Chemical graph theory

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1 Chemical graph theory Jacob Kautzky MacMillan Group eting April, 8

2 Chemical graph theory What is graph theory? Historical uses of chemical graph theory Isomer counting Chemical bonding Kinetics Modern uses of chemical graph theory

3 What is a graph? A graph is defined as a non empty set of vertices and a set of edges edge face D = 4 vertex D = rder of the graph (N) = # of vertices Size of the graph (M) = # of edges Degree of a vertex (D) = # of edges incident with a vertex N Σ D(i) = M i= N M + F = Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

4 Walks, trails, paths, and cycles d = Walk - an alternate sequence of vertices and edges, begining and ending with a vertice pen - starts and ends at different vertices Closed - starts and ends at the same vertex walk Length (l) - number of occurence of edges in a walk Trail - a walk where all edges are distinct girth = Path - a walk where all vertices are distinct Distance (d) - the length of the shortest path between vertices Cycle - a path that starts and ends at the same vertex cycle Girth - length of the smalest cycle in a graph Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

5 Connected vs disconnected and simple, general, and multigraphs components (k) Connected graph- a path between any vertices Simple graph - a graph without loops or multiple edges between two vertices Disconnected graph loop multiple edges Multigraph - a graph that allows multiple edges between vertices General graph - a graph that allows multiple edges and loops Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

6 Forests and trees isolated vertice terminal bridge Forest - an acyclic graph Bridge - an edge that when removed disconnects a graph Tree - a connected acyclic graph ooted tree - a tree with a vertice distinguished in some fashion Types of trees Spanning tree - a graph subgraph that includes all vertices of a graph with the minimal number of edges to remain connected Chain Star Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

7 Types of graphs regular graphs A regular graph is a graph where every vertex has the same valency M =.5*N*D regular graph of degree regular graph of degree 4 regular graph of degree complete graph cycle A graph where each vertice has D = N Denoted by K N M = ( ) N A graph where each vertice has D = Denoted by C N A cycle is termed even/odd if N is even/odd Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

8 Vertex coloring Assigning colors to vertices so that no two adjacent vertices have the same color Chromatic number (k) - the minimal number of colors required to color a graph Bipartite graph - a graph where the vertice set can be split into V and V where every edge connects a vertice in V to one in V k = k = 5 K, K 4, k = 4 Complete bipartite graph K s,u - every vertex in V is joined to every vertex in V Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

9 Planar graphs A graph is planar if it can be drawn such that no edges intersect Two graphs are isomorphic if there exists a : mapping from one onto the other (i.e. they are identical, but drawn differently) Planar Not planar Identifying isomorphic graphs without sampling all N! mappings is challenging and remains a problem in graph theory Planar Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

10 Planar graphs A graph is planar if it can be drawn such that no edges intersect A graph is planar if it does not contain a subgraph homeomorphic to K 5 or K, Planar Not planar Two graphs are homoemorphic if they can be obtained from the same graph by inserting new vertices of valency into its edges All planar graphs are 4 - colorable Planar Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

11 Euler Circuit the Königsberg bridge problem Is it possible to cross every bridge exactly once and start and end at the same spot? Graphical Model Examples of Euler Circuits Examples of Euler Paths An Euler circuit is a path that starts and ends at the same vertices and traverses every edge exactly once nly occurs if every vertice is of even degree An Euler path is a path that traverses every edge exactly once, but doesn t start and end at the same vertice nly occurs if every vertice besides two are of even degree Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

Hamiltonian circuit the icosian game Is it possible to start at one vertice of a dodecahedron, travel to every other vertex on the polyhedra, and then return to the

Graphical Model Hamiltonian Circuit A circuit that visits every vertex exactly once A mathematical formula for if a circuit exists has yet to be developed contains a

12 Hamiltonian circuit the icosian game Is it possible to start at one vertice of a dodecahedron, travel to every other vertex on the polyhedra, and then return to the initial vertex without visiting any vertex besides the starting one twice? Graphical Model Hamiltonian Circuit A circuit that visits every vertex exactly once A mathematical formula for if a circuit exists has yet to be developed contains a Hamiltonian circuit contains a Hamiltonian cycle contains neither If the D(i) N/ for every vertex i, then a circuit exists Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

13 Digraphs, line graphs, and weighted graphs 4 Digraph 5 Weighted Graph Line graph Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

14 Graphs can be converted to matrices and polynomials Adjacency matrix Distance matrix Characteristic Polynomial P(G) = det xi - A Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

15 Applying graph theory to chemistry Molecular Structures Polymers Graph Theory Kinetics And many more Bonchev, D.; ouvray, D. H. Chemical Graph Theory: Introduction and Fundamentals; Gordon and Brach Science Publishers S.A.: New York, 99.

16 Chemical graph theory Historical uses of chemical graph theory Isomer counting Chemical bonding Kinetics Modern uses of chemical graph theory

17 Isomer counting How can we go about counting the number of isomers possible for a given carbon number in a systematic fashion? C = C = C = C = 4 C = 5 C = 6 C = 7

18 Isomer counting the Caylee approach ecursive approach Counted the number of centric and bicentric trees centric tree Developed methods for counting rooted and unrooted trees and then limited them to 4 vertices Sucessfully counted the nubmer of C-C alkanes ather tedious process bicentric tree Gave incorrect answers for C and C alkanes Cayley, A. ep. Br. Assoc. Adv. Sci., 875, 45, 57.

19 Isomer counting the Henze-Blair approach Counting the number of alcohols on acyclic alkanes Let T N be the number of alcohols of carbon N and p N, s N, and t N be the number of primary, secondary and tertiary alcohols respectively Then T N = p N + s N + t N primary alcohols p N = T N H N can be thought of as counting H N secondary alcohols s N = { T * T N- + T * T N T (N-)/ * T N/ T * T N- + T * T N T (N-)/ * T (N+)/ + (/) * T (N-)/ * [ + T (N-)/ ] if N = even if N = odd H H H N + Et N + + (N-)/ N/ Henze, H..; Blair, C. M. J. Am. Chem. Soc. 9, 5, 4.

20 Isomer counting the Henze-Blair approach tertiary alcohols A tertiary alochol can be imagined to arise from combining alkyl radicals i, j, and k with i, j, and k carbons respectively to the CH group H H H i j k i i k i i i All possible permutations where i > j > k All possible permutations where i = j and i + k = N - All possible permutations where i = j = k and i = N - t N = { Σ T i * T j * T k i > j > k; i + j + k = N - + / * Σ T i * ( + T i ) * T k i = j; i + j + k = N - + /6 * T i * (T i + ) * (T i + ) i = j = k; i + j + k = N - This approach has been expanded to deal with structural saturated hydrocarbons, unsaturated hydrocarbons, alkynes and even stereoisomeric alcohols Henze, H..; Blair, C. M. J. Am. Chem. Soc. 9, 5, 4.

21 Isomer counting the Pólya enumeration approach Takes into account symmetry operations 4 Produces a polynomial that allows for isomer enumeration 6 5 f and g are equivalent if and only if there a permutation ɑ such that ɑ(f ) = g Zyklenzeiger: Z(A) = A Σ s j r (ɑ ɑ A r r How many ways are there to substitute a benzene ring with an group X times? Point group: D 6 Symmetry peration Permutation Cycle index term ne identity E () () () (4) (5) (6) s Two 6-fold rotations +/- C 6 (456), (654) s 6 6 Two -fold rotations +/- C (5)(64), (5)(46) ne vertical -fold rotation C (4)(5)(6) s s Three in plane binary axes C that bisect carbons Three in plane binary axes C that bisect carbons (6)(5)(4), ()(6)(45), (4)()(56) ()(4)(6)(5), ()(6)(5)(4), ()(5)()(46) s s s Pólya, G. Acta Math. 97, 68, 97.

22 Isomer counting the Pólya enumeration approach Takes into account symmetry operations 4 Produces a polynomial that allows for isomer enumeration 6 5 f and g are equivalent if and only if there a permutation ɑ such that ɑ(f ) = g Zyklenzeiger: Z(A) = A Σ s j r (ɑ ɑ A r r How many ways are there to substitute a benzene ring with an group X times? Point group: D 6 Z (D 6 ) = / { } s 6 s s 4s s s 6 Cycle index term s 6 substituting s z y = ( + x y ) z s 6 s Z (D 6 ) = / [ ( + x) 6 + * ( + x) * ( + x ) + 4 * ( + x ) + * ( + x ) + * ( + x 6 )] Z (D 6 ) = + x + x + x + x 4 + x 5 + x 6 There are unique isomers with 4 substituents s s s s Pólya, G. Acta Math. 97, 68, 97.

23 Isomer counting the Pólya enumeration approach Takes into account symmetry operations 4 Produces a polynomial that allows for isomer enumeration 6 5 f and g are equivalent if and only if there a permutation ɑ such that ɑ(f ) = g Zyklenzeiger: Z(A) = A Σ s j r (ɑ ɑ A r r How many ways are there to substitute a benzene ring with an group X times? Point group: D 6 Z (D 6 ) = + x + x + x + x 4 + x 5 + x 6 Pólya, G. Acta Math. 97, 68, 97.

24 Isomer counting the Pólya enumeration approach Takes into account symmetry operations 4 Produces a polynomial that allows for isomer enumeration 6 5 f and g are equivalent if and only if there a permutation ɑ such that ɑ(f ) = g Zyklenzeiger: Z(A) = A Σ s j r (ɑ ɑ A r r How many ways are there to substitute a benzene ring with an group X times? Point group: D 6 This method has been extended to isotopic isomers, cyclic molecules, benzenoid hydrocarbons, porphyrins, chiral and achiral alkenes, ferrocenes, clusters, and inorganic structures, among others Pólya, G. Acta Math. 97, 68, 97.

25 Isomer counting the N-tuple code Assigns each tree a unique code Start from most substituted vertice Isomorphic compounds have the same code Have been used for generation and enumeration of acyclic graphs emove vertex and incident edges, then examine the subtrees Produce the lexicographically largest code {4,,,,,,} 4 {4,,,,,,} {4,,,,,,} {,,,,,,} {,,,,,,} {,,,,,,} {,,,,,,} {,,,,,,} {,,,,,,} Knop, J. V.; Müller, W..; Jeričević, Ž.; Trinajstić, N. J. Chem. Inf. Comput. Appl. Chem. 98,, 94.

26 Chemical graph theory Historical uses of chemical graph theory Isomer counting Chemical bonding Kinetics Modern uses of chemical graph theory

NM assignment of methyl peaks in large molecules via MAGMA NM spectroscopy can probe both structure and dynamics of biomolecules at atomic resolutions thyl-tsy can be utilized to study protein

27 NM assignment of methyl peaks in large molecules via MAGMA NM spectroscopy can probe both structure and dynamics of biomolecules at atomic resolutions thyl-tsy can be utilized to study protein complexes up to MDa in molecular weight A major challenge to this approach is the need to match resonances in the NM with specific atoms Generally obtained by either monitoring assignments from smaller in tact proteins or individually changing residues and observing the NM perturbations Would it be possible to use graph theory to simplify this process? thyl Assignment by Graph MAtching Pritišanac, I.; Degiacomi, M. T.; Alderson, T..; Carneiro, M. G.; AB, E.; Siegal, G.; Baldwin, A. J. J. Am. Chem. Soc. 7, 9, 95.

NM assignment of methyl peaks in large molecules via MAGMA e i a 5 9 f b d c h Leu Val Ile 6 4 8 observed interactions deliberate error PDB g 7 e:6 d:4

28 NM assignment of methyl peaks in large molecules via MAGMA e i a 5 9 f b d c h Leu Val Ile observed interactions deliberate error PDB g 7 e:6 d:4 i:9 a: b: f:5 c: h:8 g:7 Pritišanac, I.; Degiacomi, M. T.; Alderson, T..; Carneiro, M. G.; AB, E.; Siegal, G.; Baldwin, A. J. J. Am. Chem. Soc. 7, 9, 95.

29 NM assignment of methyl peaks in large molecules via MAGMA Pritišanac, I.; Degiacomi, M. T.; Alderson, T..; Carneiro, M. G.; AB, E.; Siegal, G.; Baldwin, A. J. J. Am. Chem. Soc. 7, 9, 95.

30 Kinetic Analysis with Prim s algorithm Minimum spanning tree a subset of edges that connect a graph together with the minimal possible weight Prim s algorithm finds minimum spanning by repeatedly adding the cheapest vertice to the spanning tree Yoshimura, T.; Maeda, S.; Taketsugu, T.; Sawamura, M.; Morokuma, K.; Mori, S. Chem. Sci. 7, 8, 4475.

31 Kinetic Analysis with Prim s algorithm H H S H S H NEt cat. [h(i)((s)-binap)l ] + THF, 6, 5 h, 97% ee NEt Yoshimura, T.; Maeda, S.; Taketsugu, T.; Sawamura, M.; Morokuma, K.; Mori, S. Chem. Sci. 7, 8, 4475.

32 Kinetic analysis with Prim s algorithm Dissociative Associative Bolded edges are part of the spanning tree Green paths represent earlier in the pathway and red later Yoshimura, T.; Maeda, S.; Taketsugu, T.; Sawamura, M.; Morokuma, K.; Mori, S. Chem. Sci. 7, 8, 4475.

33 Kinetic analysis with Prim s algorithm Dissociative Proposed mechanism Bolded edges are part of the spanning tree Green paths represent earlier in the pathway and red later Yoshimura, T.; Maeda, S.; Taketsugu, T.; Sawamura, M.; Morokuma, K.; Mori, S. Chem. Sci. 7, 8, 4475.

34 Can we use graph theory to predict reactions? High-throughput computer-based reaction predictions (HTP) can be used for de novo drug design, virtual chemical space exploration, and to predict if retrosynthesis disconnections are feasible Can we use this to predict reactions? Constructed a knowledge graph from 4.4 million reactants and 8. million binary reactions Vertices represent reactants and reaction conditions with edges connecting reactants to reaction conditions and other reactants Predict reactions, products and reaction conditions Correctly predicted the right product with 67.5% accuracy on 8, reactions that were withheld from the data set Able to predict the products and reaction conditions of novel reactions with decent efficiency Segler, M. H. S.; Waller, M. P.; Chem. Eur. J. 7,, 68.

35 Can we use graph theory to predict reactions? High-throughput computer-based reaction predictions (HTP) can be used for de novo drug design, virtual chemical space exploration, and to predict if retrosynthesis disconnections are feasible Can we use this to predict reactions? Constructed a knowledge graph from 4.4 million reactants and 8. million binary reactions eactants eaction Conditions Products Vertices represent reactants and reaction conditions with edges connecting reactants to reaction conditions and other reactants A 5 Predict reactions, products and reaction conditions? B 6 Correctly predicted the right product with 67.5% accuracy on 8, reactions that were withheld from the data set Able to predict the products and reaction conditions of novel reactions with decent efficiency 4 C 7 Segler, M. H. S.; Waller, M. P.; Chem. Eur. J. 7,, 68.

36 Can we use graph theory to predict reactions? Segler, M. H. S.; Waller, M. P.; Chem. Eur. J. 7,, 68.

37 Can we use graph theory to predict reactions? Ir(ppy) (dtbbpy)pf 6 N Ph N Ph CN C Et Ph C Et H Ph Pd(PPh ) 4 AcH Ph Ph Would it be possible to develop a reaction between these components? Segler, M. H. S.; Waller, M. P.; Chem. Eur. J. 7,, 68.

38 Can we use graph theory to predict reactions? Predicted N Ph H Ph Ir(ppy) (dtbbpy)pf 6 Pd(Ph ) 4, AcH, CN N Ph Ph Discovered N Ph H Ph Ir(ppy) (dtbbpy)pf 6 Pd(Ph ) 4, HC H, CN N Ph Ph Xuan, J.; Zeng, T.-T.; Feng, Z.-J.; Deng, Q.-H.; Chen, J.-.; Lu, L.-Q.; Xiao, J. X.; Alper, H. Angew. Chem. Int. Ed. 5, 54, 65 Segler, M. H. S.; Waller, M. P.; Chem. Eur. J. 7,, 68.

39 Applying graph theory to chemistry Molecular Structures Polymers Graph Theory Kinetics And many more

40 Questions

CHEMICAL GRAPH THEORY

CHEMICAL GRAPH THEORY SECOND EDITION Nenad Trinajstic, Ph.D. Professor of Chemistry The Rugjer Boskovic Institute Zagreb The Republic of Croatia CRC Press Boca Raton Ann Arbor London Tokyo TABLE OF CONTENTS