Salt-Induced Protein Precipitation in Aqueous Solution: Single and Binary Protein Systems

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1 Macroolecular Research, Vol. 11, No. 1, pp 5-61 (00) Salt-Induced Proten Precptaton n Aqueous Soluton: Sngle and Bnary Proten Systes Sang Gon K and Young Chan Bae* Dvson of Checal Engneerng and Molecular Therodynacs Lab., Hanyang Unversty, Seoul 1-791, Korea Receved Dec. 7, 00; Revsed Jan. 4, 00 Abstract: A olecular-therodynac odel s developed for the salt-nduced proten precptaton. The proten olecules nteract through four nterolecular potentals. An equaton of state s derved based on the statstcal echancal urbaton theory wth the odfed Chew s equaton for the flud phase, Young's equaton for the sold phase as the erence syste and a urbaton based on the proten-proten effectve two body potental. The equaton of state provdes an expresson for the checal potental of the proten. In a sngle proten syste, the phase separaton s represented by flud-flud equlbra. The precptaton behavors are sulated wth the partton coeffcent at varous salt concentratons and degree of pre-aggregaton effect for the proten partcles. In a bnary proten syste, we regard the syste as a flud-sold phase equlbru. At equlbru, we copute the reduced osotc pressure-coposton dagra n the dverse proten sze dfference and salt concentratons. Keywords: proten, nteracton potentals, precptaton, phase equlbra, pre-aggregaton, urbaton, salt. Introducton In early days of proten chestry, the only practcal way of separatng dfferent types of proten was by precptatng part of a xture through the alternaton of soe proy of the solvent. Proten precptaton s the splest and the oldest practcal way to separate dfferent protens fro a soluton xture. Separaton s acheved through the addton of precptaton agents such as norganc salts, nononc polyers, polyelectrolytes, and organc solvents. 1-5 A varety of researches on the proten precptaton behavor have been studed by usng varous experental technques. Shh et al. observed the solublty of lysozye, α-chyotrypsn and bovne seru albun n an aqueous electrolyte soluton as a functon of onc strength, ph, the checal nature of salt, and the ntal proten concentraton. Coen et al. 6 studed the saltng-out phase equlbra for lysozye and α-chyotrypsn fro the concentrated aonusulfate soluton. Ther experental results suggest that the proten saltng-out ay be consdered a flud-flud phase separaton resultng n a supernatant flud phase wth a dense precptate flud phase. The degree of separaton s characterzed by the partton coeffcent, K, whch s defned as the rato of the proten concentraton n the dense phase *e-al : ycbae@hanyang.ac.kr /0/ Polyer Socety of Korea to that n the supernatant phase. Theoretcally, any researchers(verwey and Overbeek, 1948; Asakura and Oosawa, 1958; Vr, 1976; Joanny et al., 1979; De Hek and Vr, 1995; Gast et al., 198b; Grson, 198; Vctor and Hansen, 1984) 7-14 reported odels to descrbe the phase behavors of these coplex systes by usng the one-coponent ean-force potental approxaton. Mahadevan and Hall, 15,16 Vlachy and Prausntz 17,18 have used the odel to descrbe the phase behavor of aqueous globular protens n solutons at low salt concentraton, and Chew et al. 19 and Kuehner et al. 0 used a slar approach for solutons at hgh salt concentarton. Therodynac odel wth a properly chosen potental of ean force leads to a satsfactory descrpton of the phase behavor of a proten soluton. Therodynac proes and phase-separaton condtons of proten solutons descrbed by such odel have been coputed usng a nuber of dfferent satstcalechancal approxaton ethods. These ethods can be charaterzed as based on the osotc vral expanson, statstcal-echancal urbaton theory, ntegral-equaton theory, and the rando-phase approxaton. By usng the second-order Baker and Henderson urbaton theory wth the Asakura Oosawa osotc attracton 8, 1 as the donant contrbuton of the ean force potental, Gast et al. and Mahadevan and Hall 16 have studed the polyer-nduced phase separatons of aqueous collodal, nonaqueous collodal and nonadsorbed proten systes. Predctons based on ths ethod led to sold-flud phase 5

2 S. G. K and Y. C. Bae transton rather than flud-flud phase separaton observed experentally by de Hek and Vr 11 for collodal systes. However, They have shown that the urbaton theory cobned wth the Asakura-Oosawa potental s able to predct both flud-flud and sold-flud transtons when very large polyer olecules are present. Based on the experental studes of de Hek and Vr 11 for collodal systes and of Shh et al. for proten solutons, t suggests that a salt- (or polyer-) nduced proten (or collod) precptaton ay be ore approprately vewed as a phase separaton resultng n two flud phases. Grson, 1 Vlachy et al., 18 Chew et al. 19 and Kuehner et al. 0 have used the randophase approxaton to descrbe a flud-flud phase separaton for a slar ean-force potental to that used n the prevously entoned urbaton theory calculatons. The ajor advantage of the rando-phase approxaton s ts splcty; lttle coputatonal effort s requred to calculate the whole phase dagra. In ths study, we present a olecular-therodynac fraework for the proten precptaton by norganc salt. The suary of our works s as follow: (1) We descrbe the sngle proten syste wth a flud-flud phase equlbru. The equlbru odel represents the soluton as a pseudo-one coponent syste contanng only a contnuous solvent and a globular proten. Our equaton of state s the su of a hard-sphere erence contrbuton and a urbaton. The erence ter s derved by the odfed Chew s odel to descrbe the pre-aggregatng effect 0, 1 of proten. Proten-proten effectve two-body potentals are also dscussed. These potentals nclude Coulobc repulson, dsperson attracton, osotc attracton, and attractve specfc potental to represent specfc checal nteractons. The deternaton of reasonable values of degree of pre-aggregaton effect s accoplshed by correlatng our odel wth the partton coeffcent-onc strength data. 6 () We develop a olecular-therodynac odel to copute phase behavors of the bnary proten systes that contan two types of globular protens n addton to the solvent. An eq. of state s derved based on the urbaton theory. The erence ter s derved usng the splfed Chew's equaton n the flud phase and the Young's equaton n the sold phase. The urbaton equaton s gven as the sae as for the sngle proten syste for both flud and sold phases. The energy dfference between two phases s represented by the dsparty of the proten densty. The nfluence of the proten sze dfference and salt concentraton s also dscussed. elec r where r s the center to center separaton length. W () s the electrc double-layer-repulson potental, W dsp () r s the dsperson potental of Haaker, W osotc () r s an attractve nteracton due to the excluded-volue effect of the salt ons, and W specfc () r s an attractve potental between protens representng any specfc checal effects such as hydrophobc nteractons. Appendx provdes expressons for the varous potentals of ean force. Fgure 1 shows a representatve proten-proten urbaton potental of ean force, when the onc strength s 0.01 M (a), and the effect of electrc repulsve potental dsappears when the onc strength s 5 M (b). Ths eans that the electrc double-layer potental has the repulsve nteracton between partcles. At hgh onc strength, however, t can be neglgble. As the onc strength ncrease, the osotc nteracton potental greatly ncreases to the negatve drecton. At extreely low onc strength, theore, total nteracton potental can be repulsve and saltng-n regon can be observed. Equaton of State. In urbaton theory, an assebly of hard spheres s used as the erence syste, whle the Theoretcal Consderaton Proten-Proten Potentals. The effectve two-body potental between two dfferent proten olecules, W, s gven by the su of four potentals. W () r = W elec r dsp r osotc r () + W () + W () + W specfc r () (1) Fgure 1. Contrbuton to the total effectve two-body potental as a functon of r/d n the case of I = 0.01 M (a) and I = 5 M (b): ph=7, H/ = 8.9, ε sp / =, δ = 0. n, d s = n, d =.44 n, C salt =5M. 54 Macrool. Res., Vol. 11, No. 1, 00

3 Salt-Induced Proten Precptaton n Aqueous Soluton: Sngle and Bnary Proten Systes reanng nteractons are treated as urbatons; P P = ρ ρ P ρ where ρ s the densty of proten olecules, and P s the pressure. Sngle Proten Syste. In aqueous soluton, proten partcles folded to sphere and dspersed as the collodal dsperson. In ths case, proten olecules are not absolutely dspersed to only sngle type of proten partcles, but have soe porton of der or trer. Ths effect s called preaggregaton. The erence syste s gven by the odfed Chew s equaton 19 to consder pre-aggregaton effect: ρ 1 -- η = 1+ 4ω PA η ( ω ( 1 η) PA 1) where η s the packng fracton, gven by η = πρd /6, where d s the daeter of sngle proten, ω PA represents the average degree of pre-aggregaton that s reduced by the hydrophobc part of the proten surface. The urbaton ter s ρ ω PA ρu = where U s the urbaton energy per unt densty, gven (for a sngle-proten syste) by U = 4π W = j ()r r dr where W = j () r s the proten-proten effectve two-body potental defned n eq. (1). The total equaton s, theore, ρ 1 -- η = 1+ 4ω PA η ( 1 η) ( ω PA 1) 1 -- η ( 1 η) ω PA ρu The general eq. for calculatng the Helholtz energy fro a pressure-explct equaton of state s Eq. (7) can be wrtten n ters of T and ρ η ( 1 η) ATV (, ) A ( T) P N N = + dv + N V V ln V () () (4) (5) (6) (7) A Nω PA A = Nω PA + ρω PA Then, the checal potental s A µ = N ln( ρω PA ) TV, µ o d ( ρω PA) ρω PA ω PA ρω PA µ µ µ µ = = = 8ω PA η η + 4ω 41 ( η) PA η η + ( ω 41 ( η) PA 1) ln( 1 η) ( ω PA 1) η ( ω 41 ( η) PA 1) η 6η + 5η ( η) lnρ 1 ω PAρU (8) (9) (10) At equlbru, proten concentratons n the supernatant and dense-flud phases are calculated fro eqs. (6) and (10) based on the classcal equlbru condtons: µ s = µ P s = P d d (11) (1) where superscrpts s and d denote the supernatant and dense phases, respectvely. Bnary Proten Syste. Dervaton of the equaton of state for xtures follows a rgorous frst-order statstcalechancal urbaton theory based on the xture of hardspheres as a erence syste. In bnary proten syste, flud-sold phase separaton s dealt wth proten-poor phase and proten-rch phase separaton. 4 Flud Phase: The equaton of state for flud xtures s wrtten as flud P = flud, P flud, (1) The erence syste s gven by the extenson of eq. (6) wth the sple case, ω PA = ρ + = 1 + ρ x x j b g ( d ) (14) where x = N /N s the nuber fracton of olecules, g (d + ) s the par radal dstrbuton functon of hardsphere xtures at contact and b s the cobnng second vral coeffcent of hard sphere b π = d (15) Macrool. Res., Vol. 11, No. 1, 00 55

4 S. G. K and Y. C. Bae where d s the cobnng effectve hard sphere daeter d d + d = jj (16) where d and d jj are the effectve hard-sphere daeters for pure fluds and j. To obtan explct equaton of state fro eq. (14), a sutable atheatcal for 5 for g (d + ) s needed. g ( ηξ, ) ξ ξ 1 1 = η ( 1 η) ( 1 η) (17) The fnal for s o A A x ρu total = N ρx N x j b W Then, the checal potental s A µ k = N k The result s TVN,, k (5) (6) In proten xtures, the packng fracton η s defned by: η = -- ρ 4 x b ξ b b jj 1 ρ = -- x 4 k b kk b k (18) (19) For one-coponent systes and equal-segent-sze xtures, ξ = η, and eq. (14) reduces to the Carnahan-Starlng eq. for hard spheres. 6 The urbaton ter s ρ ρu total = where U total / s the total nteracton energy of all pars U total = x x j, = 1 U total (0) (1) where x s the ole fracton of the coponent and U total s the total nteracton energy for -j par. total U = 4π d + r W () r r dr () where W () r s the proten-proten effectve twobody potental for the -j par. Theore, the total equaton of state, the su of the erence ter and the urbaton ter, s gven by W = 1 + ρ x ρ x j b g ( d ) + πρ x x j r dr () The general equaton for calculatng the Helholtz energy fro equaton of state s o A A = x N N + ρ dρ x ρ ρ + ln( x ρ) (4) µ k k B T where, = ρ x b k Q k + ρ x x j b = 1, = 1 N Q ln( x N k k ρ) + 1 I Q ξ I η -- ξ = I η η N Q N k Q η N η N k Q ξ = + N ξ N k 1 I 1 = ln( 1 η), I n = I n η( 1 η) n 1 (7) (8) (9) (0) Sold Phase For the sold phase, the erence equaton of state and Helholtz energy are gven by 4 : (1) + 9.5V 5.95V 15.0 x () lnx where V * = V V 0 V 0 Nd 0 :d 0 x 1 d 11 f( α)x 1 x d 1 x, = = + + d. Here, paraeter α s the rato of saller to larger hard-sphere daeters. In ths work, d 11 û d s used so that α = d /d 11 and the functon f(α) s deterned approxately to ft the coputer-generated flud-sold coexstence curves for bnary hard-sphere xtures n the range 0.85 ú αú1. 4 η n 1 = ρ ( V 1) 1.19( V 1) V ( V 1) 5.95V 15.0 x ln( x ) A N + + = 1 V 1 = ln V ln( V ) 0.78V f( α) = ( 1 α) = 1 () 56 Macrool. Res., Vol. 11, No. 1, 00

5 Salt-Induced Proten Precptaton n Aqueous Soluton: Sngle and Bnary Proten Systes Eqs. (1) and () are based on the ft to the coputer generated copressblty factor for an one-coponent hardsphere eltng pont. The checal potental of coponent k s defned by µ k A = N The urbaton of the sold phase s the sae as that of the flud phase, but the energy s represented by the dfference of the nuber densty. The total equatons, theore, are P = (5) (6) For aqueous solutons contanng two knds of protens, the equlbru condton s (7) (8) (9) where subscrpts "1" and "" represent speces of protens. Results and Dscusson V * V * ( 1 V * ) V * 19.04V * 0.78] N V * ln( x N k k ) (4) P = ρ ( V * 1) 1.19( V * 1) V * ( V * 1) 5.95V * 15.0 x ln( x ) πρ W + x x j r dr Sngle Proetn Syste. For the precptaton of a sngle + + µ k µ k µ = k A = V * N V * ( 1 V * ) V * V * 0.78 N V * N k ln ( x ) k W + 4πρ x k r dr µ flud 1 = µ sold 1 µ flud = µ sold P flud = P sold Fgure. Effect of onc strength: ph = 7, H/ = 8.9, ε sp / =, δ = 0. n, d s = n, d =.44 n, C salt =5M. proten n an aqueous salt soluton, we exane the effect of onc strength n phase-separaton systes. The partton coeffcent, K, of the proten syste can be obtaned fro the equlbru condtons and s gven by the rato of the equlbru nuber densty of proten n the dense phase to that n the supernatant phase [K = ρ d /ρ s = η d /η s ]. Fgure shows the predcted partton coeffcent K plotted as a functon of onc strength for systes wth H/ =7, ε sp / =, δ =Å, σ on = 6.94 Å, ph = 4, σ p = 4. Å, and r = 0.08 Å for varous values of ω PA. The partton coeffcent, K, ncreases exponentally wth the onc strength. Ths dependence s coonly observed feature n saltng-out not only for protens but also for other organc substances and dssolved gases. The exponental for has been used extensvely n correlatng proten saltng-out data. Partton coeffcent ncreases wth the degree of pre-aggregaton. It agrees wth the prevous theoretcal results that large partcles separate ore effcently. Fgure shows the dependence of parttonng on squarewell paraeters δ and ε/. Both square-well depth and wdth gve large effects on the proten parttonng. It eans that specfc nteractons between proten olecules (e.g. hydrogen bondng, proten surface structure, etc.) play an portant roll n the aggregaton of proten olecules. Coen et al. 6 have conducted precptaton experents for two sall globular protens, hen-egg-whte lysozye and α-chyotrypsn n solutons of aonu sulfate at varous onc strengths and ph. Fgure 4 shows experental and calculated values of C p,super and K as a functon of onc strength for the hen-egg-whte lysozye (at ph 4). Fgure 4 represents the α-chyotrypsn data (at ph 8.) for C p,super and K as a functon of onc strength. In those calculatons, Haaker constant and the value of, r, the thckness of the hydraton/stern layer were 8.9 and 0.8 n, respectvely. These values are concdent wth values reported by Kuhner, 7 who ndcated that Haaker constant depends on the value Macrool. Res., Vol. 11, No. 1, 00 57

6 S. G. K and Y. C. Bae Fgure. Effect of odel paraeter [ε sp / (a), δ (b)]: H/ = 8.9, ε sp / =, δ = 0. n, d s = n, ph = 7, C salt =5M, d =.4 n. of the thckness of the hydraton/stern layer. As shown n Fgure 5, calculated equlbru coeffcent and supernatant concentraton are n qualtatve agreeent wth experental results of hen-egg-whte lysozye for ω PA = 1.4, ε sp / =.7 and δ = 4 Å. If the value of ω PA = 1.4 s consdered, 40% of lysozye s pre-aggregated before the parttonng s processed. The proposed odel also agrees very well wth α- chyotrypsn experental data for ω PA =1.5, ε sp / =. and δ = Å. Consderng ω PA =1.5, t ples that 5% of the α-chyotrypsn s pre-aggregated before the parttonng process. Coparng ω PA values for two odel protens presented n ths study ndcates that the effect of the specfc nteracton s ore effectve n lysozye soluton than that of α-chyotrypsn soluton. Bnary Proten Syste. For aqueous xtures of globular proten, we present reduced osotc pressure-coposton dagras wth varous onc strengths and gven proten daeters. Fgure 6 shows the effect of the salt concentraton for the systes wth H/ = 8.9, r =0.08n ε sp / = 0., δ =0.n, Fgure 4. Experental and correlated values of C p,super (a) and K (b) n the case of α-chyotrypsn n aonu sulfate at ph 8.: H/ = 8.9, r = 0.08 n, ε sp / =., δ = 0. n, d s = n, d =4.4n, ω PA = 1.5. Dark squares are experental data fro Coen et al. 6 and the sold lnes are calculated values usng the proposed odel. d s = n, d 11 =.5 n and d =.4 n. The solublty of globular protens ncreases wth the salt concentraton. The coposton dfference between flud and sold phases at the lower salt concentraton s larger than that of the hgher salt concenraton. However, the results show that the salt concentraton effect s sall. Fgure 7 shows the effect of the sze dfference between proten-1 and proten- for the gven syste wth H/ = 8.9, r =0.08n ε sp / = 0., δ = 0. n, d s = n, and C salt = M for varous d and fxed d 11 =.5 n. In the case of d 11 =.5 n and d =.4 n, there s slght coposton dfference between the flud and sold phase. The larger dsparty n sze between dsslar protens shows the lager coposton dfference between flud and sold phases at the gven coposton n the flud phase. The proten solublty decreases wth ncreasng the sze of proten-. At fxed sze of proten-1, the sze of proten- ncreases wth the ean sze of proten. It s correspondent wth the sze effect of the 58 Macrool. Res., Vol. 11, No. 1, 00

7 Salt-Induced Proten Precptaton n Aqueous Soluton: Sngle and Bnary Proten Systes Fgure 7. Theoretcal phase dagras for aqueous proten xtures wth dfferent sze d : H/ =8.9, r =0.08n, ε sp / = 0., δ =0.n, d s = n, C salt =M. Fgure 5. Experental and correlated values of C p,super (a) and K (b) n the case of hen-egg-whte lysozye n aonu sulfate at ph 4: H/ = 8.9, r = 0.08 n, ε sp / =.7, δ = 0.4 n, d s = n, d =.4n, ω PA =1.4. Dark squares are experental data fro Coen et al. 6 and the sold lnes are calculated values usng the proposed odel. Fgure 8. Theoretcal phase dagras for aqueous proten xtures wth dfferent sze d : H/ =8.9, r =0.08n, ε sp / = 0., δ =0. n, d s = n, C salt = 0.01 M. sngle proten syste, that s, large solute olecules partton ore strongly than those of sall olecules. 8 Fgure 8 shows theoretcal phase dagras for the syste wth H/ = 8.9, r =0.08n ε sp / = 0., δ = 0. n, and d s = n for varous d and fxed d 11 =.5n at the salt concentraton C salt = 0.1 M. Coparng wth each syste of Fgure 7 n the sae sze dsparty, the dfference of equlbru coposton between sold and flud phase s greater than that of Fgure 7. It agrees wth the prevous result shown n Fgure 6. Conclusons Fgure 6. Theoretcal phase dagras for aqueous proten xtures wth dfferent C salt : H/ =8.9, r = 0.08 n, ε sp / = 0. n, δ = 0. n, d s = n, d 11 =.5 n, d =.4 n. We proposed a therodynac odel to descrbe the saltnduced proten precptaton based on effectve potentals of ean force. The odel s developed based on a statstcal echancal urbaton theory and the erence ter s Macrool. Res., Vol. 11, No. 1, 00 59

8 S. G. K and Y. C. Bae derved fro the odfed Chew's equaton for the flud phase and Young's equaton for the sold phase. In a sngle proten syste, odel predctons ndcate that the electrolyte concentraton plays a prary role n affectng phase separaton. The proten partton coeffcent, K, ncreases exponentally wth the onc strength. Our results show that the preaggregaton effect of proten plays an portant role n the precptaton of protens. Calculated equlbru supernatant concentraton and partton coeffcent are n qualtatve agreeent wth experental results for both hen-egg-whte lysozye and α-chyotrypsn n solutons of aonu sulfate when the effect of the pre-aggregaton s consdered. In the bnary proten syste, we consder two effects on the phase separaton. The coposton dfference between flud and sold phases at a gven coposton of the flud phases decreases wth ncreasng the value of the dsparty n proten sze and the salt concentraton. Further, the proten sze dfference s ore effectve than that of the salt concentraton on the phase behavors of proten/salt systes. Appendx Contrbutons to the effectve two-body potentals for protens n aqueous electrolyte soluton. 1. The electrc double-layer repulson 7,7 : W elec () r z z j e [ e κ ( r d p) ] = for r*(d + r) 4πε o ε r r 1 κd κd jj (A1) k : Boltzann constant T : absolute teperature z : valence of the speces e : the unt of electron charge d : the daeter of speces d = (d +d jj )/ 4πε 0 : the delectrc perttvty of free space ε r : the relatve delectrc perttvty of water r : the effectve-sphere hydraton/stern layer κ : the nverse of the Debye length; gven by κ =(e N A I)/ (ε 0 ε) N A : Avogadros nuber I : the onc strength of the salt, gven by I =(z anρ an +z catρ cat )/ z an and z cat : the anon and caton valences, respectvely ρ an and ρ cat : the onc nuber denstes.. The attractve Haaker dsperson nteracton 7, 9 : W dsp r () H d d jj d d jj r d = ln r d r ( d d jj ) r ( d d jj ) for r> d p + r (A) H : the effectve Haaker constant for the proten-proten nteracton. The osotc attractve nteracton potental 16, 7 : W osotc r () = for d < r< d s + r (A) ρ s : the total onc nuber densty d s = (d + d s )/ d s =(z an d cat + z cat d an )/(z cat +z an ): a valence-weghted on daeter 4. The specfc nteracton 0, 7 : W specfc () s ε = sp for d p < r<( d p + δ) ε sp and δ : odel paraeters References πρ s d s d jjs (A4) (1) P. R. Foster, P. Dunhll, and M. D. Llly, Boche. Bophys. Acta, 17, 505 (1975). () R. N. Hare, W. A. Tsel, J. C. Whte, and A. Rosenberg, Bopolyers,, 761 (1984). () Y. C. Shh, H. W. Blanch, and J. M. Prausntz, Botech. Boeng., 40, 1155 (199). (4) M. Q. Nederauer and C. E. Glatz, Adv. Boche. Eng. Technol., 47, 159 (199). (5) F. Rothsten, n Proten Precptaton Process Engneerng, R. G. Harron, Ed., Dekker, New York, (6) C. J. Coen, H. W. Blanch, and J. M. Prausntz, AIChE J., 41, 1 (1995). (7) E. Verwey and J. Overbeek, Theory of Stablty of Lyophobc Collods, Elsever, Asterda, (8) S. Askura and F. Oosawa, J. Poly. Sc.,, 18 (1958). (9) A. Vr, Pure & Appl. Che., 48, 471 (1976). (10) J. F. Joanny, L. Lebler, and P. G. de Gennes, J. Poly. Sc., Poly. Phys. Ed., 17, 107 (1979). (11) H. De Hek and A. Vr, J. Collod Interf. Sc., 41, 996 (1995). (1) A. P. Gast, C. K. Hall, and W. G. Russel, J. Farad. Dscuss. Che. Soc., 76, 189 (198b). (1) M. J. Grson, J. Che. Soc. Farad. Trans., 79(), 817 (198). d s r r + d jjs d s d jjs 4 d s + d jjs d s d jjs r d s + d jjs 60 Macrool. Res., Vol. 11, No. 1, 00

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