# SAMA: A Fast Self-Adaptive Memetic Algorithm for Detecting SNP-SNP Interactions Associated with Disease

Authors: Ying Yin; Boxin Guan; Yuhai Zhao; Yuan Li
Journal: BioMed Research International
 (2020)

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## Abstract

Detecting SNP-SNP interactions associated with disease is significant in genome-wide association study (GWAS). Owing to intensive computational burden and diversity of disease models, existing methods have drawbacks on low detection power and long running time. To tackle these drawbacks, a fast self-adaptive memetic algorithm (SAMA) is proposed in this paper. In this method, the crossover, mutation, and selection of standard memetic algorithm are improved to make SAMA adapt to the detection of SNP-SNP interactions associated with disease. Furthermore, a self-adaptive local search algorithm is introduced to enhance the detecting power of the proposed method. SAMA is evaluated on a variety of simulated datasets and a real-world biological dataset, and a comparative study between it and the other four methods (FHSA-SED, AntEpiSeeker, IEACO, and DESeeker) that have been developed recently based on evolutionary algorithms is performed. The results of extensive experiments show that SAMA outperforms the other four compared methods in terms of detection power and running time.


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## Body

## 1. Introduction

The development of high-throughput sequencing technology makes it possible to analyze single-nucleotide polymorphisms (SNPs) from thousands of individuals [1, 2]. With the purpose of detecting the association between SNPs and a disease, genome-wide association study (GWAS) plays a vital role in recognizing causes of diseases [3–5]. GWAS has been successfully applied to identify numerous SNPs associated with diverse diseases, such as about 30 loci associated with schizophrenia [6–8]. However, due to the large amount of computation imposed by the high-dimensional search space, it is difficult to measure the association between SNP-SNP interactions and disease in genome-wide data [9–11].In the past few years, many methods have been raised for detecting two-locus disease models. These algorithms can be categorized into exhaustive search, stochastic search, heuristic search, and swarm intelligent optimization algorithms [12]. The exhaustive search is a method which evaluates the degree of correlation between all possible SNP-SNP interaction combinations and disease [13, 14] but is often computationally unaffordable for datasets with very large number of SNPs.The random search uses probabilistic methods to find the optimal solution [15, 16]. The heuristic search is an approximate search algorithm that speeds up the search process by reducing the search space [17, 18]. However, the two kinds of searches cannot make the commitment of finding the optimal solution all the time.In the recent years, swarm intelligent optimization algorithms arising from natural phenomena and biological system have held high attention in the detection of disease-associated SNP-SNP interactions [19–21]. For instance, FHSA-SED [22] combines the harmony search algorithm with two scoring functions for the detection of SNP-SNP interactions. AntEpiSeeker [23] detects disease-associated SNP-SNP interactions by using a two-stage ant colony optimization (ACO) [24, 25]. IEACO [26] automatically adjusts path selection strategies using information entropy to detect SNP-SNP interactions. DESeeker [27] uses a two-stage differential evolution (DE) [28, 29] algorithm to identify the SNP-SNP interaction. However, it is worth noticing that all of these methods remain defective owing to their low detection power.One promising approach for tackling the drawbacks mentioned above is to use a fast local search in the evolutionary algorithm. Hybridization of genetic algorithms (GAs) with local search (LS) has already been studied in various optimization problems [30–32]. Such a hybrid algorithm is often called a memetic algorithm (MA) [33]. Thus, we propose a fast self-adaptive memetic algorithm (SAMA) to detect two-locus SNP-SNP interactions associated with disease. In the SAMA algorithm, we improve the crossover, mutation, and selection of MA. These three improved operations are more suitable for detecting two-locus SNP-SNP interactions. Moreover, we incorporate a self-adaptive local search into the proposed algorithm to avoid premature convergence. We compare our algorithm with the state-of-the-art methods and conduct experiments on a wide range of simulated datasets and a real-world biological dataset. The results show the proposed algorithm has improved power in detecting correct SNP-SNP interactions with different disease models.The paper is organized as follows. In Section2, we introduce the problem definition of two-locus SNP-SNP interactions associated with disease and propose the SAMA algorithm. In Section 3, we describe the experiments carried out in order to determine the detection power of our method. Finally, we present the conclusion in Section 4.


## 2. Methods

### 2.1. Problem Definition

A set of SNPs is represented byS=r1,r2,⋯,rL, where r is an SNP and L is the number of SNPs. For detecting two-locus disease models, there are LL−1/2 combinations that can be selected. The value of each SNP is 0, 1, or 2, which represent the homozygous major genotype, the heterozygous genotype, and the homozygous minor genotype, respectively. A dataset contains n samples (nd cases and nu controls), and each sample has a set of SNPs. If the genotype distribution of a two-locus SNP-SNP interaction is significantly different between cases and controls, it may lead to an increase in the risk of the disease.


### 2.2. The SAMA Algorithm

It is a time-consuming task to detect SNP-SNP interactions associated with disease if all possible two-locus interactions from hundreds of thousands of SNPs are considered in a genome-wide scale. In this paper, a fast self-adaptive memetic algorithm (SAMA) is proposed to enhance the detection power of two-locus SNP-SNP interactions in an efficient way.Memetic algorithm (MA) [33] is inspired by natural system model and population evolution. By combining evolutionary algorithms with local search, it can provide a local improvement opportunity for the individuals in a genetic search. The framework of MA can be outlined as Figure 1, and this figure shows the basic structure of the MA algorithm. MA consists of two parts: genetic search and local search, where the local search part includes crossover, mutation, and selection. The SAMA algorithm follows the basic framework in Figure 1 to detect two-locus SNP-SNP interactions associated with disease, and the process is shown in Algorithm 1.Figure 1
The framework of MA.


#### 2.2.1. Initialization

The SAMA algorithm randomly generates a initial population withM individuals. An individual is expressed as xi=rp,rq1<rp,rq<L where rp and rq are SNPs, and the individual xi is generated by
(1)xi=rp←rand0,1·L,rp←rand0,1·L,where  is an upward rounding operation, rand0,1 is a random number between 0 and 1, and L is the number of SNPs in a dataset. After initialization, SAMA finds the current optimal solution xbest with the best value of fitness function. In SAMA, the χ2 test is used as the fitness function to measure the association between two-locus SNP-SNP interactions and the disease.


#### 2.2.2. Hybrid Crossover (HC)

The crossover operator, a fundamental genetic search operator, takes advantage of the information available in the search space. In the SAMA algorithm, we use a hybrid crossover (HS) to cross two individuals. HC can be considered the hybrid between the current best individual and the individuals in the current iteration. The pseudocode of HC is shown in Algorithm2.Algorithm 1: SAMA.
Input: a SNP dataset G, the maximum number of iterations Nmax, the number of individuals M, and the significance threshold θ.Output: two-locus SNP-SNP interactions with p values below the significance threshold θ.1:fori=1 to Mdo2:  Initializexi with two SNPs;3:end for4: Finds the current optimal solutionxbest;5:forj=1 to Nmaxdo6:fori=1 to Mdo7:yi← HC(xi, xbest);8:zi← DBM(yi);9:wi← SLS(zi);10:xi← Selection(wi, xi);11:end for12:  Finds the current optimal solutionxbest;13.  Calculatep value according to xbest;14:ifp value < θthen15:   Recordxbest as a two-locus SNP-SNP interaction;16:end if17:end forAlgorithm 2: HC.
Input: an individual xi=rp,rq, the current best individual xbest=rbestp,rbestqOutput: an individual yi1:r1←rand0,12:ifr1 < pc1then3:rp←rbestp4:end if5:r2←rand0,16:ifr2 < pc2then7:rq←rbestq8:end if9:yi←xiIn the algorithm, the current best individualxbest and the individual xi in the current iteration are selected as two parents. If the random number r1 between 0 and 1 is less than the crossover probability pc1, the first SNP rp in xi is replaced by the first SNP rbestp in xbestp. If the random number r2 is less than the crossover probability pc2, the second SNP rq in xi is replaced by the second SNP rbestq in xbestq. If the conditions of r1<pc1 and r2<pc2 are satisfied at the same time, xi is replaced by xbest.


#### 2.2.3. Distributed Breeder Mutation (DBM)

The mutation operator is used to randomly create the diversity of individuals in a population. We use a mutation called distributed breeder mutation (DBM) in the SAMA algorithm. DBM, inspired by the breeder genetic algorithm proposed by Muhlenbein and Schlierkamp-Voosen [34], is a robust global search based on a solid theory. The mutated individual zi is calculated by the following equation:
(2)zi=r′p←rp±range·δ,r′q←rq±range·δrpandrp∈yi,where range is the mutation set to 0.1·L, δ is calculated from a distribution which prefers a small value, and the “+” or “−” is chosen with a probability of 0.5. Thus, rp is mutated in the interval between [rp−range·δ] and [rp+range·δ], and rq is mutated in the interval between [rq−range·δ] and [rq+range·δ].If the mutated individualzi is outside the specified range 1<rp,rq<L, zi will be reinitialized. δ is computed according to the following equation:
(3)δ=∑i=115αi2−iαi∈0,1.αi is set to 0 before the mutation operation. Then, each αi is mutated to 1 with a probability of 1/16. The minimum step size is produced with a precision of rangei·2−15. Algorithm 3 gives the execution process of DBM.Algorithm 3: DBM.
Input: an individual yi=rp,rqOutput: an individual zi=r′p,r′q1: Computeδ according to (3)2: Select+ or −3: Determine the rangeδ4:r′p←rp+range·δorrp−range·δ5:r′q←rq+range·δorrq−range·δ6:r′p<1orr′p>Lthen7:Reinitializer′p8:end if9:ifr′q<1orr′q>Lthen10:Reinitializer′q11:end if


#### 2.2.4. Self-Adaptive Local Search (SLS)

Local search (LS) is a simple iterative method for finding approximate solutions. If a candidate solution has better or equal fitness, LS moves the search from the current solution to the candidate solution. If LS is applied to every solution many times, the running time is very long because the additional functional evaluations required for LS is expensive. Thus, a self-adaptive LS (SLS) is introduced, which uses a probability to reduce the number of times that are used for local search. The probability that each individual is selected to allpy the SLS operation ispzi, and the pzi is defined by
(4)pzi=1ifziisimprovedξ·pziotherwise,where ξ is the switch parameter, and zi is an individual after HC and DBM. The initial pzi of each individual is 1; hence, each individual will be selected at least once for SLS. If the fitness value of the individual zi is improved, the probability pzi that zi is selected is still 1. Otherwise, pzi is changed to ξ·pzi. If the fitness value of zi is not improved after being selected n times, this value is ξn·pzi. The pseudocode of SLS is shown in Algorithm 4.Algorithm 4: SLS.
Input: an individual zi after crossover and mutationOutput: an individual wi1:r3←rand0,12:wi←03:whiler3 < pzido4:ui← DBM(zi)5:iffitui > fitzithen6:iffitui > fitwithen7:wi←ui8:pzi←19:else10:pzi←ξ·pzi11:end if12:else13:wi←zi14:pzi←ξ·pzi15:end if16:r3←rand0,117:end while


#### 2.2.5. Elitist Selection (ES)

In the SAMA algorithm, an elitist selection is introduced to select individuals that evolve to the next iteration. After HC, DBM, and SLS, the ES operation is performed according to(5)xi=wiiffitwi>fitxixiiffitwi≤fitxi.If the fitness value of the individualwi is greater than that of the previous individual xi, xi is replaced by wi. Otherwise, xi is unchanged.


### 2.3. A Running Instance of SAMA

In this subsection, we give a running instance of SAMA in Figure2. Suppose that there are five individuals in the current population. After initialization, x1=54,63, x2=75,53, x3=107,87, x4=121,82, and x5=83,78. Among them, x4 obtains the highest fitness value, i.e., fitxi=62.8, and hence, x4 is the current optimal solution xbest (rbestp=121 and rbestq=82).Figure 2
A running instance of SAMA.First, we perform the HC operation. Supposer1≥pc1 and r2≥pc2 for x1 and x4, r2 < pc2 for x2, r1 < pc1 for x3, and r1 < pc1 and r2 < pc2 for x5. According to Algorithm 2, x1 and x4 are not changed and assigned directly to y1 and y4, whereas the other three individuals are changed. One SNP in x2 and x3 is replaced; hence, x2 is changed to y2=75,82 and x3 is changed to y3=121,87. x5 is changed to y5=121,82 because both SNPs in x5 are replaced.Next is the DBM operation. We assume thatrange·δ of y1 is 0, the range·δ of y2 and y3 is 10, and the range·δ of y4 and y5 is 15. y2 and y4 get “−”, whereas y3 and y5 get “+.” Thus, y1 is not changed and assigned directly to z1=54,63, y2 is changed to z2=65,72, y3 is changed to z3=131,97, y4 is changed to z4=106,67, and y5 is changed to z5=136,97.After completing HC and DBM, the SLS operation is executed.z1, z2, and z5 are not changed and assigned directly to w1, w2, and w5 due to r3≥pz. For z3 and z4, SLS is operated cyclically because of r3<pz. z3 is changed to w3=141,107 and z4 is changed to w4=126,87 after the DMB operation in SLS.Finally, the selection operation is performed. We suppose thatfitw1≤fitx1, fitw2≤fitx2, fitw3≤fitx3, fitw4>fitx4, and fitw5>fitx5. Thus, x1, x2, and x3 are retained to the next generation. For x4 and x5, the two individuals are replaced and assigned to the next generation.


## 2.1. Problem Definition

A set of SNPs is represented byS=r1,r2,⋯,rL, where r is an SNP and L is the number of SNPs. For detecting two-locus disease models, there are LL−1/2 combinations that can be selected. The value of each SNP is 0, 1, or 2, which represent the homozygous major genotype, the heterozygous genotype, and the homozygous minor genotype, respectively. A dataset contains n samples (nd cases and nu controls), and each sample has a set of SNPs. If the genotype distribution of a two-locus SNP-SNP interaction is significantly different between cases and controls, it may lead to an increase in the risk of the disease.


## 2.2. The SAMA Algorithm

It is a time-consuming task to detect SNP-SNP interactions associated with disease if all possible two-locus interactions from hundreds of thousands of SNPs are considered in a genome-wide scale. In this paper, a fast self-adaptive memetic algorithm (SAMA) is proposed to enhance the detection power of two-locus SNP-SNP interactions in an efficient way.Memetic algorithm (MA) [33] is inspired by natural system model and population evolution. By combining evolutionary algorithms with local search, it can provide a local improvement opportunity for the individuals in a genetic search. The framework of MA can be outlined as Figure 1, and this figure shows the basic structure of the MA algorithm. MA consists of two parts: genetic search and local search, where the local search part includes crossover, mutation, and selection. The SAMA algorithm follows the basic framework in Figure 1 to detect two-locus SNP-SNP interactions associated with disease, and the process is shown in Algorithm 1.Figure 1
The framework of MA.


### 2.2.1. Initialization

The SAMA algorithm randomly generates a initial population withM individuals. An individual is expressed as xi=rp,rq1<rp,rq<L where rp and rq are SNPs, and the individual xi is generated by
(1)xi=rp←rand0,1·L,rp←rand0,1·L,where  is an upward rounding operation, rand0,1 is a random number between 0 and 1, and L is the number of SNPs in a dataset. After initialization, SAMA finds the current optimal solution xbest with the best value of fitness function. In SAMA, the χ2 test is used as the fitness function to measure the association between two-locus SNP-SNP interactions and the disease.


### 2.2.2. Hybrid Crossover (HC)

The crossover operator, a fundamental genetic search operator, takes advantage of the information available in the search space. In the SAMA algorithm, we use a hybrid crossover (HS) to cross two individuals. HC can be considered the hybrid between the current best individual and the individuals in the current iteration. The pseudocode of HC is shown in Algorithm2.Algorithm 1: SAMA.
Input: a SNP dataset G, the maximum number of iterations Nmax, the number of individuals M, and the significance threshold θ.Output: two-locus SNP-SNP interactions with p values below the significance threshold θ.1:fori=1 to Mdo2:  Initializexi with two SNPs;3:end for4: Finds the current optimal solutionxbest;5:forj=1 to Nmaxdo6:fori=1 to Mdo7:yi← HC(xi, xbest);8:zi← DBM(yi);9:wi← SLS(zi);10:xi← Selection(wi, xi);11:end for12:  Finds the current optimal solutionxbest;13.  Calculatep value according to xbest;14:ifp value < θthen15:   Recordxbest as a two-locus SNP-SNP interaction;16:end if17:end forAlgorithm 2: HC.
Input: an individual xi=rp,rq, the current best individual xbest=rbestp,rbestqOutput: an individual yi1:r1←rand0,12:ifr1 < pc1then3:rp←rbestp4:end if5:r2←rand0,16:ifr2 < pc2then7:rq←rbestq8:end if9:yi←xiIn the algorithm, the current best individualxbest and the individual xi in the current iteration are selected as two parents. If the random number r1 between 0 and 1 is less than the crossover probability pc1, the first SNP rp in xi is replaced by the first SNP rbestp in xbestp. If the random number r2 is less than the crossover probability pc2, the second SNP rq in xi is replaced by the second SNP rbestq in xbestq. If the conditions of r1<pc1 and r2<pc2 are satisfied at the same time, xi is replaced by xbest.


### 2.2.3. Distributed Breeder Mutation (DBM)

The mutation operator is used to randomly create the diversity of individuals in a population. We use a mutation called distributed breeder mutation (DBM) in the SAMA algorithm. DBM, inspired by the breeder genetic algorithm proposed by Muhlenbein and Schlierkamp-Voosen [34], is a robust global search based on a solid theory. The mutated individual zi is calculated by the following equation:
(2)zi=r′p←rp±range·δ,r′q←rq±range·δrpandrp∈yi,where range is the mutation set to 0.1·L, δ is calculated from a distribution which prefers a small value, and the “+” or “−” is chosen with a probability of 0.5. Thus, rp is mutated in the interval between [rp−range·δ] and [rp+range·δ], and rq is mutated in the interval between [rq−range·δ] and [rq+range·δ].If the mutated individualzi is outside the specified range 1<rp,rq<L, zi will be reinitialized. δ is computed according to the following equation:
(3)δ=∑i=115αi2−iαi∈0,1.αi is set to 0 before the mutation operation. Then, each αi is mutated to 1 with a probability of 1/16. The minimum step size is produced with a precision of rangei·2−15. Algorithm 3 gives the execution process of DBM.Algorithm 3: DBM.
Input: an individual yi=rp,rqOutput: an individual zi=r′p,r′q1: Computeδ according to (3)2: Select+ or −3: Determine the rangeδ4:r′p←rp+range·δorrp−range·δ5:r′q←rq+range·δorrq−range·δ6:r′p<1orr′p>Lthen7:Reinitializer′p8:end if9:ifr′q<1orr′q>Lthen10:Reinitializer′q11:end if


### 2.2.4. Self-Adaptive Local Search (SLS)

Local search (LS) is a simple iterative method for finding approximate solutions. If a candidate solution has better or equal fitness, LS moves the search from the current solution to the candidate solution. If LS is applied to every solution many times, the running time is very long because the additional functional evaluations required for LS is expensive. Thus, a self-adaptive LS (SLS) is introduced, which uses a probability to reduce the number of times that are used for local search. The probability that each individual is selected to allpy the SLS operation ispzi, and the pzi is defined by
(4)pzi=1ifziisimprovedξ·pziotherwise,where ξ is the switch parameter, and zi is an individual after HC and DBM. The initial pzi of each individual is 1; hence, each individual will be selected at least once for SLS. If the fitness value of the individual zi is improved, the probability pzi that zi is selected is still 1. Otherwise, pzi is changed to ξ·pzi. If the fitness value of zi is not improved after being selected n times, this value is ξn·pzi. The pseudocode of SLS is shown in Algorithm 4.Algorithm 4: SLS.
Input: an individual zi after crossover and mutationOutput: an individual wi1:r3←rand0,12:wi←03:whiler3 < pzido4:ui← DBM(zi)5:iffitui > fitzithen6:iffitui > fitwithen7:wi←ui8:pzi←19:else10:pzi←ξ·pzi11:end if12:else13:wi←zi14:pzi←ξ·pzi15:end if16:r3←rand0,117:end while


### 2.2.5. Elitist Selection (ES)

In the SAMA algorithm, an elitist selection is introduced to select individuals that evolve to the next iteration. After HC, DBM, and SLS, the ES operation is performed according to(5)xi=wiiffitwi>fitxixiiffitwi≤fitxi.If the fitness value of the individualwi is greater than that of the previous individual xi, xi is replaced by wi. Otherwise, xi is unchanged.


## 2.2.1. Initialization

The SAMA algorithm randomly generates a initial population withM individuals. An individual is expressed as xi=rp,rq1<rp,rq<L where rp and rq are SNPs, and the individual xi is generated by
(1)xi=rp←rand0,1·L,rp←rand0,1·L,where  is an upward rounding operation, rand0,1 is a random number between 0 and 1, and L is the number of SNPs in a dataset. After initialization, SAMA finds the current optimal solution xbest with the best value of fitness function. In SAMA, the χ2 test is used as the fitness function to measure the association between two-locus SNP-SNP interactions and the disease.


## 2.2.2. Hybrid Crossover (HC)

The crossover operator, a fundamental genetic search operator, takes advantage of the information available in the search space. In the SAMA algorithm, we use a hybrid crossover (HS) to cross two individuals. HC can be considered the hybrid between the current best individual and the individuals in the current iteration. The pseudocode of HC is shown in Algorithm2.Algorithm 1: SAMA.
Input: a SNP dataset G, the maximum number of iterations Nmax, the number of individuals M, and the significance threshold θ.Output: two-locus SNP-SNP interactions with p values below the significance threshold θ.1:fori=1 to Mdo2:  Initializexi with two SNPs;3:end for4: Finds the current optimal solutionxbest;5:forj=1 to Nmaxdo6:fori=1 to Mdo7:yi← HC(xi, xbest);8:zi← DBM(yi);9:wi← SLS(zi);10:xi← Selection(wi, xi);11:end for12:  Finds the current optimal solutionxbest;13.  Calculatep value according to xbest;14:ifp value < θthen15:   Recordxbest as a two-locus SNP-SNP interaction;16:end if17:end forAlgorithm 2: HC.
Input: an individual xi=rp,rq, the current best individual xbest=rbestp,rbestqOutput: an individual yi1:r1←rand0,12:ifr1 < pc1then3:rp←rbestp4:end if5:r2←rand0,16:ifr2 < pc2then7:rq←rbestq8:end if9:yi←xiIn the algorithm, the current best individualxbest and the individual xi in the current iteration are selected as two parents. If the random number r1 between 0 and 1 is less than the crossover probability pc1, the first SNP rp in xi is replaced by the first SNP rbestp in xbestp. If the random number r2 is less than the crossover probability pc2, the second SNP rq in xi is replaced by the second SNP rbestq in xbestq. If the conditions of r1<pc1 and r2<pc2 are satisfied at the same time, xi is replaced by xbest.


## 2.2.3. Distributed Breeder Mutation (DBM)

The mutation operator is used to randomly create the diversity of individuals in a population. We use a mutation called distributed breeder mutation (DBM) in the SAMA algorithm. DBM, inspired by the breeder genetic algorithm proposed by Muhlenbein and Schlierkamp-Voosen [34], is a robust global search based on a solid theory. The mutated individual zi is calculated by the following equation:
(2)zi=r′p←rp±range·δ,r′q←rq±range·δrpandrp∈yi,where range is the mutation set to 0.1·L, δ is calculated from a distribution which prefers a small value, and the “+” or “−” is chosen with a probability of 0.5. Thus, rp is mutated in the interval between [rp−range·δ] and [rp+range·δ], and rq is mutated in the interval between [rq−range·δ] and [rq+range·δ].If the mutated individualzi is outside the specified range 1<rp,rq<L, zi will be reinitialized. δ is computed according to the following equation:
(3)δ=∑i=115αi2−iαi∈0,1.αi is set to 0 before the mutation operation. Then, each αi is mutated to 1 with a probability of 1/16. The minimum step size is produced with a precision of rangei·2−15. Algorithm 3 gives the execution process of DBM.Algorithm 3: DBM.
Input: an individual yi=rp,rqOutput: an individual zi=r′p,r′q1: Computeδ according to (3)2: Select+ or −3: Determine the rangeδ4:r′p←rp+range·δorrp−range·δ5:r′q←rq+range·δorrq−range·δ6:r′p<1orr′p>Lthen7:Reinitializer′p8:end if9:ifr′q<1orr′q>Lthen10:Reinitializer′q11:end if


## 2.2.4. Self-Adaptive Local Search (SLS)

Local search (LS) is a simple iterative method for finding approximate solutions. If a candidate solution has better or equal fitness, LS moves the search from the current solution to the candidate solution. If LS is applied to every solution many times, the running time is very long because the additional functional evaluations required for LS is expensive. Thus, a self-adaptive LS (SLS) is introduced, which uses a probability to reduce the number of times that are used for local search. The probability that each individual is selected to allpy the SLS operation ispzi, and the pzi is defined by
(4)pzi=1ifziisimprovedξ·pziotherwise,where ξ is the switch parameter, and zi is an individual after HC and DBM. The initial pzi of each individual is 1; hence, each individual will be selected at least once for SLS. If the fitness value of the individual zi is improved, the probability pzi that zi is selected is still 1. Otherwise, pzi is changed to ξ·pzi. If the fitness value of zi is not improved after being selected n times, this value is ξn·pzi. The pseudocode of SLS is shown in Algorithm 4.Algorithm 4: SLS.
Input: an individual zi after crossover and mutationOutput: an individual wi1:r3←rand0,12:wi←03:whiler3 < pzido4:ui← DBM(zi)5:iffitui > fitzithen6:iffitui > fitwithen7:wi←ui8:pzi←19:else10:pzi←ξ·pzi11:end if12:else13:wi←zi14:pzi←ξ·pzi15:end if16:r3←rand0,117:end while


## 2.2.5. Elitist Selection (ES)

In the SAMA algorithm, an elitist selection is introduced to select individuals that evolve to the next iteration. After HC, DBM, and SLS, the ES operation is performed according to(5)xi=wiiffitwi>fitxixiiffitwi≤fitxi.If the fitness value of the individualwi is greater than that of the previous individual xi, xi is replaced by wi. Otherwise, xi is unchanged.


## 2.3. A Running Instance of SAMA

In this subsection, we give a running instance of SAMA in Figure2. Suppose that there are five individuals in the current population. After initialization, x1=54,63, x2=75,53, x3=107,87, x4=121,82, and x5=83,78. Among them, x4 obtains the highest fitness value, i.e., fitxi=62.8, and hence, x4 is the current optimal solution xbest (rbestp=121 and rbestq=82).Figure 2
A running instance of SAMA.First, we perform the HC operation. Supposer1≥pc1 and r2≥pc2 for x1 and x4, r2 < pc2 for x2, r1 < pc1 for x3, and r1 < pc1 and r2 < pc2 for x5. According to Algorithm 2, x1 and x4 are not changed and assigned directly to y1 and y4, whereas the other three individuals are changed. One SNP in x2 and x3 is replaced; hence, x2 is changed to y2=75,82 and x3 is changed to y3=121,87. x5 is changed to y5=121,82 because both SNPs in x5 are replaced.Next is the DBM operation. We assume thatrange·δ of y1 is 0, the range·δ of y2 and y3 is 10, and the range·δ of y4 and y5 is 15. y2 and y4 get “−”, whereas y3 and y5 get “+.” Thus, y1 is not changed and assigned directly to z1=54,63, y2 is changed to z2=65,72, y3 is changed to z3=131,97, y4 is changed to z4=106,67, and y5 is changed to z5=136,97.After completing HC and DBM, the SLS operation is executed.z1, z2, and z5 are not changed and assigned directly to w1, w2, and w5 due to r3≥pz. For z3 and z4, SLS is operated cyclically because of r3<pz. z3 is changed to w3=141,107 and z4 is changed to w4=126,87 after the DMB operation in SLS.Finally, the selection operation is performed. We suppose thatfitw1≤fitx1, fitw2≤fitx2, fitw3≤fitx3, fitw4>fitx4, and fitw5>fitx5. Thus, x1, x2, and x3 are retained to the next generation. For x4 and x5, the two individuals are replaced and assigned to the next generation.


## 3. Results

To evaluate of the performance of the SAMA algorithm, we test it on both simulated and real-world biological datasets. we compare it with FHSA-SED, AntEpiSeeker, IEACO, and DESeeker on these datasets. For the simulated datasets, we adopt three two-locus disease models. For the real-world biological dataset, we run SAMA on an age-related macular degeneration (AMD) data [35].


### 3.1. Simulated Datasets

In this subsection, we carry out the experiments in three simulated disease models (Models 1-3) [36]. Model 1 is a two-locus multiplicative model in which the disease prevalence PD increases multiplicatively with the incremental presence of the disease genotype interaction. Model 2 is a two-locus threshold model, in which PD does not increase until the number of disease genotype interactions pass the threshold. Model 3 is a two-locus concrete mode that simulates the effects of SNP-SNP interactions on susceptibility to traits. In the three models, PD is set to 0.1, and the minor allele frequencies (MAF) is 0.05, 0.10, 0.20, and 0.50. The genetic heritability (h2) is 0.005 in Model 1, and h2 is 0.02 in Models 2 and 3. According to the combination of these values, 12 penetrance tables are obtained (see Table 1). 200 datasets corresponding to each penetrance table are generated using GAMETES_2.0 [37]. 100 SNPs are generated in the first 100 datasets, whereas the number of SNPs is 2000 in the other 100 datasets.Table 1
Details of three two-locus disease models.
MAF0.050.100.200.50AAAaaaAAAaaaAAAaaaAAAaaaModel 1 (PD=0.1, h2=0.005)BB0.0980.0980.098BB0.0960.0960.096BB0.0920.0920.092BB0.0780.0780.078Bb0.0980.2990.522Bb0.0960.1970.282Bb0.0920.1450.181Bb0.0780.1050.122bb0.0980.5220.912Bb0.0960.2820.408Bb0.0920.1810.227Bb0.0780.1220.142Model 2 (PD=0.1, h2=0.02)BB0.0960.0960.096BB0.0920.0920.092BB0.0840.0840.084BB0.0520.0520.052Bb0.0960.5330.533Bb0.0920.3190.319Bb0.0840.2100.210Bb0.0520.1380.138bb0.0960.5330.533Bb0.0920.3190.319Bb0.0840.2100.210Bb0.0520.1380.138Model 3 (PD=0.1, h2=0.02)BB0.0800.1920.192BB0.0720.1640.164BB0.0610.1460.146BB0.0670.1550.155Bb0.1920.0800.080Bb0.1640.0720.072Bb0.1460.0610.061Bb0.1550.0670.067bb0.1920.0800.080Bb0.1640.0720.072Bb0.1460.0610.061Bb0.1550.0670.067


### 3.2. Parameter Setting

In the experiments, we set the same maximum number of iterations for the five algorithms, that is, the maximum iteration number for datasets with200 SNPs is set to 50 and the maximum iteration number for datasets with 2000 is set to 500. The maximum number of iterations is less than the number of iterations using an exhaustive algorithm. Furthermore, the other parameters of the five compared algorithm are shown in Table 2.Table 2
Parameter setting of five algorithms.
AlgorithmParametersSAMAThe crossover probabilitiespc1 and pc2=0.8; the switch parameter ξ=0.5; the number of individuals M=500FHSA-SEDThe harmony memory considering rateHMCR =0.9; the pitch-adjusting rate PAR =0.35; the number of harmonies evaluated with Bayesian network scoring HM1=250; the number of harmonies evaluated with Gini scoring HM2=250AntEpiSeekerThe size of large SNP setslargesetsize=6; the size of small SNP sets smallsetsize=3; the weight parameters α and β=1; the pheromone evaporation rate ρ=0.05; the initial pheromone τ0=100; the number of ants M=500IEACOThe switch parameterθ is 0.001; the upper bound of negative feedback pheromone on worse paths μ=300; the weight parameters α and β=1; the parameter determining the weight of negative feedback pheromone γ=1; the number of ants M=500DESeekerThe number of SNPs in a large size SNP combinationW=6; the number of vectors M=500


### 3.3. Performance Evaluation Criteria

With the purpose of conducting the experiments comprehensively, we introduce two measurements: detection power and running time. The detection power is defined below:(6)Power=#T/#G,where #G is the datasets that are generated by the same penetrance table (#G=100 in the experiments) and #T is the number of datasets in which the two-locus SNP-SNP interaction associated with disease is detected.


### 3.4. Experiments on Simulated Datasets

Figures3 and 4 present the detection power of the five compared algorithms on the three disease models. It is indicated from the figures that the SAMA algorithm is better than or equal to FHSA-SED, AntEpiSeeker, IEACO, and DESeeker on most settings, with the exception of MAF=0.50 in Model 1 with 200 SNPs. SAMA detects all disease-associated SNP-SNP interactions on six settings for the datasets with 200 SNPs, and the algorithm detects all disease-associated SNP-SNP interactions on two settings for the datasets with 2000 SNPs. On the datasets with 200 SNPs, the other four algorithms can be comparable with SAMA because they also have good performance. On the datasets with 2000 SNPs, the detection power obtained by our algorithm is significantly greater than that of the other four algorithms, especially in Model 3. Followed by FHSA-SED and DESeeker, these two algorithms also show not bad performance. Next is IEACO. The performance of AntEpiSeeker performance is the worst in this experiment. The above analysis reveals that the proposed algorithm is more effective for detecting two-locus SNP-SNP interactions.Figure 3
Power comparison of five compared algorithms on the datasets with 200 SNPs.Figure 4
Power comparison of five compared algorithms on the datasets with 2000 SNPs.Tables3 and 4 show the running time of the five compared algorithms on the three disease models. As illustrated in the two tables, the running time of our method is less than that of the other four methods. This demonstrates that SAMA can efficiently decrease the running time in detecting two-locus SNP-SNP interactions.Table 3
Running time of five compared algorithms on the datasets with200 SNPs.
ModelMAFSAMAFHSA-SEDAntEpiSeekerIEACODESeekerModel 10.059.12±0.5310.55±0.5946.63±2.3111.21±0.7610.03±0.640.108.97±0.5110.32±0.5348.52±2.4012.45±0.819.89±0.700.209.32±0.4910.47±0.5847.71±2.2910.93±0.799.93±0.660.509.55±0.4410.62±0.6245.63±1.9913.06±0.8210.32±0.73Model 20.059.53±0.4811.04±0.6548.57±2.3710.90±0.7110.54±0.770.109.29±0.5710.86±0.6849.12±2.3011.35±0.669.98±0.690.208.86±0.4611.06±0.6446.83±2.1212.52±0.7310.74±0.650.509.22±0.5010.75±0.7046.89±2.0611.83±0.689.76±0.59Model 30.059.06±0.5510.63±0.6350.02±2.5512.04±0.7410.63±0.620.109.52±0.5911.05±0.6847.74±2.1911.67±0.8010.72±0.580.209.32±0.5110.64±0.5748.82±2.4912.42±0.699.48±0.610.509.94±0.6010.74±0.6145.90±2.0511.53±0.789.80±0.65Table 4
Running time of five compared algorithms on the datasets with2000 SNPs.
ModelMAFSAMAFHSA-SEDAntEpiSeekerIEACODESeekerModel 10.0584.63±3.7698.74±5.32431.53±11.57108.64±5.9697.56±4.970.1087.53±4.02103.63±5.67427.87±10.94109.42±6.03100.55±5.170.2090.89±3.9098.85±5.15442.35±10.52111.34±6.1299.74±5.200.5088.16±3.95101.15±4.96425.84±12.02104.44±6.04103.85±5.06Model 20.0591.48±4.1297.88±4.87435.14±12.53110.45±5.64102.66±5.070.1089.86±3.79100.56±5.04448.57±10.89102.63±6.2398.85±5.120.2089.17±4.0399.95±4.78459.84±11.78101.34±5.98105.05±5.310.5092.74±3.87100.13±4.83418.52±10.97105.65±5.95104.43±5.13Model 30.0590.63±3.9397.73±5.01451.45±12.32112.56±6.46101.89±5.440.1086.73±3.89103.54±5.21432.85±11.67109.93±6.15104.92±5.190.2087.83±4.0796.97±4.89429.50±12.02113.56±5.9699.71±5.080.5090.09±3.86101.34±5.36440.86±12.63114.37±6.07103.67±5.32


### 3.5. Experiments on a Real-World Biological Dataset

According to the results of the simulated experiments, SAMA performs well for detecting two-locus SNP-SNP interactions. In this section, we conduct experiments on a real-world biological dataset [35]. The purpose of the experiment is to detect two-locus SNP-SNP interactions associated with the disease by using the five compared algorithms. The five algorithms are run 10 times, and Figure 5 is drawn according to the obtained p values. In the figure, a solid dot has two values, one is x-value, and the other is y-value. The y-value represents the p value, and the x-value denotes the SNP-SNP interaction detected by an algorithm with a certain p value. For the SAMA algorithm, 31 solid dots are detected, that is, 31 two-locus SNP-SNP interactions are detected. It can be seen evidently that the number of solid dots found by the proposed algorithm is more than that found by the other four algorithms. Followed by AntEpiSeeker, this algorithm detects 27 solid dots. Next is DESeeker and FHSA-SED. The DESeeker algorithm detects 23 solid dots, and the FHSA-SED algorithm detects 22 solid dots. The number of interactions found by IEACO is relatively less. This algorithm only finds 21 solid dots. The above analysis shows that SAMA can detect more two-locus SNP-SNP interactions than the other algorithms under the same number of iterations.Figure 5
The number of two-locus SNP-SNP interactions detected by five algorithms.Table5 presents the two-locus SNP-SNP interactions with p values less than 1.0e-06 detected by our method. In the table, the number of two-locus SNP-SNP interactions found by the SAMA algorithm with p values less than 1.0e-08, 1.0e-07, and 1.0e-06 are 1, 9, and 21, respectively. Table 6 gives the number of two-locus SNP-SNP interactions detected by SAMA under different parameters. It can be seen from the Table 5 that rs380390 and rs1329428 are interacted with many other SNPs. The two SNPs are are located in the CFH gene, and the CFH gene has been commonly association with AMD [16, 38–40]. Furthermore, many SNPs included in detected SNP-SNP interactions are located in non-gene coding regions (NA). There are seven interactions between the CHF gene and NA when the p value is less than 1.0e-07, and there are ten interactions between the CHF gene and NA when the p value is between 1.0e-07 and 1.0e-06. The CHF gene has one interaction with the KDM4C gene, and it has two interactions with the MED27 gene. SNP rs2224762 is located in the KDM4C gene that can regulate chromosome segregation during mitosis [41]. This gene that may be associated with AMD has been reported before [22, 42]. SNPs rs7467596 and rs9328536 in the MED27 gene are related to melanoma [43], and the mutation in the MED27 gene may be associated with AMD [42]. Moreover, SAMA detected some new two-locus SNP-SNP interactions that have not been reported before. For example, rs1329428 has a interaction with rs10272438 and rs1740752 has a interaction with rs943008. SNP rs10272438 resides in the BBS9 gene which is involved in parathyroid hormone action in bones. SNP rs943008 resides in the NEDD9 gene, which is closely related to cancer. However, these two-locus SNP-SNP interactions require further examination in future studies. It can be seen from the Table 6 that the parameters we set before can find the most number of two-locus SNP-SNP interactions.Table 5
Results of two-locus SNP-SNP interactions detected by SAMA on AMD dataset.
SNP 1GeneSNP 2Genep valuesrs380390CFHrs1363688NA<1.0e-08rs380390CFHrs2224762KDM4Crs380390CFHrs555174NArs380390CFHrs1374431NArs380390CFHrs1740752NArs1329428CFHrs7467596MED27<1.0e-07rs1329428CFHrs9328536MED27rs1329428CFHrs3922799NArs1329428CFHrs10489076NArs1740752N/Ars3009336NArs380390CFHrs718263NCALDrs380390CFHrs223607NArs380390CFHrs620511NArs380390CFHrs2178692COPS7Ars380390CFHrs34512NArs380390CFHrs3853728EGFEM1Prs380390CFHrs210758NArs380390CFHrs2446023ZNF518Ars380390CFHrs2167167NArs380390CFHrs956275PPATrs380390CFHrs1896373NA<1.0e-06rs380390CFHrs1896373NArs380390CFHrs143627607DDX3Xrs1329428CFHrs10504043ANK1rs1329428CFHrs10272438BBS9rs1329428CFHrs2695214PPP3CArs1329428CFHrs78812154NArs1329428CFHrs74412587NArs1329428CFHrs1363688NArs1329428CFHrs9328536MED27rs1740752NArs943008NEDD9Table 6
Number of two-locus SNP-SNP interactions detected by SAMA under different parameters.
ξ0.10.20.30.40.50.60.70.80.9pc1 and pc2.191214171918171310.2121417202321181611.3131316192118201613.4131516202421211818.5161717233025232019.6151718242825252217.7151318252726272119.8141422283130272526.9121317232925262221


## 3.1. Simulated Datasets

In this subsection, we carry out the experiments in three simulated disease models (Models 1-3) [36]. Model 1 is a two-locus multiplicative model in which the disease prevalence PD increases multiplicatively with the incremental presence of the disease genotype interaction. Model 2 is a two-locus threshold model, in which PD does not increase until the number of disease genotype interactions pass the threshold. Model 3 is a two-locus concrete mode that simulates the effects of SNP-SNP interactions on susceptibility to traits. In the three models, PD is set to 0.1, and the minor allele frequencies (MAF) is 0.05, 0.10, 0.20, and 0.50. The genetic heritability (h2) is 0.005 in Model 1, and h2 is 0.02 in Models 2 and 3. According to the combination of these values, 12 penetrance tables are obtained (see Table 1). 200 datasets corresponding to each penetrance table are generated using GAMETES_2.0 [37]. 100 SNPs are generated in the first 100 datasets, whereas the number of SNPs is 2000 in the other 100 datasets.Table 1
Details of three two-locus disease models.
MAF0.050.100.200.50AAAaaaAAAaaaAAAaaaAAAaaaModel 1 (PD=0.1, h2=0.005)BB0.0980.0980.098BB0.0960.0960.096BB0.0920.0920.092BB0.0780.0780.078Bb0.0980.2990.522Bb0.0960.1970.282Bb0.0920.1450.181Bb0.0780.1050.122bb0.0980.5220.912Bb0.0960.2820.408Bb0.0920.1810.227Bb0.0780.1220.142Model 2 (PD=0.1, h2=0.02)BB0.0960.0960.096BB0.0920.0920.092BB0.0840.0840.084BB0.0520.0520.052Bb0.0960.5330.533Bb0.0920.3190.319Bb0.0840.2100.210Bb0.0520.1380.138bb0.0960.5330.533Bb0.0920.3190.319Bb0.0840.2100.210Bb0.0520.1380.138Model 3 (PD=0.1, h2=0.02)BB0.0800.1920.192BB0.0720.1640.164BB0.0610.1460.146BB0.0670.1550.155Bb0.1920.0800.080Bb0.1640.0720.072Bb0.1460.0610.061Bb0.1550.0670.067bb0.1920.0800.080Bb0.1640.0720.072Bb0.1460.0610.061Bb0.1550.0670.067


## 3.2. Parameter Setting

In the experiments, we set the same maximum number of iterations for the five algorithms, that is, the maximum iteration number for datasets with200 SNPs is set to 50 and the maximum iteration number for datasets with 2000 is set to 500. The maximum number of iterations is less than the number of iterations using an exhaustive algorithm. Furthermore, the other parameters of the five compared algorithm are shown in Table 2.Table 2
Parameter setting of five algorithms.
AlgorithmParametersSAMAThe crossover probabilitiespc1 and pc2=0.8; the switch parameter ξ=0.5; the number of individuals M=500FHSA-SEDThe harmony memory considering rateHMCR =0.9; the pitch-adjusting rate PAR =0.35; the number of harmonies evaluated with Bayesian network scoring HM1=250; the number of harmonies evaluated with Gini scoring HM2=250AntEpiSeekerThe size of large SNP setslargesetsize=6; the size of small SNP sets smallsetsize=3; the weight parameters α and β=1; the pheromone evaporation rate ρ=0.05; the initial pheromone τ0=100; the number of ants M=500IEACOThe switch parameterθ is 0.001; the upper bound of negative feedback pheromone on worse paths μ=300; the weight parameters α and β=1; the parameter determining the weight of negative feedback pheromone γ=1; the number of ants M=500DESeekerThe number of SNPs in a large size SNP combinationW=6; the number of vectors M=500


## 3.3. Performance Evaluation Criteria

With the purpose of conducting the experiments comprehensively, we introduce two measurements: detection power and running time. The detection power is defined below:(6)Power=#T/#G,where #G is the datasets that are generated by the same penetrance table (#G=100 in the experiments) and #T is the number of datasets in which the two-locus SNP-SNP interaction associated with disease is detected.


## 3.4. Experiments on Simulated Datasets

Figures3 and 4 present the detection power of the five compared algorithms on the three disease models. It is indicated from the figures that the SAMA algorithm is better than or equal to FHSA-SED, AntEpiSeeker, IEACO, and DESeeker on most settings, with the exception of MAF=0.50 in Model 1 with 200 SNPs. SAMA detects all disease-associated SNP-SNP interactions on six settings for the datasets with 200 SNPs, and the algorithm detects all disease-associated SNP-SNP interactions on two settings for the datasets with 2000 SNPs. On the datasets with 200 SNPs, the other four algorithms can be comparable with SAMA because they also have good performance. On the datasets with 2000 SNPs, the detection power obtained by our algorithm is significantly greater than that of the other four algorithms, especially in Model 3. Followed by FHSA-SED and DESeeker, these two algorithms also show not bad performance. Next is IEACO. The performance of AntEpiSeeker performance is the worst in this experiment. The above analysis reveals that the proposed algorithm is more effective for detecting two-locus SNP-SNP interactions.Figure 3
Power comparison of five compared algorithms on the datasets with 200 SNPs.Figure 4
Power comparison of five compared algorithms on the datasets with 2000 SNPs.Tables3 and 4 show the running time of the five compared algorithms on the three disease models. As illustrated in the two tables, the running time of our method is less than that of the other four methods. This demonstrates that SAMA can efficiently decrease the running time in detecting two-locus SNP-SNP interactions.Table 3
Running time of five compared algorithms on the datasets with200 SNPs.
ModelMAFSAMAFHSA-SEDAntEpiSeekerIEACODESeekerModel 10.059.12±0.5310.55±0.5946.63±2.3111.21±0.7610.03±0.640.108.97±0.5110.32±0.5348.52±2.4012.45±0.819.89±0.700.209.32±0.4910.47±0.5847.71±2.2910.93±0.799.93±0.660.509.55±0.4410.62±0.6245.63±1.9913.06±0.8210.32±0.73Model 20.059.53±0.4811.04±0.6548.57±2.3710.90±0.7110.54±0.770.109.29±0.5710.86±0.6849.12±2.3011.35±0.669.98±0.690.208.86±0.4611.06±0.6446.83±2.1212.52±0.7310.74±0.650.509.22±0.5010.75±0.7046.89±2.0611.83±0.689.76±0.59Model 30.059.06±0.5510.63±0.6350.02±2.5512.04±0.7410.63±0.620.109.52±0.5911.05±0.6847.74±2.1911.67±0.8010.72±0.580.209.32±0.5110.64±0.5748.82±2.4912.42±0.699.48±0.610.509.94±0.6010.74±0.6145.90±2.0511.53±0.789.80±0.65Table 4
Running time of five compared algorithms on the datasets with2000 SNPs.
ModelMAFSAMAFHSA-SEDAntEpiSeekerIEACODESeekerModel 10.0584.63±3.7698.74±5.32431.53±11.57108.64±5.9697.56±4.970.1087.53±4.02103.63±5.67427.87±10.94109.42±6.03100.55±5.170.2090.89±3.9098.85±5.15442.35±10.52111.34±6.1299.74±5.200.5088.16±3.95101.15±4.96425.84±12.02104.44±6.04103.85±5.06Model 20.0591.48±4.1297.88±4.87435.14±12.53110.45±5.64102.66±5.070.1089.86±3.79100.56±5.04448.57±10.89102.63±6.2398.85±5.120.2089.17±4.0399.95±4.78459.84±11.78101.34±5.98105.05±5.310.5092.74±3.87100.13±4.83418.52±10.97105.65±5.95104.43±5.13Model 30.0590.63±3.9397.73±5.01451.45±12.32112.56±6.46101.89±5.440.1086.73±3.89103.54±5.21432.85±11.67109.93±6.15104.92±5.190.2087.83±4.0796.97±4.89429.50±12.02113.56±5.9699.71±5.080.5090.09±3.86101.34±5.36440.86±12.63114.37±6.07103.67±5.32


## 3.5. Experiments on a Real-World Biological Dataset

According to the results of the simulated experiments, SAMA performs well for detecting two-locus SNP-SNP interactions. In this section, we conduct experiments on a real-world biological dataset [35]. The purpose of the experiment is to detect two-locus SNP-SNP interactions associated with the disease by using the five compared algorithms. The five algorithms are run 10 times, and Figure 5 is drawn according to the obtained p values. In the figure, a solid dot has two values, one is x-value, and the other is y-value. The y-value represents the p value, and the x-value denotes the SNP-SNP interaction detected by an algorithm with a certain p value. For the SAMA algorithm, 31 solid dots are detected, that is, 31 two-locus SNP-SNP interactions are detected. It can be seen evidently that the number of solid dots found by the proposed algorithm is more than that found by the other four algorithms. Followed by AntEpiSeeker, this algorithm detects 27 solid dots. Next is DESeeker and FHSA-SED. The DESeeker algorithm detects 23 solid dots, and the FHSA-SED algorithm detects 22 solid dots. The number of interactions found by IEACO is relatively less. This algorithm only finds 21 solid dots. The above analysis shows that SAMA can detect more two-locus SNP-SNP interactions than the other algorithms under the same number of iterations.Figure 5
The number of two-locus SNP-SNP interactions detected by five algorithms.Table5 presents the two-locus SNP-SNP interactions with p values less than 1.0e-06 detected by our method. In the table, the number of two-locus SNP-SNP interactions found by the SAMA algorithm with p values less than 1.0e-08, 1.0e-07, and 1.0e-06 are 1, 9, and 21, respectively. Table 6 gives the number of two-locus SNP-SNP interactions detected by SAMA under different parameters. It can be seen from the Table 5 that rs380390 and rs1329428 are interacted with many other SNPs. The two SNPs are are located in the CFH gene, and the CFH gene has been commonly association with AMD [16, 38–40]. Furthermore, many SNPs included in detected SNP-SNP interactions are located in non-gene coding regions (NA). There are seven interactions between the CHF gene and NA when the p value is less than 1.0e-07, and there are ten interactions between the CHF gene and NA when the p value is between 1.0e-07 and 1.0e-06. The CHF gene has one interaction with the KDM4C gene, and it has two interactions with the MED27 gene. SNP rs2224762 is located in the KDM4C gene that can regulate chromosome segregation during mitosis [41]. This gene that may be associated with AMD has been reported before [22, 42]. SNPs rs7467596 and rs9328536 in the MED27 gene are related to melanoma [43], and the mutation in the MED27 gene may be associated with AMD [42]. Moreover, SAMA detected some new two-locus SNP-SNP interactions that have not been reported before. For example, rs1329428 has a interaction with rs10272438 and rs1740752 has a interaction with rs943008. SNP rs10272438 resides in the BBS9 gene which is involved in parathyroid hormone action in bones. SNP rs943008 resides in the NEDD9 gene, which is closely related to cancer. However, these two-locus SNP-SNP interactions require further examination in future studies. It can be seen from the Table 6 that the parameters we set before can find the most number of two-locus SNP-SNP interactions.Table 5
Results of two-locus SNP-SNP interactions detected by SAMA on AMD dataset.
SNP 1GeneSNP 2Genep valuesrs380390CFHrs1363688NA<1.0e-08rs380390CFHrs2224762KDM4Crs380390CFHrs555174NArs380390CFHrs1374431NArs380390CFHrs1740752NArs1329428CFHrs7467596MED27<1.0e-07rs1329428CFHrs9328536MED27rs1329428CFHrs3922799NArs1329428CFHrs10489076NArs1740752N/Ars3009336NArs380390CFHrs718263NCALDrs380390CFHrs223607NArs380390CFHrs620511NArs380390CFHrs2178692COPS7Ars380390CFHrs34512NArs380390CFHrs3853728EGFEM1Prs380390CFHrs210758NArs380390CFHrs2446023ZNF518Ars380390CFHrs2167167NArs380390CFHrs956275PPATrs380390CFHrs1896373NA<1.0e-06rs380390CFHrs1896373NArs380390CFHrs143627607DDX3Xrs1329428CFHrs10504043ANK1rs1329428CFHrs10272438BBS9rs1329428CFHrs2695214PPP3CArs1329428CFHrs78812154NArs1329428CFHrs74412587NArs1329428CFHrs1363688NArs1329428CFHrs9328536MED27rs1740752NArs943008NEDD9Table 6
Number of two-locus SNP-SNP interactions detected by SAMA under different parameters.
ξ0.10.20.30.40.50.60.70.80.9pc1 and pc2.191214171918171310.2121417202321181611.3131316192118201613.4131516202421211818.5161717233025232019.6151718242825252217.7151318252726272119.8141422283130272526.9121317232925262221


## 4. Conclusion

In the paper, we propose the SAMA algorithm to detect two-locus SNP-SNP interactions associated with disease. The global search ability of SAMA is greatly increased by using HC, DBM, and EC. The self-adaptive behavior of SLS enhances the local search ability of SAMA without significantly increasing the running time. When using simulated datasets, the experimental results indicate that SAMA is more effective than FHSA-SED, AntEpiSeeker, IEACO, and DESeeker in terms of detection power and running time. When utilizing the real-world biological dataset, the experiments show that the proposed algorithm successfully detected known disease-associated SNP-SNP interactions and some new suspected interactions. However, the SAMA algorithm still has some limitations. First, the detection power of SAMA is low for the disease models with smallMAF. Furthermore, the current version of SAMA cannot detect high-order SNP-SNP interactions (SNPs>2). As far as we know, there does not exist a powerful method for detecting high-order SNP-SNP interactions in GWAS. Therefore, detecting high-order SNP-SNP interactions associated with disease has many rooms to explore in the future.



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*Source: 5610658-2020-08-25.xml*