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. 2024 Jun 13;64(13):5161–5174. doi: 10.1021/acs.jcim.4c00031

Evolutionary Multiobjective Molecule Optimization in an Implicit Chemical Space

Xin Xia †,, Yiping Liu ‡, Chunhou Zheng †, Xingyi Zhang †, Qingwen Wu †, Xin Gao §, Xiangxiang Zeng ‡,*, Yansen Su †,¶,*
PMCID: PMC11235097  PMID: 38870455

Abstract

graphic file with name ci4c00031_0007.jpg

Optimization techniques play a pivotal role in advancing drug development, serving as the foundation of numerous generative methods tailored to efficiently design optimized molecules derived from existing lead compounds. However, existing methods often encounter difficulties in generating diverse, novel, and high-property molecules that simultaneously optimize multiple drug properties. To overcome this bottleneck, we propose a multiobjective molecule optimization framework (MOMO). MOMO employs a specially designed Pareto-based multiproperty evaluation strategy at the molecular sequence level to guide the evolutionary search in an implicit chemical space. A comparative analysis of MOMO with five state-of-the-art methods across two benchmark multiproperty molecule optimization tasks reveals that MOMO markedly outperforms them in terms of diversity, novelty, and optimized properties. The practical applicability of MOMO in drug discovery has also been validated on four challenging tasks in the real-world discovery problem. These results suggest that MOMO can provide a useful tool to facilitate molecule optimization problems with multiple properties.

1. Introduction

Molecule optimization fine-tunes the structure of a promising lead compound to improve its properties while maintaining its structural features.1,2 This process holds significant promise in increasing the likelihood of the generation of drug candidates. It is also considered as a crucial step for many drug discovery applications,3,4 such as the design of antimicrobials starting from existing molecules with better toxicity against resistant pathogens, and the prompt design of drug candidates to better inhibition of severe acute respiratory syndrome coronavirus 2 proteins. Traditional molecule optimization methods, such as high-throughput screening, experimental trial-and-error or computer simulation-based approaches,5,6 are time-consuming and labor-intensive. As a consequence, there has been growing interest among chemists in efficient molecule optimization strategies that can simultaneously optimize multiple properties of lead molecules.7−9

Artificial intelligence (AI) technologies have accelerated the development of molecule optimization models,10−14 whether they operate in the discrete space or the implicit space. However, these AI-based molecule optimization models have encountered limitations in their optimization capabilities due to constraints in terms of the search strategy or training data. Specifically, in the discrete space, molecules are typically represented as sequences or graphs, which are optimized using combinatorial optimization methods, primarily reinforcement learning (RL)15,16 and evolutionary algorithms (EA).17,18 Nevertheless, both RL-based and EA-based molecule optimization models exhibit a low optimization efficiency due to the complexity and extensiveness of the solution space and a high likelihood of generating invalid molecules violating chemical valence bond constraints. Recent advancements in deep generative models (DGMs) present a promising solution for optimizing molecules in the implicit spaces.19−22 These models transform discrete molecule representations, such as sequences or graphs, into implicit spaces, enabling the automatic generation of optimized molecules through vector manipulation.23−25 However, the limited availability of training data hinders the effective training and application of these DGMs-based models for molecule optimization. More seriously, since multiple interrelated properties need to be optimized in the molecule optimization process, conventional single objective-based molecule optimization models face challenges in finding multiple high-property molecules (i.e., molecules with at least one high property value) for complex molecule optimization scenarios.26−28

The challenges of these single objective-based molecule optimization models mainly arise from the task of aggregating multiple objectives into a single objective and the difficulty of identifying nondominant molecules.29−32 On the one hand, owing to the complexity of assessing the relative importance of multiple molecular properties, it is difficult to convert multiple objectives into a single objective for optimization. On the other hand, single-objective models tend to generate a single optimal molecule under a deterministic weighted combination method (Figure 1a). Considering diverse trade-offs between the conflicting objectives, the molecule achieved by a single objective-based model always lacks novelty and diversity, where the novelty represents the fact that the molecular structures of the successful optimized molecules are novel in the existing public database. The diversity here refers to the distribution of solutions within the solution set in the objective space, which also reflects the diversity of molecules in the chemical space. That is, diverse molecules with various properties are dissimilar in the chemical space. Moreover, it is costly to obtain diverse molecules through massive experiments with replications. In comparison, Pareto-based multiobjective evolutionary algorithms (MOEAs) are well suited for exploring the chemical space and optimizing multiple molecular properties. They can approximate the Pareto optimal solution set of multiobjective optimization problems33−35 (Figure 1b). Although there are several models proposed for de novo drug discovery by multiobjective evolutionary optimization algorithms,36−40 these models mainly focus on the de novo drug discovery problem, which are not efficient in generating similar and desired molecules from a given lead compound.

Figure 1.

Figure 1

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Optimization procedures for reduction to single-objective optimization and Pareto-based multiobjective molecule optimization. (a) Reduce to single-objective optimization, where only a single solution is acquired in an optimization procedure. The lead molecule (depicted as a red star) is optimized to improve its properties following a search process (indicated by a gray arrow). The dots with different colors represent the generated molecules, and the red oval represents the set of optimal solutions to the original multiproperty optimization problem. (b) Pareto-based multiobjective optimization, which can find a set of that reveal different trade-offs between objectives in an optimization procedure. In this approach, the lead molecule (represented by a red star) is optimized by a population of molecules (depicted as oval surfaces with different colors) to gradually approach the Pareto front.

To this end, we propose a Pareto-based multiobjective molecule optimization framework, termed MOMO, for simultaneously optimizing multiple molecular properties of a certain lead compound. MOMO incorporates a specially designed Pareto-based multiproperty evaluation strategy in the implicit chemical space, enabling a comprehensive exploration of high-property molecules. To be specific, the main contributions of the paper are summarized as follows:

  • 1.

    Molecule optimization, with the purpose of fine-tuning the structure of a promising lead compound to improve its physicochemical properties, is defined here as multiobjective optimization problems. The optimization objectives include the similarity to the lead compound and the optimization of common physicochemical properties, e.g., drug-likeness and the synthetic accessibility. Successfully solving this optimization problem leads to a set of molecules with high similarity to the given lead compound and excellent physicochemical properties.

  • 2.

    To address the molecule optimization problems, a multiobjective molecule optimization method (MOMO) is designed for lead compounds with multiple molecular properties to be optimized. The proposed method optimizes molecules by evolving them in an implicit chemical space and evaluating them based on a Pareto-based multiproperty evaluation strategy.

  • 3.

    Experimental results demonstrate that MOMO effectively explores the implicit chemical space, which identifies diverse and novel solutions with various objectives trade-offs in a single run, thereby enhancing the likelihood of successful multiproperty optimization of lead compounds. In addition, the superior performance of the proposed MOMO is verified over five state-of-the-art compared methods on six molecule optimization tasks, especially for the molecules with the optimization of more than two properties.

2. Related Work

2.1. Molecule Optimization in Discrete Spaces

Traditional machine learning methods typically directly modify molecular structures on discrete molecular representations. Reinforcement learning (RL)-based methods can directly apply complex and nondifferentiable molecular properties as rewards and generate molecules through the designed actions and rewards. Existing RL-based methods normally model molecule optimization as a Markov decision process (MDP) and progressively modify molecules by adding characters on SMILES or adding/deleting an atom or bond on the molecular graph.41−43 In fact, due to the large space of substructures, searching in discrete chemical spaces via stepwise modification of molecules is inefficient.44 Evolutionary algorithms (EAs) also directly use molecular properties as objectives and generate new molecules by defining crossover and/or mutation operations based on different molecular representations.17,18 For example, STONED45 and GA+D46 conducted mutation on SELFIES strings of molecules. Molfinder47 combined crossover and mutation on molecular SMILES. Compared with modifying molecules by stepwise in RL, EA-based molecule optimization methods directly obtains the final structures of molecules, which is highly nonlinear, easily modifiable, and parallelizable.48 Although flexible and straightforward, EA methods lack prior knowledge of chemistry, leading to low efficiency when searching in discrete spaces.

2.2. Molecule Optimization in Implicit Spaces

The rise of advances in deep generative models has found wide applications in molecule optimization, including VAE,14,49 RNN,22,50 GAN,24,51 GNN,20,25 etc. DL-based approaches construct implicit spaces by learning latent features from discrete molecules. These models typically generate new molecules by decoding the new continuous vectors, which are obtained by integrating features over continuous vectors or updating continuous vectors with gradients computed from properties. For example, VJTNN19 and SCVAE12 are graph-based VAE models that capture features from molecular graphs and integrate the difference vectors between pairs of molecules to generate new molecules. QMO28 searched the implicit chemical space by gradient-updating the latent vectors of molecules, where the gradients are estimated by a zeroth-order optimization procedure on the evaluated molecular properties. These DL-based models require large amounts of training data that are not easily accessible. Moreover, the objective function needs to be specifically designed, as molecular properties are nondifferentiable and cannot be adapted to directly train the model.

2.3. Single-Objective Molecule Optimization

Molecule optimization is a challenging task where several important properties, such as drug-likeness, biological activity, and synthetic accessibility, are required to be optimized simultaneously.52 Most existing approaches either optimize a single objective with similarity constraint19,22,52 or aggregate multiple objectives into single objective optimization problems.23,28 Early molecule optimization methods such as Mol-cyclegan24 and VJTNN19 only optimized one simple property such as QED, PlogP, or DRD2 under similarity constraints. Recent works have focused on optimizing several properties; however, aggregating multiple properties into one objective leads to many limitations. For example, Molfinder47 weighted summation of QED and SA into objective functions of molecules. MSO26 combined and weighted the two binding affinity scores as an objective, which has been employed to establish a compound optimization platform.53 QMO designed a loss function for molecules by summing the hinge losses over all of the properties. In these methods, it is difficult to determine the relative importance (i.e., the weights) of optimization properties that lead to unrobust optimization performance.

2.4. Multiobjective Molecule Optimization

Since multiple conflicting properties need to be further optimized or kept for generating drug candidates in molecule optimization, this is in fact a multiobjective optimization problem. In contrast to single-objective optimization methods, advances in Pareto-based multiobjective optimization avoid measuring the importance between properties and approximate the Pareto optimal set by a single search process, which has the potential to improve the performance of multiproperty molecule optimization.54,55 Due to the huge and complex chemical space, it is necessary to incorporate chemical knowledge into multiobjective optimization methods to improve the exploration of the chemical space. While multiobjective evolutionary optimization has been applied in de novo molecule generation methods, these tasks aim to generate a large number of novel and diverse molecules from de novo, which differ from molecule optimization in optimizing a given lead compound and maintaining its structural features.56,57

3. Methods

In this section, we present the proposed multiobjective molecule optimization framework MOMO for simultaneously optimizing multiple properties of molecules. For a set of molecules to be optimized, the proposed MOMO first assigns these molecules to an implicit chemical space, and then iteratively evolves them through a designed Pareto-based multiproperty evaluation strategy to simultaneously optimize multiple molecular properties in the implicit chemical space. The main components of the proposed MOMO are the evolution strategy in an implicit chemical space and the Pareto-based multiproperty evaluation strategy. In the following, we first illustrate the multiobjective molecule optimization problems, then introduce the MOMO framework, and present these two main components of MOMO in detail.

3.1. Problem Formulation

We formulate the lead compound optimization tasks as multiobjective molecule optimization problems, where each property of a molecule is treated as an objective. Specifically, given a lead molecule x0, the multiobjective molecule optimization problem can be mathematically defined as follows:

3.1. 1

where x ∈ Ω is the solution (i.e., an optimized molecule), and Ω represents an implicit chemical space. f1, f2, ..., and fm are m objectives, i.e., multiple properties. Note that the considered properties include but are not limited to the following properties:

  • 1.

    Comprehensive evaluation indexes for molecular drug-forming properties, such as molecular drug-likeness (QED), synthetic accessibility (SA), etc., which are not target-specific.

  • 2.

    Biological activity of molecules against disease targets.

  • 3.

    Pharmacokinetic (ADME) properties,58 such as solubility, etc.

  • 4.
    Similarity Sim(x, x0) between optimized molecule x and the lead molecule x0. Here, Sim(x, x0) represents the Tanimoto similarity based on Morgan fingerprints,59 since the Tanimoto similarity has been identified as one of the best similarity indicators for measuring similarity between molecular structures.60 It is calculated as follows:
    graphic file with name ci4c00031_m002.jpg 2
    fp(x) denotes the Morgan fingerprint of the molecule x, which is obtained from RDKit.61

3.2. MOMO Framework

On the one hand, unlike the optimization objectives of molecular properties for de novo drug discovery, structural similarity is typically assessed for molecule optimization. This is due to the requirement to preserve the structural features of a lead compound when optimizing its properties. On the other hand, multiple properties, which are often conflicting with each other, require to be concurrently optimized for molecule optimization tasks. Considering that an important goal is to enhance the search efficiency in the complex and extensive molecule space, molecules are encoded in an implicit space, which facilitates efficient evolutionary processes. In addition, it is unreasonable to assign different weights to multiple molecular properties and aggregate them, as they have different magnitudes, and the predefined weights will introduce bias. To address these issues, we propose a framework for multiproperty molecule optimization, termed MOMO. Given a set of molecules to be optimized, the proposed MOMO first assigns them into an implicit chemical space and generates an initial population for each lead molecule. Further, the molecules are evolved by a designed Pareto-based multiproperty evaluation strategy to simultaneously satisfy multiple physicochemical properties and structural similarity property in the implicit chemical space. To be specific, the proposed MOMO mainly consists of the following seven steps.

  • 1.

    An encoder-decoder module is pretrained on public molecule database to construct an implicit chemical space (Figure 2a).

  • 2.

    The lead molecule is embedded into the implicit space, and the molecule’s continuous representation is obtained from the encoder.

  • 3.

    For the first generation, MOMO constructs an initial population by perturbing the input molecule’s continuous embedding vector with Gaussian noise (Figure 2b). Each individual in the population serves as a solution of the multiobjective optimization problem, i.e., the embedding vector of the optimized molecules.

  • 4.

    Evolutionary operations (i.e., selection, crossover, and mutation) are performed on the continuous vectors within the population to generate new vectors of offspring (Figure 2c).

  • 5.

    The continuous vectors from the parent population and offspring population are combined and decoded into the molecular space using the pretrained decoder.

  • 6.

    The population is updated by a Pareto-based multiproperty evaluation strategy at the discrete molecule space. Specifically, the properties of evolved molecules are acquired by the property evaluator at first61−63 and then ordered based on nondomination rank and reference point mechanism. The selection of desired molecules for the new population is achieved through the partial order and a designed dynamic acceptance probability.

  • 7.

    Steps 4–6 are iterated to search for molecules in the implicit space, and the Pareto solutions in the last generation of population are ultimately returned as the optimized molecules (Figure 2d). The detailed MOMO framework is described in Supporting Information S1.

Figure 2.

Figure 2

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Diagram illustrating the MOMO framework. (a) An encoder-decoder module is pretrained on a large molecule data set to construct an implicit chemical space and extract the continuous representation of molecules. (b) The initial population of the lead molecule is obtained by perturbing the latent vector of the input molecule with Gaussian noises. (c) Mechanisms of the evolution of molecules. In the implicit chemical space, molecules in the population perform evolutionary operations (i.e., selection, crossover, and mutation) among their latent vectors to exchange information and generate new molecular vectors. (d) The optimization framework of MOMO. MOMO optimizes the target properties of the lead molecule by evolving the molecular latent vectors in the implicit chemical space, evaluating the properties in the molecular space, and preserving the desired molecules into a new population. Molecules in the new population are continue to be encoded for evolution in the next generation. Molecules on the Pareto front of the last generation are output as the optimized molecules.

In this way, the pretrained codec constructs an implicit space that provides continuous vectors as solutions during the evolution process, thereby enhancing the validity and smoothness of evolutionary operations. In addition, the designed Pareto-based multiproperty evaluation strategy at the molecule sequence level facilitates the search for a set of diverse and novel molecules with high properties. The resulting optimal set of molecules provides a diverse trade-off among multiple properties that can provide a selection for expertise. Moreover, no additional weights are required in this process.

3.3. Evolution Strategy in the Implicit Chemical Space

In our work, the molecules evolve in an implicit chemical space. First, we construct an implicit chemical space by utilizing a pretrained encoder-decoder to obtain the continuous vectors of molecules. Subsequently, new molecules are evolved by performing evolutionary operations in the implicit chemical space.

3.3.1. Construction of the Implicit Chemical Space

MOMO employs a codec framework to construct an implicit chemical space. Specifically, the encoder E maps a molecule x into an implicit chemical space. In this way, the molecule is represented as a continuous embedding vector, i.e., z = E(x). Finally, the decoder D decodes the embedding vector z into a new molecule x′, i.e., x′ = D(z). Note that MOMO is applicable to any codec structure with multiple molecule representation (such as strings64,65 or graphs66) for the continuous representations of molecules. In this study, we utilize a pretrained model67 to obtain the implicit chemical space, where the RDKit canonical SMILES is used before encoding.

3.3.2. Evolutionary Operations

MOMO employs the mating and environmental selection strategies within the implicit chemical space to generate solutions for better performance on the multiobjective molecule optimization problems.

  • 1.

    Initial population. Given a lead molecule x0, MOMO constructs an initial population with N individuals by perturbing the molecule’s embedding vector z0 with Gaussian noises, where each individual in the initial population is the embedding vector of a perturbed molecule.

  • 2.

    Selection. The binary tournament selection operator68 is used to choose a subset of individuals with higher dominant level for evolution from the current population. Specifically, two individuals are randomly sampled from the previous population, and the one dominants the other molecule is selected for evolution. The process is repeated multiple times until the population size is equal to N.

  • 3.
    Crossover. The blending linear crossover operator69 is considered in MOMO for generating new molecules. Specifically, for an individual z1, another individual z2 is randomly selected to generate new offspring z1′ and z2′ as eq 3.
    graphic file with name ci4c00031_m003.jpg 3
    where u1, u2 ∼ uniform[0,1], and d ≥ 0 is parameter that control whether the exploration space is interpolated or extrapolated.
  • 4.

    Mutation. The mutation operator of MOMO aims to perform a local search to optimize molecules. An individual is mutated with a certain probability (pm = 0.5). To be specific, a random value r is sampled for an individual z (D-dimensional vector). If r < pm, then an integer c is randomly drawn from {1, ...., D}, and the c-th position is replaced with a random value drawn from the standard Gaussian distribution.

3.4. Pareto-Based Multiproperty Evaluation Strategy

The population is updated based on the Pareto-based multiproperty evaluation strategy at the molecule sequence level. To be specific, we employ the nondomination rank and the reference point mechanism to evaluate the partial order of molecules according to their properties and similarity values.70 Subsequently, the new population of molecules is updated based on the partial order and a designed dynamic acceptance probability.

  • 1.

    Nondomination rank. For a multiobjective optimization problem with m objectives {f1, f2, ..., fm}, the feasible solution z1 is said to dominate and be better than another solution z2 (denoted as z1 ≺ z2), if and only if ∀i ∈ {1,2, ..., m}: fi(D(z1)) ≥ fi(D(z2)) and ∃j ∈ {1,2, ..., m}: fj(D(z1)) > fj(D(z2)). If z1 is not dominated by other solutions, then it is a nondominated solution, i.e., a Pareto optimal solution for the multiobjective optimization problem. All Pareto optimal solutions constitute the Pareto front of the multiobjective optimization problem. Thus, all solutions can be sorted by Pareto ranks F = {F1, F2, ...}. For example, rank Fi dominates Fj for j > i.

  • 2.

    Reference point mechanism. The reference point mechanism is used to establish the order of individuals with the same dominant rank, and it helps to maintain population diversity by introducing widely distributed reference points. The nondominant individuals that are sparsely associated with the reference point are retained. The following main steps are used to select individuals: (1) Das and Dennis’s systematic approach71 is used to generate structured reference points that are uniformly distributed on the (m – 1)-dimensional hyperplane (a total of m objectives). (2) The objectives of individuals in the population are normalized by the max-min normalization method. (3) Individuals are associated with reference points by constructing an ideal point f* = (fMax1, fMax2, ...., fMaxm) of the population and connecting it to each reference point to form reference lines that divide the objective space symmetrically. Each individual is then associated with the reference point on the closest reference line, and the reference point associated with solution zi is denoted as R(zi). (4) Individuals are sorted based on the niche count of the reference points (denoted as ρj), which is calculated by the number of population members associated with the reference points. Individuals associated with reference points with small niche counts are selected first for the next generation. If more than one individual is associated with a reference point, an individual is randomly selected.

  • 3.

    Partial order. A solution that exhibits higher values across multiple objectives and is far from other solutions in the objective space is considered better. The partial order of solutions is defined as if feasible solution z1 dominates z2, i.e., F(z1) < F(z2), then z1 is superior to z2. If F(z1) = F(z2), and the niche count Inline graphic, then z1 is superior to z2.

  • 4.
    Update population. To ensure that the population evolves toward molecules with higher properties, MOMO preserves excellent individuals in population based on the Pareto-based multiproperty evaluation strategy. Molecule vectors in the population (Z) and offspring (Z′) are decoded into SMILES and employed to evaluate the property values. Molecules with high properties and diversity are selected for the next population. To be specific, the individuals in population and offspring are sorted first by nondomination rank and then by the reference point mechanism. Molecules are successively selected into the new population according to their rank with an acceptance probability. The acceptance probability is designed to maintain molecular diversity in the population and prevents fall into some local optimum regions. Top-ranked molecules (with high objective values) have a certain probability of being accepted into the next generation or may not be accepted. The acceptance probability is calculated as follow:
    graphic file with name ci4c00031_m005.jpg 4
    where t represents the current index of evolution epoch, pa is increasing with the iteration of t, and β controls the incremental speed, which is set as 0.3 in experiments.

During early evolutionary stages, molecules with low objective values still have a chance to enter the next generation to emphasize diversity in the population. As evolution proceeds, molecules with higher objective values are accepted to emphasize the properties and similarity. Ultimately, the population is updated as follows. Starting from the top molecule, a random value r is sampled; if r < pa, then the molecule is accepted. The next molecule is then processed until the size of the new population is equal to P.

4. Results

4.1. Benchmark Evaluation of MOMO

To demonstrate the performance of the MOMO framework, we compare the proposed MOMO with five state-of-the-art molecule optimization methods on six multiobjective molecule optimization tasks.

4.1.1. Test Tasks for Molecule Optimization

The six molecule optimization tasks we employed in this work contain multiple molecular properties to be optimized. To facilitate a comparison with traditional molecule optimization methods, we considered Task 1 and Task 2 with two objectives to be optimized. Further, Tasks 3–6 were designed to validate the performance of the proposed MOMO on complex tasks with a larger number of objectives, where Task 3 and Task 4 contain three objectives; Task 5 and Task 6 optimize four objectives simultaneously. In addition, Tasks 2–6 include the optimization of some practical properties (e.g., biological activity and protein–ligand docking), which makes these tasks more realistic and more challenging. Specifically, the properties, considered in the experiments, include the nonbiological properties, biological activity properties, and protein–ligand docking. The calculated nonbiological properties include drug-likeness (QED),72 three structural properties of an FDA-approved drug, i.e., perindopril,36 the synthetic accessibility (SA),27 and the Tanimoto similarity between an optimized molecule and the lead molecule (Similarity).60 Two bioactivity properties include the predicted molecule bioactivity against dopamine type 2 receptor (DRD2 activity)73 and the predicted inhibition score against glycogen synthase kinase-3β (GSK3β inhibitory).27 Protein–ligand docking in this work contains a simulated ligand efficiency (LE) score of a molecule to the 1SYH target,74 which is calculated by scaling the docking score by the number of heavy atoms of the ligand molecule, and the simulated docking score of a molecule to the 4LDE target (ΔE4LDE).74 Specifically, the details of six tasks are illustrated as follows:

  • Task 1: QED and Similarity. In this task, we utilized a data set19 comprising 800 molecules as lead molecules and optimized each molecule. The larger the values of these two properties, the better. The objective is to optimize the lead molecule to satisfy QED ≥ 0.9 and Similarity ≥ 0.4.

  • Task 2: Ligand efficiency and Similarity. The aim of Task 2 is to optimize the simulated ligand efficiency (LE) of the molecule to the 1SYH protein (an ionotropic glutamate receptor) and the Similarity. We employed a data set of 100 molecules selected from Nigam et al.74 The LE score is calculated by scaling the minus docking score based on the number of heavy atoms of the ligand molecule. The docking score is simulated by the TARTARUS benchmark.74 The larger both the LE score and Similarity, the better. The goal is to optimize the molecules to satisfy LE ≥ 0.6 and Similarity ≥ 0.4.

  • Task 3: QED, DRD2 activity, and Similarity. The biological activity is a crucial factor in real-world drug design. The objectives considered in Task 3 are QED, DRD2 activity, and Similarity, where the DRD2 activity is predicted by a surrogate model available on the Therapeutics Data Commons (TDC) platform.62 The data set23 employed in this task comprises 780 molecules. Larger values of these three properties are better. The goal of this task is to optimize molecules with QED ≥ 0.8, DRD2 activity ≥0.4, and Similarity ≥ 0.3.

  • Task 4: QED, ΔE4LDE, and Similarity. The optimization of protein ligands holds a pivotal role in drug design by enhancing the protein–ligand docking scores, which is more challenging than simple physicochemical properties.75 Task 4 focuses on simultaneously optimizing the simulated docking score of ligand for the protein target 4LDE, the QED value, and the Similarity, where 4LDE is a β2-adrenoceptor GPCR receptor responsible for muscle relaxation and bronchodilation. The data set for Task 4 contains 100 molecules selected from Bickerton et al.72 The docking score to the 4LDE target is simulated by the TARTARUS benchmark.72 For QED and Similarity, a larger value is better; for the docking score, smaller is better. The optimization objectives are the docking score of molecules to the 4LDE target ΔE4LDE < −10, QED ≥ 0.8, and Similarity ≥ 0.3.

  • Task 5: Three structural properties of an FDA-approved drug (i.e., perindopril) and Similarity. Note that this task is referred to the Pioglitazone task in GuacaMol.76 Specifically, this task involved four objectives: the dissimilarity of the generated molecule to perindopril (Score_dissmi), the molecular weight score of the generated molecule (score_mw), the score of the number of rotatable bonds in the generated molecule (Score_rb), and the similarity between the lead molecule and the optimized molecule. All scores ranged between 0 and 1, with higher scores indicating better performance (more detailed information can be found in ref (74)). The objective is to optimize the lead molecule to satisfy Score_dissmi ≥ 0.5, Score_mw ≥ 0.5, Score_rb ≥ 0.5, and Similarity ≥ 0.3.

  • Task 6: QED, GSK3β inhibitory, SA, and Similarity. In Task 6, it is required to simultaneously optimize biological activity, the drug-like property, synthetic accessibility (SA), and the similarity to the lead molecule, where SA denotes the probability of a molecule being synthesized, which is measured and normalized as in ref (27). The bioactivity in this task is the inhibition score against glycogen synthase kinase-3β, denoted as GSK3β inhibitory, which is evaluated by a surrogate model available on the TDC platform.63 The larger the four properties, the better. The optimization objective for this task is to optimize molecules to satisfy QED ≥ 0.7, GSK3β inhibitory ≥ 0.4, SA ≥ 0.7, and Similarity ≥ 0.2.

4.1.2. Compared Methods

MOMO was compared with three single-objective optimization methods (GA,18 MSO,26 and QMO28) and two multiobjective optimization methods in discrete spaces (Molsearch44 and Molfinder-P) on six tasks. Specifically, Molfinder-P is a Pareto domination-based version of Molfinder,47 since Molfinder is a genetic algorithm-based global optimization algorithm, where multiple properties are combined by using a weighted summation approach. Further information about the compared methods can be found in Supporting Information S1.2.

The comparison of the multiobjective methods with single-objective methods is complex, since it is difficult to precisely select the weights of several objectives and combine them into a single one. In this work, for each single-objective-based optimization model (i.e., QMO, GA, and MSO), we considered three classical objective-aggregation approaches (i.e., weighted summation, Chebyshev aggregation, and penalty-based boundary intersection method32) to combine multiple objectives into a single composite one. Specifically, the weighted summation approach is a common linear multiobjective aggregation, where five groups of uniformly distributed weights were set in our work according to the guideline in ref (32). The Chebyshev aggregation approach is a nonlinear method, and the aggregated score is the largest difference between an objective value of the solution and its ideal value among multiple objectives of the solution. The penalty-based boundary intersection approach is to minimize the distance between the reference point and the solution sought on the weight vector, where the objective weight was set to 1, the ideal points for both tasks were (1,1), and the penalty parameter was 0.2. The details of three single-objective aggregation methods are provided in Supporting Information S3.1.

4.1.3. Evaluation Measures

Due to the fact that all molecule optimization tasks considered here have multiple objectives to be optimized, we adopted not only four widely used performance indicators for optimizing molecules (i.e., success rate, novelty, average value, and the standard deviation of properties or the improvement in properties) but also a popular performance indicator for multiobjective optimization, i.e., hypervolume.

Specifically, the success rate (SR for short) is defined as the percentage of optimized molecules whose Similarity and property values exceed their respective thresholds.23 The novelty is calculated as the percentage of the successfully optimized molecules in the test sets, which are not included in the ZINC database.77 The hypervolume78 (HV) is a widely used measure in multiobjective optimization to quantify the diversity and convergence of solutions along the approximated Pareto front, where the diversity measures the molecules spread over the objective space, and the convergence refers to the proximity of the molecule solutions to the ideal Pareto front with one or more properties that are the highest. It calculates the hypervolume of the objective space that is dominated by the Pareto solutions,

4.1.3. 5

where δ is the Lebesgue measure to evaluate the hypervolume, |S| represents the number of solutions in a nondominated solution set, and vi represents the hypervolume between the reference point and the i-th solution in the solution set. Further information about the data sets, compared methods, evaluation metrics, and objectives can be found in Supporting Information S1.

4.2. Comparisons on Test Tasks

In this experiment, we evaluated the performance of a single-objective model in multiple forms using three kinds of objective-aggregation approaches. Figure 3 plots the best values of success rate (SR) and hypervolume (HV) of three single-objective methods in multiple forms, as well as the SR and HV values of the proposed MOMO and a considered Pareto-based multiobjective molecule optimization method on six optimization tasks. More detailed experimental results of single-objective methods in seven forms are shown in Tables S8 in Supporting Information S.3.2. From the figure and tables, the following three observations can be obtained.

Figure 3.

Figure 3

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Performance evaluation of MOMO and compared methods on six tasks. (a) Success rate (SR) and hypervolume (HV) of MOMO and compared methods on Task 1 with a QED threshold of 0.9 and a Similarity threshold of 0.4. (b) SR and HV on Task 2 with an LE threshold of 0.6 and a Similarity threshold of 0.4. (c) SR and HV of optimized molecules of Task 3 with a QED threshold of 0.8, a DRD2 activity threshold of 0.4, and a Similarity threshold of 0.3. (d) SR and HV on Task 4 with a QED threshold of 0.8, a ΔE4LDE threshold of −10, and a Similarity threshold of 0.3. (e) SR and HV of the molecules optimized by various algorithms in Task 5 with three structure properties of a perindopril threshold of 0.5 and a Similarity threshold of 0.3. (f) SR and HV of Task 6 with a QED threshold of 0.9, a GSK3β inhibitory threshold of 0.4, an SA threshold of 0.4, and a Similarity threshold of 0.2. Note that each single-objective model was only tested on Task 1 for multiple forms because of the complexity of selecting weights of more than two objectives. On Tasks 2–6, for three single objective-based optimization methods (i.e., GA, MSO, QMO), a comprehensive objective is achieved by the sum of several normalized objectives, where each normalized objective has the weight of one.

First, the proposed MOMO achieved overall better performance in comparison with the five considered molecule optimization methods on six tasks, which demonstrated its superior ability to optimize multiple properties of lead compounds. Specifically, as shown in Figure 3, MOMO outperformed all the considered compared methods with higher values of SR and HV on six multiproperty optimization tasks. For example, the SR of MOMO surpassed that of the single-objective optimization methods by 32% and outperformed the multiobjective optimization method by 18% on Task 3 (Figure 3c). Further, Figure S1 presents the property values of the optimized molecules’ property values achieved by different methods on six tasks, averaging over all test lead molecules with the same parameters. It can be found that the MOMO model overall achieved the largest average value of the properties. Note that Molsearch modifies molecules in a step-by-step manner, which tends to generate molecules with high Similarity but low physicochemical properties. Moreover, to compare the stability of the methods for different molecules, we measured the standard deviation of the property values or improvements and the evaluation indicators among the optimized molecules. As shown in Tables S1–S6, MOMO achieved overall higher properties while maintaining a low standard deviation. It is concluded that the performance of the proposed MOMO is relatively stable for optimization of different molecules. Furthermore, we evaluated the novelty of optimized molecules achieved by different methods. The novelty of MOMO and the compared methods on six tasks are shown in Table S7. It is found that MOMO achieved the highest novelty of successfully optimized molecules for each test set. These experimental results indicated that MOMO gains the overall best performance in terms of optimized property values, diversity, novelty, and success rate.

Second, the proposed MOMO is more suitable for optimizing a large number of molecular properties. The six designed tasks involve the simultaneous optimization of two, three, or four objectives. The performance of all molecule optimization methods considered here deteriorated considerably as the number of optimized objectives increases due to the fact that the tasks become more complex. However, the proposed MOMO exhibited relatively robust performance on Tasks 3–6, which showed significantly better performance than all the compared. Specifically, on Task 1, all of the considered methods achieved a high success rate (SR), where MOMO achieved the highest SR value (i.e., 0.95), followed by QMO (i.e., 0.93), and the values obtained by the other three methods are above 0.80. For the complex tasks with three or four objectives, MOMO also obtained the highest SR value. In contrast, the values of SR achieved by the compared methods decrease sharply. For example, QMO resulted in an SR value of 0.46 on Task 3, ranking the third. In addition, on Tasks 4–6, the values of SR were less than 0.2, which made QMO rank the last. Especially on tasks with four objectives, MOMO demonstrated a larger performance improvement. For example, MOMO achieved a 15% improvement in SR value compared to the best-performing compared model on Task 5 (Figure 3e) and a 56% improvement on Task 6 (Figure 3f). These findings demonstrated MOMO’s promising performance to optimize complex tasks with a greater number of objectives.

Third, MOMO has a prominent advantage in optimizing realistic molecular properties. For example, on Task 4, which involved the optimization of the protein–ligand docking, the SR value of molecules optimized by MOMO was 4.3 times higher than that of the considered single-objective methods and 11% higher than that of the considered multiobjective compared methods (Figure 3d). In addition, the optimized molecules of MOMO achieved a higher docking score with the 4LDE protein (Figure S1d). To visually illustrate the effectiveness of the proposed MOMO on Task 4, Figure 4 plots the top docking poses of the 4LDE protein target with the lead molecule and a MOMO-optimized molecule. As can be seen from the figure, the original lead molecule formed only one hydrogen bond to dock with two amino acid residues, whereas the optimized molecule established three hydrogen bonds with the protein, thereby improving the docking score. Additionally, the QED of the optimized molecule was improved by 0.18 while maintaining a certain level of similarity with the lead molecule. Furthermore, on Task 6 with the optimization of both biological activity and synthetic accessibility of molecules, MOMO also achieved the highest SR and HV values (Figure 3f). These findings demonstrated MOMO’s promising performance to optimize complex tasks with more realistic properties, such as protein–ligand docking, bioactivity, and synthetic accessibility.

Figure 4.

Figure 4

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Top docking posture of the small molecule ligand and 4LDE protein. The left figure is the complete target-ligand docking posture. The scaled area is the specific docking position of the small molecule hydrogen bond with the protein amino acid residue, and the name of the amino acid residue and the length of the hydrogen bond are marked in the figure. (a) Docking posture of the lead molecule with the 4LDE target. The QED value of the lead molecule was 0.724, and the docking score with the 4LDE target was −7.8. (b) The docking posture of the optimized molecule with the 4LDE target, the QED of the optimized molecule was improved to 0.908, the docking score with the 4LDE target was optimized to −10.5, and the Similarity with the lead molecule was maintained to 0.3.

From the above results, we conclude that the proposed MOMO is a promising molecule optimization model, especially for those with complex molecular properties to be optimized. The promising performance of the proposed MOMO may be attributed to the Pareto-based multiproperty evaluation strategy in an implicit chemical space, since it facilitates a comprehensive search in the implicit chemical space to identify high-property molecules under various trade-offs of relative importance.

4.3. Effectiveness of the Evolution in the Implicit Chemical Space

As shown in Figure 3, molecule optimization methods searching in the implicit space outperformed those in the discrete space on most tasks. Specifically, for the three single-objective optimization methods, the two methods (MSO and QMO) in the implicit space outperformed the GA model in a discrete space for most tasks. Additionally, MOMO outperformed the multiobjective evolutionary methods (Molsearch and Molfinder-P), which search molecules in discrete spaces on all six tasks. That is mainly because the evolution in an implicit chemical space enhances the smoothness and efficiency of exploration, thereby improving the performance of molecule optimization. These findings indicated the efficiency gained from exploration within an implicit space.

To further analyze the search capability of these multiobjective optimization methods in different chemical spaces, we visualized the increase in the hypervolume of the Pareto solutions (Figure S2). It can be observed that MOMO showed competitive improvements in the properties of the molecules from the approximated Pareto optimal solution set in each population, which further demonstrated the search ability in the implicit space.

4.4. Effectiveness of Pareto-Based Multiproperty Optimization

Table S8 presents the performance of single-objective methods (i.e., QMO and MSO) in seven forms on Task 1 by using three kinds of objective-aggregation approaches. As can been seen from Table S8, for single-objective methods, the optimal aggregation of objectives is difficult to be determined. No single-objective aggregation approach could achieve the best performance for all single-objective optimization methods. For example, MSO achieved the best SR on Task 1 with the Chebyshev aggregation approach, while QMO achieved the best SR on this task using the weighted summation approach. This observation highlighted the inherent challenge of effectively integrating multiple subobjectives into a single comprehensive one. A suitable objective-aggregation method may be achieved by repeating plenty of experiments to evaluate all of the possible strategies. However, it is impractical to exhaustively test all possible strategies to determine the optimal one.

To further illustrate the effectiveness of the Pareto-based multiproperty evaluation strategy in MOMO, we performed an ablation experiment by comparing the indicator values of QMO, MOMO, and MOMO_S, where MOMO_S is the modified MOMO model by replacing the evaluation of molecules based on Pareto-based dominance as the evaluation by comparing a weighted aggregation of the considered properties (Figure S3). As can be seen from the figure, MOMO_S did not exhibit a significant improvement in terms of the success rate in comparison to QMO, but the properties, dominant area, and diversity were largely improved. It is indicated that the population-based evolution in the implicit space outperformed the individual-based gradient update used in QMO. Compared with MOMO_S, MOMO achieved a better success rate and properties, and the Pareto solution set exhibited improved convergence and distribution. That is mainly because the Pareto-based multiproperty evaluation strategy aids in optimizing multiple properties of molecules and facilitates the discovery of a set of diverse solutions in a single run. Above results demonstrated the superior performance of the Pareto-based evaluation strategy in multiproperty optimization tasks.

4.5. Examples of Search Capability and Molecule Evolution by MOMO

To analyze the search capability of MOMO, we presented an example of a molecule from Task 1 that was evolved by MOMO (Figure 5). The lead molecule was represented by a star, and its molecular structure was displayed adjacent to it. The dots with various colors indicated the molecules generated by MOMO in different generations. The red dots corresponded to the molecules on the final optimized Pareto front, and their corresponding molecular structures were illustrated nearby. The results in Figure 5 indicated that MOMO can ultimately find optimized molecules with diverse preferences of properties, whether it be a high Similarity, high QED, or a successful molecule that considers both Similarity and QED. It is demonstrated that MOMO possesses the ability to explore multiple directions and search for molecules with higher objective values as the number of generations increases. The designed Pareto-based multiproperty evaluation strategy enables MOMO to identify the optimal set of solutions with different relative importance among multiple objectives.

Figure 5.

Figure 5

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Example to demonstrate the search capability of MOMO. The evolution of molecular properties through MOMO on the task is plotted. The red star is the lead molecule. The dots in various colors represent molecules generated in different evolutionary generations of MOMO. The final optimized molecules on the Pareto front are displayed alongside the red dots.

To further illustrate the molecule evolution during the search process, Figure 6 depicts a trajectory example of a molecule in Task 1. The three rows presented the evolutionary paths of three distinct progenitors from the lead molecule toward the Pareto front in different directions. The shaded region represented the dissimilarity region of the molecule concerning the parent molecule. From Figure 6, the first row generated a successful molecule with an extremely high QED and relatively low similarity. The second row produced a molecule with both high QED and similarity. The third row led to a molecule with low QED but quite similar to the lead molecule. The structural modification process demonstrated that molecular evolution in the implicit space is smoother and more tractable than structure-based modifications. Additionally, the Pareto-based multiproperty evaluation strategy guides the evolutionary process, enabling different evolutionary paths to reach optimized molecules with various preferences. These indicated that given a lead molecule, MOMO can gradually optimize the molecule in different directions until it discovers a diverse set of molecules that satisfy different objectives.

Figure 6.

Figure 6

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Example trajectory of a molecule’s evolution on task to demonstrate the molecular modification procedure of MOMO. The figure consists of three rows representing the evolutionary processes of the three progenitors from the lead molecule to the Pareto front in different directions. The shaded region in each row shows the modified substructure of the improved molecule compared with the original one.

5. Conclusions and Limitations

In this paper, we formulated the lead compound optimization as a multiobjective optimization problem. Further, we proposed a multiobjective molecule optimization framework, namely, MOMO, to optimize a molecule’s simulated properties while preserving its structural features, particularly for molecules with a large number of simulated properties or realistic simulated molecular properties to be optimized. The proposed MOMO achieves the optimization of molecules by the Pareto-based multiproperty evaluation strategy to guide the evolution of molecules in an implicit chemical space. MOMO requires the input of molecular SMILES sequences for optimization without additional data or label information. The application domain of MOMO may be different in different molecular optimization tasks, which depends on the applicability scope of certain employed molecular property measures. Experimental results on both benchmark problems and complex molecule optimization tasks have demonstrated the superior performance of MOMO in molecule optimization by assessing the optimized property, diversity, novelty, and success rate, and MOMO has been shown to be well suitable for the molecule optimization tasks with a large number of simulated properties or challenging properties to be optimized.

Our work has demonstrated the effectiveness of the Pareto-based multiproperty evaluation strategy in an implicit chemical space. This strategy enables the simultaneous optimization of multiple properties and the generation of diverse and novel molecules, which also facilitates smooth and efficient exploration of the chemical space. Despite this, more efficient and faster methods for complex molecular optimization tasks are required. There are many promising directions to investigate for future works. For example, refining the evolution process to explore more diverse and better distributed Pareto fronts can enhance the performance of complex molecule optimization tasks. Moreover, using more precise structural similarity measures or adding structural constraints may prevent unrealistic molecule generation. Furthermore, the proposed molecule optimization model is suggested to apply in some practical and complex scenarios, such as optimizing known antimicrobial peptides and drug discovery against new virus.

Data Availability Statement

The data sets and codes used in this project are updated and available at https://github.com/ahu-bioinf-lab/MOMO-master.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jcim.4c00031.

  • Additional experimental details, materials, methods, results, and analysis (PDF)

Author Contributions

Y.S. conceived the study. X.X. constructed the databases and developed the codes. X.X. and Y.S. implemented experiments and analyzed the results. X.X., Y.S., X. Zhang, C.Z., X. Zeng, X.G., Q.W., and Y.L. wrote and critically revised the manuscript.

This work was supported by the National Key Research and Development Program of China (2021YFE0102100), the Natural Science Foundation of China (grant nos. 62322301, 62172002, 62122025, 61872309, U19A2064, and 62106073), and the Natural Science Foundation of Hunan Province of China (grant no. 2021JJ40116).

The authors declare no competing financial interest.

Supplementary Material

ci4c00031_si_001.pdf (450.1KB, pdf)

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

ci4c00031_si_001.pdf (450.1KB, pdf)

Data Availability Statement

The data sets and codes used in this project are updated and available at https://github.com/ahu-bioinf-lab/MOMO-master.

Articles from Journal of Chemical Information and Modeling are provided here courtesy of American Chemical Society

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