“Topological Index Calculator” A JavaScript application to introduce quantitative...
1
“Topological Index Calculator” A JavaScript application to introduce quantitative structure-property relationships (QSPR) in undergraduate organic chemistry Irvin J. Levy, Departments of Chemistry & Computer Science, Gordon College, Wenham, MA 01984, [email protected] Steven D. Granz, Departments of Mathematics & Computer Science, Gordon College Since the development of the Wiener Index, numerous topological indices have been described. These methods convert molecular structure to a mathematical representation (a chemical graph) and then define computations to be performed on the resulting graph. Statistical correlations between those results and physical properties serve as a predictive tool. In organic chemistry, students are taught the relationship between molecular structure and boiling point but generally do not investigate the phenomenon because tools to support the tedious calculations are lacking. We have developed a JavaScript program, "Topological Index Calculator," which computes key indices rapidly. Use of JavaScript benefits instructors who may wish to modify or extend the program's capabilities and students who may want to use the tool easily both in and out of the laboratory. With this program, students may work cooperatively to develop correlations between topological indices and physical properties of alkanes. Background: A topological index is a value that is dependent on the molecular structure of a molecule. They are used to approximate physical properties of molecules, such as the boiling point. To get a better understanding of how indices are used, we will examine how to calculate the Wiener Index of a molecule. Two very important graph-theoretical matrices are the adjacency matrix and the distance matrix. Both of these can be used to find the Weiner Index of a molecule. The adjacency matrix A of a labelled connected graph G with N vertices, is a square symmetric matrix of order N. It is defined as: A ij = 1; if vertices i and j are adjacent = 0; otherwise The distance matrix D of a labelled connected graph with N vertices, is a square symmetric matrix of order N. It is defined as: D ij = l ij ; if i ≠ j = 0; otherwise where l ij is the length of the shortest path (the distance) between the vertices i and j in G. The Wiener Index is defined as one-half the sum of the elements of the distance matrix. N N W = 1/2∑ ∑ D ij i=1 i=1 For example: What is the Wiener Index of 2,3-dimethylbutane? Adjacency Matrix: 2,3-dimethylbutane 0 1 0 0 0 0 1 0 1 0 1 0 A = 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 Distance Matrix: 2,3-dimethylbutane 0 1 2 3 2 3 1 0 1 2 1 2 D = 2 1 0 1 2 1 3 2 1 0 3 2 2 1 2 3 0 3 3 2 1 2 3 0 Wiener Index: 2,3-dimethylbutane 0 + 1 + 2 + 3 + 2 + 3 + 1 + 0 + 1 + 2 + 1 + 2 + 2 + 1 + 0 + 1 + 2 + 1 + 3 + 2 + 1 + 0 + 3 + 2 + 2 + 1 + 2 + 3 + 0 + 3 + 3 + 2 + 1 + 2 + 3 + 0 = 58 Wiener Index = 58 / 2 = 29 Results: Topological indices can be calculated quickly using the “Topological Index Calculator.” This information can easily be used to create an index equation by plotting the experimental boiling point vs. the index computed for a set of molecules and performing a linear regression analysis on the data. For example, data in the table below can be used to generate index equations for alkanes. N: Polarity: Wiener: Balaban: Odd-even: VDI: Harary: Randic ethane 2 0.0000 1.0000 1.0000 2.0000 1.0000 1.0000 1.000 propane 3 0.0000 4.0000 1.6330 3.5000 2.1634 2.2500 1.414 2-methylpropane 4 0.0000 9.0000 2.3238 4.5000 3.3677 3.7500 1.732 butane 4 1.000010.0000 1.9747 5.2222 2.9131 3.6111 1.914 2,2-dimethylpropane 5 0.000016.0000 3.0237 5.0000 4.5895 5.5000 2.000 2-methylbutane 5 2.000018.0000 2.5395 6.4444 3.8514 5.2222 2.270 pentane 5 2.000020.0000 2.1906 6.8194 3.4010 5.0347 2.414 2,2-dimethylbutane 6 3.000028.0000 3.1685 7.1667 4.8758 7.0833 2.560 2,3-dimethylbutane 6 4.000029.0000 2.9935 7.8889 4.6112 6.9444 2.642 2-methylpentane 6 3.000032.0000 2.6272 7.9167 4.1451 6.7083 2.770 3-methylpentane 6 4.000031.0000 2.7542 8.2639 4.1925 6.7569 2.808 hexane 6 3.000035.0000 2.3391 8.4967 3.7591 6.4983 2.914 . . . . . . . . . . . . . . . . . . 4-ethyl-2-methylhexane 9 8.000098.0000 3.3074 12.72285.092511.7364 4.201 3,5-dimethylheptane 9 8.0000 100.0000 3.2230 12.79225.084911.7017 4.201 2,5-dimethylheptane 9 7.0000 104.0000 3.0608 12.71945.056211.5958 4.163 2,6-dimethylheptane 9 6.0000 108.0000 2.9147 12.30615.029011.5003 4.125 2-methyloctane 9 6.0000 114.0000 2.7467 12.94334.728011.2633 4.270 3-methyloctane 9 7.0000 110.0000 2.8766 13.12474.754911.3540 4.308 4-methyloctane 9 7.0000 108.0000 2.9548 13.05534.747911.3887 4.308 3-ethylheptane 9 8.0000 104.0000 3.0922 13.23674.790311.4794 4.346 4-ethylheptane 9 8.0000 102.0000 3.1753 13.16724.788011.5142 4.346 nonane 9 6.0000 120.0000 2.5951 13.41204.443411.0289 4.414 2,2,3,3,4-pentamenthylpentane 10 15.0000 108.0000 4.4038 13.08336.690914.7917 4.193 Boiling Point vs. Wiene 0 50 100 150 200 250 300 350 400 450 500 0 20 40 60 80 100 120 140 160 180 Weiner Inde Boiling Point vs. Randi 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4 5 6 Randic Inde Index equations created for particular indices to predict approximate boiling point of molecules: N: BP = 177.38 ln(N) + 24.742 Average Error: 2.30% Polarity: BP = 10.16 (Polarity Index) + 323.6 Average Error: 4.63% Wiener: BP = 56.81 ln(Wiener Index) + 157.99 Average Error: 2.85% Balaban: BP = 25.684 (Balaban Index) + 324.95 Average Error: 7.69% Odd-Even: BP = 156.97 ln(Odd-Even Index) + 1.9792 Average Error: 4.53% Vertex Degree Distance: BP = 45.453 ln(VDD Index) + 313.74 Average Error: 8.95% Harary: BP = 15.036 (Harary Index) + 220.98 Average Error: 3.21% Expected vs. Calculated Boi with Wiener Index Equati 200 250 300 350 400 450 500 200 250 300 350 400 450 500 Calculated BP Expected vs. Calculated Boil with Randic Index Equati 200 250 300 350 400 450 500 200 250 300 350 400 450 500 Caculated BP Future Directions: use the tool to verify values found in the literaure develop new indices with better approximations of the boiling point • combine current indices with one another • develop unique index References: Cao, C. "Topological Indices Based on Vertex, Distance and Ring: On Boiling Points of Paraffins and Cycloalkanes." J. Chem. Inf. and Comp. Sci., 2001, 41, 4. Mihalic, Z. "A Graph-Theoretical Approach to Structure- Property Relationships." J. Chem. Educ. 1992, 69, 9. Trinajstic, N. Chemical Graph Theory. Vol II. Florida: CRC Press, 1983. Abstract: N: Polarity: Wiener: Balaban: Odd-even: VDI: Harary: Randic ethane 2 0.0000 1.0000 1.0000 2.0000 1.0000 1.0000 1.000 propane 3 0.0000 4.0000 1.6330 3.5000 2.1634 2.2500 1.414 2-methylpropane 4 0.0000 9.0000 2.3238 4.5000 3.3677 3.7500 1.732 butane 4 1.000010.0000 1.9747 5.2222 2.9131 3.6111 1.914 2,2-dimethylpropane 5 0.000016.0000 3.0237 5.0000 4.5895 5.5000 2.000 2-methylbutane 5 2.000018.0000 2.5395 6.4444 3.8514 5.2222 2.270 pentane 5 2.000020.0000 2.1906 6.8194 3.4010 5.0347 2.414 2,2-dimethylbutane 6 3.000028.0000 3.1685 7.1667 4.8758 7.0833 2.560 2,3-dimethylbutane 6 4.000029.0000 2.9935 7.8889 4.6112 6.9444 2.642 2-methylpentane 6 3.000032.0000 2.6272 7.9167 4.1451 6.7083 2.770 3-methylpentane 6 4.000031.0000 2.7542 8.2639 4.1925 6.7569 2.808 hexane 6 3.000035.0000 2.3391 8.4967 3.7591 6.4983 2.914 . . . . . . . . . . . . . . . . . . 4-ethyl-2-methylhexane 9 8.000098.0000 3.3074 12.72285.092511.7364 4.201 3,5-dimethylheptane 9 8.0000 100.0000 3.2230 12.79225.084911.7017 4.201 2,5-dimethylheptane 9 7.0000 104.0000 3.0608 12.71945.056211.5958 4.163 2,6-dimethylheptane 9 6.0000 108.0000 2.9147 12.30615.029011.5003 4.125 2-methyloctane 9 6.0000 114.0000 2.7467 12.94334.728011.2633 4.270 3-methyloctane 9 7.0000 110.0000 2.8766 13.12474.754911.3540 4.308 4-methyloctane 9 7.0000 108.0000 2.9548 13.05534.747911.3887 4.308 3-ethylheptane 9 8.0000 104.0000 3.0922 13.23674.790311.4794 4.346 4-ethylheptane 9 8.0000 102.0000 3.1753 13.16724.788011.5142 4.346 nonane 9 6.0000 120.0000 2.5951 13.41204.443411.0289 4.414 2,2,3,3,4-pentamenthylpentane 10 15.0000 108.0000 4.4038 13.08336.690914.7917 4.193 http://www.math-cs.gordon.edu/ courses/topo/
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Transcript of “Topological Index Calculator” A JavaScript application to introduce quantitative...
“Topological Index Calculator”A JavaScript application to introduce quantitative structure-property relationships (QSPR) in undergraduate organic chemistry
Irvin J. Levy, Departments of Chemistry & Computer Science, Gordon College, Wenham, MA 01984, [email protected] D. Granz, Departments of Mathematics & Computer Science, Gordon College
Since the development of the Wiener Index, numerous topological indices
have been described. These methods convert molecular structure to a
mathematical representation (a chemical graph) and then define
computations to be performed on the resulting graph. Statistical correlations
between those results and physical properties serve as a predictive tool.
In organic chemistry, students are taught the relationship between molecular
structure and boiling point but generally do not investigate the phenomenon
because tools to support the tedious calculations are lacking.
We have developed a JavaScript program, "Topological Index Calculator,"
which computes key indices rapidly. Use of JavaScript benefits instructors who
may wish to modify or extend the program's capabilities and students who may
want to use the tool easily both in and out of the laboratory. With this program,
students may work cooperatively to develop correlations between topological
indices and physical properties of alkanes.
Background:A topological index is a value that is dependent on the molecular structure of a molecule. They are used to approximate physical properties of molecules, such as the boiling point. To get a better understanding of how indices are used, we will examine how to calculate the Wiener Index of a molecule.
Two very important graph-theoretical matrices are the adjacency matrix and the distance matrix. Both of these can be used to find the Weiner Index of a molecule.
The adjacency matrix A of a labelled connected graph G with N vertices, is a square symmetric matrix of order N. It is defined as:
Aij
= 1; if vertices i and j are adjacent
= 0; otherwise
The distance matrix D of a labelled connected graph with N vertices, is a square symmetric matrix of order N. It is defined as:
Dij
= lij; if i ≠ j
= 0; otherwise
where lij is the length of the shortest path (the distance) between the vertices i and j in G.
The Wiener Index is defined as one-half the sum of the elements of the distance matrix.
N NW = 1/2∑ ∑ Dij
i=1 i=1
For example: What is the Wiener Index of 2,3-dimethylbutane?
Adjacency Matrix: 2,3-dimethylbutane
0 1 0 0 0 0 1 0 1 0 1 0 A = 0 1 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
Distance Matrix: 2,3-dimethylbutane
0 1 2 3 2 3 1 0 1 2 1 2 D = 2 1 0 1 2 1 3 2 1 0 3 2 2 1 2 3 0 3
3 2 1 2 3 0
Wiener Index: 2,3-dimethylbutane
0 + 1 + 2 + 3 + 2 + 3 +1 + 0 + 1 + 2 + 1 + 2 +2 + 1 + 0 + 1 + 2 + 1 +3 + 2 + 1 + 0 + 3 + 2 +2 + 1 + 2 + 3 + 0 + 3 +3 + 2 + 1 + 2 + 3 + 0 = 58
Wiener Index = 58 / 2 = 29
Results:Topological indices can be calculated quickly using the “Topological Index Calculator.” This information can easily be used to create an index equation by plotting the experimental boiling point vs. the index computed for a set of molecules and performing a linear regression analysis on the data. For example, data in the table below can be used to generate index equations for alkanes.
N: Polarity: Wiener: Balaban: Odd-even: VDI: Harary: Randic:ethane 2 0.0000 1.0000 1.0000 2.0000 1.0000 1.0000 1.0000propane 3 0.0000 4.0000 1.6330 3.5000 2.1634 2.2500 1.41422-methylpropane 4 0.0000 9.0000 2.3238 4.5000 3.3677 3.7500 1.7321butane 4 1.0000 10.0000 1.9747 5.2222 2.9131 3.6111 1.91422,2-dimethylpropane 5 0.0000 16.0000 3.0237 5.0000 4.5895 5.5000 2.00002-methylbutane 5 2.0000 18.0000 2.5395 6.4444 3.8514 5.2222 2.2701pentane 5 2.0000 20.0000 2.1906 6.8194 3.4010 5.0347 2.41422,2-dimethylbutane 6 3.0000 28.0000 3.1685 7.1667 4.8758 7.0833 2.56072,3-dimethylbutane 6 4.0000 29.0000 2.9935 7.8889 4.6112 6.9444 2.64272-methylpentane 6 3.0000 32.0000 2.6272 7.9167 4.1451 6.7083 2.77013-methylpentane 6 4.0000 31.0000 2.7542 8.2639 4.1925 6.7569 2.8081hexane 6 3.0000 35.0000 2.3391 8.4967 3.7591 6.4983 2.9142. . . . . . . . .. . . . . . . . .4-ethyl-2-methylhexane 9 8.0000 98.0000 3.3074 12.7228 5.0925 11.7364 4.20193,5-dimethylheptane 9 8.0000 100.0000 3.2230 12.7922 5.0849 11.7017 4.20192,5-dimethylheptane 9 7.0000 104.0000 3.0608 12.7194 5.0562 11.5958 4.16392,6-dimethylheptane 9 6.0000 108.0000 2.9147 12.3061 5.0290 11.5003 4.12592-methyloctane 9 6.0000 114.0000 2.7467 12.9433 4.7280 11.2633 4.27013-methyloctane 9 7.0000 110.0000 2.8766 13.1247 4.7549 11.3540 4.30814-methyloctane 9 7.0000 108.0000 2.9548 13.0553 4.7479 11.3887 4.30813-ethylheptane 9 8.0000 104.0000 3.0922 13.2367 4.7903 11.4794 4.34614-ethylheptane 9 8.0000 102.0000 3.1753 13.1672 4.7880 11.5142 4.3461nonane 9 6.0000 120.0000 2.5951 13.4120 4.4434 11.0289 4.41422,2,3,3,4-pentamenthylpentane 10 15.0000 108.0000 4.4038 13.0833 6.6909 14.7917 4.1934
Boiling Point vs. Wiener Index
0
50
100
150
200
250
300
350
400
450
500
0 20 40 60 80 100 120 140 160 180
Weiner Index
Boiling Point vs. Randic Index
0
50
100
150
200
250
300
350
400
450
500
0 1 2 3 4 5 6
Randic Index
Index equations created for particular indices to predict approximate boiling point of molecules:
N: BP = 177.38 ln(N) + 24.742 Average Error: 2.30%
Polarity: BP = 10.16 (Polarity Index) + 323.6 Average Error: 4.63%
Wiener: BP = 56.81 ln(Wiener Index) + 157.99 Average Error: 2.85%
Balaban: BP = 25.684 (Balaban Index) + 324.95 Average Error: 7.69%
Odd-Even: BP = 156.97 ln(Odd-Even Index) + 1.9792 Average Error:
4.53%
Vertex Degree Distance: BP = 45.453 ln(VDD Index) + 313.74 Average Error: 8.95%
Harary: BP = 15.036 (Harary Index) + 220.98 Average Error: 3.21%
Randic: BP = 184.73 ln(Randic Index) + 150.09 Average Error: 1.45%
Expected vs. Calculated Boiling Point with Wiener Index Equation
200
250
300
350
400
450
500
200 250 300 350 400 450 500
Calculated BP (K)
Expected vs. Calculated Boiling Point with Randic Index Equation
200
250
300
350
400
450
500
200 250 300 350 400 450 500
Caculated BP (K)
Future Directions:use the tool to verify values found in the literaure
develop new indices with better approximations of the boiling point
• combine current indices with one another• develop unique index
References: Cao, C. "Topological Indices Based on Vertex, Distance and Ring: On Boiling Points of Paraffins and Cycloalkanes." J. Chem. Inf. and Comp. Sci., 2001, 41, 4.
Mihalic, Z. "A Graph-Theoretical Approach to Structure-Property Relationships." J. Chem. Educ. 1992, 69, 9.
Trinajstic, N. Chemical Graph Theory. Vol II. Florida: CRC Press, 1983.
Abstract:
N: Polarity: Wiener: Balaban: Odd-even: VDI: Harary: Randic:ethane 2 0.0000 1.0000 1.0000 2.0000 1.0000 1.0000 1.0000propane 3 0.0000 4.0000 1.6330 3.5000 2.1634 2.2500 1.41422-methylpropane 4 0.0000 9.0000 2.3238 4.5000 3.3677 3.7500 1.7321butane 4 1.0000 10.0000 1.9747 5.2222 2.9131 3.6111 1.91422,2-dimethylpropane 5 0.0000 16.0000 3.0237 5.0000 4.5895 5.5000 2.00002-methylbutane 5 2.0000 18.0000 2.5395 6.4444 3.8514 5.2222 2.2701pentane 5 2.0000 20.0000 2.1906 6.8194 3.4010 5.0347 2.41422,2-dimethylbutane 6 3.0000 28.0000 3.1685 7.1667 4.8758 7.0833 2.56072,3-dimethylbutane 6 4.0000 29.0000 2.9935 7.8889 4.6112 6.9444 2.64272-methylpentane 6 3.0000 32.0000 2.6272 7.9167 4.1451 6.7083 2.77013-methylpentane 6 4.0000 31.0000 2.7542 8.2639 4.1925 6.7569 2.8081hexane 6 3.0000 35.0000 2.3391 8.4967 3.7591 6.4983 2.9142. . . . . . . . .. . . . . . . . .4-ethyl-2-methylhexane 9 8.0000 98.0000 3.3074 12.7228 5.0925 11.7364 4.20193,5-dimethylheptane 9 8.0000100.0000 3.2230 12.7922 5.0849 11.7017 4.20192,5-dimethylheptane 9 7.0000104.0000 3.0608 12.7194 5.0562 11.5958 4.16392,6-dimethylheptane 9 6.0000108.0000 2.9147 12.3061 5.0290 11.5003 4.12592-methyloctane 9 6.0000114.0000 2.7467 12.9433 4.7280 11.2633 4.27013-methyloctane 9 7.0000110.0000 2.8766 13.1247 4.7549 11.3540 4.30814-methyloctane 9 7.0000108.0000 2.9548 13.0553 4.7479 11.3887 4.30813-ethylheptane 9 8.0000104.0000 3.0922 13.2367 4.7903 11.4794 4.34614-ethylheptane 9 8.0000102.0000 3.1753 13.1672 4.7880 11.5142 4.3461nonane 9 6.0000120.0000 2.5951 13.4120 4.4434 11.0289 4.41422,2,3,3,4-pentamenthylpentane 10 15.0000108.0000 4.4038 13.0833 6.6909 14.7917 4.1934
http://www.math-cs.gordon.edu/courses/topo/
