Biochimica et b.
`.
`'
`t
`l O ph YS l Ca ·
`~1c a. Internat .
`.
`of b.
`ional .Jo1.1rn ~
`1
`.
`ioc1em1stry and
`.
`b1ophysics
`. BML lst & 2nd Floore
`• J ~ .,,.
`UC San Die g O
`R~ce i ved on: 1216-09-97.
`
`Page 1 of 11
`
`CSL EXHIBIT 1051
`CSL v. Shire
`
`

`

`BIOCHIMICA ET BIOPHYSICA ACTA
`International Journal of Biochemistry, Biophysics
`& Molecular Biology
`
`Editor-in-Chief: Peter C. van der Vliet (Utrecht, The Netherlands)
`
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`(including Reviews on Bioenergetics)
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`
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`
`G.H. Lorimer (Wilmington, DE)
`Y. Lindqvist (Stockholm)
`
`T.E. Kreis (Geneva)
`J. Avruch (Charlestown, MA)
`
`Gene Structure & Expression
`
`P.C. van der Vliet (Utrecht)
`
`Molecular Basis of Disease
`
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`
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`Authors should submit their manuscripts (four copies) to one of the appropriate Executive Editors at either of the
`following Editorial Offices:
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`Copyright © 1997 Elsevier Science B.V. Ail rights reserved
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`E) The paper used in this publication meets the requirements of ANSI/NISO 239.48-1992 (Permanence of Paper).
`
`0167-4838/1997/$] ?.0Ü
`
`_......
`
`Page 2 of 11
`
`

`

`ELSEVIER
`
`Protein Structure & Molecular Enzymology
`
`Vol. 1339, No. 2; 23 May 1997
`
`Contents
`
`BIOCHlMICA ET BIOPH YSICA ACTA
`
`BB':)
`
`BBA is cired in: Currenr Conrenrs Life Sciences - Bio/ogica/ Absrracrs - Chemica/ Abstracts - Index Chemirns - Index Medicus - CSA Absrracrs -
`Excerpra Medica (EMBASE) - R~ference Update - Current Awarene.u in Biologico / Sciences (CA BS)
`
`Information for Contributors
`
`uii
`
`Short sequence-paper
`
`Cloning and characterization of a protein phos(cid:173)
`phatase type 1-binding subunit from smooth mus(cid:173)
`cle similar to the glycogen-binding subunit of
`liver
`K. Hirano, M. Hirano and D.J. Hartshorne ( USA) 177
`
`Regular papers
`
`Characterization and mutational studies of equine
`infectious anemia virus dUTPase
`H. Shao, M.D. Robek, D.S. Threadgill,
`l.S.
`Mankowski, C.E. Cameron, F.J. Fuller and S.L.
`Payne (USA)
`Characterization of the affinity of the GM2 activator
`protein for glycolipids by a fluorescence de(cid:173)
`quenching assay
`N. Smiljanic-Georgijeu, B. Rigat, B. Xie, W. Wang
`and D.J. Mahuran (Canada)
`Identification of /3 2-glycoprotein I as a membrane(cid:173)
`associated protein
`in kidney: purification by
`calmodulin affinity chromatography
`D.A. Klœrke, R. R(/Jjkjœr, L. Christensen and /.
`Schousboe ( Denmark)
`Characterization of the interaction between /32-gly(cid:173)
`coprotein I and calmodulin, and identification of a
`binding sequence in /3 2-glycoprotein I
`R. R(/Jjkjœr, D.A. Klœrke and/. Schousboe (Den-
`mark)
`Site-specific modification of rabbit muscle creatine
`kinase with sulfuydryl-specific fluorescence probe
`by use of hydrostatic pressure
`N. Tanaka, T. Tonai and S. Kunugi Uapan )
`Target size analysis of an avermectin binding site
`from Drosophila melanogaster
`A. Pomes, E. Kempner and S. Rohrer (USA)
`
`181
`
`192
`
`203
`
`217
`
`226
`
`233
`
`239
`
`247
`
`253
`
`Glutathione alters the mode of calcium-mediated
`regulation of adenylyl cyclase in membranes from
`mouse brain
`J. Nakamura and S. Bannai Uapan)
`Structure of membrane glutamate carboxypeptidase
`N.D. Rawlings and A.J. Barrett (UK)
`Acetylcholinesterases from Elapidae snake venoms:
`biochemical, immunological and enzymatic char(cid:173)
`acterization
`Y. Frobert, C. Créminon, X. Cousin, M. -H. Rémy,
`1.-M. Chatel, S. Bon, C. Bon and J. Grassi
`( France)
`Purification and characterization of two new cy(cid:173)
`tochrome P-450 related to CYP2C subfamily from
`rabbit small intestine microsomes
`Y. Shimizu, E. Kusunose, Y. Kikuta, T. Arakawa,
`K. lchihara and M. Kusunose ( Japan)
`High sait concentrations
`induce dissociation of
`dimeric rabbit muscle creatine kinase. Physico(cid:173)
`chemical characterization of
`the monomeric
`species
`F. Couthon, E. Clottes and C. Vial ( France)
`Characterisation of a xylanolytic amyloglucosidase
`of Termitomyces clypeatus
`A.K. Ghosh, A.K. Naskar and S. Sengupta (lndia) 289
`Kinetic study of the suicide inactivation of latent
`polyphenoloxidase from iceberg lettuce ( Lactuca
`satiua) induced by 4-tert-butylcatechol
`in
`the
`presence of SDS
`S. Chazarra, J. Cabanes, J. Escribano and F.
`Garda-Carmona (Spain)
`Concentration and temperature dependence of vis(cid:173)
`cosity in lysozyme aqueous solutions
`K. Monkos ( Poland)
`
`268
`
`277
`
`297
`
`304
`
`continued
`
`Page 3 of 11
`
`

`

`Efficient purification, characterization and partial
`amino acid sequencing of two a-1,4-glucan lyases
`from fungi
`S. Yu,
`T.M.I.E. Christensen, KM. Kragh,
`K. Bojsen and 1. Marcussen (Denmark)
`Photoaffinity
`labeling of peroxisome proliferator
`binding proteins
`in
`rat hepatocytes; dehy(cid:173)
`droepiandrosterone sulfate- and bezafibrate-bind(cid:173)
`ing proteins
`H. Sugiyama, J. Yamada, H. Takama, Y. Kodama,
`T. Watanabe, T. Taguchi and T. Suga (Japan)
`
`311
`
`Effects of profilin-annexin I associat10n on some
`properties of both profilin and annexin I: modifi(cid:173)
`cation of the inhibitory activity of profilin on
`actin polymerization and inhibition of the self-as(cid:173)
`sociation of annexin I and its interactions with
`liposomes
`M.-T. Aluarez-Martinez, F. Porte, 1.P. Liautard
`and J. Sri Widada (France)
`
`Cumulative Contents, Vol. 1339
`
`321
`
`Author Index
`
`331
`
`341
`
`344
`
`Page 4 of 11
`
`

`

`ELSEVIER
`
`Biochimica et Biophysica Acta 1339 (1997) 304-310
`
`BB
`
`Concentration and temperature dependence of viscosity in lysozyme
`aqueous solutions
`
`Karol Monkos *
`
`Department of Biophysics, Silesian Medical Academy, H. Jordana 19, 41-808 'Zabrze 8, Poland
`
`Received 25 September 1996; revised 2 January 1997; accepted 9 January 1997
`
`Abstract
`
`The paper presents the results of viscosity determinations on aqueous solutions of hen egg-white lysozyme at a wide
`range of concentrations and at temperatures ranging from 5°C to 55°C. lt has been proved that, at each fixed concentration,
`the viscosity-temperature dependence may be quantitatively described by the modified Arrhenius formula. On the basis of
`the generalized Arrhenius formula, the parameters of the Mooney approximation were calculated. It has been concluded that
`lysozyme molecules in aqueous solution behave as hard quasi-spherical particles. By applying an asymptotic forrn of the
`generalized Arrhenius formula, such rheological quantities as the intrinsic viscosity and Huggins coefficient were
`calculated.
`
`Keywords: Lysozyme; Viscosity; Arrhenius formula; Activation energy; Huggins coefficient
`
`1. Introduction
`
`Hen egg-white lysozyme is a .well-known enzyme
`that acts as a glycoside hydrolase. This small globular
`protein consists of two functional domains located on
`each side of the active site cleft and contains both
`helices and regions of {3 sheet, together with loop
`regions, turns and disulfide bridges [l]. Its structure,
`dynamics and hydration have recently been studied
`extensively by a wide range of experimental tech(cid:173)
`niques including 1 H, 13C and 15N-NMR spectroscopy
`[ 1-6], dielectric spectroscopy [7-9], Fourier trans(cid:173)
`form infrared spectroscopy [10,11] and X-ray crystal(cid:173)
`lography [12]. Sorne sophisticated theoretical meth(cid:173)
`ods have also been applied to tho_se problems in the
`
`• Corresponding author.
`
`literature [13-15].
`lysozyme
`the
`In a solution
`molecules are surrounded by water. lt participates in
`stabilizing the protein structure and stimulating the
`activity of active site. Comparison of solution NMR
`parameters with those predicted by the crystal struc(cid:173)
`tures of the protein has shown the very close similar(cid:173)
`ity of the structure of lysozyme in solution and in
`crystals. Although recent X-ray diffraction studies of
`dehydrated lysozyme crystals have revealed numer(cid:173)
`ous small displacements in the positions of individual
`atoms, the overall conformation does not differ greatly
`from that of the fully hydrated protein [16].
`As is known, size and shape of proteins have an
`important influence on their hydrodynamic properties
`in solution. Such properties have not been sufficiently
`studied for lysozyme solutions. This is especially the
`case for the viscosity of lysozyme solutions. In the
`present study, the results of viscosity determinations
`
`0167-4838/ 97 / $17.00 Copyright © 1997 Elsevier Science B.V. Ali rights reserved. ·
`Pll S0167-4838(97)00013-7
`
`Page 5 of 11
`
`

`

`K. Monkos / Biochimica et Biophysica Acta 1339 ( 1997) 304-310
`
`305
`
`on aqueous solutions of hen egg-white lysozyme at a
`, wide range of concentrations and at temperatures
`ranging from 5°C to 55°C are presented. On the basis
`of these measurements, a generalized Arrhenius for(cid:173)
`rnula [ 17] connecting the viscosity with temperature
`and concentration of the dissolved proteins is dis(cid:173)
`cussed. At a fixed temperature the generalized Arrhe-
`1 nius formula gives the Mooney approximation [18],
`i.e., the viscosity-concentration relationship, and al-
`1 · Jows calculation of the coefficient S and the self-
`1 crowding factor K, as well, in Mooney's formula. At
`J Jow concentrations, an asymptotic form of the gener(cid:173)
`J alized Arrhenius formula is presented.
`
`2. Materials and methods
`
`2.,J. Materials
`
`Hen egg-white lysozyme was purchased from
`Sigma Chemical Company and was used without
`l further purification. From the crystalline form the
`1 material was dissolved in distilled water and the
`l solution was treated to remove dust particles with
`filter papers. The samples were stored at 4°C until
`l just prior to viscometry measurements, when they
`were warmed from 5°C to 55°C. The pH values of
`'. such prepared samples were about 7 .0 and varied
`slightly with different protein concentrations.
`
`2.2. Viscometry
`
`The viscosity of lysozyme solutions was previ(cid:173)
`ously investigated by Lefebvre [ 19]. The author has
`reported results mainly for lysozyme in fully dena(cid:173)
`tured state, i.e. , in the random coil conformation. For
`1 the solutions of lysozyme in the native state, the
`author has shown that the flow behaviour was New(cid:173)
`tonian for shear rates from O to at least 128.5 s - 1 and
`up to a concentration of at least 370 kg / m3
`. This is
`J important information because it justifies the use of a
`• viscometer, which is a simple and convenient experi(cid:173)
`. mental tool, for viscosity measurements of lysozyme
`solutions.
`Viscosity was measured with an Ubbelohde-type
`capillary microviscometer with a flow time for water
`of 28.5 s at 25°C. The microviscometer was placed in
`a waterbath controlled thermostatically at 5°C to
`
`55°C ± 0.05°C. The same viscometer was used for all
`measurements and was mounted so that it always
`occupied the same position in the bath. Flow times
`were recorded to within 0.1 s. Solutions were temper(cid:173)
`ature-equilibrated and passed once through the capil(cid:173)
`Iary viscometer before any measurements were made.
`For most concentrations the viscosity measurements
`were made from 5°C to 55°C in 5°C intervals. Such a
`range of temperatures was chosen because above
`55°C the thermal denaturation of lysozyme occurs.
`Solutions densities were measured by weighing
`and protein concentrations were deterrnined by a dry
`weight method in which samples were dried at high
`temperature for several hours. This method was re(cid:173)
`cently successfully applied to other globular proteins
`too [20,21]. The viscosities of the lysozyme solutions
`were measured for concentrations from 24.9 kg/m3
`up to 342.6 kg/m3
`•
`
`3. Results and discussion
`
`3.1. Generalized Arrhenius formula
`
`For glass-forming liquids the temperature-depen(cid:173)
`dent change of viscosity is analysed according to the
`Williams-Landel-Ferry equation, which tums out to
`be applicable in the range from glass transition tem(cid:173)
`perature Tg to about Tg + 100°C [22]. At higher tem(cid:173)
`peratures an equation of the Arrhenius form is widely
`used:
`
`(1)
`
`where, E, R and Tare viscosity, activation energy of
`viscous flow, gas constant and absolute temperature,
`respectively. However, it gives a good approximation
`to the experimental values only in relatively narrow
`temperature ranges. Viscosity data, when taken at a
`sufficiently wide range of temperatures, show that the
`dependence of ln 1J on T- 1 is not linear, i.e., Eq. (1)
`is not a good one. Very recently we have proved, on
`the basis of the results of viscosity determinations on
`aqueous solutions of bovine serum albumin, that the
`most useful relation connecting the viscosity with
`temperature is a somewhat modified Arrhenius for(cid:173)
`mula. It describes the viscosity-temperature depen-
`
`Page 6 of 11
`
`

`

`306
`
`K. Monkos / Biochimica et Biophysica Acta 1339 ( 1997) 304-310
`
`dence from the neighbourhood of solution freezing(cid:173)
`point up to the vicinity of the temperature where the
`protein' s thermal denaturation occurs and bas the
`form [17):
`
`77 = exp( -B +DT+ :; )
`
`(2)
`
`where B and D are parameters and Es is the activa(cid:173)
`tion energy of viscous flow of solution. The above
`formula is identical to the Arrhenius relation in Eq.
`(1), if the pre-exponential factor A= exp (-B + DT).
`Fig. 1 shows the results of lysozyme solution viscos(cid:173)
`ity measurements at the highest concentration we
`studied here. As seen, a curve obtained by using the
`relation from Eq. (2) gives a very good fit to the
`experimental points over the whole range of tempera(cid:173)
`tures. For the smaller concentrations a situation is the
`same. For all concentrations the parameters B, D and
`Es were calculated by using the least squares method.
`It is worth noting that the experimental values of
`water viscosity ( when given in centipoise) agree very
`well with those calculated on the basis of Eq. (2)
`when B = Bw = 25.94, D = Dw = 0.02 K - 1 and Es
`= Ew = 32.01 kJ /mol.
`An activation energy of viscous flow E can be
`interpreted within the frame of applications of the
`absolute rate theory to the process of flow [23). In
`this theory E is identified as the activation energy for
`the jump of a molecule from one equilibrium position
`in the liquid to the next or as a minimum energy
`required for a molecule of the solution to escape the
`
`80
`
`70
`
`IO
`
`'b ~
`f' 40
`"' 0 :i: 30
`
`;; 211
`
`10
`
`0
`
`t[Cl
`
`Fig. 1. Temperature dependence of the viscosity of hen egg-white
`lysozyme aqueous solution for the concentration c = 342.6
`kg / m3
`• (e ) Experimental points; the curve shows the fit ob(cid:173)
`tained by using Eq. (2) with B = 122.94, D = 0.169 K- 1 and
`E. = 169.39 kJ / mol.
`
`~
`
`100
`
`1SO
`200
`C [kg/m"J]
`
`250
`
`300
`
`350
`
`Fig. 2. Plot of the solution activation energy E, versus concentra(cid:173)
`tion. (e) experimental points are obtained by using the least
`squares method; the curve shows the fit according to Eq. (3) with
`a= 7.956· 10 5 kg/m3, v = 2.592 · I0 - 3 m 3 / kg, Ew = 32.01
`kJ / mol and EP = 3.97·10 4 kJ / mol.
`
`influence of its neighbouring molecules [24). Fig. 2
`shows that Es is a monotonically increasing fonction
`of concentration. One can easily explain this fact. In
`a streamline flow of a solution molecules of both
`water and dissolved proteins take part. Therefore, the
`activation energy Es should be a superposition of the
`activation energy of water Ew and protein EP
`molecules. As bas been shown in our earlier work
`[17), this leads to the following relation:
`
`( Ep - Ew) + Ew
`
`Es =
`
`C
`
`Π-
`
`/3c
`
`where
`
`(3)
`
`(4)
`
`(5)
`
`and
`/3= av-1
`The quantities c, Pw, v , MP and Mw denote the
`solute concentration and water density in kg / m3. the
`effective specific volume of a protein and the molec(cid:173)
`ular masses of the dissolved proteins and water,
`respectively. The effective specific volume is the
`constant of proportionality between the effective mo- .
`Jar volume and the molar mass of a macrosolute [25).
`The molecular weight of hydrated ben egg-white
`lysozyme is 14 320 Da [26). So, in this case a ==
`7.956 X 10 5 kg / m3. In Eq. (3) the activation energy
`of dissolved particles EP and the effective specific
`volume of a particle v must be taken into account as
`two adjustable parameters. The parameters were cal-
`
`Page 7 of 11
`
`

`

`K. Monkos / Biochimica et Biophysica Acta 1339 ( 1997) 304- 310
`
`307
`
`[27], in the case of aqueous solutions of globular
`proteins, the most useful functional form describing
`the dependence of relative viscosity on concentration
`is that of Mooney [1 8]:
`
`1Jr = exp[ l ~:<P]
`
`{10)
`
`where <P
`is the volume fraction of the dissolved
`particles, K is a self-crowding factor and S denotes
`the parameter which, in general, depends on the
`shape of the dissolved particles and on hydrodynarnic
`interactions of proteins in solution. The volume frac(cid:173)
`tion <P = NA V c / M where NA, V and M are Avo(cid:173)
`gadro' s number, the hydrodynamic volume of one
`dissolved particle and the molecular weight, respec(cid:173)
`tively. One can easily show that Mooney's relation,
`given in the above form, is identical with Eq. (8), if
`the parameter
`MW
`S=---
`
`and the self-crowding factor
`
`) MP
`Mw
`(
`K= v- PwMp NAV
`
`( 12)
`
`Both coefficients can be calculated when the hydro(cid:173)
`dynamic volume of the dissolved proteins is known.
`· Hen egg-white lysozyme molecules can be treated as
`prolate ellipsoids of revolution with the main axes
`a = 4.5 nm and b = 3 nm [26]. It gives the hydrody(cid:173)
`namic volume of hen egg-white lysozyme V= 2.12
`X 10- 26 m 3
`. Fig. 3 shows the numerical values of
`the parameter S obtained from Eq. (11). As is seen
`this parameter decreases monotonically with increas(cid:173)
`ing temperature from S = 3.425 (at t = 5°C) up to
`s = 2.697 (at t = 55°C).
`The Mooney formula describes the viscosity-con(cid:173)
`centration dependence from dilute up to concentrated
`solutions and the parameters S and K do not depend
`on concentration. For dilute solutions, i.e., in the
`limit <P - 0, one can transform the Eq. (10) into the
`form: 1Jr = 1 + S<P. This is the well-known relation
`developed by Einstein, who has proved that for hard
`spherical particles immersed in a solution, S = 2.5.
`
`culated by using the least squares method too. The
`obtained values are as follows : EP = 3.97 X 10 4
`kJ/ mol and v = 2.592 X 10 - 3 m 3/kg. As seen in
`) fig. 2, Eq. (3) gives good approximation to the
`1 experimental values then. Differences in the values of
`l activation energy of water (Ew = 32.01 kJ/mol),
`
`J'
`
`lysozyme ( EP = 3.97 X 10 4 kJ / mol) · and bovine
`1 serum ~lbumin (_EP = 5.374 X 10 5 kJ/ mol) [17] show
`th1s quant1ty depends strongly on molecular
`· that
`weights and dimensions of the molecules . To estab-
` lish the ~xact dependence between those qua~tities,
`the expenmental values of EP for more protems are
`needed.
`{ The coefficients B and D from Eq. (2) depend on
`1 concentration exactly in the same way as Es- Both B
`and D increase monotonically with increasing con-
`
`tions as for Es:
`C
`( Bp - Bw) + Bw
`
`j
`
`B =
`1
`
`a -
`
`(3c
`
`l. centration. Therefore, one can write the same rela(cid:173)
`l and
`
`( 6)
`
`(7)
`
`c
`
`D =
`
`( Dp - Dw) + Dw
`
`a -
`(3c
`1
`where v, BP and v , DP are adjustable parameters.
`l The above relations give a good fit to the experimen(cid:173)
`, ta! values for v = 2.615 X 10 - 3 m3 /kg, BP = 2.642
`X 10 4 and for v = 2.6 X 10- 3 m 3 / kg, DP = 41.97
`K- 1
`• The three values of the effective specific vol(cid:173)
`, urne obtained above differ each other only slightly
`1 and give the average value ( v ) = 2.602 X 10- 3
`m3 / kg. By substituting Eq. (3), (6) and (7) into Eq.
`' (2), one can obtain the following relation for relative
`viscosity of a solution:
`1J
`
`1),= -
`1Jo
`
`1
`
`'1
`
`a -
`
`[-(Bp-Bw)+(Dp-Dw)T
`c
`=exp{
`(3 c
`+ Ep ;TEW]}
`where 170 denotes viscosity of water:
`
`1/0 = exp( - Bw + DwT + ;; )
`
`(8)
`
`(9)
`
`At a fixed temperature, Eq. (8) describes the viscos(cid:173)
`ity-concentration dependence of a solution. On the
`other hand, as has been shown in our earlier work
`
`Page 8 of 11
`
`

`

`308
`
`K. Monkos/ Biochimica et Biophysica Acta 1339 ( 1997) 304-310
`
`508
`
`Simha [28] has extended the technique employed by
`Einstein for a suspension of nonspherical particles.
`The viscosity contribution caused by a suspension of
`nonspherical particles depends on their orientation
`with respect to the direction of flow of the fluid . This
`orientation is modified by two effects: (1) the orienta(cid:173)
`tion of the principal axis of the particles to the flow
`direction by the flow and (2) the rotational Brownian
`motion produced as a reaction of the particles to the
`random heat motions of the solvent. The latter effect
`acts against the first one. Simha showed that in the
`case when the Brownian motion prevails and the
`orientation of particles is completely at random, the
`factor S depends, in a very complicated way, on the
`axial ratio p = a/ b of the dissolved particles. How(cid:173)
`ever, for ellipsoids of revolution for which 1 < p <
`15, the asymptotic formula can be used [29]:
`s = 2 .5 + 0.4075( p - 1 ) 1
`(13)
`'
`Hen egg-white lysozyme has the axial ratio p = 1.5
`and the above formula gives S = 2.643 then. This
`value agrees well with the experimental one in the
`high temperature limit (t = 55°C), as could be ex(cid:173)
`pected. At the same time, it indicates that the hen
`egg-white lysozyme molecules in aqueous solution
`behave as hard quasi-spherical particles.
`The self-crowding factor K (Eq. (12)), in turn,
`does not depend on temperature. Substitution of the
`volume V= 2.12 X 10 - 26 m 3 into Eq. (12) gives the
`numerical value K = 2.91. It is worth noting that in
`his original work (18]. Mooney showed, on the basis
`of purely geometric arguments, that for rigid spheri(cid:173)
`cal particles, values of K should lie between 1.35
`and 1.91. There are no theoretical estimations of the
`self-crowding factor for ellipsoids of revolution. In
`this case the values of the factor K should, undoubt(cid:173)
`edly, lie in a wider range than for spherical particles.
`Therefore, all experimental values of K are valuable.
`It is not possible to obtain the exact value of the
`parameters S and K when the hydrodynamic volume
`of the dissolved proteins is not known. In this case,
`one can obtain only the ratio of K / S by the method
`proposed by Ross and Minton (30]. The authors
`transformed Mooney ' s formula (Eq. (10)) into the
`forrn with only one parameter K/S, but then it is
`necessary to know the intrinsic viscosity. The ratio
`K / S was obtained by the authors for two sets of data
`for human hemoglobin. However, as has appeared,
`
`s
`
`10
`
`20
`
`30
`t[C)
`
`40
`
`50
`
`so
`
`Fig. 3. The parameter S, given by Eq. (11), as a function of
`temperature .
`
`the fit of the modified Mooney 's formula to the
`experimental points was not the best one, especially
`in the moderately concentrated region. The problem
`was discussed in detail in our previous paper [21 ) and
`an alternative treatment of the problem has been
`proposed there.
`
`3.2. Low concentration limit
`
`At low concentrations, the relation between the
`solution viscosity and the concentration may be ex(cid:173)
`pressed by the polynominal [31]:
`
`T/sp = [71] +k,[71] 2c+k2 [T1] 3c2 +k3[T/] 4c3+ ...
`C
`
`(14)
`
`where
`.
`[ l 1
`T/sp
`T/ = 1m -
`c--+ 0 C
`is the intrinsic viscosity, T/sp = T/r - 1 is the specific
`viscosity, the dimensionless parameter k I is the Hug(cid:173)
`gins coefficient and k2 , k 3 are the higher coefficients
`of the expansion. In our case the generalized Arrhe(cid:173)
`nius Eq. (8) can be expanded in the power series of
`concentration, too. An expansion identical to that in
`Eq. (14) can be obtained from:
`
`[ ~] - : [- ( B, - B.) + ( D, - D. )T + E, ;/• 1
`(15)
`
`k1
`
`- ~[-(B,,- B, ) + (D:~ D. )T+ E, ; /• + ll
`
`(16)
`
`Page 9 of 11
`
`

`

`K. Monkos / Biochi111ica et Biophysica Acta J 339 ( 1997) 304-31 O
`
`309
`
`fable 1
`The numerical values of the intrinsic viscosity [ T]] and the Huggins coefficient k 1 for hen egg-white lysozyme calculated from Eq. (15)
`} and Eq. (16)
`) tlC]
`l [r,l X 10 3[m 3 / kg]
`k 1
`
`5
`
`3.05
`1.35
`
`10
`
`2.94
`1.39
`
`15
`
`2.83
`1.42
`
`20
`
`2.74
`1 .45
`
`25
`
`2.66
`1.48
`
`30
`
`2.59
`1.50
`
`35
`
`2.54
`1.53
`
`40
`
`2.49
`1.54
`
`45
`
`2.45
`1.56
`
`50
`
`2.42
`1.57
`
`SS
`2.41
`1.58
`
`The higher coefficients of expansion have a more
`complicated form and are omitted here. As seen, both
`intrinsic viscosity and the Huggins coefficient depend
`J on temperature. This is the case for the higher coeffi(cid:173)
`l cients of expansion, too. The numerical values of [Y/]
`'i and k 1 for hen egg-white lysozyme are presented in
`Fig. 4 and in Table 1. It is worth noting that the
`numerical value of the intrinsic viscosity calculated
`
`l from Eq. (15) at t = 25°C ([r,] = 2.66 X 10 - 3 m3 /kg)
`agrees very well with the value given in the literature
`l at the same temperature ([Y/] = 2. 7 10 - 3 m3 /kg)
`l [19].
`· A lot of numerical values of k I have been pub(cid:173)
`J 1ished in the literature ([31] and refs. therein). The
`lack of agreement for different liquids and solutions
`1 ensues from the fact that the intermolecular hydrody(cid:173)
`namic interactions can be different and, in each case,
`1 k1 should be studied in detail separately. Thus, the
`' experimental values of k I are desirable. There are no
`theoretical estimations of the higher coefficients of
`expansion in Eq. (14). However, on the basis of the
`l generalized Arrhenius formula (8), expansion in the
`, power series of concentration, the coefficients k 2 , k 3
`and so on can be calculated. As has appeared, they
`
`0,0032
`
`1,6
`
`~ ·;;;
`
`Q
`
`·! ~ 0,0028
`·-....,
`.5 è
`.5 .. ..c <"'
`
`_; _ 0,0028
`
`1 I
`
`0,0024
`
`1,55 C
`"' ·c
`1,5 E
`~ ...
`1,45 f
`"ëi,
`""
`t:Ê .. .c
`
`1,35 <"'
`
`1,4
`
`15
`
`25
`
`35
`
`45
`
`55
`
`t[CJ
`
`' Fig. 4. The intrinsic viscosity [ 1j] (-) and the Huggins coeffi-
`cient k 1
`) , given by Eq. ( 15) and Eq. ( 16), respec-
`(
`. lively, as a function of temperature.
`
`are connected with the Huggins coefficient k I in the
`following way:
`
`")
`k = k- -
`1
`2
`
`1
`-
`12
`
`1
`1
`k = k - -k + -
`4 1
`24
`
`3
`
`1
`
`3
`
`1
`1
`k = k 4 - -k 2 + -k -
`2 1
`6 1
`
`4
`
`1
`
`(17)
`
`( 18)
`
`( 19)
`
`1
`-
`80
`
`independently of temperature. The above relation(cid:173)
`ships show that any property of a solution which
`determines the value of k I will also determine the
`value of k 2 , k 3 and so on. In other words, the Eqs.
`( 17)-( 19) provide a test for any theoretical treatment
`of dilute solution viscosity.
`..-·
`
`4. Conclusions
`
`The viscosity of hen egg-white lysozyme solutions
`at temperatures up to 55°C and in a wide range of
`concentrations at neutral pH may be quantitatively
`described by the generalized Arrhenius formula (8).
`This formula enables the calculation of parameters in
`Mooney's approximation. The value of the parameter
`S at high temperature limit indicates that lysozyme
`molecules in aqueous solution behave as hard quasi(cid:173)
`spherical particles. The asymptotic form of the gener(cid:173)
`alized Arrhenius formula for small concentrations
`enables the calculation of the intrinsic viscosity and
`the Huggins coefficient. Both quantities depend on
`temperature. The higher coefficients of expansion in
`the power series of concentration k 2 , k 3 and so on,
`are connected with the Huggins coefficient and the
`Eqs. (17)-( 19) provide a test for any theory of dilute
`solution viscosity .
`
`Page 10 of 11
`
`

`

`310
`
`References
`
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`
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`Page 11 of 11
`
`