> Wr- IC/8U/136INTERNAL REPORT

(Limited distribution)

International Atomic Energy Agency

and

Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

COOPERATIVITY AMD BINDING ISOTHERMS

OF THE AMYLOSE-IODINE-TRIIODIDE COMPLEX •

Attilio Cesaro

Laboratory of Macromolecular Chemistry, BBCM, University of Trieste, Italy,

Julio C. Benegas **

International Centre for Theoretical Physics, Trieste, Italy

and

Daniel R. Ripoll

Institute, de Matematica Aplicada, Escuela de Fisica,Universidad Nacional de San Luis, Argentina.

ABSTRACT

A simple model has been presented which ia Cased on some wide-ly accepted characteristics ot the dmylose-iodme-trnodidecomplex and is capable 01 cot relating a number ot experimentalfeatures with molecular parameters. A substantial improvementis obtained removing weakness and arbitrariness of sone adjust -abJi: parameters ot previous models. The proposed model assu-mes that sequences or bound species are initiated by tmodideions and ptopaqated 1. principally > by iodine. The theory isdeveloped by using the matrix method ot Zlmn and Bragg andassuming that the cooperative interaction is of electrostaticnature. The model reproduces most of the experimental data andpredicts a un i te length or the bound polyiodine chain for anylength ot the host poiymet.

KIRAHARE - TRIESTE

August 1981*

* To be submitted for publication.

** Permanent address: Ins t i tu t e de Matematica Aplicada, Escuela deFis ica , Universidad tJacional de San Luis, Argentina.

-1-

INTfcODULTIuN

One or the beat known example- or inclusion complex in aqueous

solution is due to the reaction ol Amylose with iodine in the

presence 01 iodide to give a blue complex L1J. A lull under-

standing of the aqueous complex: has btjen elusive mainly i,r<.'HUSO

°!" the large number or independent variables that must be

controlled and because it may exist in various states 01 cia-

gregation, which depend upon Kinetic as well at, equilibrium

factors L1-4.J. As a consequence, contlictmcr models appeared

in the past years as an outgrowth ot ottorts to interpret hy-

drodynamic L1-5J and spectroscopic data Lb-yJ. Although a vlir.

iety O I helical structures have been derived 11 am the solid

State x-ray fiber diffraction analysis of araylose L1U-14J, the

helical character is hatdly maintained in solution. In u o t ,

in the absence ot completing agents, amylose behaves, above a

given chain-length, as a random coil LlbJ. However, the local

conformation ot an amylose chain is also a mattei ot dispute.

Differences in the model proposed for the contotmation in so-

lution can be ascribed to the extent to which ctyutalline

helical conformationis) is (ate) retained in solution LlbJ.

Host of recent studies have shown that, because the lotational

angles ol the glucosidic linkage can fluctuate in a cjuite l;ur,>~

conrormationul space L17. 1BJ, com lauratlwul average may

take account ot the macroscopic equilibrium, and dynamic, (;to-

perties tiy.203.

STRUCTURE AMD KNEftUtTlCS OF THE UlSiiULVfcD CUHtLEX

In the dissolved amyiose-iodine complex, there «ov sew. lii.Ue

'Kmiii. that iodine resides within the annular cavity or a

more or less regular helical amylose chain i figure 1 J , as

early emerged ftom the work ot Fiundle and cowork.ees on the

crystalline LjiOJ and dissolved LzlJ complex. Again, the he-

Lical character ot the dissolved complex does not irapLy chat

the completed polymer adopt a tigid, tod-like contotmationj

indeed, the hydrodynamic volume ot the polymer decreases upon

the comptexation with iodine Ll,.l,bJ. Although all the spec-

troacopic (Jtopetties ot the dissolved complex suggest that

iodine upet:iey aie arrayed in linear sequences within the ca-

vity, neither the distc Lbutiori ot iodine chain lengths nor

even the mean chain length has been unambiguously established.

The complex ptevents strong absorption and circular diciucic

bands neat 600 nm Li<i--4J, the wavelength or these bands tieing

a strong tunction ot the mean degree ot polymerization x ot

the amylose chains. This tltect saturates above x - 10U where

the waveienqth ot the maximum absorption, Amax , achieves an

asymptotic upper Limit near tj40 nm. These chancres have been

related tu the changes in chain length oi the polyiodine ar-

rays £iJ, Jt'J. suggesting some analogy with the Kuhn model tor

the polyenes L/bJ. The value ot Xmax also depends aomewhaL on

the degreeot saturation ot the complex and more so on che con-

centration ot iodide ion L1J. The jtoichiometry of the complex

continues to be <* matter ot debate. The stoichiometry ot the

complex can be expressed in terms ot the ratio k ot bound

triicdide to total bound iodine as R - LljJ /( tljjJ ^ Ll^ J ).

It has now been repeatedly contitmed that k c<±nnot be zero L1J;

that is, iodide lor other negative) ions are mandatoty tor the

development ot the blue colour. Although aome treatments ot

the complex assume that the bound species is iJiK = li, a c-i.n-

inl analysis ot the existing data reveals no evidence tor a

structure of 3uch high charge density, at least in solution LI,

2 7-JOJ, The Boat recent works on the suinect, using a variety

ot techniques, favours R * 0. j L30J or « * u.b tU,31J. These

and other results suggest also that h may depend somewhat on

such variables as mean polymer chain length, iodide and/or

salt concentration, and the physical state of the complex

(i.e. crystalline or dissolved). As a matter ot tact, the

complex seems to exist in a variety ot different compositions

which, however, in the limit ot an overall value ot ot H = 1,

do not produce the other known optical properties typical ot

the blue complex. The cooperativity of the iodine binding pro-

cess has been accepted for many years LJ^i,3JJ. The original

proposals ot a direct charge transfer interaction between io-

dine and the oxygen atoms of the polyaaccharide annulut> LJ. J4,

J5J have not been successfully corroborated by studies on the

-eyelodextrin-pentaiodide structure LJ1J, and on o( -cyclodex-

trin crystallized with Ijin the absence ot i' LJbj. The en-

thalpy ot compiexation is found to be constant in the range ot

reaction conditions which leave unchanged tJOJ, and it vanes

with amylose chain length in a way that mimics the dependence

of A max on the degree ot polymerization Li4J. Therefore the

rather large enthalpy change ica. -/I kJ^mol ot bound 1̂ ,' inuat

sustain its largest contribution t rum the cooperative liitiuac-

tions between the atom ot the linear bound iodine chains ana a

much smaller contribution from interactions ot the bound spe-

cies with the polymer chain. Host or the enthaipic data on

- I 4 -

complex loriition LiOJ were derived trom the van t Hoft plot

ot the apparent equilibrium constant as a tunction ot the tem-

perature, A H W H , and range trom -42 to -0/ fcj/mol (ot bound

molecular iodine>- UntorLunately, despite the number ot fac-

tors which attest the properties ot the complex, too often the

experimental conditions are poorly reported, making comparison

very difficult. Direct calorimetrIC measurements have been

reported by Takahashi and Ono LJUJ, by Cesaro and firant LJJ

and by Cesaro et al. CJOJ. Given the cooperativity ot the

complex formation, one would also expect a dependence of the

isosteric heat of binding on the degree ot compiexation, ti, as

well as on temperature. The direct microcalorimetrlc determi-

nations tJOJ have indeed ahown that the integral heat of reac-

tion ttom d = u to o.A i A H , , M J IS ca. U kJ/moi more nega-

tive than the value of iiHe^.^ under same experimental condi-

tions, a fact which supports the believe of a cooperative pro-

cess. Concerning the energetics of the compiexation, a further

aspect too otten neglected is that the helical sequences ot

sites for the iodine binding cannot be considered preformed,

as the helical character ot the uncompiexed chain is highly

transitory and flexible. Therefore, a ptocess whirh requires

•''»'" segments arrayed in helical sequences, such as tor

the present case, must involve a conformational transition

with, at ieat>c, an entropy contribution not negligible a prio-

ri .

The above summary of the known properties has been w<-r,-,::-,:iry

in order to generate an understanding ot the framework,

of f a d s on which a theoretical model can be developed.

THE MODEL

A statistical-mechanical model has been developed tor the

Anylose-lodine-trilodide complex which follows the genera-

lization of Cee&ro, Konlc and tit ant L40J, on the basis ot

the Zimm and bragg C ktJ matrix model tor the helix-coil tran-

sition in polypeptides. In this paper the assumption is made

that the cooperative 4 or stacking) interaction is or a limited

effective range, which is further assumed to be electrostatic

in nature. Since the presence ot charged species is mandatory

for the initiation ot the complex, it is postulated that the

process begins with the binding of 1J to some helical conlor-

mational sites ot am/lose. The initial ion induces the gro-

wing of the complex by further condensation or polarizabie

neutral lj molecules driven mainly by electrostatic ion-induced

dipole interaction. This hypothesis is substantiated by the

strong intermolecular forces existing between iodine molecules

giving tor example an heat ot sublimation ot I5.tst> Kcal.mole,

which 13 43% ot the dissociation energy ot the gaseous diato-

mic iodine molecule at 0 K L421. The distance dependence ot

this interaction also follows the requirement ot shoe t eitec-

tive range.

According to the characteristics ox the complex in the i>o-

lid state C20,21J, it is assumed that only helical ^untit-

mations or Amylose can bind iodine and that approximately

one helical turn constitutes a "binding site", tor ^ufpose

of modelling, each "site" has the sane average number ot glu-

cose units, no matter which particular conrormationai state is

associated to each dimeric unit, provided that the overall

chain trajectory is helical. Thus, the following states Land

corresponding statistical weights* can Oe identified tor a

generic site in the amylosic chain:

1 : tree helical site

bA : I| bound to helical site

bi: i A bound to helical site neighbour to I3

b^: lj bound to helical site, second neighbour to ij

in addition to the statistical weight, c, ot a coil site and a

which denote the beginning of a helical sequence.

Furthermore,the stacking interaction is considered to be ef-

fective only up to the second neighbe-urto the initiating 17 ion.

since the strong distance depend nee ot the ion-induced dipole

interaction makes this interaction very small tor longer <<•'•'••

t. M l 11 •<.-.:.

The model cot relates three consecutive sites, and lor

each site there are live statistical weights, living a ma-

trix of statistical weights ot size zb X <;b. fly eliminating

the redundancy C44J it reduces to the following y X y matrix }J:

\

u =

1

0

su

1

1

0

1

u

1

0

3

0

0

1

0

1

u

cu

c

0

c

c

0

c

0

cuc

0

0

c

D

c

0

ut>1

uXb<

0

0

^

0

0

0

u

0

*>*

0

0

u

0

0

^ b j

0

0

0

0

0

b ,

0

u

u

u0

b 30

0

<J

b0

Sb

u0

0

u

u

0

(1)

'•('

The partition function can be written as follows:

where

P U y

I? = 1 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 )

a n d U =

010110

10 /

where N is the total number ot sites or the polysacchande

The thermodynanic averages or physical interest »•<''• obt.ai

by taking the appropriate derivatives ot L;. i'\>r inutnii

the average number ot bound 1 is obtained as

2UZ. 5 U, T

with i!i = o I-U = 2&iU^., an(1 u 1 3 ttle matrix tor vector > with all- ele-

aenta zero, dimilatly tor the avetage values corteaponding

to the other statistical weights.

-fl-

AND DETliHMlNATlONS OF FREE FAHAMETEKS

the--a

The determination ot the values ot tree parameters has been

carried out ttora the litting of the expetlmental data of

binding liiotherms of Amylose (x = 1S00I at 2bl'C and C^

For thi3 purpose it is necessary to remind that the sta-

tistical weighta Lot the bound states are proportional to

the concentration ot the ttee species:

b, = k,

b = k,

b = k,

where k^ and k̂ '' are the i calculated) contributions ot

the stacking electrostatic interaction at the first and second

sites neighbouring an initiating site with l^ bound. LI3 JF and

CI3Jfai:e the tesfjtrctive concentration ot free species. In the

same set at equations, the intrinsic association constant for

binding 01 1 to the amylose site is denoted by k.i , while the

association constant tor binding an 12 molecule to the initiated

romi'l'-.'x is denoted by k}. The tree parameters are the as-

sociation constants (K, and ka) and the statistical weighta tor

coil (cj and tor the beginning ot the helical sequence MS).

Figure 1 shows the experinental binding isotherms LJOJ and the

corresponding theoretical tit. This result corresponds to a

polyaaccharide chain of about i5(J sites tot which the end ef-

tect on the polymer is not important. Figure 3 shows the ex-

perimental data obtained for the dependance of Amax on the

degree ot polymerization, x L45J. In the same figure is re-

ported the curve calculated with the proposed model for the

-9-

variation ot the average length of the complex, L = in, <-na+n3i /

n., , with the number of sites, N. To obtain this result, a sta-

tistical weight, S4, has been introduced to take into account

the unfavourable binding of the site at"t*>e ends or tiie theoretical

chain. The relationship recognized in the literature between

the S i z e Of d i s p e r s i o n centres and >max L i . 2J can- therefore,b e verified with the present model and provide some fur-

ther insight of the complex structure. In tact, the theoreti-

cal results are expressed by the following equation, relating

the average length of the polyiodlne chain to the number 01

sites 01 the theoretical chain:

L-1 - O 2 O + 6.90 N -' ibi

whereas the experimental data are empiri-

cally titted by the equation!

1.56 IO"3 + 10.25 It)'* 1 /)

In order to get a mapping of the theoretical curve onto the

experimental points, the functional farm

AMAX = f (L) 18J

can be obtained by eliminating x from equations ibl and <7)

and imposing a linear relation between x and N

x = A N + B

with A - 6 and B = IS.

1 9)

To check: the predictions of the model, other experimental

-10-

data are used. The literature values of Amax are plotted in

figure 4 as a function of total iodide concentration, Li" Jt .

The same figure shows the theoretical results f or Amax v^. LI

J , which have been obtained from the calculated variation of "L

vs_. C1~J, by using equation (3). As a further check, the de-

pendence of the iodine binding capacity, 1BC (defined as the

percentual ratio of 1 bound by the poly saccharide under satu-

rating conditions L U ) , with the degree of polymerization has

been investigated irigure 5). The predicted dependence can be

calculated from the results ot our model and using equation <<(>

to transform the number of available sites into total number

of glucose units. The theorical prediction is shown in !i«-

"'•f S by the continuous line. Finally, it is interesting

to note that the stoichiometry of the complex tor the pre-

sent mudel is given by

R = n* / ( n + n «-n\ — ~L~ *

that is, the average charge per bound site is the inverse ot

the average ienght L. This relation is implicit in the as-

sumptions and provides the functional dependence of K upon

x, total iodide concentration and other physico-chemical '-iri

phlttt; (i.e. as 1/ i.) .

-11-

DISCUSSION AND CONCLUSIONS

The aim of this paper has mainly been to exploit some pen-

ernl consequences of a mechanical statistical model for a coo-

perative binding process on a linear chain mulecule and there-

fore to explain some experimental results obtained on the Amy-

lose-Iodine-Iodide complex in aqueous solution trom a molecu-

lar point of view. The results here obtained can provide some

insight into the physical mechanism involved in this kind ot

processes. The use of a generalization ot the well known Zimm

and Bragg matrix treatment for the helix-coll contormational

transition in polypeptides, and the assumption that the icoo-

perative) interaction between neighbouring sites of a short

effective range, allow us to describe theoretically several foa-

tures ot the complex. Following the aoove described formalism,

five statistical weights are defined tor a generic site, but

only three ot them are free parameters and are obtained by

fitting the experimental data tot the isotherm of the complex.

From the values ot these tree parameters it Is inferred that

the random coil site is preferred to the ordered helical one in

solution").

"1 These conclusions refer to the overall site conformation

made up in the model by six glucose units, each of which is

allowed to move in the conformations! energy surface.

The values obtained tor the intrinsic association constants

(for Ij and the further binding of i^ I are high enough to

account for the driving mechanism of the formation ot the

—J P—

inclusion complex and provides the energy necessary tor the

random coil to helix contorraatiunal transition. In the fr;une-

wni-k of our model, once the complex has been initiated by

toinding an 13 , it grows rapidly by addition ot i_j to an ?>v-

i-mr.e length tnat depends upon the degree of polymerization of

the poiysaccharide. To obtain a good agreement between the

theoretical trend of L vs the number ot sites, N, and the ex-

perimental ^max vs x, the "end-etfects" ot the polymer had to

be considered. This quite general behaviour ot cooperative

systems tJJJ has been reached in cwo steps. The additional

statistical weight S 1 r introduced tor the sites at the ends of

the chain, shows the more unfavourable binding of Iodine to

these siLes with respect to any internal chain sites. The

determination of the free parameter has been done independently

"•' the others, which have been obtained trom the isothera

of an Amylose sample (x = ibUO) toe which end ettects are not

important. The second step has been to correlate the physical

x with the theoretical "number ot binding sites". By assuming

a simple proportionality, x oc ti, the relationship Xmax - f(N)

provides a very "good agreement between the experimental data

and the calculations, it is worthwhile to comment on the pa-

rameters A and B ot equations (8). The first point is that

there are 6 alucose units per binding site <A=bj, which is the

value accepted toe the complex in the solid 3tate. Secondly ib

glucose units do not participate to the binding process th-

roughout the whole range ot DP (B=lb(. The straighttorward

Interpretation is that, at both ends, about /-B glucose units

do not bind Iodine. These units may not reach the helical

coniormation necessary tor a e r v m g as binding sites and provi-

de tor the Junction zones. The concept ot helical regions

which eventually fold back to form a molecular crystallite is

consistent with the present model and parallels same other

experimental observations on phase separation which have not

been discussed here. By limiting to the behaviour- ot the dilute

complex, both the fundamental order-disorder theory and the

careful scrutiny ot the experimental evidences provide compel-

ling reasons tor the adoption of the model presented here.

Acknowledgment

This work has been carried out with the financial

of the Italian Ministry ox Public Education iMFIt through

the University of Trieste and ot the National kesearch Council

Of Argentina i C o n i c e t ) . One o!1 the authors; (J.I'.B.) is giMtuful to

Professor Abdus '"alfun, the International Atomic Kner^y Agency and UWKLUXI ['or

financial support and hospitality at the rnterntaionu.1 Centre for Theureticu]

Physics, Trieste.

l i

4 t

b i

61

1 1 )

12 t

U)

14)

REFERENCES

Banks, H. and Jtoenwood. I'.T. Jtatch and Its Components

University tress: Edinburgh. 1'J/b.

ts. Carbohvdr. kee. id'/tt. i>J., 41.

Cesafu, A. and brant. E'.A. biopolvmec s l'J/b. 16, JOJ,

Dintiis, F.H., fleckwith, A.C.. Babcock. G.E. and Tobin,

Macromoleculea is/6, '±, 471 and +7b.

Senior, M.B. and H<±moi i, E. biopoivwei s iy"/3, Li. *>$.

, M. Chem. Lett. 1^/2, ,'->.

M.E., kimai, L.. Kilponeri, h.G. and Gill, 0.

J. Am. i.'hem.̂ oi". 1 y < - . ±±. lii'.*-£.

ft.

Teitelbdum. k.C, kuby, J.L. and Marks, T.J. J .Am.Chem.

Soc. L'Jiti. luU, jtib.

Handa, T. and ¥a.iLina, H. biopolymers ia79, 1^, 8/3.

trench. A.D. and Murphy, '/.ij. t-olvmer I-J , 7, 18,

t'rench. A.D. dnd-Murphy, V.C.. Cat bohvdc • Kea. 1

Chu, J.ti.C. and Jettrey, u.A. A«'ta Crystall. iy>

Wu, H.-C. and ̂ arko, A. biopoivmera iy;b. oj.. /

Winter. W.T. and darko. A. Biopolvmers iy/4, ij.

iarko. A. ilia bilujki, A. Carbohvdi . hes. U O u ,

Zj_,

1UJ8.

144?.

r±. li.

M . , W o d a , H . a n d K a m a r a , T . fiiopolvmers i s . B , 1 7 .

I / j Jordan, K.C. and Btant. D.A. Mact omoiev:ul^s 1^80, ij_, 4yl.

181 Jordan. H.C., Brant, U.A. and L'esaro, A. BiQPolvmers i'lili.

19i Cesaro, A. in Mew Developments in Industrial Folysacchatides

ed. l.C.M. Dea, tiordon and breach. New 'fork, UtS4.

- 1 I 4 -

! 13 i Wm*-

2U.

21

22

23

25

26

Rundie , H.t;. J.Aa.L.'tiea.Soc. li>47. fe9, i'/bi*.

Rundle , k . E . and Baldwin. R.k. J.Am.Chem.aoc. i y 4 J , &£, 5b4.

B a i l e y , J.M. and Whelan, W.J. J. b io l . i ' hem. i J i i . j j b . ift>9.

P iannenmue l l e r , B . , Mairerhoeler, H. and l ichulz , fi.C.

MaKromol.enem. 1*69, 121 . 1 4 / .

bank.3, W. , Greenwood, C.T. and Kahn, K.li. iJarbod'/dr • Rea.

1 9 / i , I Z , -2b.

Kuhn. H. J.Chem.t'hys. iy4y, W , 1198.

Ono, S., Tsuchihashi, ij. and Kuge, T. J.A

JU.

32.

Ji .

J4.

J6.

Jb.

37,

36.

39.

Gilbert, G.A. and MaiMOtt J.V.K. Ttans. Faraday Sor..

194B. 44, B4.

Knutson, U.A.. Cluskey J.K. oind Dintzis F.h. Cafbohydf.kes-

19B2, Jjn, 117.

Cronan, C.L. and Schneider, K.W. J. Fhvs.Chem. Ut>J. 2i,

J99O.

Cesaro , A. , J e i i a n , t . and S a u l e , i3. fliopolvmet s . U'tSO,

1 1 , 14yi .

Noltemeyet, H. and Saenger, W. Mature. Ulb, Ibv. o2y.

Rundle. R.L., Foster, J.T. and Baldwin, k,k. J.Am.Chew.doc.

1944, 6b, illb.

i s t e l n , U.S. and Kundle, R.E. J .-Jhem. Phya. Iy4u , i b . i yb .Murakami. H. J .Chem.^hvs. iat>4, 22, Jb7 .G t l l t i t h s , D.W. and Bender , M.L. Adv-'-atcii . 1'J/J, i J , iUa.

McMullan, R.K., iiaenger, W., Fayos, J. and Moot:, D.

Carbohvdr.Hes. 1973, J±, ill.

achneider, F.W., Cronan, C,L. and Podder, J.K.

1368, 71

Takahashi, K. and Ono, i>. J.blochem. iy/2, i_i, Iu41.

Brant, U.A. Quart.key.Biophv3. ly/b, 2< 5*/'

-16-

40. Cesaro, A., Konic, w. and brant, D.A. In Solution Properties

or ^olvsaccharldes brant, D.A. tH, ACS Synpoaium Series

N.15U, Washington J.y81, Chap. J2.

41, ilimm, B.H. and Braory, J.K. J .Chem.Phvs. iyb9, il, b2b.

•il. Jhiriey, D.A. and Giaique, W.E. J.Am.L'hem.^oc. i!#59, Bl.

4 //a.

4J. foland. U. and ijcheiaga, H.A. Theory Qi Helljc-Coil Trans i t ions

in Biouolvniera Academic f-'ress. New York,

-17-

flGUKE CAPTION

Figure 1. Schematic picture ot the dissolved Amyloae-iodine-

triiodide complex in aqueous solution. Conformation

of uncompleted chain is a "snapshot" taken from

reference 18.

Figure 2. Degree of complexation t> vs In t [ If, the natural

logarithm ot the total concentration ot tree iodine.

Experimental data it rum reterence 40): O

The continuous line is the theoretical result with

c = i.b; k, = 400; kx = 800; S ^ 10. The calculated

stacking constants are: kjt - 4ibO; k^ * i4b.

Figure 3. Dependence ot AmA, upon the decree ot polymerization

of anylose, x. Experimental data (from reference 6i)

: O • The continuous line la the theoretical result

with SA - 0.1 (see text).

Figure 4. Dependence ot Amay upon the logarithm uf the iodide

concentration, in Ll~Jt . Experimental data itrom

reterence 2 6 )i O . The continuous line is the

theoretical result.

Figure 5. Dependence of the iodine binding capacity, IBC, upon

the degree ot polymerization, x. Experimental data

(from reterence 21): 0 • '1'he continuous line is

the theoretical result.

dEoU

u

o-

o-

0o

o o- i f i -

II1 ;

oMB

1

p Ol

to

01

Hco

o

CO

O

eno

too

120

X

500

o

^°

099

\

600

~i

°\o

o

)

=3

3

-20- -21-

oin10

o©in

- TOf

o

S

oCM

inQO

oo

om

-< # w * 1