Multimodal multiobjective optimization with structural network control principles to optimize personalized drug targets for drug discovery of individual patients

Abstract

Structural network control principles provided novel and efficient clues for the optimization of personalized drug targets (PDTs) related to state transitions of individual patients. However, most existing methods focus on one subnetwork or module as drug targets through the identification of the minimal set of driver nodes and ignore the state transition capabilities of other modules with different configurations of drug targets [i.e. multimodal drug targets (MDTs)] embedding the knowledge of previous drug targets (i.e. multiobjective optimization). Therefore, a novel multimodal multiobjective evolutionary optimization framework (called MMONCP) is proposed to optimize PDTs with network control principles. The key points of MMONCP are that a constrained multimodal multiobjective optimization problem is formed with discrete constraints on the decision space and multimodality characteristics, and a novel evolutionary algorithm denoted as CMMOEA-GLS-WSCD is designed by combining a global and local search strategy and a weighting-based special crowding distance strategy to balance the diversity of both objective and decision space. The experimental results on three cancer genomics data from The Cancer Genome Atlas indicate that MMONCP achieves a higher performance including algorithm convergence and diversity, the fraction of identified MDTs, and the area under the curve score than advanced algorithms. Additionally, MMONCP can detect the early state from the difference between the target activity and toxicity of MDTs and provide early treatment options for cancer treatment in precision medicine.

Introduction

Accurate prediction of drug targets is essential for drug discovery. In precision medicine, it is the basis of finding efficient drugs of individual patients but a big challenge for discovering suitable personalized drug targets (PDTs). The personalized genomic profile of individual patients makes it possible to optimize PDTs [1, 2]. Current large samples-based static models face challenges in performance and false negatives [3, 4], since the identification of effective drug targets for individual patients using personalized omics data is limited by the scarcity of personalized sample information. Advances in network science have promoted the development of methods using sample-specific gene interaction networks. Methods such as CPGD [1] and SCS [5] have been developed to optimize PDTs on personalized gene interaction networks (PGINs). The above tools usually contain two steps: (i) constructing a PGIN, and (ii) optimizing PDTs by selecting the targeted driver genes. Commonly used methods for constructing PGIN include Paired-SSN [6], LIONESS [7], and SSN [8]. To optimize PDTs, structural network control principles provide a theoretically precise depiction of the manner in PGIN where state transitions can be accomplished by appropriately selecting driver genes [9].

Nevertheless, these traditional methods based on network control, such as the Maximum Matching Set (MMS)-based control methods (MMS) [10], Minimum Dominating Set-based control method (MDS) [11], Nonlinear Control of undirected network Algorithm (NCUA) [6], and Directed Feedback Vertex-based control method (DFVS) [12], are subject to two primary limitations: (i) these methods pose computational challenges in optimizing the set of driver nodes within large-scale networks featuring nonlinear dynamics, but it is possible to approximate efficient solutions [9]. (ii) These methods primarily concentrate on system control through the identification of the minimal set of driver nodes. They are constrained single-objective optimization problems but neglect the knowledge of previous drug targets during the optimization process. To solve these problems, multiobjective optimization-based structural network control principles (MONCP) [13] were developed by simultaneously minimizing the number of driver nodes and maximizing the information of prior-known drug targets in PGIN. Based on this, a set of Pareto-optimal solutions or a trade-off solution curve could be obtained, allowing decision-makers to make a selection and provide additional potential pathways for personalized therapy in the field of precision medicine. Subsequently, in our another previous work to improve the performance of MONCP, we proposed a knowledge-embedded multitasking constrained multiobjective Evolutionary Algorithm (KMCEA) by analyzing the characteristics of the problems and mining relevant knowledge to design specific strategies [14]. Furthermore, in our recent work, we also applied MONCP on temporal network observability model to identify the disease predictive biomarkers [15].

Though the above methods have been gradually optimized for optimizing PDTs, they all ignore a fact that multiple sets of PDTs are equivalent in the information available about prior-known drug targets and the number of driver nodes, but they have differences in their configurations. These different sets of PDTs may provide varying biological functions [i.e. multimodal drug targets (MDTs)], as shown in Fig. S1. Therefore, this study proposed a novel multimodal multiobjective optimization framework with network control principles (called MMONCP) to optimize PDTs. The demonstration of MMONCP is illustrated in Fig. 1. MMONCP firstly considers the optimization of PDTs in individual patients as a constrained multimodal multiobjective optimization problem (CMMOP).

Figure 1.

Demonstration of MMONCP. The MMONCP aims to discover multiple MDTs on the PGIN. Several sets of drug targets are called MDTs, if they are equivalent in MMONCP in the information available about prior-known drug targets (red color) and the number of driver nodes, but they have differences in their configurations. The PGIN was constructed with Paired-SSN by integrating their gene expression data of paired samples of an individual patient and gene mutation data. Then, a constrained multimodal multiobjective evolutionary algorithm with structural network control principles is used to optimize MDTs according to the structure of the network.

Then, a new CMMOEA called CMMOEA-GLS-WSCD is developed by integrating a combined global and local search (GLS) strategy and a weighting-based special crowding distance (WSCD), which considers the diversity of both the decision space and the objective space. In detail, CMMOEA-GLS-WSCD employs a multitask framework, where the main task is used for global search by solving CMMOP, and the two auxiliary tasks are used for local search by solving derivative constrained multimodal single-objective optimization problems (CMSOP). Meanwhile, a combined GLS strategy is designed to achieve a balance of GLS, where the angle-based niching parent selection method [16] is adopted for local search and the enhanced binary tournament selection is utilized for global search [17]. Additionally, to balance the diversity of the objective space and decision space, WSCD is proposed in environmental selection. Different from traditional methods, MMONCP not only considers simultaneously minimum driver nodes (the first objective) and maximum prior-known drug–target information (the second objective) embedding the knowledge of previous drug targets, but also identifies Pareto-optimal set (PS) or a trade-off curve (i.e. distribution of PDTs in two objective space) that can be equivalent in terms of the two objectives, but they may differ in terms of their gene or protein configurations. Finally, the effectiveness of MMONCP is validated on datasets obtained from TCGA, including breast invasive carcinoma (BRCA), lung adenocarcinoma (LUAD), and lung squamous cell carcinoma (LUSC). The experimental results demonstrate that MMONCP outperforms current CMMOEAs including convergence, diversity, and the fraction of MDTs in PGIN of patients. In addition, we found that our MMONCP can detect the early state in stage ia from the difference in drug activity and toxicity of different sets of MDTs in the BRCA cancer dataset. Additionally, the functional differences of MDTs are analyzed at this stage (i.e. stage ia for BRCA cancer), and it is verified that our MMONCP can provide different functional targets for precise treatment of cancer.

Material and methods

Structural network control theory

Comprehending the underlying dynamics of molecular networks from the perspective of control theory is the key to understanding cancer progression. Specifically, the progression of cancer from a normal state to a disease state is formulated as a structural network control problem [9]. The goal of structural network control principles in PGINs is to find a set of personalized driver genes (denoted as PDGs) with the minimum size, based on a proper understanding of the molecular network structure. The state of these genes can be changed by input signals, thereby changing the state of the entire network. It is noteworthy that PDTs can be viewed as PDGs that are targeted by drug activation in PGINs. This type of problem can be summarized as follows:

(1)

where represents the state of the gene at a specific time point in the molecular network (PGIN in this work), and is the adjacency matrix of the molecular network. denotes the driving that has controllers with genes. Common controllers include the genetic or environmental factors that can provide input signals.

Intrinsic structure and dynamic propagation are crucial in controlling the process of the molecular network. However, how to represent the particular functional patterns of the molecular network is still unknown. By considering the network structure, structural network control principles can be exploited to analyze the controllability of complex networks by identifying a set with the minimum number of driver nodes.

Traditional principles include MMS [10], MDS [11], NCUA [6], and DFVS [12], where MMS and DFVS are directed networks and the other two are undirected networks. According to the way of control, there are three types of MMS-based control methods, including MMS-based full controllability [10], MMS-based target controllability [18], and MMS-based constrained target controllability [19]. These MMS-based control methods focus on the controllability of complex networks with linear or local nonlinear characteristics. However, nonlinear network dynamics are prevalent, particularly concerning individual patient heterogeneity of PGINs. MMS-based control methods may have difficulty in presenting a complete view. MDS assumes that each edge is dual-directional and each driver node has the autonomous capability to control the edges it is associated with. Moreover, non-driver nodes become controllable when they are in close proximity to a minimum of one driver node. In short, every node within MDS has autonomous control over each of its outbound edges. In the past few years, the research emphasis in network control has transitioned from linear dynamics to nonlinear dynamics [20]. DFVS is applied to direct networks with nonlinear dynamics and known network structures, where the functional form of the governing equations cannot be random but must have certain properties. To achieve effective feedback control, it is necessary to manipulate the FVS (Feedback Variable Set), which associates with the feedback loop within the network. However, DFVS cannot be directly applied to undirected networks . Therefore, NCUA is developed for the network control problem of undirect networks with nonlinear dynamics based on FVS. NCUA is the same as MDS in that the edges are bidirectional. It is noteworthy that the network in NCUA can be transformed into a bipartite graph . If is one of the nodes connected to , the edge between and will be put into . The set of nodes, which is not dominate by other nodes in , should be chosen from . By adding other objectives (i.e. maximize the information of prior drug targets), MONCP can be modified according to traditional principles. Note that MONCP under the framework of MDS, NCUA, and DFVS have different constraints but the same objectives.

Constrained multimodal multiobjective optimization theory

CMMOPs are widespread in real life, such as the site selection problem [21]. Generally, a CMMOP can be described as follows:

(2)

(3)

where is an objective vector and represents the number of objectives. Decision vector, denoted as , has decision variables. The th inequality constraint can be represented as , while the th equality constraint can be represented as . and represent the number of inequality constraints and equality constraints. Generally, the extent to which a solution satisfies a constraint can be represented by its score of constraint violation (CV), which can be calculated as follows:

(4)

where is a tolerance value used to relax the equality constraints. A solution with a CV value of 0 is called a feasible solution, while a solution with a nonzero CV value is called an infeasible solution.

The infeasible solution with a smaller CV value is superior to the infeasible solution with a larger CV value. Among a feasible solution and an infeasible solution , dominates . The concept of dominance between two feasible solutions and is expressed as follows: if for every and for at least one , then dominates . A solution that is not dominated by any other solution and satisfies all constraints is called a Pareto optimal solution. The collection of all Pareto optimal solutions forms a set known as the constrained Pareto optimal solution set (CPS), and the mapping of this set in the objective space is called the constrained Pareto front (CPF).

Different from multimodal multiobjective optimization problems, the solutions must be confined to the feasible region in CMMOPs. That is to say, the search space for constrained multimodal multiobjective optimization is not the entire search space, but rather within a feasible region. The illustration of CMMOPs is given in Fig. 2, where the shaded area represents the feasible region. In short, a CMMOP can be defined as a scenario where various solutions in the CPS map to a single point in the CPF.

Figure 2.

Illustration of constrained multimodal multiobjective problem. The constrained Pareto solution sets (CPSs) in the decision space (the left) are called CMMOPs if in the feasible domain (gray and yellow areas), they are correspond to the same constrained Pareto front (CPF) in the objective space (the right).

Recently, several tools have been developed to tackle constrained multimodal single-objective optimization problems (CMSOP). Few researches have been conducted in CMMOP due to its high complexity. Liang et al. [21] developed a differential evolutionary algorithm called CMMODE to solve CMMOPs by employing a specification mechanism to obtain multiple equivalent solutions in CPF, and they developed an enhanced environment selection strategy. Ming et al. [22] presented a coevolutionary method CMMOCEA to solve CMMOPs, which contains two populations: considers both constraints and multimodality, disregards the constraints, and both of them consider the diversity of the objective space and decision space. However, these above methods focus on test suits that are continuous and have fewer decision variables. When solving MMONCP, it may be challenging to achieve better performance.

CMMOPs can be regarded as one type of constrained multiobjective optimization problems (CMOPs) considering multimodality or multimodal multiobjective optimization problems (MMOPs) with constraints. Therefore, it is necessary to design appropriate strategies to handle constraints, develop a multimodal mechanism, and learn how to balance them. In this section, related content of CMOPs and MMOPs will be introduced in the following.

Constrained multiobjective optimization

As shown in Eqs. (2–3), CMOPs have the same formulas as CMMOPs. Recently, many CMOEAs have been developed, which can be classified into three types based on the form of the algorithm, including multipopulation, multistage, and other types. CMOEAs employ two or more populations, and an effective result can be obtained by cooperation between several populations. For these CMOEAs, multiple populations have different tasks, and cooperative evolution can be called “multitask.” More precisely, it is called multiform optimization in the latest evolutionary transfer optimization survey paper [23]. Particularly, multiform optimization exploits the alternate form of an optimization problem with single target. The search experiences derived from various formulations of a single optimization problem may lead to improved problem-solving outcomes for the target problem, as different formulations can capture diverse properties or landscapes of the problem encountered. Instead of conducting an evolutionary search on the complex search space of the original problem formulation, the search could be carried out on alternative formulations with simpler search spaces. Transferring the valuable knowledge acquired in the simpler search space back to the original space can guide the search toward high-quality solution areas, leading to improved optimization performance.

For instance, Tian et al. [24] developed a coevolutionary CMOEA (i.e. CMMO), where the main population is based on CMOP and the auxiliary population focuses on an unconstrained problem. Also, a weak cooperation of two populations is adopted, and this is a popular method. Qiao et al. [25] presented a dual-population CMOEA, where the main and auxiliary population employ different mechanisms to handle constraints. Meanwhile, a scheme for dynamic resource allocation was developed to assign varied resources to populations. CTAEA [26] and MTCMO [27] are popular CMOEAs with multipopulation. CMOEAs with multistage are used to balance objectives and constraints at different stages. For example, Fan et al. [28] developed a CMOEA PPS including two stages, where constraints are not considered at the first stage to evolve the population toward the UPF, and all constraints are considered at the second stage for convergence to CPF. Liu et al. [29] introduced a two-stage CMOEA (ToP) that employs the search for single-objectives in the first stage and the search for multiple objectives in the second stage. Additionally, there are some other types of CMOEAs, such as NSGAII-CDP [30], MOEA/D-CDP [31], etc.

However these CMOEAs still have limitations in solving our MMONCP due to discrete feasible areas. Furthermore, MMONCP is a CMMOP that is supposed to consider the diversity of the decision space. To sum up, it is challenging for these CMOEAs to achieve good performance when solving MMONCP.

Multimodal multiobjective optimization

An MMOP can be defined as a problem that requires optimizing multiple (two or more) objectives, and it has multiple solutions in the decision space that correspond to the same point in the objective space (i.e. multimodal solutions). Establishing a species protection mechanism is crucial for mining and preserving multimodal solutions, thereby enhancing the diversity of solutions in the decision space. This is typically accomplished by setting up a niche through clustering [32], employing a ring topology [33], and enhancing the diversity of the decision space [34]. Multimodal multiobjective evolutionary algorithms (MMEAs) using clustering include MMOEA/DC [35] using clustering in both decision space and objective space, MMODE_CSCD [36] using a special crowed distance based on clustering, etc. MO_Ring_PSO_SCD [33] is a representative algorithm using ring topology to generate niche. Additionally, many MMEAs exploit the diversity of decision space to solve MMOPs, such as HREA [34], DN-NSGAII [32], etc.

Despite the excellent performance demonstrated by the aforementioned multimodal optimization algorithms in simulation tests, they cannot be directly utilized for biomedical problems due to their lack of tailored strategies for biological data. Particularly, the optimization algorithm is faced with a significant challenge due to the intricate nature of biomarker recognition in dynamic networks. In addition, there are many constraints in the decision space, and these MMEAs fail to handle constraints.

M‌MONCP framework

MMONCP not only provides different treatment programs but may also be tailored to specific biological functions not addressed by other drug targets. Therefore, this paper extends MDT to network control principles and formulates the problem of MMONCP in individual patients. MMONCP is a discrete, multimodality problem with constraints on the decision space, so it may be difficult to consider the diversity of the decision space while finding multiple discrete feasible regions. To address this issue, this paper develops a novel CMMOEA (denoted as CMMOEA-GLS-WSCD), which the specific process is shown in Fig. 3, by a combined GLS strategy and a WSCD strategy. In our CMMOEA-GLS-WSCD, a combined GLS strategy is used by combining the angle-based niching parent selection method and the enhanced binary tournament selection method to balance local and global search. Meanwhile, CMMOEA-GLS-WSCD simultaneously considers the diversity of the decision space and objective space. When the crowding distance of the decision space is always greater than that of the objective space in basic special crowding distance, the objective space will have poor diversity. To achieve this, WSCD strategy is proposed in this paper in the process of environmental selection.We have put the source code and data at GitHub repository for its web availability.(https://github.com/WilfongGuo/MMONCP).

Figure 3.

The method explanation of CMMOEA-GLS-WSCD. (A) The overall workflow of CMMOEA-GLS-WSCD. (B) The main framework of the combined global and local search strategy. (C) The main framework of the weighting-based special crowding distance.

Objective functions of MMONCP

In fact, MMONCP is the MONCP by considering multimodality. Therefore, MMONCP has the same objective functions as MONCP:

(5)

(6)

subject to

(7)

where represents the number of decision variables (i.e. the number of genes in the PGIN), and denotes the th gene. For binary decision variable , if the th gene is identified, =1; otherwise, =0. represents the number of constraints. represents the label of the th PDT, and it is set to 1 if the th PDT is prior-known; otherwise, it is set to 0. denotes the first objective of minimizing the number of genes in the PGIN, and represents the second objective of minimizing the maximum prior-known drug–target information. For Eq. (9), which is a summary form, MDS, NCUA, and DFVS have different constraints (network controllability), as shown in Eqs. (10–12), respectively.

The form of MDS is given below:

(8)

where represents the set of adjacent nodes of . This equation indicates that at least one of two adjacent nodes must be identified.

For NCUA, the constraints in the following form are used to ensure the network is controllable:

(9)

for an undirected network , its bipartite graph is represented as , which is described in Section II. is set to 1 if the th node belongs to the cover set in to cover the nodes in .

For DFVS, the concept of FVS allows for determining the driver nodes in a directed network based on its cycle structure and source nodes, which can be described as an integer linear programming (ILP) problem:

(10)

where is set to 1 when node is included in FVS.

Designing constrained multimodal multiobjective evolutionary algorithm

Algorithm 1 provides the detailed process of CMMOEA-GLS-WSCD. The constraint equations are formed according to the PGIN using Eq. (10), Eq. (11), or Eq. (12), in which different equations represent different MONCPs (line 1). The main population () and two auxiliary populations ( and ) are randomly initialized with corresponding sizes (lines 2 and 3). The following steps are conducted in each iteration until the termination condition is satisfied. First, the non-dominance level is obtained by non-dominated sorting, and the WSCD of is calculated by Eqs. (14–15) (line 5). The diversity () of and are evaluated based on the Hamming distance to the nearest individual in the decision space (line 6). Then, a combined GLS strategy is used in to generate offspring by combining the angle-based niching parent selection method and the enhanced binary tournament selection method (line 7). The specific details of this are presented in Algorithm 2. According to , the mating pool is selected, and offspring of and of are generated using a binary GA (lines 8-9). Finally, for the main population, solutions are selected by using the environmental selection method in [21], from            . Unlike the original version, WSCD is employed to balance the diversity of decision space and objective space (line 11). For two auxiliary populations, solutions are selected by comparing the fitness of             and             (lines 12-13). denotes the final population and output.

Weighting-based special crowding distance

Special crowding distance (SCD), proposed by Yue et al. [33], considers the diversity of both decision space and objective space. However, of SCD cannot reflect the distribution and convergence of the decision space in MMONCP, which is a discrete large-scale problem. Hamming distance is usually employed to measure the diversity in discrete problems [37–39], and it is also used in our work. When there are many individuals of the population in the later stages of evolution that are far away in the decision space but very close to the objective space, the SCD is set to . This implies that SCD only considers the crowding distance in the decision space. In this case, these solutions will be retained, leading to poor diversity of the objective space. To achieve a balance of diversity in both the decision space and objective space, WSCD is proposed and shown as follows:

(11)

(12)

where and represent the crowding distance and the average crowding distance in the decision space, respectively. Similarly, and represent the crowding distance and the average crowding distance in the objective space, respectively. In this work, Hamming distance is utilized to reflect the crowding distance of the decision space, and the calculation method for is given in [21]. denotes the number of individuals in the population. is set to 1 if is less than or equal to , and otherwise, it is set to 0. is set in the same way as above. represents an average score by calculating the ratio of the average crowding distance of the decision space to the average crowding distance of the objective space. represents an individual score of the -th individual by calculating the ratio of the crowding distance of the decision space to the crowding distance of the objective space. If is larger than , it implies that the crowding distance of the decision space of the -th individual is much greater than that of the objective space. When there are many individuals of this type, the diversity of the objective space will be poor, indicating that it is necessary to improve the diversity of the objective space. Therefore, the proportion of the individuals with being larger than is set to , and the proportion of the individuals with being less than or equal to is set to .

This paper considers the number of individuals whose crowding distance in the decision space is less than that in the objective space, to adjust the crowding distance weight of the objective space and decision space. To achieve this, the ratio of crowding distance in the decision space and the objective space is calculated for each individual, and the relative ratio of the crowding distance of all individuals is compared with the threshold value . If the number of individuals whose crowding distance in the decision space is smaller than that in the objective space decreases, the crowding distance weight in the decision space increases adaptively, and otherwise, the crowding distance weight in the decision space decreases adaptively.

Combined global and local search strategy

To achieve a balance of global search and local search, a combined search strategy is proposed in this paper, and its details are given in Algorithm 2. This method includes two steps: local search and global search. Local search is used to help the population find multiple discrete feasible regions and maintain the diversity of the objective space, and global search is used to obtain multimodal solutions and maintain diversity.

For local search, it starts by using the angle-based niching parent selection method. The objective values of individuals in are calculated and transformed into a new coordinate axis (lines 1-2). The angle value of every two individuals is obtained based on (line 3). Then, two parents are chosen from the neighbors with the smallest values (line 5), and the GA operator is employed to generate offspring based on these parents (line 6).

For global search, an enhanced binary tournament selection method [17] using WSCD and decision space-based niching are employed to select better individuals for evolutionary calculation. Mating pool is selected according to the following guidelines (lines 8–22): (1) two solutions are selected from all individuals in the current population, (2) two solutions with higher Pareto levels will preferentially enter the mating pool, and (3) if the Pareto levels of the two solutions are the same, then those with larger SCD values will preferentially enter the mating pool. Furthermore, the GA operator is employed to generate offspring.

Computational complexity

The main factors that contribute to the time complexity of MMONCP (or CMMOEA-GLS-WSCD) at each generation including initializing the population, evaluating individuals, generating offspring, and updating the population for all three population. These factors have complexities of , , , and , respectively, in the main population. Thus, the computational complexity in the main population can be estimated as . Similarly, the computational complexity in the auxiliary population can be approximated as . and denote the sizes of the main and auxiliary populations, respectively. represents the number of iterations, and represents the number of genes in the PGIN. In conclusion, the overall computational complexity of MMONCP is in the PGIN for an individual patient.

Experimental

Data

In the experiment, three gene expression datasets of cancer patients from TCGA are used, namely BRCA, LUSC, and LUAD. The paired samples of individual patients, including normal samples and tumor samples, are obtained from the same patients to construct the PGIN. The specific process can be found in the Section S-I in supplementary materials. Meanwhile, single nucleotide variations are employed to form the reliable edges of the PGIN. BRCA has 112 patients, and their numbers of genes range from 1581 to 1780; LUAD has 57 patients, and their numbers of genes range from 1987 to 2342; and LUSC has 47 patients, and their numbers of genes range from 2100 to 2344. Note that the gene number of the used datasets is less than the actual number of cancer genes in individual patients. This is because not all genes in the process of cancer transformation change, and only genes with changed functions are used. Also, the prior known reliable gene interactions are collected from existing biological datasets with incomplete information.

To describe the definition of “well-estimated drug targets”, we firstly introduce the source of data. In the experiment, the prior known reliable gene interactions as “well-estimated drug targets” are collected from existing DrugCombDB (http://drugcombdb.denglab.org). DrugCombDB is a comprehensive database dedicated to integrating drug combinations from various data sources. In addition, 21,000 FDA approved or literature-supported interactions of drugs and drug targets collected from external databases and text-mining followed by manual curations can be downloaded.

To show the performance of MMONCP in discovering clinical combinatorial drugs, combinatorial drug and gene interaction networks are collected in this work. Then, a network between gene and drug is constructed according to the interactions between combinatorial drugs and genes [40]. This interaction network includes 342 combinatorial drugs, where 122 combinations of drugs have been found to be effective in treating cancer. Meanwhile, drug toxicity data (https://toxric.bioinforai.tech/home) and drug activity data (https://www.ebi.ac.uk/chembl/) are collected to explore the biological information of PDTs.

Parameter settings of MMONCP

Since there are numerous genes in a PGIN (1581 to 1780 for BRCA, 1987 to 2342 for LUSC, and 2100 to 2344 for LUAD), the main population size is designated as 300 (), while the auxiliary population size is specified as 90 (). This setup is the same as that of the recent work [13]. The maximal number of function evaluations is set to 20,0000 [41]. To enhance result reliability, MMONCP is executed independently 30 times for each patient.

Algorithms for comparison

Since there are few CMMOEAs so far, MMONCP is compared with two CMMOEAs, two MMOEAs, and one CMOEA for solving MMONCP to show its effectiveness. Specifically, the two CMMOEAs are CMMODE [21] and CMMOCEA [22], which are described in Section II. The two MMOEAs are DN-NSGA-II [32] and MP-MMEA [42], where DN-NSGA-II employ the niche based on decision space to solve MMOPs and MP-MMEA is used to solve large-scale MMOPs by using multiple sub-populations. To ensure the effectiveness of MMOEA on MMONCP, the CDP method [30] is used in these two MMOEAs, and they are denoted as CDN-NSGA-II and CMP-MMEA, respectively. The one CMOEA is LSCV-MCEA, which is used to optimize PDTs in recent work [13]. To ensure fairness, the crowding distance of the decision space is considered in LSCV_MCEA. itionalitionally, some methods based on cancer-specific driver genes and traditional structural network control methods, including CPGD [1], ActiveDriver [43], OncoDriveFM [44], and DriverML [45], are taken for comparison. Our method has been compared with traditional methods, including MDS [11], NCUA [6], and DFVS [12], for MDT optimization in the PGIN. It is noteworthy that all algorithms adopt the particular parameter settings of the original article and the same shared parameters with our method. The statistical analysis is conducted using the Wilcoxon rank-sum test [46] with a significance level of 0.05.

Performance metrics

To verify the effectiveness of all algorithms on the three cancer datasets, four performance metrics are used in this work.

• Hypervolume (HV) [47]: It is obtained by computing the volume bounded by the discovered solutions and the reference points. A higher HV value signifies superior performance in both diversity and convergence aspects. In this paper, the reference points are set to [1.1, 1.1].

• Inverted Generational Distance (IGD) [48]: It mainly reflects the diversity and convergence of the algorithms by evaluating the gap between the obtained solutions and the Pareto optimal solutions. However, the PF and PS are unknown for our MMONCP. This study considers non-dominated solutions of the union set, and the Pareto optimal solutions of MMONCP are obtained by combining solutions of all algorithms [13]. Different from HV, a smaller IGD indicates better performance.

• Area Under Curve (AUC): It is used to assess the effectiveness of utilizing PDTs in personalized drug discovery. A larger AUC indicates better performance in discovering clinical combinatorial drugs. It is calculated by the predicted probability and the true label in the CAC, and the specific calculation details are shown in Section S-II in the supplementary materials:

• multiSet Value (MSV) [49]: It is applied in this paper to compare the multimodality of all algorithms on the three cancer datasets. The calculation is performed in the following manner:

  (13)

  where represents the number of genes in each PGIN, and denotes the number of multimodal solutions in the true PF. The true PF is unknown, and its calculation is the same as that of IGD. denotes the number of multimodal solutions obtained by algorithms that must be in the true PF. A larger MSV indicates higher multimodality.

Results and discussions

Performance of discovering clinical combinatorial drugs

In order to demonstrate the superiority of MMONCP in identifying clinical combinatorial drugs, we apply AUC to evaluate the performance of MMONCP relative to other algorithms across the three cancer datasets. The highest AUC value for each individual in each algorithm is recorded, and the average results of 30 runs in MDS, NCUA, and DFVS are shown in Fig. 4. It is noteworthy that since CMMODE cannot find a feasible solution, it is not shown in Fig. 4. The results suggest that only LSCV_MCEA has a similar performance to MMONCP on the three cancer datasets. In simpler terms, MMONCP demonstrates superior performance compared with the other methods. In particular, MMONCP performs better than LSCV_MCEA in MDS across the three cancer datasets and in NCUA in the LUSC dataset. Therefore, the results presented here confirm the effectiveness of our proposed MMONCP algorithm in identifying clinically effective combinatorial drugs.

Figure 4.

The boxplot of AUC for MMONCP and other evolutionary algorithms.

Additionally, MMONCP is compared with some traditional methods, including methods based on cancer-specific driver genes and traditional methods based network in discovering clinical combinatorial drugs. The specific results are shown in Fig. 5, where MMONCP_MDS, MMONCP_MDS, and MMONCP_DFVS represent MMONCP in MDS, NCUA, and DFVS, respectively. In addition, we also conducted rank sum tests on the results of various methods at a significance level of 0.05 to perform statistical analysis and provide corresponding p-value of Fig. 5 in section S-V of supplementary material. It can be found that MMONCP outperforms other methods in terms of AUC value across the three cancer datasets, indicating MMONCP performs better in discovering clinical combinatorial drugs.

Figure 5.

The comparison results of AUC indicator on three cancer datasets for MMONCP and other traditional methods.

Detection of functional drug targets in early stage

The development of cancer is generally categorized into four stages, from I to IV. Stage IV, often referred to as advanced cancer, is characterized by the proliferation of cancer cells and their metastasis to other tissues. Consequently, effectively detecting early warning signs during the progression of cancer is crucial for both diagnosis and prognosis. According to data from the TCGA database, patients with BRCA, LUSC, and LUAD exhibit different cancer stages. To ensure statistical significance, we merged the stages with fewer than three patients into their nearest former stages. The final staging for BRCA patients includes I, IA, IIA, IIB, IIIA, IIIB, IIIC, and IV. For LUSC and LUAD, the stages are IA, IB, IIA, IIB, III, and IV. This consolidation allows for a more robust analysis of cancer progression across these patient groups.

In this work, the biological significance of MMONCP is verified from the perspective of toxicity and activity on the BRCA dataset. Drug and gene interaction networks are also collected from DrugCombDB. In this work, the total activity or toxicity of the gene attached to drugs is considered as the two biological activity and toxicity function indexes which reflects the regulation function of genes in the drug-gene interaction network [50]. Firstly, toxicity values and activity values are calculated for MDTs. That is, MDT could be defined as multiple sets of PDTs which are equivalent in the information available about prior-known drug targets and the number of driver nodes, but they have differences in their configurations. Then for each MDT along the PS, the difference between the maximum toxicity and the minimum toxicity or maximum activity and minimum activity is recorded and we choose the maximum difference among all MDTs along the PS as the toxicity or activity value of an individual patient. Next, this work obtains the clinical stage information of all patients on the BRCA cancer dataset from TCGA and calculates the average difference of each stage. Finally, the line chart of toxicity and activity is shown in Fig. 6. From Fig. 6, it can be seen that the average difference of activity (toxicity) curve has a maximum (minimum) value in stage ia. These results demonstrate that the MDTs of our MMONCP can detect the early state from the difference of activity (toxicity) curve for MDTs and provide early cancer treatment options in precision medicine.

Figure 6.

The average difference of nine stages using activity or toxicity on BRCA dataset.

To further investigate the biological significance of these drug targets, the patients of stage ia are taken for validation. Firstly, the MDT with the largest difference in toxicity or activity on PS in each patient is recorded. Then, within this MDT, the gene sets of the patients in stage ia with maximal activity, minimal activity, maximal toxicity, and minimal toxicity are obtained and merged respectively. Finally, the merged gene sets of all the patients in stage ia are used for KEGG pathway enrichment analysis (https://david.ncifcrf.gov/), as shown in Fig. S2. From Fig. S2, it can be seen that there are some differences in gene sets with maximal activity and minimal activity Fig. S2(A-B), such as focal adhesion and Human papillomavirus infection. Within the gene sets with maximal toxicity and minimal toxicity Fig. S2(C-D), similar observations can be obtained (e.g. human papillomavirus infection is prominent in PDTs with minimal toxicity). These results suggest that MMONCP can provide different functional targets for the precise treatment of cancer.

It is worth noting that LUSC and LUAD have different clinical stage information.The specific results are shown in Fig S3. We can find that the average difference of activity curve of LUSC and LUAD has a maximum (minimum) value in stage ib and iia, respectively. Meanwhile the average difference of toxicity curve of LUSC and LUAD has a minimum value in stage ib and ia, respectively. These results demonstrate that average difference of activity curve varies in different cancer datasets and the similar prediction results are observed in BRCA and LUSC datasets.

Analysis of algorithm performance

Comparison of constrained multimodal multiobjective evolutionary algorithms

To demonstrate the advantages of MMONCP for optimizing PDTs compared with several other methods, HV and IGD are calculated. The average HV and IGD values on each dataset are presented in Fig. 7. The result of CMMODE is 0 in Fig. 7(A–C) because this algorithm fails to find feasible solutions. Therefore, CMMODE is removed in Fig. 7(D–F). From the results of Fig. 7, it can be seen that MMONCP has higher HV and lower IGD values under the framework of MDS and NCUA than other algorithms. Under the framework of DFVS, MMONCP is more efficient than most algorithms, except LSCV_MCEA. This is mainly because there are intermediate variables ( and in Eq. (12)) and DFVS focuses on directed networks, leading to a more complicated problem under the framework of DFVS which needs to achieve a more proper balance between the diversity of the decision space and objective space.

Figure 7.

The box plot of HV and IGD for MMONCP and other evolutionary algorithms.

To include specific numerical improvements, we conducted rank sum tests on the results of various methods HV and IGD at a significance level of 0.05 to perform statistical analysis, shown in the Tables SI–SVI. We can find that MMONCP almost has higher HV or lower IGD. In 112 BRCA patients, 49 LUSC patients, and 57 LUAD patients than other CMMOEAs under the framework of MDS and NCUA. For example, under the frame of MDS, the mean HV of CDN-NSGA-II, CMP-MMEA, CMMOCE, CMMODE, LSCV_MCEA and MMONCP are 0.343, 0.230, 0.391, 0.000, 0.540 and 0.545 on BRCA. On LUSC data, the mean HV of these algorithms are 0.347, 0.231, 0.389, 0.000, 0.542 and 0.551. On LUAD data, the mean HV of these algorithms are 0.346, 0.231, 0.385, 0.000, 0.541 and 0.551. There are no feasible solutions in CMMODE so the HV is set to 0.To further investigate the multimodality of MMONCP, MSV is used in this work by calculating the proportion of PDTs in all individual patients. The results are illustrated in Fig. 8. Figure 8 demonstrates that MMONCP can provide more MDTs than other methods, especially in MDS and NCUA. However, LSCV_MCEA performs better than MMONCP under the framework of DFVS because there are more solutions in the true PF ( has a larger value than that in MMONCP). The results demonstrate that MMONCP generally provides notable benefits compared with other methods in effectively optimizing PDTs, which helps to explore the multimodal characteristics of PDTs.We have compared calculation time of different algorithms on three cancer data. All the experiments used a specialized computer with Intel(R) Xeno(R) Gold 6230 CPU and 640 GB RAM. The results of the comparison are presented in Figure S3, which leads to the following conclusions: The results of the comparison are presented in Fig. S4, which leads to the following conclusions: As shown in Fig. S4, under the framework of MDS, NCUA, and DFVS, although our MMONCP have slight worse performance compared with CMMOCEA, CMPMMEA, and CMMODE, the computation time of these algorithms remains at the same order of magnitude.

Figure 8.

The boxplot of MSV for MMONCP and other evolutionary algorithms.

Ablation experiments

MMONCP is designed to optimize PDTs in the PGIN by using specific strategies. There are two main strategies in MMONCP, including a combined GLS strategy and WSCD. In order to validate the efficacy of the two parts, two variants of MMONCP are generated: one is MMONCP using traditional special crowding distance instead of WSCD, denoted as , and the other is MMONCP with common decision space based on the niching method instead of using combined GLS strategy, denoted as . The convergence and diversity of the two variants of MMONCP are compared by calculating HV and IGD. As shown in Fig. 9, MMONCP has a higher HV and lower IGD values than and , especially in MDS and DFVS, indicating that combined GLS strategy and WSCD are effective in solving MMONCP.

Figure 9.

Ablation experiments of MMONCP with other two variants.

In addition, we verify whether these two strategies are effective for MMONCP to find PDTs, shown in Fig. 10. We can find that MMONCP obtains more multimodal solutions in almost all cases, except under the framework of DFVS on the LUAD dataset. The results demonstrate that MMONCP can optimize PDTs effectively. For example, compared with MMONCP under the framework of MDS, the mean HV of MMONCP without the combined global and local search strategy and the WSCD dropped by about 0.008 and 0.013 on BRCA, respectively.

Figure 10.

The comparison results of MSV indicator.

The statistical insights into the characteristics of the network

The characteristics of the network is considered to help the reader better understand the underlying network structure and its influence on the outcomes. Firstly, we considered the frequency of each node identified in Pareto Sets as the controllability of MDS, NCUA, and DFVS. Then the Pearson correlation coefficient (PCC) between the controllability of different methods and Closeness Centrality (CC), Betweenness Centrality (BC), Degree Centrality (DC), Local Clustering Coefficient(LCC) was calculated and shown in Fig. S5. From Fig. S5, we can obtain the following results and conclusions, (i) for the controllability of MDS, Closeness Centrality (CC) has the larger positive correlation coefficient than other network structure parameters. Meanwhile for the controllability of DFVS and NCUA, the Degree Centrality (DC) has the larger positive correlation coefficient. These results show that Closeness Centrality (CC) has relative larger influence on the controllability of MDS while Degree Centrality (DC) has relative larger influence on the controllability of DFVS and NCUA. (ii) The influence of network structure on the controllability varies for different cancer individual patients. For example, for MDS controllability, Degree Centrality (DC) has positive correlation coefficient in BRCA cancer datasets while Degree Centrality (DC) has negative correlation coefficient in LUNG cancer datasets.

Conclusion

This study proposes a multimodal multiobjective optimization based network control framework MMONCP to optimize PDTs by considering multiple equivalent drug targets with the same information of prior-known drug targets and number of driver nodes but differing in gene or protein configurations. Our study is the first time to introduce the evolutionary multimodal muti-objective optimization into the field of network control principles. To solve MMONCP, a novel evolutionary algorithm, called CMMOEA-GLS-WSCD, is designed to optimize PDTs that can support drug targets in clinical applications. Meanwhile, a multitask framework is applied to CMMOEA-GLS-WSCD, which includes two local auxiliary tasks with a single-objective. Besides, a combined GLS strategy is developed to improve the search convergence in objective and decision space. Additionally, the WSCD is proposed to balance the diversity of objective space and decision space where the traditional special crowding distance is improved by dynamically weighting the distance in the decision space and the distance in the objective space.

The advantages of MMONCP over other algorithms are verified on BRCA, LUSC, and LUAD. The results indicate that MMONCP achieves better performance in convergence and diversity than other algorithms. Then, the advantages of MMONCP in finding PDTs are verified, especially in MDS and NCUA. In comparison with certain advanced algorithms, MMONCP have a better performance in effectively optimizing the PDTs for predicting clinical combinatorial drugs. Furthermore, our MMONCP can detect the early state from the difference in target activity and toxicity of MDTs in the BRCA cancer dataset. To sum up, this study provides decision-makers with additional therapeutic options for the early treatment of cancer patients. In future work, we will design more effective strategies to optimize PDTs under the framework of MMONCP. Also,

Accurate prediction of drug–target interactions is is a very promising research direction for drug development, yet current deep learning models face challenges in performance and false negatives [3]. Several models like EADTN [51] and MRBDTA [52] have been proposed, incorporating techniques such as ensemble modeling, feature adaptation, and transformer-based architectures. In the future, multimodal multiobjective optimization models could provide clues for enhancing prediction accuracy, interpretability, and reliability in in predicting interactions and aiding drug repurposing, with validation in both experiments and case studies [4]. In addition, wet-lab experiments (maybe animal studies) in real-world drug application scenarios are persuasive, compared with other methods.

Key Points

• The problem of MMONCP in individual patients is formulated. MMONCP aims to find multiple equivalent sets of PDTs with the same information of prior-known drug targets and the number of driver nodes but different configurations.

• To solve MMONCP, a novel CMMOEA called CMMOEA-GLS-WSCD is developed, where a combined GLS method is proposed to achieve a balance of GLS, and WSCD is developed to balance the diversity of the objective space and decision space.

• The experimental results on datasets obtained from TCGA show that MMONCP outperforms current advanced methods including algorithm convergence, diversity, and the area under the curve score of enriching clinical combinatoral drugs. Furthermore our MMONCP can detect the efficient functional drug targets of early state in stage ia.

 

Conflict of interest: The authors have declared no competing interests.

Funding

This paper was supported by the National Natural Science Foundation of China (62476253,62002329), and Henan Province Natural Science Foundation (242300421401), and China postdoctoral science foundation (2021M692915), and Open Research Fund of State Key Laboratory of Digital Medical Engineering (2024-K07), and Frontier Exploration Projects of Longmen Laboratory (LMQYTSKT031), and Henan Province Scientific Research Joint Fund Industrial Category Major Program (245101610001).