Degradation of Glycogen by Alkali - ACS Publications

always to be susceptible to anaerobic degradation by hot concentrated alkali. ... apparent alkali stability of glycogen after preliminary exposure to ...
1 downloads 0 Views 956KB Size
MARJORIER. STETTEN AND HOWARD M. KATZEN

2912

Vol. 83

THE NATIONAL INSTITUTE OF ARTHRITISA N D METABOLIC DISEASES, NATIONAL INSTITUTES OF HEALTH, HEALTH SERVICE, U N I T E D STATES DEPARTMEKT OF HEALTH, EDUCATION AND WELFARE, BETHESDA, MARYLAND]

[ COXTRIBUTION FROM PUBLIC

Degradation of Glycogen by Alkali BY MARJORIE R. STETTEN AND HOWARD M. KATZEN RECEIVED JAYUARY 9, 1961 Glycogen isolated from animal tissues by extraction methods in which all contact with alkali is avoided has been shown always t o be susceptible t o anaerobic degradation by hot concentrated alkali. The rapid initial decrease in molecular weight has been followed by light scattering methods. The chief products formed have been identified as a polydisperse series of relatively stable polysaccharide acids, corresponding to “glycogen” isolated by classical alkali methods and small amounts of free isosaccharinic acid. The quantities of isosaccharinic acid split off from glycogen samples of known averagc molecular size are in the same general order as would be predicted from calculations based upon idealized molecules of similar size having the highly branched tree-like structure indicated by enzymological studies. Borohydride reduction of the terminal aldehyde group has been found to render the glycogen molecule stable to anaerobic alkali degradation. The apparent alkali stability of glycogen after preliminary exposure to alkali is due to the replacement of the reducing end of each molecule by a saccharinic acid residue.

That glycogen is relatively stable in hot concentrated alkali has been an accepted fact since the early isolation procedures were described by Bernard’ and by Pfluger.2 Purified samples of glycogen isolated after alkaline digestion (KOH method) are usually found to have molecular weights between 1 and 8 x lo6 when determined by the methods of light scattering or ultracentrifugal analy~is.3-~On the other hand, when tissues are extracted with cold trichloroacetic acid (TCA method) the average size of the glycogen isolated is usually far higher, values up to 100 X lo6 being Still larger particles can be obtained following aqueous extraction.9, lo There is considerable evidence in support of the opinion that the reported molecular weights of glycogen isolated by acid methods are not artificially and that these samples are more nearly representative of native glycogen than are those isolated by the more commonly employed hot alkali methods. The studies reported here were undertaken in an attempt to shed some light on the nature of the changes which occur in glycogen when i t is treated with hot concentrated alkali, simulating the conditions used in the isolation of glycogen from tissues. I n the past few years there has been a considerable revival of interest in the effects and mechanism of action of alkalies on carbohydrates. Excellent reviews of this subject have been written by Sowden12 and by Whistler and BeMiller.13 The principal steps involved in the action of an alkali on the reducing portion of a carbohydrate molecule are generally accepted to be formation and ionization of an enediol, 0-elimination of a hy(1) C. Bernard, Compl. rend., 44, 578 (1857). ( 2 ) E. F. W. Pfluger, “ D a s Glycogen,” Bonn, Germany, 1905, p . 53. (3) C. T. Greenwood, A d z a i i c e s in Cauboliydrale Chein., 7 , 289 !1952). (4) C. T . Greenwood, i b i d . , 11, 333 (1956). ( 2 ) D. J . Manners, i b i d . , 12, 2G1 (1957). (6) AI. R . S t e t t e n , H. M. Katzen and D. S t e t t e n , J r . . J . A i d . Chc~it.. 222, 587 (1950). (7) X. R . S t e t t e n , H. A I . Katzen a n d D. S t e t t e n , Jr.. i b i d . , 232, 475 (1958). ( 8 ) XI’, A. J. Bryce, C. T. Greenwood a n d I. G. Jones, J . Chem. S o c , 3816 (1958). (!J) A. Lazarom., Anal. Rec , 8 4 , 31 (1942). (10, S. A . Orrell, J r . , a n d E . Bueding, J . Alii Chewi. S o r . . 80, 3 h 0 0 (1!4.78). ( l l j S. R. Erlander and D. French, ibid., 80, -1413 (1958). (12; J C . Sowden, Aiisances in Carbohydrate Cheni., 1 2 , 35 (19.573. :13) R. I,, Whistler and J. 5.Be?ililler, i b i d . , 13, 289 (1958).

droxyl or alkoxy1 group and rearrangement of the molecule to an a-dicarbonyl form which then undergoes an internal Cannizzaro type of reaction to yield a saccharinic acid. An 0-glycosyl or 0-alkyl anion is more readily extruded from the sugar enediol anion than is a hydroxyl ion. The use of alkaline degradation methods for studying the structure and branching patterns of naturally occurring polysaccharides, proposed by Corbett, Kenner and Richards14 and by Whistler and Corbettlj has yielded many interesting results. Detailed studies of the acidic products formed from the starch-glycogen group of polysaccharides have been restricted to the unbranched component, amylose.16 Because of the high degree of branching of amylopectin and glycogen, changes taking place during alkali degradation are quantitatively relatively slight. These changes, however, are shown here to be qualitatively similar to those occurring with other polysaccharides. Results Turbidity Measurements in Alkaline Solutions. -The use of alkali resistant Teflon reaction vessels has made i t possible to follow the change in turbidity which occurs when polysaccharides are TABLE I GLYCOGEN SAMPLES Sample

Source

Method of isolation

Subsequent treatment

Molecular wt. X

10-6

Rabbit liver TCA ... 34 Rabbit liver KOH ... 10.7 Rabbit liver TCA ... 13c Sample I11 10 N NaOH‘ 5.8 Sample I11 0.1 NHCl* 6.9 Rabbit liver TC.4 0 . 1 N HClh n.91 1’11 Saniple VI 10 :V XaOH’’ .57 TI11 Rabbit liver TC.A 0 . 1 S HC1’ .lY IS Sample VI11 10 A T KOHa . I C S Sample VI11 BorohJ-dride 1!)C The decrease in turbidity of the a At 100’ for 3 hr. glycogen solution with time of heating a t 100’ in 0.1 N HC1 was followed until, based on previous experience, approximately the desired average molecular size was reached. The time required varied between 5 and 30 minutes for the samples used. Estimated molecular weights. I I1 111 IV V VI

(1-I) U- A I . C o r b e t t , J . Kenner and G. X . Richards, Cheni. ?PI n d . ( L o I L d o n ) , $62 (1Y53) (15) R . L. Whistler a n d W. hI. C o r b e t t , J . A m . Chein. Sac., 7 8 , 1003 (1966) :1G) G. Xachell and G . X. Richards, J . C h i n . S o c . , 1199 (1958).

DEGRADATION OF GLYCOGEN BY ALKALI

July 5 , 1961

8

2913

)I

80

1 2

4

6

8

20

48

HOURS.

Fig. 1.-Alkaline degradation of a single sample of glycogen (TCA method, Sample I, mol. wt. 34 X lo6) studied V NaOH, 25', atmosphere under various conditions: A, 10 i of air; B, 0.1 IV NaOH, looo, atmosphere of air; C, 10 N NaOH, loo", atmosphere of nitrogen; D, 10 iV NaOH, loOD, atmosphere of air; E, 10 N NaOH, loo', oxygen bubbling through solution.

heated in alkaline solution, without resorting to isolation and purification of each sample a t each point in time as was done previo~sly.~JThe glycogen samples used were of various average molecular sizes, ranging from about 100,000 to 34,000,000, prepared from rabbit liver by either acidic or alkaline isolation methods as summarized in Table I. Glycogen was heated a t 100' in dust-free 10 N NaOH. Aliquots, removed a t suitable time intervals, were added to 0.1 ill KCl for turbidity measurements. In Fig. 1 the results obtained with a single glycogen sample (I) of average molecular weight 34 X lo6 (TCX method) studied under different experimental conditions are plotted as per cent. of the initial turbidity of the solution against time. Each curve is the average of a t least two separate runs. Curve D shows the typical rapid initial decrease, followed by a gradual continuous decline in turbidity, observed when glycogen (TCA method) was heated a t 100' in concentrated alkali under normal atmospheric conditions. The solubility of oxygen in hot concentrated alkali is negligible." When the same reaction was carried out in an atmosphere of N2, using tank N2 which had been freed of 02 carefully, the initial decrease in turbidity observed (curve C) was almost as rapid as in air. With a stream of 0 2 bubbling through the glycogen solution in hot concentrated alkali, the turbidity fell much further and after a few hours little polysaccharide remained (curve E). It thus appears that two reactions are involved. The first reaction, independent of 02,presumably is the result of successive rapid degradative steps starting a t the existent aldehyde end of the molecule. The other reaction, rapid only under 02,is probably one of oxidation followed by splitting anywhere in the large polysaccharide molecule, producing new terminal aldehyde groups which can in turn serve as new points for rapid attack by alkali upon the m o l e c ~ l e . ' ~That heat is required for the initial rapid reaction is shown by the fact that the decrease in turbidity in concentrated alkali is very slow at room temperature (curve Xj. (17) "International Critical Tables," S e w York, N. Y., 1925, Vol. 3, p . 271.

?rIcGrau~-Hill Book

Co.

20L'

L '

1 2

'

'

4

'

'

'

*

.

D '

'

'

HOURS.

"

"

'

"

1

'

20

Fig. 2.-Comparison of the effect of hot 10 N NaOH upon glycogen samples of comparable size obtained by acidic and by alkaline isolation methods: A, rabbit liver glycogen isolated after KOH digestion, mol. wt. 5.8 X lo6 (Sample IV); B, rabbit liver glycogen isolated after TCA extraction, mol. wt. 6.9 X lo6 (Sample V ) ; C, rabbit liver glycogen isolated after KOH digestion, mol. wt. 10.7 X 106 (Sample 11); D, rabbit liver glycogen isolated after TCA extraction, mol. wt. 34 X lo6 (Sample I). Light scattering measurements were made by removing aliquots a t various time intervals and measuring turbidity after addition to dust-free 0.1 N KCl.

When hot dilute alkali (0.1 N NaOH) is employed, the initial reaction is somewhat slower than with hot concentrated alkali, but the second reaction is more prominent, persumably because of the greater solubility of 0 2 in dilute alkali. The over-all breakdown of the polysaccharide thus proceeds further (curve B). In contrast, when glycogen isolated by the KOH method was heated with concentrated alkali, a rapid initial decrease in turbidity was not observed. Figure 2 shows the results of parallel experiments with KOH glycogen (curve C) and TCA glycogen (curve D). To eliminate the possible effect of the relatively great difference in average molecular size of the samples used, two samples of glycogen with about the same molecular weight were prepared from a single rabbit liver. On treatment with hot concentrated alkali, the KOH glycogen (curve A) did not undergo the initial rapid reaction exhibited by the TCA glycogen (curve B). At later times all of the glycogen samples, regardless of the method of preparation, showed the continual slow decrease in molecular size attributed to a small amount of unavoidable oxidation. The very rapid initial stage of the reaction was further studied by a slightly different method. Hot concentrated alkali was added to a concentrated solution of the polysaccharide in a Pyrex cylindrical cuvette and the turbidity changes measured directly without taking aliquots. Because of the compensating complications due to cooling of the solution in the cuvette and the production of light scattering material from the glass, the reaction could only be followed for a short time by this method. Figure 3 shows that

MARJORIE R.STETTEN AND HOWARD M. KATZEN

2914

Vol. 83

which they O C C U ~ . ~ * Whereas .~~ a-l,G linkages are somewhat less susceptible than a- 1,4 linkages to acid hydrolysis, their relative scarcity in glycogen 90 >' 1\ I makes this difference unimportant.lg It is therefore assumed that the random mild acid hydrolytic split to which glycogen was subjected in decreasing the average size of the molecules did not greatly change the structural pattern of the polysaccharide. After the mild acid hydrolysis, the glycogen sample used for isolation of saccharinic acid was purified by repeated reprecipitations from aqueous solution with ethanol and was shown by chromatographic examination to be free of detectable glucose, saccharinic acids and low molecular weight oligosaccharides. It had an average molecular weight of 0.91 x lo6 (sample VI). Four grams of 70' 5 IO I5 20 this glycogen was heated a t 100' in 10 N NaOH in a MINUTES, Teflon vessel for 18 hr. After removal of cations Fig. 3.-Comparison of the initial effect of hot 10 N NaOH by chromatography on ion exchange resin and on the turbidity of solutions of glycogen isolated by acidic precipitation of the bulk of the polysaccharide by or alkaline methods: A, 0.57 X lo6(alkali method) (Sample ethanol, the alcohol-soluble products were concen, VII); B, 0.91 X 108 (acid method) (Sample VIII); C, trated by evaporation of the solution and repeated 5.8 X lo6 (alkali method) (Sample IV); D, 6.9 X lo6 precipitation with ethanol. (acid method) (Sample V). Turbidity measurements were These products were separated by paper chromade directly upon solutions of glycogen and hot alkali matography. Material eluted from the areas reacting in a glass cuvette. corresponding to the location of known isosaccharinic lactone was purified by repeating the the difference in rate of alkaline degradation be- chromatography step and by rejection from the tween glycogens of comparable size prepared by eluate of all material insoluble in small volumes of acidic and alkaline methods is immediately ap- ethanol and of ethyl acetate. The product, a t parent. first an oil, then crystalline, was recrystallized Mild acid hydrolysis, splitting glucosidic bonds twice from ethyl acetate. Five mg., melting point in a random fashion, will introduce new aldehyde 93-94', with crystalline form identical with that of groups which can serve as points of attack in sub- isosaccharinic lactone, was obtained. The lactone sequent alkali treatment. A sample of glycogen was converted to the anilide20.21which was found (TCA method), which during treatment with 10 N to have identical chromatographic properties with NaOH a t 100' showed the degradative effect illus- those of the anilide of isosaccharinic a ~ i d . ~ ~ m ~ ~ trated in curve D (Fig. 2), was isolated after this Polysaccharide Acids.--Some of the material alkaline treatment and purified. One portion of which did not migrate from the origin of the this polysaccharide, on retreatment with concen- preparative chromatographic papers gave a positive trated alkali in exactly the same way, exhibited no test for lactones. This material, eluted with water rapid initial degradation. The remainder of the and precipitated with ethanol, was free of monorecovered sample was heated for a short time in 0.1 saccharinic acids before hydrolysis, but after cornN HC1 until the average size, measured by tur- plete hydrolysis by dilute acid was shown chrobidity, was somewhat diminished. After isola- matographically to yield, along with glucose, contion, this polysaccharide was shown to be once siderable isosaccharinic acid and some material again subject to an appreciable rapid alkali deg- with properties suggestive of metasaccharinic acid. radation. Since presumably only one monosaccharinic acid Characterization of Products : Isolation of Iso- residue is liberated on hydrolysis of each polysacsaccharinic Acid.-Many preliminary investiga- charide acid molecule, the ease with which this tions of the products of the action of concentrated residue can be detected varies inversely with the alkali on glycogen showed that an appreciable molecular weight of the material studied. The amount of the acidic compound with chromato- overwhelming amount of glucose produced on total graphic properties resembling those of isosaccharinic acid hydrolysis of glycogens (KOH method) of OH CH,OH average molecular weight larger than about a million prevented even chromatographic detection of acid HOOC-C-CH,-CHOH-CH20H was pro- the relatively small amount of monosaccharinic duced. Since the yield of saccharinic acid was rela- acid liberated. With samples having molecular tively greater when glycogen of low average molecu- weights in the range of a few hundred thousand, lar weight was degraded, a sample of glycogen of purified until completely free of the monosacmolecular weight about one million was prepared (18) M. L. Wolfrorn, E. N. Lassettre and A. N. O'Neill, J. A m . from TCA rabbit liver glycogen by a mild con- Chem. Sot.. 73, 595 (1951). trolled treatment with 0.1 N HC1. It has been (19) M.A. Swanson and C. F. Cori, J . B i d . Chcm., 173,797 (1948). (20) B. Sorokin, J . grakl. Chcm., (21 37, 318 (1888). observed that, as a first approximation, the aL.M.Utkin and G. 0. Grabilina, Doklody Akad. Nauk S.S.S.K.. 1,4 and a-1,6 linkages of polysaccharides are hy- 93,(21) 301 (1953). drolyzed a t a uniform rate regardless of their posi(22) J. W.Green, J. Am. Clicm. Soc., 76,5791 (1954). tion in the chain or the length of the chain in (23) J. W.Green, ibid., 7 8 , 1894 (1956). 100

I\

* A

'

[ "

1

charinic acid produced during the alkali treatment, total acid hydrolysis yielded a mixture in which, after removal of the bulk of the glucose, isosaccharinic acid readily could be demonstrated chromatographically. In control experiments with glycogen (TCA method) of comparable molecular size, acid hydrolysis liberated no saccharinic acid. Quantitative Estimation of Isosaccharinic Acid Produced by Alkaline Degradation of Glycogens of Various Sizes.-To estimate the amount of saccharinic acid produced, 10-mg. portions of each of a series of glycogen samples (TCA method) of different average molecular size were subjected to identical conditions of alkaline degradation and separation of products. The alcohol-soluble products, after lactonization, were separated by descending paper chromatography with ethyl acetate :acetic acid :water (10 :1.3:1)24 and visualized by reaction with hydroxylamine and FeCt. 25 The quantity of isosaccharinic lactone formed was estimated by visual comparison of the lavender color with that produced from known quantities of the reference compound similarly developed and stained. The results, given in Table 11, showed that the yieid of isosaccharinic acid from glycogens isolated by acid methods increased as the average niolecular size of the glycogen decreased. TABLE I1 FREE ISOSACCHARINIC ACID 1'ROnUCED BY HEATING 10 MG. OF EACH GLYCOGEN SAMPLE IN 30y0 KO13 FOR 3 HOURS QUANTITY O F

Samples prepared by acid isolation methods. Approximate mol. wt. Sample

VI11 VI V I11

I

2915

DEGRADATION OF GLYCOGEN BY ALKALI

July 5, 1961

x lo-'

0.19 0.91

6.9 13 34

glycogen (TCA method), none was produced from glycogen (KOH method). Reduced Glycogen.--A sample of glycogen of molecular weight 0.19 X lo6 was treated with sodium borohydride,26 converting the terminal TABLE I11 COMPARISON OF YIELDSOF FREE ISOSACCIIARINIC ACID PRODUCED FROM GLYCOGENS OF COMP.4RAUI.E AVERAGE MOLECULAR SlZE PREPARED BY DIFFERENT h>~ETHODS

Sample

Quantity of isosaccharinic acid lactone, y Calcd. from Produced Table IV

600,500 100 50,60 30 25

Fig. 4.-Idealized glycogen molecule having 5 tiers of glucosyl residues: 0 represent those glucosyl residues which are joined in the single unbranched a-1,4linked chain containing the free reducing end of the polysaccharide molecule (R). Dash lines mark the limits of the successive tiers.

300 60

15 8

3

The same quantitative alkali degradation method applied to glycogens isolated by the Pfliiger method or subsequently treated with concentrated alkali in the course of the preparation of the sample showed little or no free isosaccharinic acid split off. In Table I11 are given results obtained with glycogens of comparable size but differing in the method of preparation. Prior introduction of a terminal acidic group in the polysaccharide rendered the molecules stable to subsequent degradation by alkali. After 3 hr. of heating glycogen (KOH method) with concentrated KOH, a very small amount of free isosaccharinic acid could sometimes be detected, probably due, as in the light scattering studies, to secondary reactions resulting from the fact that the conditions were not strictly anaerobic. When hot alkali treatment was limited to 15 minutes, only the effect of the initial rapid reaction taking place from the reducing end of the molecule was evident. I n the time in which appreciable isosaccharinic acid was liberated from (24) hZ. Moilanen and H. Richtzenhain, Arla Chem. Scaxd., 8 , 704 (1Q54). ( 2 5 ) XI. Abdel-hkher a n d P. S m i t h , J . Am. Chcm. Soc., 73, 5859 (1951).

Alethurl of prepn.

A,pproximate mol. at.

X

10-6

Quantity of isosacchariiiic acid produced from 10 nig. of glycogen i n 30% KOII a t loo", y I n 15 min. In 3 hr.

Acid 6.9 30 60 Alkali 5.8 00 10 Acid 0.91 100 Alkali .57 0 Acid .19 400 600 illkali .10 0 25 Borohydride .19 0 20 a I.e., less than 5 7 , the smallest amount detectable by the methods used.

V IV VI VI1 VI11 IX X

aldehyde group into an alcohol. The quantity of isosaccharinic acid produced by hot concentrated alkali treatment of 10 mg. of this reduced glycogen was compared with that produced in the same way from the original sample and from a sample previously treated with hot alkali. The results, samples VIII, IX and X of Table 111, show that glycogen is rendered stable to anaerobic alkali degradation by prior conversion of the terminal aldehyde group into either a carboxy acid or an alcohol. In contrast, such reduced glycogen samples readily are degraded by hot dilute alkali in the presence of oxygen.? Discussion All of the results are consistent with the picture that the principal action of concentrated alkali upon glycogen starts with a.n attack on the one free aldehyde group (represented by R in Fig. 4) of each molecule. The fact that the quantities of the various products formed are very different from those obtained from the a-1,4 linked polysac(26) M. Abdel-Akher, J. K. Hamilton a n d F. Smith, ibid., 7S, 4691 (1951).

MARJORIE R. STETTEN AND HOWARD M. KATZEN

2916

Vol. 83

the currently accepted mechanism of alkali degradation of carbohydrates resulting from many diverse s t ~ d i e s ,the ~ ~principal .~~ changes which such a molecule probably undergoes in the absence of oxidizing conditions and in the presence of hot concentrated alkali are as follows: 1. The glucose residue a t the reducing end of the molecule (R) is converted by way of its 1,2-enediol to the 2,3enediol of fructose. 2 . The succeeding elimination of the anionic group 0 to the carbonyl group results in the expulsion from C-4 of the remainder of the glycogen molecule as an alkoxy anion. Kenner, et al., have shown that 0-glucosyl anions Fig. 5 .-Products of the anaerobic action of concentrated are more readily extruded from sugar enediol alkali on an idealized glycogen molecule: 0 represent free anions than are hydroxyl ions.32 The possibility of isosaccharinic acid molecules ; 0 represent the terminal course exists that the occasional elimination a t the isosaccharinic acid residue, to the carbon-6 of which are 1,2-enediol stage of the hydroxyl group a t C-3 inattached polysaccharide residues with the branching patstead of the glycosyl anion a t C-4 might occur, retern of glycogen. A , R, C and D are the polydisperse sulting in the production of a terminal metasacseries of polysaccharide acids identical with "glycogen" charinic acid group which would stop further degisolated by alkaline methods. radation, as has been observed with the longer 1,4 charide amylose16 and the $-1,4 linked cell~lose?~linked chains of cellulose33 and amylose.16 3. can be attributed to the occurrence of appreciable The monosaccharide, remaining after expulsion of l,(i linked branching in glycogen. the bulk of the molecule! rearranges by way of an The branching pattern of glycogen probably a,P-dicarbonyl intermediate to give principally varies appreciably both between and within mole- D-glucoisosaccharinic acid along with acidic scission cules but for purposes of calculation we shall as- products. 16,34 4. The polysaccharide portion, misume that an average typical glycogen molecule nus one glucosyl residue, nom has another free has a regular multiple branched, tree-like struc- reducing group exposed and can undergo a similar ture28 with an average of 4 a-1,3 linked glucosyl series of reactions. By successive reactions, called residues between the residues linked by a-1,B by LVhistler'3 a peeling process, all of the glucosyl bonds at the branch points and an average of 8 residues in the straight chain would be converted a-1,4linked glucosyl residues per outer hai in.^^^^ to isosaccharinic acid molecules, indicated by the 9relatively small molecule of this structure, hav- black circles in Fig. 3. Those glucosyl residues ing 5 tiers. is represented in Fig. 4.30 Rased upon with a side chain attached a t C-6 would yield isosaccharinic acid indicated by 8 , with a polysac( 2 7 ) G. K , Richards a n d H . 1%. S c p h t i ~ i i .J . C h r i n SOC.. l i 9 2 (1937). charide structure still attached. ( 3 8 ) K. 11. J l e y c r a n d I[. Iirit,, C / J L I ; I;. l ~ / o ,24, .;i.i ( I n i l ) . If it now be assumed that, i n the absence of (29) J . Larner, R. I l l i n g w x t h , G. 'r. Cori and C . 1,. Cori, .I. Eiol. oxidizing conditions, no alkali sensitix-e linkages occur in the polysaccharide molecule other thall uted bg- I)e\Vitt S t e t t e n , Jr. alculation of Maximum >f I 7 fiers removed i r < > mt h r reducinr: ~ n d i,t I < i l i o i i i t h a t 17 u i n i l ~ l r t e , i tiers are t h e fireatest number pro\ ideii b y tliia m,,