Mathematical model for determining octanol-water partition coeffi cient of alkanes

Transcription

1 Mathematical model for determining octanol-water partition coeffi cient of alkanes Jelena Sedlar University of Split, Croatia 2nd Croatian Combinatorial Days, Zagreb, Croatia September 27-28, 2018 Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

2 Introduction Chemical molecules are the subject of the study of chemistry and they consist of: atoms, bonds. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

3 Introduction Chemical molecules are the subject of the study of chemistry and they consist of: atoms, bonds. Graph theoretical model of a molecule is a graph in which: atoms are represented by vertices, bonds are represented by edges. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

4 Introduction The example of molecules represented by a graph can be: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

5 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

6 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

7 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

8 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

9 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

10 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

11 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

12 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

13 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

14 Introduction The example of molecules represented by a graph can be: isomers of pentane, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

15 Introduction The example of molecules represented by a graph can be: isomers of pentane, isomers of octanes, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

16 Introduction The example of molecules represented by a graph can be: isomers of pentane, isomers of octanes, isomers of nonanes,... Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

17 Introduction Chemical properties of the molecules which are of interest to people are various, for example: boiling point, melting point, heat capacities, density, heat of vaporization, enthalpy of formation, motor octane number, molar refraction, total surface area, octanol-water partition coeffi cient, molar volume, entropy, acentric factor. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

18 Introduction Chemical properties of molecules are usually presented in a table. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

19 Introduction Obtaining chemical properties of the molecules can be done: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

20 Introduction Obtaining chemical properties of the molecules can be done: experimentally in a laboratory - expensive; Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

21 Introduction Obtaining chemical properties of the molecules can be done: experimentally in a laboratory - expensive; mathematically - cheap. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

22 Introduction Obtaining chemical properties of the molecules can be done: experimentally in a laboratory - expensive; mathematically - cheap. Mathematical model for obtaining chemical properties is based on: deriving topological index of a graph which correlates well with the chemical property of a molecule. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

23 Introduction Obtaining chemical properties of the molecules can be done: experimentally in a laboratory - expensive; mathematically - cheap. Mathematical model for obtaining chemical properties is based on: deriving topological index of a graph which correlates well with the chemical property of a molecule. Wiener index of a graph is the first and maybe the best studied example, it is defined as W (G ) = u,v V (G ) d(u, v) and correlates well with a boiling point of paraffi nes. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

24 Introduction In 1972 a paper [1] was published, in which approximate formulas for the total π electron energy (E) were derived. [1] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

25 Introduction In 1972 a paper [1] was published, in which approximate formulas for the total π electron energy (E) were derived. [1] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) In the formulas for the total π electron energy the folowing terms ocured: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

26 Introduction In 1972 a paper [1] was published, in which approximate formulas for the total π electron energy (E) were derived. [1] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) In the formulas for the total π electron energy the folowing terms ocured: the first Zagreb index M 1 (G ) = deg(v) 2 v V (G ) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

27 Introduction In 1972 a paper [1] was published, in which approximate formulas for the total π electron energy (E) were derived. [1] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) In the formulas for the total π electron energy the folowing terms ocured: the first Zagreb index M 1 (G ) = deg(v) 2 v V (G ) the Forgotten index F (G ) = deg(v) 3 v V (G ) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

28 Introduction The predictive ability of the F index was tested in [2]: using a dataset of octane isomers, found at concurring with a suggestion of the International Academy of Mathematical Chemistry. [2] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

29 Introduction The predictive ability of the F index was tested in [2]: using a dataset of octane isomers, found at concurring with a suggestion of the International Academy of Mathematical Chemistry. [2] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) The result of the test showed that both F and M 1 have similar predictive ability, i.e. both: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

30 Introduction The predictive ability of the F index was tested in [2]: using a dataset of octane isomers, found at concurring with a suggestion of the International Academy of Mathematical Chemistry. [2] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) The result of the test showed that both F and M 1 have similar predictive ability, i.e. both: correlate well with entropy and acentric factor; Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

31 Introduction The predictive ability of the F index was tested in [2]: using a dataset of octane isomers, found at concurring with a suggestion of the International Academy of Mathematical Chemistry. [2] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) The result of the test showed that both F and M 1 have similar predictive ability, i.e. both: correlate well with entropy and acentric factor; correlate badly with all other properties. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

32 Introduction The example of correlation of F and M 1 with (log of) octanol-water partition coeffi cient is presented below (from [2]). Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

33 Introduction In order to improve the predictive ability of these indices, a simple linear model is devised (in [2]): M 1 + λf where λ ( 20 λ 20) is fitting parameter. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

34 Introduction In order to improve the predictive ability of these indices, a simple linear model is devised (in [2]): M 1 + λf where λ ( 20 λ 20) is fitting parameter. The model gave: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

35 Introduction In order to improve the predictive ability of these indices, a simple linear model is devised (in [2]): M 1 + λf where λ ( 20 λ 20) is fitting parameter. The model gave: insignificant improvement for all physico-chemical properties but the octanol-water partition coeffi cient; Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

36 Introduction In order to improve the predictive ability of these indices, a simple linear model is devised (in [2]): M 1 + λf where λ ( 20 λ 20) is fitting parameter. The model gave: insignificant improvement for all physico-chemical properties but the octanol-water partition coeffi cient; a major improvement for the octanol-water partition coeffi cient. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

37 Introduction The correlation coeffi cient is the best for λ = Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

38 Introduction The experimental and calculated values of (log of) the octanol-water partition coeffi cient for λ = Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

39 Introduction The questions that arise are the following: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

40 Introduction The questions that arise are the following: is M F a good model for (log of) the octanol-water partition coeffi cient of all alkanes (and not just octanes); Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

41 Introduction The questions that arise are the following: is M F a good model for (log of) the octanol-water partition coeffi cient of all alkanes (and not just octanes); can a coeffi cient λ = 0.14 be derived from the structure of the molecule? Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

42 The elimination of a free factor λ. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

43 The elimination of a free factor λ. Note that for octanes (n = 8) we have λ = 0.14 = Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

44 The elimination of a free factor λ. Note that for octanes (n = 8) we have λ = 0.14 = {n = 8} Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

45 The elimination of a free factor λ. Note that for octanes (n = 8) we have λ = 0.14 = {n = 8} 1 n 1 Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

46 The elimination of a free factor λ. Note that for octanes (n = 8) we have λ = 0.14 = {n = 8} 1 n 1 so maybe M 1 (G ) 1 n 1 F (G ) would be a more appropriate term for modeling the octanol-water partition coeffi cient. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

47 Lanzhou index Lzh(G ) of a graph G is defined by Lz(G ) = (n 1)(M 1 (G ) 1 F (G )) = n 1 = (n 1)M 1 (G ) F (G ). Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

48 Lanzhou index Lzh(G ) of a graph G is defined by Lz(G ) = (n 1)(M 1 (G ) 1 F (G )) = n 1 = (n 1)M 1 (G ) F (G ). Practical formula for Lanzhou index obtained by inserting the formulas for M 1 (G ) and F (G ) is Lz(G ) = d 2 (u)(n 1 d(u)) = u V Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

49 Lanzhou index Lzh(G ) of a graph G is defined by Lz(G ) = (n 1)(M 1 (G ) 1 F (G )) = n 1 = (n 1)M 1 (G ) F (G ). Practical formula for Lanzhou index obtained by inserting the formulas for M 1 (G ) and F (G ) is Lz(G ) = d 2 (u)(n 1 d(u)) = d 2 (u) d(u). u V u V Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

50 Lanzhou index Lzh(G ) of a graph G is defined by Lz(G ) = (n 1)(M 1 (G ) 1 F (G )) = n 1 = (n 1)M 1 (G ) F (G ). Practical formula for Lanzhou index obtained by inserting the formulas for M 1 (G ) and F (G ) is Lz(G ) = d 2 (u)(n 1 d(u)) = d 2 (u) d(u). u V u V We also define Lz(G ) = u V d 2 (u) d(u). Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

51 Justification of the introduced index. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

52 Justification of the introduced index. Note that: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

53 Justification of the introduced index. Note that: for octanes models M F and M 1 1 n 1 F are equivalent, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

54 Justification of the introduced index. Note that: for octanes models M F and M 1 1 n 1 F are equivalent, the performance has to be checked on another group of alkanes (nonanes, dekanes,...). Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

55 Justification of the introduced index. Note that: for octanes models M F and M 1 1 n 1 F are equivalent, the performance has to be checked on another group of alkanes (nonanes, dekanes,...). This was done for nonanes aaaand... Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

56 Justification of the introduced index. Note that: for octanes models M F and M 1 1 n 1 F are equivalent, the performance has to be checked on another group of alkanes (nonanes, dekanes,...). This was done for nonanes aaaand... Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

57 Remark. Now that the new index is chemically justified, we can gladly do the math work. :-) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

58 Remark. Now that the new index is chemically justified, we can gladly do the math work. :-) The math work is finding: extremal graphs, extremal trees, extremal unicyclic graphs,... Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

59 General graps Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

60 General graps Proposition. Let G be a graph on n vertices. Then 0 Lz(G ) 4 n(n 1)3 27 The left bound is obtained if and only if G is either complete or empty graph. The right bound is obtained if and only if n 1 (mod 3) and G is r regular with r = 2 3 (n 1). Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

61 General graps Proposition. Let G be a graph on n vertices. Then 0 Lz(G ) 4 n(n 1)3 27 The left bound is obtained if and only if G is either complete or empty graph. The right bound is obtained if and only if n 1 (mod 3) and G is r regular with r = 2 3 (n 1). Lz(G ) = u V d 2 (u)(n 1 d(u)) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

62 General graps Proposition. Let G be a graph on n vertices. Then 0 Lz(G ) 4 n(n 1)3 27 The left bound is obtained if and only if G is either complete or empty graph. The right bound is obtained if and only if n 1 (mod 3) and G is r regular with r = 2 3 (n 1). Lz(G ) = u V d 2 (u)(n 1 d(u)) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

63 Trees Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

64 Trees A double star S k,l is a tree obtained from K 2 by attaching k 1 leaves to one of its vertices and l 1 leaves to the other one. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

65 Trees A double star S k,l is a tree obtained from K 2 by attaching k 1 leaves to one of its vertices and l 1 leaves to the other one. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

66 Trees A double star S k,l is a tree obtained from K 2 by attaching k 1 leaves to one of its vertices and l 1 leaves to the other one. A balanced double star BDS(n) on n vertices is a double star in which the diference between k and l is the smallest possible, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

67 Trees A double star S k,l is a tree obtained from K 2 by attaching k 1 leaves to one of its vertices and l 1 leaves to the other one. A balanced double star BDS(n) on n vertices is a double star in which the diference between k and l is the smallest possible, hence BDS(n) = S n/2,n/2 or BDS(n) = S (n 1)/2,(n+1)/2. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

68 Trees A double star S k,l is a tree obtained from K 2 by attaching k 1 leaves to one of its vertices and l 1 leaves to the other one. A balanced double star BDS(n) on n vertices is a double star in which the diference between k and l is the smallest possible, hence BDS(n) = S n/2,n/2 or BDS(n) = S (n 1)/2,(n+1)/2. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

69 Trees Proposition. Let T n be a tree on n 15 vertices. Then Lz(S n ) Lz(T n ) Lz(BDS(n)). The lower bound is achieved if and only if T n = S n, while the upper bound is achieved if and only if T n = BDS(n). Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

70 Trees It remains to consider the trees with less than 15 vertices. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

71 Trees It remains to consider the trees with less than 15 vertices. We say that a tree T n = BDS(n) is excessive if Lz(T n ) > Lz(BDS(n)). Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

72 Trees It remains to consider the trees with less than 15 vertices. We say that a tree T n = BDS(n) is excessive if Lz(T n ) > Lz(BDS(n)). If Lz(Tn) Lz(BDS(n)), we say that T n is weakly excessive. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

73 Trees It remains to consider the trees with less than 15 vertices. We say that a tree T n = BDS(n) is excessive if Lz(T n ) > Lz(BDS(n)). If Lz(Tn) Lz(BDS(n)), we say that T n is weakly excessive. It can be proved that: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

74 Trees It remains to consider the trees with less than 15 vertices. We say that a tree T n = BDS(n) is excessive if Lz(T n ) > Lz(BDS(n)). If Lz(Tn) Lz(BDS(n)), we say that T n is weakly excessive. It can be proved that: number of excessive and of weakly excessive trees is finite, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

75 Trees It remains to consider the trees with less than 15 vertices. We say that a tree T n = BDS(n) is excessive if Lz(T n ) > Lz(BDS(n)). If Lz(Tn) Lz(BDS(n)), we say that T n is weakly excessive. It can be proved that: number of excessive and of weakly excessive trees is finite, there are no excessive nor weakly excessive trees on more than 14 vertices. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

76 Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

77 Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

78 Degree bound graphs Let Tn denote the set of all trees on n vertices with maximum degree at most. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

79 Degree bound graphs Let Tn denote the set of all trees on n vertices with maximum degree at most. Note that: Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

80 Degree bound graphs Let Tn denote the set of all trees on n vertices with maximum degree at most. Note that: = 3 is the smallest interesting case, Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

81 Degree bound graphs Let Tn denote the set of all trees on n vertices with maximum degree at most. Note that: = 3 is the smallest interesting case, for = 4 we obtain the class of chemical trees. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

82 Proposition. Let n 8 be an integer and T n T 3 n. Then 4n 2 18n + 20 Lz(T n ) 5n 2 27n (n 7) 1 ( 1)n. 2 The left inequality is sattisfied if and only if T n = P n. The right inequality is sattisfied for any tree without vertices of degree 2 if n is even, and for any tree with exactly one vertex of degree 2 if n is odd. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

83 Proposition. Let n 8 be an integer and T n T 4 n. Then 4n 2 18n + 20 Lz(T n ) 6n 2 + O(n). The left inequality is sattisfied if and only if T n = P n. The maximum value of Lz(T n ) is achieved for any tree having the largest possible number of vertices of degree 4 for a given n. Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

84 Proposition. Let n 8 be an integer and T n T 4 n. Then 4n 2 18n + 20 Lz(T n ) 6n 2 + O(n). The left inequality is sattisfied if and only if T n = P n. The maximum value of Lz(T n ) is achieved for any tree having the largest possible number of vertices of degree 4 for a given n. The details of the proofs can be seen in [3]. [3] D. Vukičević, Q. Li, J. Sedlar, T. Došlić: Lanzhou Index, MATCH Commun. Math. Comput. Chem. 80 (2018) Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27

85 In the end Thank you for your attention! Jelena Sedlar (Croatia) Mathematical model September 27-28, / 27