Quadratic descriptors and reduction methods in a two-layered model for compound inference

Abstract

Compound inference models are crucial for discovering novel drugs in bioinformatics and chemo-informatics. These models rely heavily on useful descriptors of chemical compounds that effectively capture important information about the underlying compounds for constructing accurate prediction functions. In this article, we introduce quadratic descriptors, the products of two graph-theoretic descriptors, to enhance the learning performance of a novel two-layered compound inference model. A mixed-integer linear programming formulation is designed to approximate these quadratic descriptors for inferring desired compounds with the two-layered model. Furthermore, we introduce different methods to reduce descriptors, aiming to avoid computational complexity and overfitting issues during the learning process caused by the large number of quadratic descriptors. Experimental results show that for 32 chemical properties of monomers and 10 chemical properties of polymers, the prediction functions constructed by the proposed method achieved high test coefficients of determination. Furthermore, our method inferred chemical compounds in a time ranging from a few seconds to approximately 60 s. These results indicate a strong correlation between the properties of chemical graphs and their quadratic graph-theoretic descriptors.

1 Introduction

In recent years, extensive studies have been done on the design of novel molecules using various machine learning techniques (Lo et al., 2018; Tetko and Engkvist, 2020; Gartner III et al., 2024). Computational molecular design also has a long history in the field of chemo-informatics and has been studied under the names of quantitative structure-activity relationship (QSAR) (Cherkasov et al., 2014) and inverse quantitative structure-activity relationship (inverse QSAR) (Miyao et al., 2016; Ikebata et al., 2017; Rupakheti et al., 2015). This design problem has also become an important topic in both bioinformatics and machine learning.

The purpose of QSAR is to predict chemical activities from given chemical structures (Cherkasov et al., 2014). In most QSAR studies, a chemical structure is represented as a vector of real numbers called features or descriptors, and then a prediction function is applied to the vector, where a chemical structure is given as a chemical graph, which is defined by an undirected graph with an assignment of chemical elements to vertices and an assignment of bond multiplicities to edges. A prediction function is usually obtained from existing structure–activity relationship data. To this end, various regression-based methods have been utilized in traditional QSAR studies, whereas machine learning-based methods, including artificial neural network (ANN)-based methods, have recently been utilized (Ghasemi et al., 2018; Kim et al., 2021; Catacutan et al., 2024).

Conversely, the purpose of inverse QSAR is to predict chemical structures from given chemical activities (Miyao et al., 2016; Ikebata et al., 2017; Rupakheti et al., 2015; Miyao and Funatsu, 2024), where additional constraints may often be imposed to effectively restrict the possible structures. In traditional inverse QSAR, a feature vector is computed by applying some optimization or sampling method on the prediction function obtained by standard QSAR, and then chemical structures are reconstructed from the feature vector. However, the reconstruction itself is quite difficult due to the vast number of possible chemical graphs (Bohacek et al., 1996). In fact, inferring a chemical graph from a given feature vector, except for some simple cases, is an NP-hard problem (Akutsu et al., 2012). Due to this inherent difficulty, most existing methods employ heuristic methods for the reconstruction of chemical structures and thus do not guarantee optimal or exact solutions. On the other hand, one of the advantages of ANNs is that generative models, such as autoencoders and generative adversarial networks, are available. Furthermore, graph-structured data can be directly handled by using graph convolutional networks (Kipf and Welling, 2016). Therefore, it is reasonable to try to apply ANNs to inverse QSAR (Xiong et al., 2021). Various ANN models, including recurrent networks, autoencoders, generative networks, and invertible flow models, have been applied in this context (Segler et al., 2018; Yang et al., 2017; Gómez-Bombarelli et al., 2018; Kusner et al., 2017; De Cao and Kipf, 2018; Madhawa et al., 2019; Shi et al., 2020). However, the optimality or exactness of the solutions provided by these methods is not yet guaranteed.

A novel two-phase framework based on mixed-integer linear programming (MILP) and machine learning methods has been developed to infer chemical graphs (Shi et al., 2021; Zhu et al., 2022b; Ido et al., 2021; Azam et al., 2021a). The first phase constructs a prediction function for a chemical property, and the second phase infers a chemical graph with a target value of the property based on the function . For a chemical property , we define as a data set of chemical graphs such that the observed value of property for every chemical graph is available.

In the first phase, we introduce a feature function for a positive integer , where the descriptors of a chemical graph are defined based on local graph structures by using a two-layered model (Shi et al., 2021) so that the inverse of can be modeled by MILP in the second phase. The prediction function aims to produce an output based on the feature vector for each . This output serves as a predicted value for the real value .

The task of the second phase is to infer desired chemical graphs. This phase consists of three steps. For a given set of rules called topological specification and a range of the target property value, the aim of the first step is to infer chemical graphs that satisfy the rules and (see Figure 1 for an illustration). For this, we formulate an MILP that represents (i) the computation process of from a chemical graph in the feature function ; (ii) that of from a vector in the prediction function ; and (iii) the constraint , where denotes the set of all chemical graphs that satisfy the rules in . Given an interval with , we solve the MILP to find a feature vector and a chemical graph such that and (if the MILP instance is infeasible, this suggests that does not contain such a desired chemical graph). See Zhu et al. (2022a) for a full description of the framework.

FIGURE 1

In the second and third steps of the process, we employ different techniques to generate additional desired chemical graphs based on the initial solution . These techniques include a dynamic programming algorithm (Azam et al., 2021b) and a neighbor search method (Azam et al., 2021a). The dynamic programming algorithm operates by decomposing into trees and generating their isomers. These isomers are then combined to produce a set of isomers that belong to the desired chemical graph space . These isomers are referred to as the recombination solutions of . The idea of the neighbor search method is to generate new desired chemical graphs that have a slightly different feature vector than that of . For this purpose, we solve an augmented MILP obtained by with an additional set of linear constraints. The resulting graphs obtained through this process are referred to as the neighbor solutions of .

Contribution: The feature vector of a chemical graph in this framework consists of the descriptors that extract the local information of the interior and exterior parts of the graph obtained from the two-layered model. These simple descriptors play a crucial role in deriving compact MILP formulations for the inference of a desired chemical graph. However, for certain chemical properties, the prediction functions constructed based on the feature function could not achieve evaluation scores in the acceptable range. To improve the evaluation score, we introduce new quadratic descriptors (or . This drastically increases the number of descriptors, which would take extra running time for learning or lead to overfitting of the data set. Moreover, computed quadratic descriptors cannot be directly formulated as a set of linear constraints in the original MILP. For this, we introduce a method of reducing a set of descriptors into a smaller set that delivers a prediction function with a higher performance. We also design an MILP formulation to represent a quadratic term . Based on the same MILP formulation proposed by Zhu et al. (2022b), we implemented the framework to treat the feature function with quadratic descriptors. From the results of our computational experiments on more than 40 chemical properties, we observe that our new method of utilizing quadratic descriptors has improved the performance of a prediction function for many chemical properties.

2 Quadratic descriptors and their approximation in an MILP

In the framework with the two-layered model (Shi et al., 2021), the feature vector consists of graph-theoretic descriptors that are mainly the frequencies of atoms, bonds, and edge configurations in the interior and exterior of the chemical compounds. See Zhu et al. (2022a) for the details of the interior and exterior of chemical compounds and all the descriptors , which are called linear descriptors, respectively. There are some chemical properties for which the performance of a prediction function constructed with this feature vector remains rather low. To improve the learning performance in the two-layered model, we introduce quadratic terms (or ) and as a new descriptor, where we assume that each is normalized between 0 and 1 by using the min-max normalization. Note that computing quadratic descriptors cannot be directly formulated as a set of linear constraints in the original MILP used in the two-layered model. Therefore, this section introduces an MILP formulation that approximates the product of two descriptors in a MILP.

Given two real values and with and , the process of computing the product can be approximately formulated as the following MILP. First, regard as an integer with a binary expression of bits, where denotes the value of the -th bit. Then, compute , which becomes the -th bit of .

Note that the necessary number of integer variables for computing for one pair of and is . In this article, we set in our computational experiment. The relative error by in the above method is at most , which is approximately .

3 Methods for reducing descriptors

The number of descriptors in the two-layered model increases drastically due to the quadratic descriptors. More precisely, if denotes the set of linear descriptors, and denotes the set of quadratic descriptors constructed over a data set for a property , then the total number of descriptors are . This large number of descriptors would increase the running time to construct prediction functions or lead to overfitting of the data set. To address these issues, we propose methods for reducing descriptors in this section.

Let be a given set of chemical compounds, be a descriptor set obtained using the two-layered model, and be the size of a set of selected descriptors.

For given , and a real number , we denote by Des-set-LLR the subset of such that for and the hyperplane constructed by the LLR procedure LLR.

3.1 A method based on lasso linear regression

Because the lasso linear regression finds some number of descriptors with in the output hyperplane , we can reduce a given set of descriptors by applying the lasso linear regression repeatedly. Choose parameters and so that LLR can be executed in a reasonable running time when and . Let be an integer for the number of descriptors that we choose from a given set of descriptors.

In this article, we set , , and if in our computational experiment.

3.2 A method based on the backward stepwise procedure

A backward stepwise procedure (Draper and Smith, 1998) reduces the number of descriptors one by one, choosing the one removal that maximizes the learning performance and outputs a subset with the maximum learning performance among all subsets during the reduction iteration.

For a subset and a positive integer , let denote the median of the coefficient of determination test score of a prediction function by MLR constructed during times 5-fold cross-validations. We define a performance evaluation function for an integer such that . The backward stepwise procedure with this function is described as follows.

Based on the lasso linear regression and the backward stepwise procedure, we design the following method for choosing a subset of a given set of descriptors. We are given a set of 17 real numbers and a set of 16 real numbers close to each number . The method first chooses a best parameter to construct a prediction function by LLR and then chooses a subset for each by the backward stepwise procedure. The procedure takes iterations, which may take a large amount of running time. We introduce an upper bound on the size of an input descriptor set for the backward stepwise procedure. Let , , and be integer parameters that control the number of executions of cross-validations to evaluate the learning performance in the method.

In the computational experiment in this article, we set {, }, for each , , and .

4 Results and discussion

With our new method of choosing descriptors and formulating an MILP to treat quadratic descriptors in the two-layered model, we implemented the framework for inferring chemical graphs and conducted experiments to evaluate the computational efficiency.

4.1 Chemical properties

In the first phase, we used the following 32 chemical properties of monomers and ten chemical properties of polymers.

For monomers, we used the following data sets: biological half-life (BHL), boiling point (Bp), critical temperature (Ct), critical pressure (Cp), dissociation constants (Dc), flash point in closed cup (Fp), heat of combustion (Hc), heat of vaporization (Hv), octanol/water partition coefficient (Kow), melting point (Mp), optical rotation (OptR), refractive index of trees (RfIdT), vapor density (Vd) and vapor pressure (Vp) provided by HSDB from PubChem (2022); electron density on the most positive atom (EDPA) and Kovats retention index (Kov) provided by Jalali-Heravi and Fatemi (2001); entropy (ET) provided by Duchowicz et al. (2002); heat of atomization (Ha) and heat of formation (Hf) provided by Roy and Saha (2003); surface tension (SfT) by Goussard et al. (2017); viscosity (Vis) provided by Goussard et al. (2020); isobaric heat capacities liquid (LhcL) and isobaric heat capacities solid (LhcS) provided by Naef (2019); lipophilicity (Lp) provided by Xiao (2017); flammable limits lower of organics (FlmLO) provided by Yuan et al. (2019); molar refraction at 20° (Mr) provided by Ponce (2003); and solubility (Sl) provided by ESOL (MoleculeNet, 2022), and energy of highest occupied molecular orbital (Homo), energy of lowest unoccupied molecular orbital (Lumo), the energy difference between Homo and Lumo (Gap), isotropic polarizability (Alpha), heat capacity at 298.15 K (Cv), internal energy at 0 K (U0), and electric dipole moment (mu) provided by ESOL (MoleculeNet, 2022), where the properties from Homo to mu are based on a common data set QM9.

The data set QM9 contains more than 130,000 compounds. In our experiment, we used a set of 1,000 compounds randomly selected from the data set. For the Hv property, we removed the chemical compound with the compound identifier (CID) as an extremal outlier from the original data set.

For polymers, we used the following data provided by Bicerano (2002): experimental amorphous density (AmD), characteristic ratio (ChaR), dielectric constant (DieC), dissipation factor (DisF), heat capacity in liquid (HcL), heat capacity in solid (HcS), mol volume (MlV), permittivity (Prm), refractive index of polymers (RfIdP), and glass transition (Tg), where we excluded from our test data set every polymer whose chemical formula could not be found by its name in the book (Bicerano, 2002). A summary of these properties is given in Tables 1, 2. We remark that the previous learning experiments for { ChaR, RfIdP} based on the two-layered model proposed by Azam et al. (2021a) and Zhu et al. (2022c) excluded some number of polymers as outliers. In our experiments, we do not exclude any polymer from the original data set as outliers for these properties.

4.2 Experimental setup and setting data set

We executed the experiments on a PC with a Core i7-9700 (3.0 GHz; 4.7 GHz at the maximum processor and 16 GB RAM DDR4 memory. Prediction functions were constructed using scikit-learn version 1.0.2 with Python 3.8.12, and MLPRegressor and rectified linear unit (ReLU) activation function were used for learning based on ANN. The prediction functions constructed by different machine learning models were evaluated using the coefficient of determination . We performed 10 rounds of 5-fold cross-validation, calculated the training and testing scores for each cross-validation, and recorded the median of the testing scores across all 50 cross-validations.

For each property , we first selected a set of chemical elements and then collected a data set on chemical graphs over the set of chemical elements. To construct the data set , we eliminated chemical compounds that did not satisfy one of the following: the graph is connected, the number of carbon atoms is at least four, and the number of non-hydrogen neighbors of each atom is at most four.

We set a branch-parameter to be 2, introduce linear descriptors defined by the two-layered graph in the chemical model without suppressing hydrogen, and use the set of linear and quadratic descriptors (see Zhu et al. (2022a) for the details). We normalize the range of each linear descriptor and the range of property values by using the min–max normalization.

Among the above properties, we found that the median of the test coefficient of determination of the prediction function constructed by LLR (Zhu et al., 2022b) or ALR (Zhu et al., 2022c) exceeds 0.98 for the following nine properties of monomers (resp., three properties of polymers): EDPA, Hc, Ha, Hf, LhcL, LhcS, Mr, Vd, and U0 (resp., HcL, HcS, and MlV). We excluded the above properties from further analysis because they already achieved excellent predictive performance, and further improvement would not provide additional insights. The remaining 23 chemical properties of monomers and seven chemical properties of polymers were used in the following experiments.

4.3 Results on the first phase of the framework

For methods (i)–(iii), we use the same implementation described in Zhu et al. (2022b) and Zhu et al. (2022c) and omit the details.

In method (iv), we apply LLR-Reduce to compute of size 5000 if for a monomer property . In the other case, we set . Then, apply Select-Des-set to get , which is used to construct prediction functions based on MLR.

The computation times of finding and constructing functions by our method (iv) were in the ranges of seconds and seconds, respectively.

Tables 3, 4 show that method (iv) significantly increased the learning performance of several properties and achieved the best scores among methods (i)–(iii) for 43 of 47 data sets. We also noticed that most of the selected descriptors in are quadratics, which confirms the effectiveness of the proposed quadratic descriptors.

4.4 Results on the second phase of the framework

To execute the second phase, we used a set of seven instances , , , and based on the seed graphs prepared by Zhu et al. (2022b). We here present their seed graphs (see Zhu et al. (2022a)) for the details of , , , and ).

The seed graph of (resp., of ) is illustrated in Figure 2.

FIGURE 2

Instance is introduced to infer an intermediate graph , which preserves

- the core part of : CID 24822711 given in Figure 3A; and

FIGURE 3

- the frequencies of all edge-configurations of : CID 59170444 given in Figure 3B.The seed graph of this instance is depicted in gray in Figure 3A).

Instance has been introduced in order to infer a chemical graph such that

- is monocyclic (where the seed graph of is given by in Figure 2(i)); and

- the frequency vector of edge configurations in is a vector obtained by merging those of chemical graphs : CID 10076784 and : CID 44340250 in Figures 3C, D, respectively.

4.4.1 Solving an MILP for the inverse problem

We executed the stage of solving an MILP to infer a chemical graph for two properties {Bp, Dc}.

Figure 4A illustrates the chemical graph inferred from with of Bp in Table 5.

FIGURE 4

Figure 4B (resp., Figure 4C) illustrates the chemical graph inferred from with (resp., with ) of Dc in Table 6.

In this experiment, we prepared several different types of instances: instances and have restricted seed graphs, the other instances have abstract seed graphs, and instances and have restricted sets of fringe trees. From Tables 5, 6, we observe that an instance with many variables and constraints takes more running time than those with a smaller size in general. All instances in this experiment are solved in a few seconds to approximately 60 s with our MILP formulation.

4.4.2 Generating recombination solutions

Let be a chemical graph obtained by solving the MILP for the inverse problem. We here execute a stage of generating recombination solutions of such that .

We execute an algorithm for generating chemical isomers of up to 100 when the number of all chemical isomers exceeds 100. For this, we use a dynamic programming algorithm (Zhu et al., 2022b). The algorithm first decomposes into a set of acyclic chemical graphs. Next, it replaces each acyclic chemical graph with another acyclic chemical graph that admits the same feature vector as that of , and finally, it assembles the resulting acyclic chemical graphs into a chemical isomer of . The algorithm can compute a lower bound on the total number of all chemical isomers without generating all of them.

From Tables 5, 6, we observe the running time and the number of generated recombination solutions in this stage.

The chemical graph in , , and admits a large number of chemical isomers in some cases, where a lower bound -LB on the number of chemical isomers is derived without generating all of them. For the other instances, the running time for generating up to 100 target chemical graphs in this stage is less than 0.03 s. For some chemical graphs , the number of chemical isomers found by our algorithm was small. This is because some of the acyclic chemical graphs in the decomposition of have no alternative acyclic chemical graph other than the original one.

4.4.3 Generating neighbor solutions

Let be a chemical graph obtained by solving the MILP for the inverse problem. We executed a stage of generating neighbor solutions of .

We select an MILP for the inverse problem with a prediction function such that a solution of the MILP admits only two isomers in the stage of generating recombination solutions; that is, instance for property Bp with and instances , and for property Dc with .

In this experiment, we add to the MILP an additional set of two linear constraints on linear and quadratic descriptors as follows. For the two constraints, we use the prediction functions constructed by R-MLR for properties { Lp, Sl} with in Table 3.

Let denote the set of descriptors selected in the construction of prediction function for properties { Bp, Dc} with and { Lp, Sl} with in Table 3, and let { Bp, Dc} denote the union .

The branch-and-bound method for solving an MILP sometimes takes an extremely large execution time for the same size of instances. We introduce a time limit to bound an entire running time to skip such instances when testing the feasibility of neighbors in . From Table 7, we observe that some number of neighbor solutions of the solution to the MILP could be generated for each of the four instances.

5 Conclusion

In the framework of inferring chemical graphs, the descriptors of a prediction function were mainly defined to be the frequencies of local graph structures in the two-layered model, and such a definition was important to derive a compact MILP formulation for inferring a desired chemical graph. To improve the performance of prediction functions in the framework, this article introduced a multiplication of two of these descriptors as a new descriptor and examined the effectiveness of the new set of descriptors. For this, we designed a method for reducing the size of a descriptor set to not lose the learning performance in constructing prediction functions and gave a compact formulation to compute a product of two values in an MILP. From the results of our computational experiments, we observe that a prediction function constructed by our new method performs considerably better than the prediction functions constructed by the previous methods for several chemical properties. Furthermore, the modified MILP, with the computation of quadratic descriptors, was able to infer desired chemical graphs with up to 50 non-hydrogen atoms.

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