predominate and may contribute 11 to of the total. The percentages of ion current due to similar reactions in the thiopropanoates, thiobutanoates, and thiopentanoates are not listed in Tables I to IV. I n these cases, the secondary breakdown is unimportant and frequently cannot' be dist'inguished from isobaric ions from R2. LITERATURE CITED
(1) ASThI Committee E-14 File of Uncertified Mass Spectra, A. H. Struck, chairman, Perkin-Elmer Corp., Sorwalk, Conn. (2) Beynon, J. H., "Mass Spectrometry
and Its Applications to Organic Chemistry," p. 382, Elsevier,New York, 1960. (3) Black, D. R., McFadden, W. H., Corse, J. W., J . Phys. Chem. 68, 1237 (1964). ( 4 ) Buttery, R. G., Black, D. R., Kealy, Mary P., J . Chromatog., i n press. (5) Buttery, R. G., Black, I). R., Kealy, Mary P., McFadden. W. H.. Xature 202, 701 (1964). (6) Dinh-Nguyen, Xg, Ryhage, R., Stallberg-Stenhagen, S., Stenhagen, E., Arkiv Kemi 18, 393 (1961). (7) Friedel, R. A., Shultz, J. L., Sharkey, A. G., Jr., ANAL.CHEM.28, 926 (1956). (8) Levy, E. J., Stahl, W. A., Ibid., 33, 707 (1961). (9) Ralls, J. W., lIcFadden, W. H., Seifert, R . M., Black, D. R., Kilpatrick, P. W., J . Food Sci., in press.
(10) Ryhage, R., Stenhagen, E., Arkiv Kemi 13, 523 (1959). (11) Sharkey, A. G., Jr., Shultz, Janet L., Friedel, R. A,, AKAL. CHEM.31, 87 11957). (12) Wenzel, R. W.,Jr., Reid, E. E., J . A m . Chem. Soc. 59, 1089 (1937). RECEIVED for review November 16, 1964. Accepted January 18, 1965. Western Regional Research Laboratory is a laboratory of the Western Utilization Research and Development Division, Agricultural Research Services, U. S. Department of Agriciilture. Reference to a company or product name does not imply approval or recommendation of the prodnct by the U.S. Department of Agriculture t o the exclusion of others that may be suitable.
Reactivity of Blue Tetrazolium with Nonketol Compounds JOSEPH E. SlNSHElMER and EDWARD F. SALIM' College of Pharmacy, The University of Michigan, Ann Arbor, Mich.
b The scope and reactivity of the reduction of blue tetrazolium by polyhydric phenols, quinones, thiols, and active hydrogen compounds have been examined. The similarity of structural reactivity relationships in these reactions to reactions of the same compounds with oxygen suggests an anion intermediate and a free radical mechanism for tetrazolium reductions by nonketol compounds. Base strength, common ion, and solvent effect studies confirm the importance of the anion intermediate. Oxidative cleavage of the quinone ring is advanced as an explanation of high sensitivity of quinones and hydroquinones in the reduction of tetrazolium salts. Many of the compounds studied are sufficiently reactive to permit the extension of quantitative tetrazolium methods developed for ketol compounds to the determination of nonketol reducing agents.
T
ETRAZOLIUM SALTS have found extensive use for the determination of steroid ketols (7) and reducing sugars (9). Model ketol compounds have been used to investigate the scope of this reaction ( 8 ) . However, the reactivity of tetrazolium in nonketol reducing systems has received only limited study. Rosenkrantz (1.4) has reported the reactivity of some phenols, 5-hydroxyindoles, and catecholamines with blue tetrazolium and also has compared (15)
Present address, Drug Standards Laboratory, American Pharmaceutical Association, Washington, D. C 566
ANALYTICAL CHEMISTRY
the react.ivity of a-tocopherol, a-tocopherylhydroquinone, and a-tocopherylquinone. The reduction of tetrazolium salts as a qualitat'ive test for polyhydroxy phenols and thiols has also been reported ( 1 ) . The utility of nonketol reductions of tetrazolium salts for the analysis of pharmaceutical compounds has been discussed in a previous paper from these laboratories ( 1 7 ) . It is convenient to classify such reductions into four groups -polyhydroxy compounds, quinones, thiols, and act'ive hydrogen compounds. The purpose of this investigation is to study representat'ive model compounds to evaluate the significance of tetrazolium salts for the analysis of nonket,ol compounds as well as to gain an understanding of the mechanism of reaction and the abilit'y to predict reactivity with tet'razolium salts. Reactions of act'ive hydrogen compounds (261, t'hiols (19), and hydroquinones (5, 21) in basic solutions with oxygen appeared to vary with the st,ructure of the reducing agents in a manner similar to t.he reactions of nonketol compounds with blue tetrazolium. This prompted a comparison of tetrazolium reactions to auto-osidat,ions as well as a study of such fact,ors as base strength, common ion effect, and solvent effect. which influence the formation of anions in tetrazolium reaction niixt,ures. EXPERIMENTAL
Spectrophotometric measurements were made with a Reckman Model DU spectrophotometer equipped with Bakelite thermospacers.
All calculated absorbance values for the forniazan produced are based on concentrations of reducing agent as represented in the total reaction mixture. Reagents. Reagents and test compounds mere of C.P. grade or solids were recrystallized to constant melting point and liquids fractionally distilled under nitrogen. Blue Tetrazolium Solution. Twenty five milligrams of purified (8) blue tetrazolium (Dajac Laboratories) was dissolved in 10 ml. of ethanol. Basic Solutions. I n general, the basic solution consisted of 10 ml. of 10% aqueous tetramethylammonium hydroxide solution which had been diluted to 100 ml. with absolute ethanol. Where required, alkali metal hydroxide solutions were prepared by adding .01 equivalent weight of alkali metal to 100 ml. of absolute ethanol which had been distilled over sodium. General Procedure. From 10 to 400 pg. of test compound were dissolved in 10 ml. of absolute ethanol and allowed to equilibrate to room temperature for the anion studies or to 30.5 i 0.1" C. for the rate of color development studies. Two milliliters of blue tetrazolium reagent solution were added, followed by 2 ml. of basic solution. -\bsorbances a t fixed intervals compared to reagent blanks were determined a t 530 mp. Reactions nere carried out in the thermostatically controlled cell compartment of the spectrophotometer at 25.0 + 0.1" C. for the anion studies and at 30.0 f 0.1" C. for the color development studies. Where necessary, the reaction could be quenched with 1 ml. of glacial acetic acid. Dimethylformamide Procedure. About 10 mg. of test compound,
Mechanism of Nonketol Reduction Py,wotechol
Hydroquinone
of Tetrazolium Salts.
Initial studies with blue tetrazolium indicated t h a t ,700 active hydrogen compounds containing a single strong electron-withdraw,600 - . P-B.nroq":none ing group produced reduction of the ,___-------t -Butylhydroquinone salt. Inductive effects are of im/ ,500 I,(-Dihydro"yn.phfh.I.n. portance. -1 series of alkyl-substiy tuted acetophenones and a second 2 series of p - dimethylaminopropio400 phenones (18) as outlined in Table I1 300 1.4 Nmphihoquinonr showed a n increase in tetrazolium reduction with electron-withdrawing subTatram.lhy1.D bcnroquinone stituents and a decrease with electron contributing substituents over the parent compounds. I n addition, because 2-phenylacetophenone (Table I) reduces blue tetrazolium, 2-methyl-2-phenyl0 20 40 60 60 100 120 acetophenone containing an additional TIME [MINUTES\ electron contributing group was synFigure 2. Rate of color developthesized ( 1 1 ) and tested for reaction ment for 1 X 10-jM solutions of with blue tetrazolium. As anticipated, typical hydroquinones and quinones the absorbance for 2-methyl-2-phenylwith blue tetrazolium acetophenone (Table I) was less than that of the parent compound. ing group would increase ionization of From these studies, it was predicted the compound and allow for greater that compounds such as benzoylacetoreactivity. 'These compounds, however, nitrile, 1,3-dipheny1-1,3-propanedione, as well as others listed in Table I and ethyl acetoacetate would be more which cont'ain electron-withdrawing reactive to tetrazolium salts. The groups were poorly reactive when treated with blue tetrazolium. addition of a second electron-withdraw-
------------~
I-Chloropyrototcchol
1
TIME [MINUTES1
Figure 1 . Rate of color development for 1 X 10-5M solutions of model compounds with blue tetrazolium
accurately weighed, were dissolved in 10 ml. of dimethylformamide (DMF). Two milliliters of blue tetrazolium reagent solution were added followed by 2 ml. of 0 . 1 s sodium ethoxide solution. T h e reaction was allowed to proceed for 40 minutes a t 2 5 O C. and quenched by the addition of 1 nil. of glacial acetic acid. The absorbance compared to a reagent blank was measured a t 530 mp. Those compounds producing less color than the blank were measured by comparing absorbance of the reagent blank to the sample as the reference solution. RESULTS AND DISCUSSION
Color development for model compound systems was compared using the general procedure with tetramethyiammonium hydroxide and a t 30" C. Absorbance values for these compounds were calculated on the basis of the color which would be produced by a molar concentration of reducing agent in the total reaction mixture. T h e absorbance values a t 60 minutes calculated in this manner are reported in Table I and serve as a comparison of the reactivity of the model compounds. Figures 1 and 2 show the relationship of color development with time for typical compounds. Most aliphatic thiols, catechols, and hydroquinones in this and a previous study (1'7) reached and maintained a maximum plateau of color within 2 hours, while t'he aromatic thiols and active hydrogen compounds generally continued to develop color. Two escept,ions to these patterns were diphenylacet'onitrile and phenylacetonitrile (Table I) in which formazan production reached a niaximum followed hy formation of colorless products. M o d e l Compounds.
Table 1.
Absorbances for Molar Solutions with Blue Tetrazolium a t
Compound Absorbance Active hydrogen compounds Acetonitrile 0.04 Acetophenonea 5 Benzoylacetonitrile 23 2-Benzylpyridine 2 Cy rlo hexanone 29 Cycloserine 376 Diethy lmalonate 19 Diethyl phenylmalonate 416 I>iphenylacetonitrile* 7710 trans- 1,4-Diphenyl-2butene- 1,Cdione 8060 1 Diphenylmethane 1,3-Dipheny1-1,3-propane31 dione 1,3-Dipheny1-2-propanone 3990 18 Ethyl acetoacetate 44 Ethyl benzoylacetate 615 Ethyl cyanoacetate 1180 Malononitrile 1320 ALIesityl oxide 2-hZethyl-2-phenylacetophenone 2840 2-llethylpiperidine 40 2,4-Pent anedione 32 Phenylacetamide 167 Phenylacetonit r i b 392 2-Phenylacetophenone 5450 3 l-Phenyl-1,3-butanedione 13600 Phenyl-2-propanone Prouionamide 0 4 Triphenylmethane 0 7 Polyhydric Phenols 4-Benzylresorcinol t-Britylhydroqiiinone 4-t-Biitylpyrocatechol
60 Minutes
Compound Absorbance 4-Chlororesorcinol 854 1,4-lhhydroxj naphthalene 47200 2,3-llihydroxynaphthalene 6200 3,5-Diisopropylpj rocatechol 39800 Hydroqiunone 80200 3-Isopropj Ips rocatechol 75600 4-Isoprop>lpj rocatechol 53600 l l e t h l lhi droqiiinone 79900 3-Methylp) rocatechol 73800 4-lIethylpyrocatecho1 12100 5-lIethylresorcino1 SO20 Nit rocat echo1 26000 Phlorogliicinol 11300 Pyrocatechol 81800 Resorcinol 7280 Quinones ilnthraqiunone 29 p-Benzoqmnone 55i00 2-llethyl-3-aminophenyl-
1,4-napht hoc1iiinone Tetramethyl-p-benzoqiiinoned 1,4-Xaphthoqiiinone Thiols Benzenethiol 1-Butanethiol p-Chlorobenzenethiol Et hanedithiol Et hanet hiol 2-llercaptoethanol 2-Xaphthalenethiol 1-Propanet hiol
9770 2-propanet hi 51900 0-Toliienethiol 50300 m-Toluenet hiol 4-Cy clohexy lresorcinol 8250 o-Toliienethiol Chlorohydroqiiinone 62700 p-Toliienethiol 4-Chloropyrocatechol 58100 Toliiene-3,4-dithiol a See Table I1 for related compounds. * A maximiim absorbance of 1160 was recorded at 2 minutes. c A maxinium absorbance of 524 was recorded at 5 minutes. Generonsly supplied by William S. Brinigar, Yale 1-niversity.
VOL. 37,
2670 1O'LO() 25000 417 11100 310 24200 8790
9250 66 1 11'200 6820 13400 638 973 1310 24400
NO. 4, APRIL 1965
0
567
action of ferrous salts on aqueous hydrogen peroxide, oxidized organic compounds of the type R - H through either a chain or nonchain process. MacKinnon and Waters ( 6 ) suggested that the reaction depended on the oxidation-reduction potential of the radical R . generated. They also demonstrated the reducing capacity of these radicals with tetrazolium salts. Comparison of this information with data obtained from the screening procedure with blue tetrazolium suggests an anion intermediate and a free radical mechanism for tetrazolium reduction by nonketol compounds. Thus, it is postulated that to reduce a tetrazolium salt, a compound must produce anions, which in turn are not stabilized to the extent of interfering with their conversion to radicals. The following experiments were designed to evaluate the importance of an anion intermediate in reductions of tetrazolium salts: base strength studies, common ion effect, and solvent effect. Representative nonketol compounds were tested for color development n ith a series of 0.1N ethoxide bases of increasing strength. The rate of oxidation of n-butyl mercaptan had been observed to increase 15 ith alkoxide bases in the order Cs > R b > K > li’a (18). The results summarized in Table I11 indicated an increase in color development for hydroquinone, I-cysteine, and 2-phenylacetophenone with increasing base strength in this same order. The data coincide with expected greater ionization of the compounds with stronger bases. While a study of ketoltetrazolium reactions was beyond the scope of this investigation, benzoin did not give a systematic increase with base strength and the weakest base, sodium ethoxide, was of sufficient strength to yield maximum results. These same compounds were teited for common ion effect in a manner similar to Weissberger and coworker?, study of the auto-oxidation of keto15
Table 11. Inductive Effects. Absorbances of Molar Solutions with Blue Tetrazolium after 60 Minutes
Substituent
Abaorbance B-l)imetlivla mi n o p r o Acetophenones piophenonesa __
Eone p-Hy droxy o-Hy droxy whlethoxv &Bromo p-Chloro m-Sitro
235
5 -2 6
...
-0.4b 27 0.9 153 4 5 381 6 1 357 15 9 2160 v-Nitro 21 3 1430 b-Xitro 46 3 Supplied by J. H. Burckhalter ( I d ) . * These values less than corresponding reagent blank.
Russell, hloye, and Nagpal (16) reported that the rate of oxidation of active hydrogen compounds in basic solution was dependent not only on the degree of conversion to carbanions but also on the relative stability of the carbanion and subsequent radical formed. They observed that further substitution of nitro, cyano, or carbonyl groups served to increase the stability of the carbanion but to decrease its ease of oxidation. Weissberger and coworkers ( 5 , 2f) noted the role of an anion intermediate in the formation of free radicals for molecular osidation of hydroquinones, while Wallace and Schriesheim (18, f9), investigating the oxidation of mercaptans, stressed the importance of the anion structure in the reaction’s rate-determining step. Moreover, these workers also reported (19) that stabilization of the anion resulted in inhibition of free radical formation. l l e r z and Ifraters (10) observed that free hydroxy radicals, produced by
Table 111.
Absorbances for 1
X 10-5M Solutions with Blue Tetrazolium and Ethoxide Bases
Compound Hydroquinone
1-Cysteine
2-Phenylacetophenone
Benzoin
568
Base NaOEt KOEt RbOEt CsOEt NaOEt KOEt RbOEt CsOEt KaOEt KOEt RbOEt CsOEt XaOEt KOEt RbOEt CsOEt
ANALYTICAL CHEMISTRY
5
0.581 0.668 0.779 0.813 0.030 0.033 0.060 0.071 0.010 0.011 0.035 0.080 0.196 0.122 0.169 0,207
Time, minutes 15 30 0.587 0,587 0.704 0.708 0.785 0.788 0,819 0.829 0.050 0.055 0.086 0.129 0.157 0.123 0.134 0.168 0,019 0.026 0.027 0.041 0.051 0.063 0.141 0.167 0.244 0.245 0.217 0.235 0.227 0.239 0.244 0.245
60 0,590 0.711 0 . T90 0.829 0 062 0.137 0.182 0.198 0.034 0.055 0 079 0.173 0.265 0,232 0.241 0.245
Table IV. Absorbances a t 60 Minutes for 1 X lO-’M Solutions with Blue Tetrazolium and Sodium Ethoxide for Common Ion Effect
Compound Benzoin Cysteine 2-Phenylacetophenone Hydroquinone
Absorbance, sodium chloride Absent Present
Per cent depres-
sion
0 266 0 062
0 265 0 048
23
0 034 0 590
0 028 0 551
18 7
(20). Samples tested by the general procedure with 0.1N sodium ethoxide and a common ion in the form of 10 mg. of sodium chloride in 0.1 ml. of water produced from 7 to 23’% depression of tetrazolium reduction compared to samples tested with sodium chloride (Table IV). The presence of additional sodium ion would suppress ionization of the compounds and retard reduction of the tetrazolium salt. Again benzoin was atypical and did not show color suppression by common ion effects. A series of compounds with reported ionization constants in water ( I S ) was studied for common ion effect. Diethylmalonate, ethyl acetoacetate, and 2,4-pentanedione were tested in the general procedure with 10 mg. of sodium chloride in 0.1 ml. of water and 0.1N sodium ethoxide, and the results are summarized in Table V. In this series, 2,4-pentanedione, with it’s greatest degree of ionization, was not affected in color development by the common ion to the estent of the other less ionized compounds, while diethylmalonate with its small ionization was heavily influenced by a suppression of carbanion formation. An investigation was undertaken to study compounds which reduced blue tetrazolium very weakly in absolute ethanol. This lack of reactivity could be attributed either to a low conversion to anions or to anions which were overstabilized by additional electron-withdrawing groups. Wallace and Schriesheim (18) had noticed the increased rate of mercaptan osidation by molecular oxygen in dipolar solvents such as DXIF with sodium alkoside as the base, They suggested that greater solvation by dipolar solvents of sodium cations promoted greater ionization of the mercaptans. DMF was used as the solvent for compounds listed in Table VI with 0 . 1 s sodium ethoside as the base. I t was predicted that the greatest increase in blue tetrazolium reductions under these conditions would occur for compounds with a low conversion to anions in alcoholic solution rather
Table V.
Absorbances a t 60 Minutes for 1 X 10-’M Solutions with Blue Tetrazolium and Sodium Ethoxide for Common Ion Effect
Compound Diethylmalonate Ethyl acetoacetate 2,PPentanedione a
Absorbance, sodium chloride Absent Present 0.666 0.147 0,300 0.147 0.276 0.383
Per cent depression 78 51 28
Ionization constants, K2 10-’6 2 . 1 x 10-11 1 . 0 x 10-Q
In water, as reported by Pearson and Dillon ( 1 3 ) .
than for stabilized ions. Cnfortunately, D M F reacts with blue tetrazolium when it is used as the solvent in the general procedure and results in a blank with an absorbance in the order of 3.5. Thus, in general, the more weakly ionized compounds did show an increase in absorbance over a DMF blank solution. However, the more strongly ionized compounds would appear to inhibit the DMF-tetrazolium reactions and led to a decrease in absorbance when compared to the D M F blank. This interference with the DMF blank is reported in Table VI as negative values. Also included in this table are the water-catalyzed ionization constants reported by Pearson and Dillon ( I S ) for some of these compounds. Except for the weakly ionized acetonitrile, an increase in ionization constant does agree with a decrease in formazan production in the presence of D M F . Hydroquinone and p-Benzoquinone Systems. T h e reduction of blue
tetrazolium b y hydroquinone a n d its derivatives was highly characteristic. There was a n initial rapid rise to a stable color as evidenced by plateaus with high absorbance values. The absorptivity of these plateaus varied appreciably with changes in structure as shown in Figure 2. An explanation for this high ratio could lie in the reduction of blue tetrazolium by the p-benzoquinone arising from the oxidation of the hydroquinone. The reduction of tetrazolium salts by two quinones, menadione and phytonadione, in basic solution as reported in our previous study (1’7) lent support to this concept. The two previously reported quinones together with those listed in Table I indicate that reduction of tetrazolium salts by
quinones in basic solution is a general reaction, although as illustrated in Figure 2, rate and extent of reduction again varied with structure. The extent of such reduction by quinones would of necessity be less than the corresponding hydroquinones with their available hydrogens. Oxidative cleavage of the quinone ring appears to be of significance in the reduction of tetrazolium salts by quinones. Euler (3, 4) has postulated that such cleavage, with the formation of reductones, is important in the reduction of 2,6-dichlorophenol-indophenol in basic solution. Reductones reduce tetrazolium salts (2) and, consequently, the two reductones formed by oxidative cleavage of p-benzoquinone would explain the reduction of blue tetrazolium. Euler (3, 4)has also shown that each fused ring to a quinone will prevent formation of reductones and thereby limit its reduction of 2,6dichlorophenol-indophenol. Therefore, as shown in Table I in this study, 1,4napthoquinone produced half the absorbance of p-benzoquinone while anthraquinone was relatively unreactive with blue tetrazolium. LITERATURE CITED
(11,Cheronis, N. D., Entrikin, J. B.,
Semimicro Qualitative Organic Anal2nd ed., p. 245, Interscience, ew York, 1957. (2) Euler, H. S‘., Hasselquist, H., Arkiv. Kemi.3, 139 (1951). (3) Euler, H. V., Hasselquist, H., Chimia 3, 211 (1949). (4) Euler, H. V., Hasselquist, H., “Reduktone Ihre chemischen Eigenschaften und biochemischer Wirkunger,” Sammulune chemischer und chemischtechnigcher S’ortrage, Neune Folge Heft 50, p. 27, Ferdinand Enke, Stuttgart, 1950.
YJ”
Table VI. for 1 X
Absorbances a t 40 Minutes Solutions with Blue Tetrazolium in DMF
1O%4
Absorbance 2 122 0 267 0 216 0 188 0 092 0 017 -0 068
Ionization constants, K,a
Compound 6-Mercaptopurine 2-Benzylpyridine Diethylmalonate 10-1 Diphenylmethane Triphenylmethane Acetonitrile Propionamide 1,3-Diphenyl-1,3DroDanedione -0 074 Et‘hyl’acetoacetate -0.380 2 . 1 X lo-” 1-Phenyl-1,3butanedione -0.418 4 X 10-lo Benzoylacetonitrile -1.110 2,4-Pentanedione - 1.136 1 . 0 X a In water, as reported by Pearson and Dillon ( 1 3 ) .
(5) LuS’alle, J . E., Weissberger, A., J . Am. Chem. Soc. 69, 1567 (1947).
( 6 ) MacKinnon, D. J., Waters, W. A., J . Chem. Soc. 1953, p. 323. (7) Mader, W. J., Buck, R. R., ANAL. CHEM.24, 666 (1952). (8) Manni. P. E.. Sinsheimer. J . E.. Ibid., 33, 1900 (1961). (9) Mattson, A. M., Jensen, C. O., Ibid., 22, 182 (1950). (10) Merz, J. H., Waters, W. A , , J . Chem. SOC.,1949, p. S15. (11) hleyer, I-., Oelkers, L., Ber. 21, 1297 11888). 112) Nobles. ’W. I,.. Burckhalter. .J. H..’ J . Am. Pharm. Asso;. 47, 77 (1958). (13) Pearson, R. G., Dillon, R. L., J . Am. Chem. SOC.75, 2439 (1953). (14) Rosenkrantz. H.. Arch. Bzochem. Bzophys. 81, 194 (1959). (15) Rosenkrantz, H., J . Baol. Chem. 223, 54 (1956). (16) Russell, G. A., Moye, A. J., ISagpal, K., J . Am. Chem. SOC.84, 4154 (1962). (17) Salim, E. F., llanni, P. E., Sinsheirner, J. E., J . Pharm. Scz. 53, 391 (1964). (18) Wallace. T. .J.. Schriesheim. A.. J . Org. Chem. 27, 1514 (1962). (19) Wallace, T. J., Schriesheim, A., Bartok, W., Ibzd., 28, 1311 (1963). (20) Weissberger, A., Strasser, E., hlainz, H.. Schwarze. W.. Ann. 478. 112 11930). (21) ’Weissberger, A., Thomas, D. S . , Jr., LuValle, J. E., J . Am. Chem. SOC. 65, 1489 (1943). ~
7
,
~
RECEIVED for review December 14, 1964. Accepted January 25, 1965. Abstracted in part from a thesis submitted by Edward F. Salim, Lilly Endowment Fellow, to the Graduate School, University of Michigan, Ann Arbor, in partial fulfillment of the Ph.D. requirements.
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