Enumeration of Permutational Isomers: The Porphyrins
R. E. Tapscott and D. Marcovich University of New Mexico Albuquerque, NM 87131
R. L. C. Pilgrim has enumerated the isomers possihle for a limited numher of combinations of substituent groups on the eight pyrrole sites of a porphyrin ring and hasdisc&sed errors made in isomer countings for this system in the past ( 1 ) . However, Pilgrim's work, like that of the others discussed in his article, employs a brute-force, direct counting method for isomer enumeiaGon. Since this method can lead to errors (1,2) and is unfeasible for enumeration of isomers with more complex combinations of groups on the porphyrin ring, we wish to draw attention to another procedure, based on combiuatorial mathematical principles, and to employ it in counting porphyrin isomers. We will also indicate a method for simplifying the calculations involved. Combinatorics is a field of mathematics which "counts, enumerates, examines, and investigates the existence of configurations" (3). It is not surprising therefore that comhinatorics is anolicahle to the oroblem of isomer enumeration ( 4 ) , one of theLmostpowerful Eombinatorial techniques being that based on PBlya's theorem (5).This is the method we will employ to count the isomers obtainable for all possible combinations of substituents on a porphyrin ring. Since the application of P6lya's theorem to other systems has been discussed in the literature (6-9) onlv the essentials are oresented here.
3
7
essay
Isomers of Porphyrlns and Pblya's Theorem The isomers of the porphyrins can be considered to be permutational isomers-isomers which differ only by the permutation of groups (ligands) on a molecular framework (skeleton). The molecular skeleton for the porphyrins is shown in the figure, where the sites of attachment for the suhstituents are numbered 1-8. The ligands may he any groups, such as methyl, ethyl, propionic acid, hydrogen, etc., which can occupy the numbered pyrrolic positions. The actual numbers of the various ligands present correspond to a partition of the total numher of ligands. We can represent a partition of an integer d by pd = Id,, de da
. . .I
(1)
where xdi=d i
(2)
For the porphyrins, d = 8. A partition (dl, dz, d s . . .I can be used to represent a compound with d l suhstituents of the first type, d2 substituents of the second type, etc. For example, the partition 14,2,21 might represent the combination of ligands present in mesoporphyrins (4 methyl, 2 propionic acid, and 2 ethyl groups) or protoporphyrins (4 methyl, 2 propionic acid, and 2 vinvl . erouns). " .. The problem of permutational isomer counting is merely the prohlem of determining the numher of distinct arrangements of ligands on a skeleton for a particular combination of ligands. This problem becomes more tractable when one realizes that the numher of isomers is a function only of the unordered oartition and skeleton and not of the actual t . v.~ e s of ligands present (as long as they can he treated as point ligands). For porphyrins, we need only determine the numher for all possihle unordered partitions of eight. Generating functions are known for the partitions of an integer (3) and the numher of partitions possihle, N(pd), for selected integers, d, is given in Table 1. There are 22 partitions for d = 8. Now the total numher of ordered permutations of d letters 446 1 Journal of Chemical Education
Molecular skeleton of the porphyrins.
Table 1. Number of Unordered Partitions for Various Integers d 1 2 3 4 5 6 7 8 9 10 11 12 13 14
M P ~ 1 2 3 5 7 11 15 22 30 42 56 77 101 135
d
4P d )
15 16 17 18 19 20 30 40 50 60 70 80 90 100
176 231 297 385 490 627 5.604 37.338 204,226 966,467 4,087,968 15,796,476 56.634.173 190.569.292
with d l of the first kind, dz of the second kind, etc., is given by d! = d,!dl!di!.
..
(3)
This eouation. therefore. eives the total numher of lieand assignments tothe skeletksites for a partition {dl,d2, d3y . .I. Denendin~unon the svmmetrv of the molecular skeleton. hohever, not'all of these assigdments may be distinct since some may differ only by a rotation of the molecule. The basic problem of isomer enumeration is to list all of the distinct assienments without double counting. - PBlva's . theorem is one approach to this problem. If one writes down all of the n assignments of ligands to skeletal sites, one finds that they can be grouped into classes according to whether or not specific assignments are equivalent under the rotation operations possihle for the molecular skeleton. This is expressed in more precise terms by stating that we are counting the number of equivalence classes (or schemata) of assignments r$ relative to G, where G is the rotational suhgroup of the skeleton.' There is an equivalence relation between b1and b2if 41 = 4% grC (4) PBlya's theorem can he used to generate the numher of
schemata for all partitions simultaneously starting with a function Z(G) known as the cycle index. This is obtained from the cycle structures of the skeletal site permutations associated with G. The porphyrin skeleton has D4h symmetry with a rotational subgroup of D4. The point group operators, corresponding skeletal permutations, and the cycle structure of each are presented in Table 2. The cycle index is then given (7)by
Table 2.
Operations of D,, Permutations of Substitvent Sites, and Cycle Structures
Permutation
Operation
Cycle Structure
1
2 0 4 ) = -(fl+2fi + 5fd) (5) ID41 where lD41 is the order of the rotational subgroup (eight). The substitutions f i = ~ j + ~ i + c i + ~ i + ~ i + ~ i + ~ i + ~( 6i )
into eqn. (5) give polynomials which, upon expansion and collection of like terms, give a resultant polynomial whose coefficients are the number of schemata (isomers) for all possible partitions. For example, a term such as A2B2C4 represents the partition 12, 2, 4) and its coefficient gives the numher of isomers possible for this partition. Unfortunately, expansion of the polynomials obtained upon substituting eqn. (6) into eqn. (5) is unfeasible if carried out by hand using standard algebraic multiplication method^.^ For example, a complete expansion of the first polynomial (A + B + C D E F G HI8 involves the summation of as many as terms if like terms are not combined in the intermediate steps. Moreover, the final polynomial obtained in this step will contain 15!/8! = 6435 terms (the number of ways of taking 8 things 8 a t a time allowing for repetition). A computer program may be used to carry out the expansion ( 1 0 ) ; however, the following approach simplifies the problem sufficiently that a hand calculation becomes tractable. In eeneral. it is not necessarv to carrv alone all of the terms of tbc pdynomials since sweral terms may repres~mtidentical unordered oarritions. Thus. the exl~nnaionof I' riws 64:13 to thk'% different terms, but drily 22 of thes&orrespkding unordered oartitions of eieht-need he considered. In an expansion of any polynomiafi~ B C +. . .Id, the coefficient of any term corresponding to a partitionpd = Idl, d2. d3 . . .I is given by an expression identical to that of eqn. (3) since all ordered permutations of the d letters in the various terms are obtained. For example, the coefficients of all terms corresponding t o p s = 13,3,2)in f f (terms suchas A%3C2, C3D2E3 etc.) are given by 8!/3!3!2! = 560. This analysis permits the evaluation of the coefficients for all distinct partitions of fy, fi,and f: in eqn. (5) after substitution by eqn. (6). Table 3 gives the numher of isomers possible for all 22 partitions of eight ligands on the porphyrin ring. The results have been checked using the method of Lunn and Senior ( l l ) , who determine equivalence classes among assignments using a slightly different mathematical approach. The results also agree with those of Pilgrim ( I ) for porphyrins corresponding
Table 3. Number of Isomers Possible for Permutations of Eight Substituents on the Pyrrolic Positions of the Porphyrln Ring Number of Pd
11.1.1,1.1.1.1.1~ ~2.1.1.1.1.1.11 12,2.1.1.1,11 13,1.1,11.11 12.2.2.1.11
Number of
Isomers
Pd
Isomers
5040 2520 1260 640 630
13.3.21 14.2.21 (5.1.1.11 14.3.11 15.2.11
70 60 42 35 21
+ + + + +
+ +
One can use the full point group of the skeleton in order tomake assignments equivalent under improper rotations. This means, of course. that anv enantiomerie oairs will be counted as one isomer since
to the partitions 14,41 (etioporphyrins, coproporphyrins, and uroporphyrins) and 14, 2, 21 (protoporphyrins, deuteroporphyrins, mesoporphyrins, and hematoporphyrins). However, the count can now be extended to more complex porphyrins-either naturally occurring, such as the 14,2, 1, 1)chlorocruoroporphyrins (12) with 105 isomers and the 13,2,1, 1, 1)heme a prosthetic group of cytochrome a (13) with 420 isomers, or synthetic, as in the 14,2,1,11 myoglobin model of Chang and Traylor (14). Summary
The use of combinatorial methods permits the counting of permutational isomers in systems that are sufficiently complex that direct counting is tedious andlor likely to be in error. The use of P61ya9stheorem has been illustrated for the porphyrin system, where the isomers for all possible ligand partitions have been counted. We wish to thank the National Institutes of Health for financial suooort of research on isomerism in metal ion complexes of 'biological interest through grant numher GM22435. Llterature Cited Pilgrim, R. L.C., J. CHEM EDUC.,SI. 316119141. Blackman, D..J.CHRM.EOUT., 50,258 (19781. Berge, C,"Principlea of Combinatorics? Acsdemie Pies?, New York, 1971. Ruuvray. D, H., Chemistry, 45.6 119721. P6lya. G..Acla Moth.. 68, 145 (19371. Hi1l.T. F.,J, Chem Phys., 11.294 (19431. Kmnedy,B. A . McQuarrie, D. A,. and Riubaker. C. H.,lnaip. Chem.. 3,265 119641. Emel9nnuva.N. V.,and K~vurhoi,l.V..Zh.SLr~kt. Khim.. 9.881 (1R81and reforoncps
therein.
McDaniel. D. H..Inorg. Chem.. 11.2678 119721. J. CHRM. EDIIC., 49,479 11872). Lunn. A. C., and Senior, J. K.. J. P h w Chsni., 33.1027 (19291. Fa1k.l. E.."Purphyrinr and M~iallopsrphyrinr."Elsevier,New Yurk, 1964.p.102. Ref. I i Z l , p . 98. ChanhC. K., andTray1or.T. G.,Pioc. Nai. Acnd Sci. USA. 70,2M7 119711. TapscutL,R. E.,
Volume 55, Number 7, July 1978 1 447
