Modified Structural Constraints for Candidate Molecule Generation in

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Modified Structural Constraints for Candidate Molecule Generation in Computer Aided Molecular Design Xinyu Liu, Yuehong Zhao, Pengge Ning, Hongbin Cao, and Hao Wen Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04621 • Publication Date (Web): 02 May 2018 Downloaded from http://pubs.acs.org on May 5, 2018

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Industrial & Engineering Chemistry Research

Modified Structural Constraints for Candidate Molecule Generation in Computer Aided Molecular Design Xinyu Liu1, Yuehong Zhao1, 2Pengge Ning1, Hongbin Cao1 and Hao Wen1 1

State Key Laboratory of Multiphase Complex Systems, Division of Environmental Engineering and

Technology, Institute of Process Engineering, Chinese Academy of Sciences, No.1, 2nd North Street, ZhongGuanCun, Beijing 100190, China 2

University of Chinese Academy of Sciences, No. 19A, YuQuan Road, Beijing 100049, China

ABSTRACT Computer-aided molecular design (CAMD) has attracted much attention in the past 30 years. Generation of candidate molecular structure satisfying a set of structural constraints is an important part of such problem. However, the commonly used structural constraints proposed by Odele and Macchietto [Odele O. and S. Macchietto. Computer Aided Molecular Design: A Novel Method for Optimal Solvent Selection. Fluid Phase Equilib. 1993, 82, 47.], cannot provide sufficient and necessary conditions for generating cyclic molecules. In this paper, the sufficient and necessary conditions for molecular generation were presented. According to these conditions, some modifications of the conventional constraints were made, and mathematical proofs demonstrated that the modified constraints can perfectly generate structurally feasible molecules with no more than two rings. Moreover, some new constraints were proposed to help generate feasible aromatic

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molecules with less than two rings. Finally, several cases were presented to validate the correctness and applicability of the proposed modifications via comparing the results of the CAMD problems using the constraints before and after modification, and the new constraints were applied to design solvents for liquidliquid extraction. It is shown that, the modifications presented in this paper can improve the reliability of results of CAMD, and additionally, the conditions of molecular generation proposed in this paper can be easily used to derive structural constraints for any type of target molecules.

1. INTRODUCTION Computer-aided molecular design (CAMD) is a reverse engineering procedure to generate molecules that meet various physical property requirements. Throughout the years, many different CAMD methods have been developed, applied and extended in solving a wide range of chemical product design problems. One of the initial works in this area was carried out by Gani and Brignole1, they proposed an generate and test method to identify potential quality solvents for the separation of aromatic and paraffinic hydrocarbons. Later, Odele and Macchietto2 introduced a mixed-integer nonlinear programming (MINLP) technique to solve the CAMD problem, and reported applications in solvent selection for liquid-liquid extraction and multicomponent gas absorption. Duvedi and Achenie3 developed a MINLP model to design environmentally safe refrigerants, in which, ozone depletion potential (ODP) values were taken into account to exclude refrigerants which would deplete the earth's protective ozone layer. Maranas4 presented a systematic analysis framework for transforming a class of CAMD problems with nonlinear structure-property functionalities into equivalent mixed-integer linear programming (MILP) problems, and their framework was applied to design polymers. Karunanithi and Achenie5 presented a CAMD framework for crystallization solvent design, in which, decomposed method was introduced to solve the MINLP problem. Chavali and coworkers6 utilized CAMD method to design transition metal catalysts. Specifically, they used connectivity indices to build the structureproperty correlations, and solved the problem with both tabu search and outer approximation methods. Mehrkesh and Karunaithi7 developed a computer-aided ionic liquid design model to help design of novel ionic


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liquids as thermal fluids for solar energy storage. Struebing and coworkers8 developed a quantum mechanical CAMD (QM-CAMD) methodology to design solvents for accelerated reaction kinetics. Ten and coworkers9 integrated both safety and health aspects in their model and designed solvents for gas sweetening progress. Yamamoto and Tochigi10 used neural networks (NN) method to design foaming agents. Jonuzaj and coworkers11, 12 applied generalized disjunctive programming (GDP) method into CAMD to solve mixture design problems. Austin and Sahinidis13 developed a COSMO-based method which can design solvents for material separation as well as reaction rates optimization. In their proposed framework, group contribution method was used to predict the inputs of COSMO equations, and decomposed method was used to solve the computer-aided mixture design problem. Most of the published works were focused on developing more efficient algorithms and introducing different property prediction models to expand the applicability of CAMD, only a few papers talked about the generation of molecules. Commonly, the generation of molecular structure can be classified into two stages: (1) composition design, of which the result is a set of groups that can form target molecules; (2) structure design, of which the result is a complete molecular structure, including groups as well as the connection relationships between the groups. Early works of group-contribution-based CAMDs were mainly focused on composition design. Gani and Nielsen14 proposed a four-steps CAMD algorithm involving (1) preselecting of groups, (2) generating feasible molecular structures, (3) screening molecules satisfying property targets, and (4) rating the molecules by some performance indices or by solving a optimization problem to determine the best compound. In order to generate feasible molecular structures, they classified groups into different categories, and proposed the primary group connection rules for different group sets. Odele and Macchietto2 introduced a mathematical optimization model to design the solvents or solvent mixtures. In their paper, two basic rules that feasible molecules must obey were proposed: (a) the valency (number of free attachments) of a molecule must be zero, and (b) no two adjacent groups are linked by more than one bond. And they deduced widely used mathematical constraints to generate molecules along with the limitations on molecular size. However,


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connection details between groups are not considered in their work. Thus, the constraints may fail under some specific conditions. Sahinidis and Tawarmalani15 developed an algorithm that guarantees globally optimal solutions to the MINLP problem for CAMD, and identified a number of novel potential refrigerants through their method. Their paper also discussed the structural constraints proposed by Odele and Macchietto2 and proposed by Joback and Stephanopoulos16, in which they pointed out that both are not sufficient for eliminating all infeasible group combinations, and made some improvements to tighten the structural feasibility constraints of Joback and Stephanopoulos’ proposition16. Their work helps eliminate some infeasible group combinations which satisfy the previous constraints. However, there was no further mathematical justification that the new constraints are sufficient and necessary conditions for generating structurally feasible molecules. Accompanied by the development of various quantitative structure-property relationships (QSPRs), later works of CAMD were able to do structure design. Churi and Achenie17 developed a novel model that gives nearly completely information about molecular structure, in which, connection relationships between groups are considered, and this model was illustrated with a refrigerant design example. Chavali and lin6 represented the structure of target molecules with adjacency matrixes, in order to guarantee the feasibility of generated molecules, the constraints from a network flow problem were included in their paper. Samudra and Sahinidis18 proposed a three-stage optimization-based framework of CAMD: (1) composition design, (2) structure determination, and (3) extended design. In their work, composition identification and structure determination are decoupled to achieve computational efficiency. However, according to the results of case studies, many molecular compositions generated in the first stage were structurally infeasible, and these molecules cannot form feasible structures during the second stage of molecular design. Zhang and Cignitti19 proposed a new generic mathematical programming formulation for CAMD. Their work utilized Odele’s equations2 as structural feasibility constraints to generate group combinations, and applied Churi’s equations17 to obtain the connection relationships between different groups. Later in 2016, Austin and Sahinidis20 provided a slightly more general formulation of Odele’s equations2 for group contribution based


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CAMD. In their work, the generation of aromatic rings were considered by introducing the related constraints. However, some drawbacks exists when doing structure design in CAMD. First, structure duplications, different groups connected in different sequences may corresponds to the same molecule, which makes the solution space of molecular design problem increases greatly. Thus some researchers used Odele’s constraints in structure design to reduce the solution space and to enhance computational efficiency18,19. Second, because of the limitation of capability of QSPRs, CAMD cannot sufficiently predict the properties of isomers, even though they can be generated in structure design procedure. Therefore, most of the CAMD applications focus on composition design. And Odele’s proposition is still the most widely used structural constraints in CAMD applications5, 9, 19, 21, 22, 23, 24 because of its concision and efficiency. Odele’s structural constraints can be presented as eq 1 ~ 4.

1, for acyclic molecule  j 2  v j n j  2m, m  0, for monocyclic molecule  1, for bicyclic molecule 

n i j

i

 n j v j  2  2,

(1)

for all groups j

(2)

n lj  n j  n uj

(3)

nmin   n j  nmax

(4)

j l

u

Where, n j and v j are the number and valency of group j, n j and n j are the lower and upper bounds for the number of group j appearing in one molecule, nmax and nmin are the maximum and minimum number of structural groups allowed to make up a feasible structure. For acyclic molecules, Odele’s constraints were sufficient and necessary conditions, but eq 2 was redundant in this case. It can be demonstrated by graph theory, and a mathematical proof was performed in Appendix I. However, for cyclic molecules, these constraints may fail under some specific conditions. An example was shown in section 2.1 to illustrate this problem. To generate structurally feasible molecules, in this paper, the sufficient and necessary conditions for


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molecular generation were proposed, and modifications were made to the conventional Odele’s structural constraints. The new constraints were able to generate acyclic, monocyclic and bicyclic molecules simultaneously. In addition, some constraints were formulated considering the special chemically feasible structure of aromatic compounds. This paper was organized as follows: Section 2 presented the sufficient and necessary conditions and the modifications that were made to the commonly used structural constraints. Section 3 introduced three cases to test the validity of the new structural constraints, and in the third case, we designed extractants for de-phenol of wastewater. Section 4 provided some relevant conclusions.

2. IMPROVEMENTS OF ODELE’S STRUCTURAL CONSTRAINTS 2.1 Odele’s structural constraints According to Odele and Macchietto2, if a collection of groups is used to generate feasible molecules, two rules must be obeyed: (a) All groups connect into one molecule and the molecule has zero valency, (b) No two adjacent groups are linked by more than one bond. These two rules give the criterion for judging whether a molecule is structurally feasible or not. However, equations 1 ~ 4 may fail when generating some specific monocyclic or bicyclic molecules. Below is an example to illustrate this problem: The goal was to generate 4-group monocyclic or bicyclic molecules using groups included in set G = {CH3, CH2, CH, CCL, OH}. Because the number of groups used was small, all feasible group combinations can be enumerated manually, making it easy to examine the results. The outcomes were shown in Table 1. As we can see, 5 feasible monocyclic molecules existed, however, 7 monocyclic molecules were generated by Odele and Macchietto’s constraints, in which three infeasible group combinations (2 CH3, 1 CCl, 1 CH), (1 CH, 1 CCl, 2 OH) and (1 CH3, 1 CH, 1 CCl, 1 OH) were improperly generated. While, one feasible group combination (4 CH2, cyclobutane) was missed. For bicyclic molecules, two feasible molecules were missed, they were (2 CH2, 2 CH) and (2 CH2, 2 CCl). The structure of feasible molecules that cannot be generated by Odele and Macchietto’s constraints were shown in Figure 1.


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Table 1. Comparison of molecules generated through Odele’s structural constraints and molecules enumerated manually Molecule type

Enumerated

Generated (1 CH3 2 CH2 1 CH)

Monocyclic

(1 CH3 2 CH2 1 CH)

(2 CH2 1 CCl 1 OH)

(2 CH2 1 CCl 1 OH)

(1 CH3 2 CH2 1 CCl)

(1 CH3 2 CH2 1 CCl)

(2 CH2 1 CH 1 OH)

(2 CH2 1 CH 1 OH)

(2 CH3 1 CCl 1 CH)

(4 CH2)

(1 CH 1 CCl 2 OH) (1 CH3 1 CH 1 CCl 1 OH)

(2 CH2 2 CH)

Bicyclic

(2 CH2 2 CCl)

(2CH2 1 CCl 1CH)

(2CH2 1 CCl 1CH)

Monocyclic

Bicyclic

Figure 1. Structurally feasible molecules missed by Odele’s structural constraints Considering these failure situations mentioned above, we proposed the sufficient and necessary conditions for generating cyclic molecules with C rings: (a) All the groups generated are able to connect into a integral zero-valency structure:

 v j  2n j  2  2C

(5)

j

Where, n j and v j are the number and valency of group j, C is the number of rings in target molecule, and for acyclic molecules, C equals 0. (b) In the generated group combination, there exists groups that can form a simplest target structure with C rings. Condition (a) ensures all groups can connect into one structure as well as the valency of target molecule be zero, condition (b) ensures no two adjacent groups are linked by more than one bond. Which means, for


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example, if a monocyclic molecule is to be generated, there are at least three groups with more than 2 bonds exist in the group combination, because a cyclic structure must have at least 3 vertices (assuming each group represents a vertex), moreover, if a group combination satisfy condition (a) and there are more than three groups with 2 or more bonds, there must be a one ring feasible structure that can be formed by all the groups in the group combination. The demonstration of these two conditions was shown in Appendix III. Conditions (a) and (b) provided a convenient way to judge whether a set of groups can form a structural feasible molecules, and according to these conditions, one can easily derive the structure constraints for any type of target molecules with C rings. In the following, modified structural constraints were specially formulated for the generation of monocyclic, bicyclic and aromatic molecules.

2.2 Structural constraints for molecule with rings To illustrate new constraints, five sets were defined:

G  i i is the typeof groups used in CAMD G 2  i i  G and vi  2 G 3  i i  G and vi  3 G 4  i i  G and vi  4 G C  i i  G and ni  1 Where G is the initial group set used to generate molecules, i is the group used in the group set, and vi is the valency of group i . There is one thing to note, GC is a dynamic set, it represents the groups appeared in a generated group combination.

2.2.1 Structural constraints for monocyclic molecules As is known, a feasible group combination must obey that all the groups are able to connect into one molecule and the valency of this molecule is zero. And this can be guaranteed by the monocyclic form of eq 1:

 2  v n j

j

0

(6)

j


8

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To have a monocyclic molecule, it also needs to ensure that there is a ring in a molecule and the molecule generated satisfies condition (b), then the following linear constraint was added:

n

i

 3, i  G 2

(7)

i

These two equations are sufficient and necessary to generate structurally feasible monocyclic molecules. And the mathematical justification is shown in Appendix II.

2.2.2 Structural constraints for bicyclic molecules For a bicyclic molecule, it should include groups that can form typical simplest structure of target molecule as shown in Figure 2. Considering eq 1 ensures that all the groups can be connected together and determines the number of independent loops that be formed in the molecule. The important thing is to build the constraints to help generate molecules including groups that can form the simplest structure for bicyclic molecules. The finished constraints are listed below.

Figure 2. Simplest structure for bicyclic molecules

The bicyclic form of eq 1 is shown as below:

 2  v n j

j

 2

(8)

j

For the case shown in Figure 2a, constraints needed are

n

i

 2, i  G 3

(9)

n

 6, i  G 2

(10)

i

i

i

For the case shown in Figure 2b, constraints needed are


9


n

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i

 1, i  G 4

(11)

n

 5, i  G 2

(12)

i

i

i

For the case shown in Figure 2c, constraints needed are

n

i

 2, i  G 3

(13)

n

 4, i  G 2

(14)

i

i

i

Equations 8  14 provide sufficient and necessary conditions to generate bicyclic molecules, which can be proved similarly as that was proved for monocyclic molecules. However, these constraints are independent to each other. In order to develop general structural constraints that can unify all these three cases, we introduced other constraints that every structurally feasible molecule must satisfy:

 ni  v j  1,

i, j  G C

(15)

i

Equation 15 means that for every group appears in the molecule, the number of other groups existed is larger than the number of free attachments the group has, making sure that no two adjacent groups are linked by more than one bond. The left-hand side of equation 15 was a summation of the total number of groups in a generated molecule, and the right-hand side was specified for each group j appeared in the molecule. It can be found that, under the condition of eqs 8 and 15, equations 9  14 can be expressed via one equation as eq 16.

n

i

 4, i  G 2

(16)

i

In summary, to generate acyclic, monocyclic and bicyclic molecules simultaneously, the constraints needed are as follows:

1, for acyclic molecule  j 2  v j n j  2p , p  0, for monocyclic molecule  1, for bicyclic molecule 

(17)


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0, for acyclic molecule  i ni  q , i  G 2 , q  3, for monocyclic molecule , 4, for bicyclic molecule 

(18)

n

(19)

i

 v j  1, i, j  G C

i

nlj  n j  nuj

(20)

nmin   n j  nmax

(21)

j l

u

Where n j and v j are the number and valency, respectively, of groups of type j, n j and n j are the lower and upper bounds for the number of groups of type j appear in one molecule. nmax and nmin are the maximum and minimum number of groups allowed to make up a feasible molecule.

2.3 Structural constraints for aromatic compounds Though the constraints proposed above were valid for general molecules, it may fail when generating compounds with aromatic rings. Because all constraints above were formulated to regulate the total number of groups and free attachments of a molecule, connection relationships between groups were not considered. However, for an aromatic compound, aromatic groups must be connected to each other to form a 6-group ring. Therefore, more constraints were needed to make sure that the aromatic molecules generated were chemically feasible. In this section, only molecules having independent aromatic rings were considered, molecules with fused aromatic rings were ignored considering its complexity. To illustrate this problem, two more sets were defined:

G a  i i is the typeof aromatic groups used in CAMD G na  i i is the typeof non - aromatic groups used in CAMD According to the sufficient and necessary conditions provided in section 2.1, to generate a feasible aromatic molecule, two constraints need to be satisfied. First, the group combination should satisfy eq. 5. Second, there should be a sub set of groups that can form one of the simplest aromatic structure. Just like the


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constraints for bicyclic molecules, the key problem is to build the constraints to help generate group combinations that can form the simplest structure of aromatic molecules. Typical structure of aromatic compounds with less than two rings were shown in Figure 3.

Figure 3. Typical structure for aromatic molecules

Obviously, there are some common constraints that all typical aromatic structures should satisfy. First, each aromatic ring must have six aromatic groups,

n

i

 6nac , i  G ac

(22)

i

Where nac is the number of aromatic rings in the molecule. Then, the number of aromatic rings must less than the number of rings in the target molecule, thus:

nac  nc

(23)

Where nc is the total number of rings in the molecule. Similar to bicyclic molecules, for each type of aromatic molecules in figure 3, the structure constraints can be easily built. However, some differences should be noted, (1) if there are more than one rings or some nonaromatic groups appearing in an aromatic molecule, every aromatic ring should have free attachments in order to connect with other parts of the molecule; (2) if there are some non-aromatic groups appearing in an


12

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aromatic molecule, non-aromatic groups should not be connected into an aromatic ring. These 2 special cases were taken into account by controlling the number of aromatic groups that have more than 2 free bonds which can be used as bridges for aromatic rings to connect with other parts of the molecule. For example, to generate structure type c in Figure 3, we need at least two groups with more than three free bonds, one of them must be an aromatic group, and we need at least three non-aromatic groups with more than two free bonds to form a non-aromatic ring. This can be easily transformed into mathematical inequalities. Considering all the substructures shown in Figure 3, four independent sets of inequalities were built. and these inequalities can be integrated into equations 24 and 25:

 n  1.01n i

ac

 1  0.0001 n j , i  Ga , j  Gna , vi  3

i

(24)

j

3  ( ni  1)   n j   nk  3  100  ( p  1), i

j

i, j  G a , k  G na , vi  3,

k

v j  4, vk  2

(25)

Where p is the parameter representing the type of target molecule generated, and p = 1, 0 or -1 for acyclic, monocyclic and bicyclic molecules, respectively. The left-hand side of eq 24 is the number of groups that can provide attachments for aromatic rings to connect into the molecule, and the right-hand side of this equation is the number of attachments needed for the molecule generated. Equation 24 helped eliminate infeasible aromatic molecules which satisfy Odele’s constraints, the structure shown in Figure 4a is an example. Equation 25 was used to guarantee the feasibility of generated bicyclic aromatic molecules with chemical bridging or non-aromatic ring, and it also made sure that the non-aromatic ring has at least 3 groups. Therefore, all the generated molecules will satisfy rule (b) as described in section 2.1. An example of infeasible structures that can be eliminated by eq 25 was shown in Figure 4b. Besides, aromatic rings must exist if aromatic groups exist, and vice versa. It was described as eqs 26 and 27.

nac  exia i , i  G a

(26)


13


nac   exia i , i  G a

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(27)

i

Where exia i is a binary variable that represents whether group i exists or not in the generated molecule. And exia i  1 when aromatic group i exists, exia i  0 when no aromatic group i exists.

a

b

Figure 4. Examples of infeasible structures that can be eliminated by constraints for aromatic compounds generation

3. RESULTS AND DISCUSSION Three cases of molecular generation were displayed in this section. The first one was used to demonstrate the validity of the modified structurally feasible constraints for non-aromatic molecules generation. The second one was used to test the constraints for aromatic molecules generation. In the third one, a practical solvent design problem for liquid-liquid extraction to remove phenol in waste water was presented.

3.1 Test of modified constraints for non-aromatic molecules generation In our work, a C++ program was developed to enumerate all structural feasible molecules using a set of groups as building blocks. This program was based on idea of testing every possible group connection, which is time-consuming. Results were then used as base to test the correctness and reliability of group combinations generated using the constraints developed by Odele and the modified constraints proposed in this paper.


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In the test, a set of non-aromatic groups, G = {CH3, CH2, CH, C, OH, CH3OH, CHO, CH3COO, CH2COO, HCOO, CH3CO, CH2CO, HCO, COOH, COO}, were chosen to generate molecules. Except constraints of molecule generation, other limitations include: (a) each group appears less than twice in one molecule, and (b) the molecule contains less than 6 groups. To illustrate the differences between structural constraints before and after modification, and to validate the modified constraints, molecules were generated in three different approaches: I. Enumerated all the possible group combinations that have less than 6 groups, and tested their structural feasibility. II. Enumerated all group combinations subject to the conventional structure constraints (eqs 1, 2, 3 and 4) and tested their structural feasibility. III. Enumerated all group combinations subject to the modified constraints (eqs 17, 18, 19, 20 and 21) and tested their structural feasibility. The results were summarized in Table 2. It can be seen that the number of feasible molecules generated by approach I and III are same. However, according to the outcomes of approach II, there were 44 monocyclic structures and 386 bicyclic structures actually structurally feasible but did not satisfy the conventional constraints. At the same time, 112 structures that satisfied the conventional constraints were not structurally feasible molecules. It can be concluded that the modified structural constraints are more reliable than the conventional ones. They generated correct cyclic molecules and eliminate infeasible ones.

Table 2. Number of combinations and molecules generated in three different ways Structure type

Number of group combinations Approach I

Acyclic Monocyclic

−

Bicyclic Total

42128

Number of molecules generated

Approach II

Approach III

Approach I

Approach II

Approach III

5077

5077

5077

5077

5077

2265

2309

2309

2153

2309

406

792

792

406

792

7748

8178

8178

7636

8178


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3.2 Test of modified constraints for aromatic molecules generation In this test, a simple example for designing aromatic compounds was used to illustrate the validity of the constraints (eqs 24 and 25). The Goal was to generate 12-group aromatic molecules with less than two rings. The groups used for molecule generation was G = {CH3, CH2, AC, ACH, ACCH, ACCH2}. Given that functional groups ACCH and ACCH2 each can appear less than twice in one molecule, molecules were generated by two sets of constraints separately. The first set included eqs 17 to 21, and 22, 23, 26, 27. The second set included complete structure constraints, which are eqs 17 to 27. For the first set, 49 aromatic molecules were generated, in contrast, 47 aromatic molecules were generated subject to the set two. The difference is caused by eqs 24 and 25, which eliminate two infeasible group combinations showed as below. m = −1, nac = 2 11 ACH, 1 ACCH m = 0, nac = 1

6 ACH, 6 CH2

Possible corresponding structures of these group combinations were shown in Figure 5. Obviously, these two structures both were infeasible. Therefore, equations 24, 25 can effectively help to prevent generation of infeasible molecules as shown in Figure 5.

Figure 5. Possible infeasible structures that can be eliminated by eqs 24 and 25

3.3 Solvent molecule design for liquid-liquid extraction Finally, the modified structural constraints and the conventional structural constraints developed by Odele


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were used to design solvents for extracting phenol from waste water separately. The groups chosen for this CAMD problem was presented as set G = {CH3, CH2, CH, C, C=C, ACH, ACCH3, OH, CH3CO, CH2CO, CH3COO, CH3O, CH2O, CH−O, COOH, COO, ACOH}. Our goal was to design solvent with large distribution coefficient mpart for phenol, and acyclic, monocyclic and bicyclic compounds were all considered as potential candidates. The constraints on properties and molecule size for molecule design were given in Table 3. N G is the total number of group j, N F is the total number of functional groups. Tb and T f are the boiling point and fusing point at regular pressure, respectively. MW is the molar mass of target molecules, and phro is the relative density, for which water is 1000. The parameters regarding solvent performance include distribution coefficient mpart, solvent loss sl, solvent power sp, and selectivity β.

Table 3. Lower and upper bonds for specified needs Properties need

Lower

Upper

NG

2

8

NF

0

2

Tb

335 K

468 K

Tf

−

298 K

phro

−

950

MW

74

170

mpart

0.5

−

sl

−

0.005

sp

0.5

−

β

13

−

In order to get suitable groups for molecular generation, 30 commonly used extractants in industrial practice were collected based on literature review, the structure of each molecule was analyzed and split, thus, the initial groups were obtained. While lower and Upper bound of pure component properties were also identified via analyzing these commonly used extractants’ property, the range of properties regarding solvent performance were identified according to Song25.


17


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Detailed model for CAMD is listed as below.

Objective function

f obj  max mpart

(28)

Structural constraints

1, for acyclic molecule  j 2  v j n j  2p , p  0, for monocyclic molecule  1, for bicyclic molecule 

(29)

 ni  41  p ,

(30)

i  G2

i

n

 v j  1, i, j  G c

i

(31)

i

nil  ni  niu

(32)

2   ni  8

(33)

i

n

i

 6nac , i  G a

(34)

i

nac  1  p

n

i

(35)

 1.01nac  1  0.0001 ni , i  Gna , vi  3

i

(36)

i

3  ( ni  1)   n j   nk  3  100  ( m  1), i

j

i, j  G a , k  G na , vi  3,

k

v j  4, vk  2

(37)

nac  exia i , i  Ga

(38)

nac   exia i , i  Ga

(39)

i


18

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Property constraints

0  102.425 lg  niT f  298

(40)

i

335  102.425 lg  niTb  468

(41)

i

74   ni gwi  170

(42)

i

 n gw  0.00435   n V i

i

i

l

 950

(43)

i i

i

1

 S,B 1

 A, S

 0.005

(44)

 0.5

(45)

 A, B  MWB  0.5  A, S  MWS

(46)

 B, S  13  A, S

(47)

Where, n j and v j are the number and valency of groups of type j. In this case, S, A, B represent solvent, phenol, water, respectively. gwi is the group weight, Vi l is the liquid volume for group i, and

 i,j is the

infinite activity coefficient. Equation 30 is a modified expression of eq 18, which ensures that each ring in the target molecule has at least 4 groups. Equations 40 ~ 43 are group contribution equations defined in Cosntantinou and Gani2. The decomposed method proposed by Karunanithi and Achenie4 was used to solve this MINLP problem, and 1819 feasible molecules were generated at the rate of 0.19s per structure. Finally, 183 molecular structures satisfying all the constraints were obtained using the modified constraints, many commonly used solvents for phenol extraction appear in the design results, in which MIBK (groups: 2 CH3, 1 CH, 1 CH2, 1 CH3CO) ranked


19


Page 20 of 29

No. 41, n-butyl acetate (groups: 2 CH3, 3 CH2, 1 COO) ranked No. 49, cyclohexanone (groups: 4 CH2, 1 CH2CO) ranked No. 37. Some of the results were shown in Table 4. Table 4. some designed extractants for phenol extraction Possible structures

Number of rings

mpart

1

25.648

0

21.458

1

16.238

0

15.702

0

14.679

0

8.830

2

1.937

In order to illustrate the differences between conventional constraints and the modified constraints, molecules were designed again using eqs 1 ~ 4 as the basic structural constraints, besides we set the number of aromatic groups equals six times the number of aromatic rings, just like the constraints formulated by Austin and Sahinidis13 , and all other conditions were same. However, only 177 molecules were generated, Compared to the results of modified constraints, there were 8 feasible molecules that were incorrectly eliminated and 2 infeasible molecules that were incorrectly generated. Table 5 shows 8 feasible molecules that were incorrectly eliminated by the conventional structure constraints, of which cyclohexanone is one of the conventional compounds that are used to extract phenol in industrial practice. Table 6 shows 2 infeasible molecules that were incorrectly generated by the conventional constraints. This example shows that the conventional structural constraints may miss some useful candidate molecules, and the modified constraints


20

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can give more reliable results.

Table 5. Feasible molecules that were incorrectly eliminated by the conventional structure constraints Possible structures

Number of rings

mpart

1

25.744

1

22.367

1

16.238

1

13.025

1

12.600

1

10.892

1

10.115

1

10.064

4. CONCLUSION The structural constraints proposed by Odele and Macchietto2 are widely used in CAMD for generation of possible target molecules, and these constraints work well, especially for generating acyclic molecules. However, it may fail under some conditions, i.e., generating cyclic or aromatic molecules. This paper discusses the possible reasons why conventional constraints may lead to generate structurally infeasible molecules and


21


Page 22 of 29

eliminate feasible candidates. Then, the sufficient and necessary conditions to generate structurally feasible molecules were proposed, which provides a convenient way to derive structural constraints for any type of target molecules. According to these conditions, modified structure constraints for non-aromatic and aromatic molecules were derived, and mathematical proofs of the new constraints were carried out. To validate the correctness and applicability of the modified constraints, three cases of molecular generation were carried out. The results showed that, compared to the commonly used constraints proposed by Odele and Macchietto, the modified constraints proposed in this paper improved the reliability and correctness for CAMD problem, furthermore, it can help design feasible cyclic and aromatic molecules.

Table 6. Infeasible molecules that were incorrectly generated by the conventional constraints Group combinations

Number of rings

mpart

6 ACH, 1CH3, 1CH-O

1

2.578

1 ACCH3, 5 ACH, 1CH3, 1CH-O

1

2.174

AUTHOR INFORMATION Corresponding Author * E-mail: [email protected]. Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS We would be grateful for the financial support from the National Natural Science Foundation of China (Grant No. 2156112001) and Major Science and Technology Program for Water Pollution Control and Treatment of China (Grant No. 2015ZX07202-013).


22

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APPENDIX Appendix I: Justifications of Odeles’ constraints for acyclic molecules Odele’s constraints are presented in section 1 as eqs 1 ~ 4. For acyclic molecules, Odeles’ constraints are sufficient and necessary conditions to generate structurally feasible molecules. Nevertheless, just eqs 1, 3 and 4 can guarantee to generate the reasonable acyclic molecule, because eq 2 can be derived from eq 1: For m = 1, equation 1 become

 2  v n j

j

 2 , deform this equation, then subtract n j from both side,

j

 n  2  n v   n j

j j

j

j

j

j

 n   v 1n  v  2n  2 i

i

i j

i

j

j

i j

As vi  1 ,we get eq 2

 n  v i j

i

j

 2n j  2 for all group j

where i and j are different aliases for group types in the candidate group set. That means, for acyclic molecules, eq 1 is able to guarantee that all the groups connect into one zero valency molecule, also it makes sure that no two adjacent groups are linked by more than one bond. To demonstrate this problem, we introduce a variation of the well-known Euler equation in graph theory:

l  sen

(A1)

In this equation, l is the total number of in dependent loops, s is the number of “subgraphs”, e is the total number of edges, and n is the total number of vertices. Obviously, for the molecule structure generated:

e

1  n jv j 2 j

(A2)

n  nj

(A3)

j

Then we prove this from two aspects: First, necessity. The groups of an acyclic molecule must obey eq (1). For acyclic molecules, l = 0, s = 1,


23


Page 24 of 29

equation A1 becomes

0 1  e  n

(A4)

substitute e, n with eqs A2, A3, and do some transformation, we can get

 2  v n j

j

2

(A5)

j

which is the acyclic form of eq 1. Second, sufficiency. The group combinations satisfy eq 1 must be able to form a structurally feasible acyclic molecule. From eq 1, we can easily get that

n v j

j

is even, which is the sufficient and necessary condition

j

for this series of valences to be “graphic”. In other words, there must be a graph that v1 , v2 ,, vn  are the degrees for each vertex. Then Euler rule can be applied in this situation, apparently, s = 1. From eqs A1, A2, A3 and Eqn. A5, the acyclic form of eq 1, it is derived that l = 0, which means the corresponding graph for

v1 , v2 ,, vn  having no loops. Appendix II: Justifications of modified constraints for monocyclic molecules The modified constraints discussed here are displayed in section 2.2.1 as eqs 6 and 7. To illustrate this problem, we need to use eqs A1, A2 and A3 again. And the proof is performed as follows: First, necessity. Monocyclic molecules must satisfy eqs 6 and 7. Obviously, in a monocyclic molecule, there are at least three groups with more than two free attachments, which means eq 7 must be satisfied. Similar to acyclic molecules, given l = 1, s = 1, it is easy to get eq 6 from eqs A1, A2 and A3. Second, sufficiency. Group combination satisfying eqs 6 and 7 must be able to form a structurally feasible monocyclic molecule. Simplify eqs A1, A2, A3 and 6, eliminate n j and v j , we get

ls

(A6)

To generate a molecule, we may first connect all the groups with more than 2 attachments into a loop, and two bonds are used of each group, then for all the one-attachment group left, including the circumstance that this molecule does not have one-attachment group, there are two cases:


24

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All groups can be connected into the loop. Because the sum of valences for all groups is even, every attachment should be connected. In this case, s = 1, l = 1, only one independent loop exists, which has more than three groups, and this ensures that no two adjacent groups are linked by more than one bond, a feasible molecule is generated. There are some groups that can’t be connected into the loop, then, these one-attachment groups connect to each other in pairs. In this case, s > 1, however, l = 1, l < s. Apparently, it conflict with eq A6, and this case does not exist. In short, equations 6 and 7 are necessary and sufficient conditions to generate structurally feasible monocyclic molecules.

Appendix III: Justifications of sufficient and necessary conditions for generating molecules with C rings The sufficient and necessary conditions to generate cyclic molecules with C rings are presented in section 2.1, it contains two sub-conditions: Ⅰ.

 v j  2n j  2  2C

(A7)

j

Where, n j and v j are the number and valency of group j, C is the number of rings in target molecule. Ⅱ. In the group combination generated, there exists groups that can form a simplest target structure with C rings. First, necessity. Molecules with C rings must satisfy conditions Ⅰand Ⅱ. Obviously, for a feasible target molecule that have C rings, it satisfies condition Ⅱ. And similar to acyclic molecules, given l = C, s = 1, it is easy to get eq A7 from eqs A1, A2 and A3. Second, sufficiency. Group combination satisfying conditions Ⅰand Ⅱ must be able to form a structurally feasible target molecule with C rings. Simplify eqs A1, A2, A3 and A7, eliminate n j and v j , we get


25


l  s  C 1

Page 26 of 29

(A8)

Because these groups satisfy condition Ⅱ, to generate a molecule, we may first connect all the groups with more than 2 attachments into a C loop target structure with free attachments, then for all the oneattachment group left, including the circumstance that this molecule does not have one-attachment group, there are two cases: All groups can be connected into the structure. Because the sum of valences for all groups is even, every attachment should be connected. In this case, s = 1, l = C, this structure generated has C independent loops, each loop has more than three groups, and no two adjacent groups are linked by more than one bond, it is a feasible molecule. There are some groups that can’t be connected into the loop, then, these one-attachment groups connect to each other in pairs. In this case, s > 1, however, l = C, l < s + C - 1. Apparently, it conflict with eq A8, and this case does not exist. In summarize, conditions Ⅰand Ⅱare necessary and sufficient conditions to generate structurally feasible molecules with C rings.

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(18) Samudra A P, and Sahinidis N V, Optimization-Based Framework for Computer-Aided Molecular Design, AIChE J. 2013, 59, 3686 (19) Zhang L, Cignitti S and Gani R. Generic Mathematical Programming Formulation and Solution for Computer-Aided molecular Design. Comput Chem Eng. 2015, 78, 79. (20) Austin N D, Sahinidis N V and Trahan D W. Computer-aided molecular design: An introduction and review of tools, applications, and solution techniques. Chemical Engineering Research and Design. 2016, 116: 2. (21) Karunanithi A T, Achenie L E K and Gani R. A New Decomposition-Based Computer-aided Molecular/Mixture Design Methodology for the Design of Optimal Solvents and Solvent Mixtures. Ind Eng Chem Res. 2005, 44, 4785. (22) Buxton A, Livingston A G and Pistikopoulos E N. Optimal Design of Solvent Blends for Environmental Impact Minimization. AIChE J. 1999, 45, 817. (23) Karunanithi A T, Achenie L E K and Gani R. A Computer-aided Molecular Design Framework for Crystallization Solvent Design. Chem Eng Sci. 2006, 61, 1247. (24) Cignitti S, Zhang L and Gani R. Computer-aided Framework for Design of Pure, Mixed and Blended Products. 12th International Symposium on Process Systems Engineering and 25th European Synposium on Computer Aided Process Engineering. Copenhagen, Denmark. 2015. (25) Song J. Computer-aided molecular design of environmental friendly solvents for separation process. [D], Tianjin University, Tianjin, 2008.


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Table of Contents


29

Modified Structural Constraints for Candidate Molecule Generation in

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