

Finally, we can get the gene feature after encoding the gene network.

We obtained the gene feature in the last section. Then, we can input the gene feature and label into SVM to get the relationship between the gene and AMD. The workflow of SVM is shown in Figure 1.

First, we used Z-score normalization to process the gene feature. Then, we constructed a Lagrangian function to obtain the values and dualized the original problem. Sequential minimal optimization (SMO) algorithm was used to solve the dualization problem. Finally, we can obtain the classification model and output the prediction results.

We obtained 34 known AMD-related genes from GWAS data. We constructed a gene network which has 239 nodes (genes). We did 10-cross validation by SVM-RW and tested the performance of SVM-RW. The area under the curve (AUC) of SVM-RW is shown in Figure 2.

SVM-RW achieved AUC of 0.927 in identifying AMD-related genes. We compared the results of SVM-RW with several other methods. The results are shown in Table 1.

After verifying the effectiveness of SVM-RW, we randomly selected 34 genes as negative samples and built a final SVM model. SVM-RW predicted 65 novel genes as AMD-related genes.

If we use g to denote a genetic variant (such as a SNP), x as the expression level of a gene, and y as the trait, then the two-step least-squares (2SLS) estimate of the effect of x on y from an MR analysis can be denoted as:

where E^z⁢y and E^z⁢x indicate the least-squares estimates of y and x on z, respectively, and E_xy indicates the effect size of x on y free of confounding from non-genetic factors. Then the sampling variance of the 2SLS estimate of E_xy can be denoted as:

where n denotes the sample size, Px⁢y2 indicates the proportion of variance in the explanation of y by x, and Pz⁢y2 is the proportion of variance in the explanation of y by z. Therefore, we use the statistic T_MR to test the significance of E_xy; T_MR can be denoted as:

where T_MR=χ12 . Based on the suggestion that the power of detecting E_xy can be significantly increased using a two-sample MR analysis (Inoue and Solon, 2010; Pierce and Burgess, 2013), if GWAS and eQTL datasets share the same population, we can use unbiased estimates ϵ^z⁢x to replace E_zx. We therefore have

where E^z⁢y is the estimate of a SNP effect from a GWAS for a trait, and ϵ^z⁢x is the estimate of a SNP effect on the expression level of a gene from an eQTL study. The sampling variance of E^x⁢y can be approximately computed by the Delta method (Lynch and Walsh, 1998) as:

where cov⁢(ϵ^z⁢x,E^z⁢y) is 0. Based on experience, we can replace them by their estimates in practice, indicated as an approximate χ² test statistic of:

where z_xy is the z statistic of the GWAS and z_zx is the z statistic of the eQTL study.

In an MR analysis, E_xy is interpreted as the effect of a phenotype on the gene expression without considering non-genetic confounders. We first collected GWAS summary data and blood eQTL data from available online studies. We first collected a GWAS summary dataset composed of 12,711 advanced AMD cases and 14,590 controls of European descent from the study by Han et al. (2020); 707 Caucasian AMD patients and 2,014 controls from the study by Yan et al. (2018); and 14,034 cases, 91,214 controls, and 11 sourced from datasets including the International AMD Genomics Consortium, IAMDGC, and UKBB from the study by Winkler et al. (2020). The distribution of the above datasets is shown in Figures 3A,B.

Then summary data-based Mendelian randomization (SMR) analysis is implemented on the blood eQTL data and GWAS data; in this paper, we identified 48 SNPs regulating 45 genes (including 41 coding genes and four non-coding genes) resulting in AMD susceptibility. The workflow is shown in Figure 4.

For the first GWAS datasets consisting of 12,711 AMD cases and 14,590 controls from European cohorts, in total we found 3,872 SNPs coexist in both GWAS data and eQTL data; 43 of 3,872 SNPs are significant and regulate 44 genes in gene expression level. In the second GWAS dataset, we found 714 SNPs coexist in both GWAS dataset and eQTL dataset, with none significant. In the third GWAS dataset, we found 1,149 SNPs coexist both in GWAS dataset and eQTL dataset, with one significant regulating one gene in gene expression level. The distribution of the p-value of SNPs regulating genes tested by SMR is shown in Figures 5A–C. A Supplementary Table 1 indicates the p-values of significant SNPs regulating genes tested by SMR; the last line resulted from GWAS dataset 3, and the rest resulted from GWAS dataset 1.

Age-related macular degeneration has been described as a partly genetic disease (Heiba et al., 1994; Stone et al., 2004). Recently, a unifying hypothesis is that immune response gene polymorphisms modulate susceptibility to AMD. Human leukocyte antigen (HLA) polymorphisms, encoded within the major histocompatibility complex (MHC), are the most polymorphic within the human genome. In AMD, researchers detected intense HLA-DR immunoreactivity in not only soft but also hard drusen (Mullins et al., 2000). In the study of Goverdhan et al. (2005), considering the effect of smoking, age, and body mass index (BMI), HLA alleles B^∗4001, DRB1^∗1301, and Cw^∗0701 were found to be related to AMD, which is consistent with our results displayed in Table 1.

In a study by Gu et al. (2013), they researched P2RX7 and P2RX4 genes in 744 AMD patients and 557 Caucasian controls and reached a conclusion that a rare functional haplotype of the P2RX4 leads to loss of innate phagocytosis and confers increased risk of AMD. P2RX7 and P2RX4 damage the normal scavenger function of macrophages and microglia through interaction, making individuals susceptible to AMD.

Figure 6 shows the gene interaction network produced from the results of SMR on AMD. Based on the interaction network, the HLA class intensively interacted and is significantly associated with AMD.

The cluster consisting of DDX39B (aka BAT1), PRRC2A (aka BAT2), and SKIV2L are genes found in the class III region of the MHC (MHC Class III). These genes encode RNA-binding proteins with clear roles in post-transcriptional gene regulation and RNA surveillance. They are likely to have important functions in immunity and are associated with autoimmune diseases (Schott and Garcia-Blanco, 2020). Early work by immunologists have shown that DDX39B promoted gene expression of anti-inflammatory pathways (Allcock et al., 2001). Therefore, understanding the genes interactions may help speculate on the proposed AMD mechanisms and immunotherapy.