Page 1 ofll
`
`CSL EXHIBIT 1050
`
`CSL V. Shire
`
`Page 1 of 11
`
`CSL EXHIBIT 1050
`CSL v. Shire
`
`

`

`© 1996, Elsevier Science B.V. Ali rights reserved.
`
`•
`
`.,,,.:;-:-,,-r'?--< -
`., . ... -..
`Il\-··
`. 1 1-{>"' "~!(
`. ~ i'J l ' . •
`~~-;*
`...... •
`~'.'"- '\
`_;.:
`.:,. '·dJi -;:, I
`. . ? ~
`.. ..... .
`~·
`\~'- i- ","•f. •.}. ~j
`
`No part of this publication may be reproduced, stored in a retrievsfs?em or transnùtted, in any form or by any means, electronic, mechanical
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`Printed in the United Kingdom
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`Page 2 of 11
`
`

`

`Vol. 18, Nos. 1,2
`
`Biological
`l\1acrotnolecules
`
`51 RI.;( TliMI
`
`I UNCl ION ANI> INII MAC! IONS
`
`CONTENTS
`
`February 1996
`
`This journal is abstracted/indexed in Current Contents; Polymer Contents; Index Medicus; Food Science and Technology Abstracts;
`Excerpta Medica; EMBASE
`
`Ab initio tertiary-fold prediction of helical and non-helical protein chains using a genetic algorithm
`T. Dandekar, P. Argos (Germany) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`Mixed iota-kappa carrageenan gels
`M.J. Ridout, S. Garza, G.J. Brownsey, V.J . Morris (UK; Spain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`Characterization and solution properties of a new exopolysaccharide excreted by the bacterium Alteromonas sp. strain
`1644
`L. Bozzi, M . Milas, M . Rinaudo (France) .. .. ... ... .... .. ....................... . .............. ..... . .
`Chemical modification of silk fibroin with N-acetyl-chito-oligosaccharides
`Y. Gotoh, M. Tsukada, S. Aiba, N. Minoura (Japan) ................................................. .
`Dielectric studies of proton transport in air-dried fully calcified and decalcified bone
`E. Marzec, L. Kubisz, F. Jaroszyk (Poland) ......... ... ... . ......... . .......................... .... .. .
`Efîect of pressure on the deuterium exchange reaction of cr-lactalbumin and /3-lactoglobulin
`N. Tanaka, S. Kunugi (Japan) ............................... . .... . ................................. .
`The interaction of sodium dodecyl sulfate and urea with cat-fish collagen solutions in acetate bufTer: hydrodynamic
`and thermodynamic studies
`C. Rose, A.B. Mandai (lndia) ............ .. ........................................................ .
`Conformational transition of insulin induced by n-alkyltrimethylammonium bromides in aqueous solution
`C. Pombo, G. Prieto, J.M. del Rio, F. Sarmiento, M.N. Jones (Spain; UK) ..................... . ........ .
`Viscosity of bovine serum albumin aqueous solutions as a fonction of temperature and concentration
`K. Monkos (Poland) . ... .. . ........ .... . ... ........... . . . ........................... ..... ......... .
`Relations between aggregative, viscoelastic and molecular properties in gluten from genetic variants of bread wheat
`J. Hargreaves, Y. Popineau, M . Comec, J. Lefebvre (France) .......... . . ... ...... .. .................... .
`Probing structure-activity relationship in diamine oxidase -
`reactivities of lysine and arginine residues
`M.A. Shah, S. Tayyab, R. Ali (India) ................................................ ....... . ........ .
`Solution and gel rheology of a new polysaccharide excreted by the bacterium Alteromonas sp. strain 1644
`L. Bozzi, M . Milas, M . Rinaudo (France) ............. . ....... .. .................................. .. . .
`Dependence of the content of unsubstituted (cellulosic) regions in prehydrolysed xanthans on the rate of hydrolysis
`by Trichoderma reesei endoglucanase
`B.E. Christensen, O. Smidsrpd (Norway) ............................................... ... ........ ... .
`Molecular dynamics simulations of hybrid and complex type oligosaccharides
`P.V. Balaji, P.K. Qasba, V.S.R. Rao (USA) .......... . ............... . ............ .. ................ . .
`Single crystals of V amylose complexed with glycerol
`S.H.D. Hulleman, W. Helbert, H. Chanzy (The Netherlands; France) ............. ...... .. .............. . .
`Muscle contraction: the step-size distance and the impulse-time per ATP
`C.R. Worthington, G.F. Elliott (USA; UK) ....... .. ...................... . ... .. ..................... .
`Rapid size distribution and purity analysis of gastric mucus glycoproteins by size exclusion chromatography/multi angle
`laser light scattering
`K. Jumel, 1. Fiebrig, S.E. Harding (UK; USA) . .... . ................ ... .. ......... ........... . ........ .
`FGF protection and inhibition of human neutrophil elastase by carboxymethyl benzylamide sulfonate dextran
`derivatives
`A. Meddahi, H. Lemdjabar, J.-P. Caruelle, [~_,.Barritault, W. Homebeck (France) ......... . .............. . .
`Lymphocyte activation efîect of (1-6)-2,5-anhydro-D-glucitol and it derivatives with 3,4-di-0-methyl and sulfate
`groups
`T. Kakuchi, S. Umeda, T. Satoh, K. Yokota, T. Yuhta, A. Kikuchi, S. Murabayashi (Japan)
`
`Short communication
`Utilization of DNA as functional materials: preparation of filters containing DNA insolubilized with alginic acid gel
`K. Iwata, T. Sawadaishi, S.-1. Nishimura, S. Tokura, N. Nishi (Japan) .................................. .
`
`€) The paper used in this publication meets the requirements of ANSI/NISO 239.48-1992 (Permanence of Paper)
`
`1111111111111111 IIII ~I Ill 1111111111111111
`
`0141-8130(199602)18:1/2;1-8
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`19-26
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`27-31
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`33-39
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`41-53
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`55-60
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`61-68
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`05048
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`Page 3 of 11
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`

`

`International Journal of Biological Macromolecules
`18 (1996) 61-68
`
`Biological
`Maaomolecules
`
`Viscosity of bovine serum albumin aqueous solutions as a function of
`temperature and concentration
`
`K. Monkos
`
`Department of Biophysics, Silesian Medica/ Academy, H. Jordana 19, 41-808 Zabrze 8, Po/and
`
`Received 10 March 1995; revision received 30 May 1995; accepted 2 June 1995
`
`This paper presents the results of viscosity determinations on aqueous solutions of bovine serum albumin (BSA) at a wide range
`concentrations and at temperatures ranging from 5°C to 45°C. On the basis of these measurements a general formula connecting
`relative viscosity 'Ir with temperature T and concentration c of the dissolved proteins was established:
`
`-
`(
`c
`:exp - - -B+DT+ - -
`[
`RT
`a-{3c
`quantities a, {3, B, J5 and AË are described in the text below. A simple substantiation of the formula was also given. This rela(cid:173)
`givcs immediately the Mooney approximation and allows the prediction of the values of the parameter Sand a self-crowding
`r Kin this approximation. By applying an asymptotic form of the formula such rheological quantities as the intrinsic viscosity
`Huggins coefficient were calculated.
`
`- AË)]
`
`l,ywords: Viscosity; Bovine serum albumin; Arrhenius formula; Huggins coefficient
`
`Viscosity of liquids is highly dependent on tempera(cid:173)
`. Even for simple liquids viscosity-temperature rela(cid:173)
`ips are quite complex. This has been evidenced by
`large number of empirical expressions for this de(cid:173)
`ence which have appeared in the literature (1,2].
`glass forming liquids the Williams-Landel-Ferry
`tion [3] is widely used, which turns out to be ap(cid:173)
`ble in the range from glass transition temperature
`to about T8 + l00°C. At higher temperatures the
`rature-dependent change of viscosity is usually
`ysed according to an equation of the Arrhenius
`
`'1, t:.E, Rand Tare viscosity, activation energy of
`
`{l)
`
`l-8130/96/$15.00 © 1996 Elsevier Science B.V. Ali rights reserved
`l 0141-8130(95(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3))01057-2
`
`viscous flow, gas constant and absolute temperature,
`respectively. The pre-exponential factor A is considered
`to be independent or approximately independent oftem(cid:173)
`perature. Equation (l) is still widely used for different li(cid:173)
`quids (4-7). Avery convenient way of data presentation
`in this case, consists of plotting the viscosity versus r 1
`in a log-normal plot. In relatively narrow temperature
`ranges the dependence of ln r, on 1 1 is then linear.
`However, viscosity data when taken at a sufficiently
`wide range of temperatures, even for such a simple liq(cid:173)
`uid as water, shows that this relationship is only approx(cid:173)
`imately linear. Another empirical equation that is more
`accurate over wider ranges of temperature is the
`Vogel-Fulcher's expression (discussed in detail below),
`successfully established for many liquids (8,9).
`As far as is known, very little attention has been paid
`to the analysis of the viscosity-temperature relationship
`for protein solutions. In the present study, the results of
`viscosity determinations on aqueous solutions of bovine
`serum albumin (BSA) at a wide range of concentrations
`
`Page 4 of 11
`
`

`

`62
`
`K. Monkos / International Journal of Biological Macromolecules 18 ( 1995) 61-68
`
`T0 [K]
`
`220
`
`200
`
`180
`
`160
`
`140
`
`550
`
`450
`
`350
`
`150
`
`100
`
`Fig. 1. Plots of the parameters Z ( x) and T0 <•) in the Vogel(cid:173)
`Fulcher's fonction versus concentration; the curves show the fit ac(cid:173)
`cording to expressions (3) and (4), respectively.
`
`relationship over a broad temperature range can be
`described by the Vogel-Fulcher's expression:
`
`2-
`
`(2)
`
`1/ = W exp ( -
`)
`T- T0
`where W, Z and T0 are parameters. As it has appeared,
`the fonction may well fit the results of the viscosity
`measurements for BSA aqueous solutions in the whole
`range of concentrations. However, the adjustable par(cid:173)
`ameters W, Z and T0 appeared to be dependent on con(cid:173)
`centration and, for each fixed concentration, had to be
`calculated separately. The calculations were made by
`applying a non-linear regression procedure in the com(cid:173)
`putational program STATGRAPHICS (version 5.0).
`The results are shown in Figs. 1 and 2. As we can see
`each parameter depends on concentration in a different
`
`lnW
`
`0
`
`-1
`
`-2
`
`-3
`
`are presented. On the basis of these measurements an
`analysis of applicability of the Vogel-Fulcher's expres(cid:173)
`sion is conducted. Because of some faults of such an
`based on an equa(cid:173)
`analysis, an alternative treatment -
`tion of the Arrhenius form -
`is proposed. As a result,
`a generalized Arrhenius formula connecting the viscos(cid:173)
`ity with temperature and concentration of the dissolved
`proteins is established. The obtained formula gives, in
`the natural way, the Mooney approximation [10). At the
`same time it allows prediction of the coefficient S and
`the self-crowding factor K, as well, in Mooney's for(cid:173)
`mula. At low concentrations, the intrinsic viscosity and
`the Huggins coefficient from the generalized Arrhenius .
`formula are determined.
`
`2. Experimental
`
`2.1. Materials
`BSA (Cohn fraction V) was purchased from Sigma
`Chemical Company and was used without further puri(cid:173)
`fication. From the crystalline form the material was
`dissolved in distilled water and then filtered by means of
`filter papers in order to remove possible undissolved
`fragments. The samples were stored at 4°C until just
`prior to viscometry measurements, when they were
`warmed from 5°C to 45°C. The pH values of such pre(cid:173)
`pared samples were about 5.2, i.e. in the neighbourhood
`of the isoelectric point of BSA and changed only in(cid:173)
`significantly during the dilution of the solutions.
`
`2.2. Viscometry
`Capillary viscosity measurements were conducted
`using an Ubbelohde microviscometer placed in a water(cid:173)
`bath controlled thermostatically at 5°C to 45°C ±
`0.05°C. The same viscometer was used for ail measure(cid:173)
`ments 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 temperature-equilibrated
`and passed once through the capillary viscometer before
`any measurements were made. For most concentrations
`the viscosity measurements were made from 5°C to 45°C
`in 5°C intervals. Such a range of temperatures was
`chosen because the secondary structure of BSA is rever(cid:173)
`sible only below 45°C [11). Beyond this temperature,
`BSA !oses the reversibility of secondary structure in the
`thermal denaturation. Therefore, we will call the tem(cid:173)
`perature 45°C the high temperature limit.
`Solution densities and concentrations were measured
`by weighing as described previously [12). The viscosities
`of the BSA solutions were measured for concentrations
`from 17.6 kg/m 3 up to 363.4 kg/m 3
`.
`
`3. Results and discussion
`
`3.1. Voge/-Fulcher's function
`As was mentioned above, the viscosity-temperature
`
`Fig. 2. Plot of the parameter Win the Vogel-Fulcher's fonction versus
`concentration in a log-normal plot; a straight line shows the fit accord·
`ing to relation (5).
`
`Page 5 of 11
`
`

`

`K. Monkos I International Journal of Biological Macromolecu/es 18 ( 1995) 61-68
`
`63
`
`..-nner. Functions, which give the best fit to the data in
`f'JIS· 1 and 2 are as follows:
`
`(3)
`wbere a 1 = 558.9K, a2 = 2.78 Km 3kg- 1 and a 3 = 5.19 x
`10-3 K.m6kg-2
`
`(4)
`wbere b1 = 139.9K, b2 = 0.48 Km 3kg- 1 and b3 = 7.04
`X 10-4 K m6 kg-2.
`
`(5)
`
`wbere d1 = 3.71 and d2 = 1.36 x 10-2 m 3 kg- 1 and W
`11 given in centipoise. lt is obvious that the parameters
`ai, b1 and d1 define the viscosity of water.
`The Vogel-Fulcher's expression (2) in conjunction
`with functions (3), (4) and (5) gives a complete descrip(cid:173)
`tion of the viscosity as a fonction of temperature and
`concentration for our samples. However, the scheme
`clescribed above has several faults: (i) none of the par(cid:173)
`ameters in expression (2) has a clear physical meaning,
`(û) it is difficult to ex plain the concentration dependence
`of the parameters Z, T0 and W and (iii) it is necessary
`to calculate five numerical constants (a 2, a3, b2, b3, d2)
`to describe the viscosity as a fonction of temperature
`and concentration for a given protein. Taking this into
`consideration, an alternative treatment of the problem
`bas been proposed below.
`
`40
`
`30
`
`20
`
`/
`
`/
`
`10 j__ _
`
`___J___•:C../----'-- - - ' - - ----'----'----'--
`3.1
`3 2
`3 3
`3 4
`3 5
`3 .6
`r -•. 10'[K-')
`
`3.2. Generalization of the Arrhenius formula
`The results of the viscosity measurements for BSA
`aqueous solutions have shown that the plot of ln 7/ ver(cid:173)
`sus 1 1, for each fixed concentration, is not linear (see
`Fig. 3). This is especially seen in the case of higher con(cid:173)
`centrations. To establish an exact relationship between
`7/ and T, for a fixed concentration, the viscosity for the
`concentration c = 291 kg/m 3 was measured at temp(cid:173)
`eratures ranging from 5°C to 45°C in 2°C intervals.
`Then, it was assumed that: (i) relation of the type ln
`7/ = ln F + GIT is folfilled, and (ii) the quantities F and
`G are quasi-constant between temperature t and t + 2°C
`(i.e. between 5°C and 7°C, 7°C and 9°C and so on). The
`obtained values of ln F and G were considered to be
`exact in the middle of each temperature range (i.e. for
`6°C, 8°C and so on). As has appeared, both ln F and G
`are linear fonctions of temperature: -ln F = ô1 -
`'YI T
`and G = ô2 -
`-y2T, where ô1, -y 1, ô2 and -y2 are constants.
`Finally, the viscosity of a solution, for each fixed con(cid:173)
`centration, could be written in the following way:
`
`!::.Es)
`7/ = exp - B + DT+ RT
`(
`
`(6)
`
`where B = ô1 + -y2, D = 'Yi and !::.Es = Rô2•
`The above formula is identical to the Arrhenius rela(cid:173)
`tion ( 1 ), if the pre-exponential factor A = exp(-B +
`D1). The calculation of numerical values of the coeffi(cid:173)
`cients B, D and !::.Es by using the method described
`above is troublesome. This method was used only for
`the determination of the exact relation between viscosity
`and temperature (equation (6)). Numerical values of the
`coefficients B, D and !::.Es for ail measured concentra(cid:173)
`tions were calculated by applying a method described in
`the Appendix. The results are presented in Figs. 4 and
`5. As seen in Fig. 6 a curve obtained by using the rela(cid:173)
`tion (6) gives a perfect fit to the experimental points over
`the whole range of temperatures. lt is also worth noting
`that the experimental values of water viscosity (when
`given in centipoise) agree very well with these calculated
`on the basis of relation (6) when B = Bw = 25.94,
`D = Dw = 0.02 K -I and !::.Es = !::.Ew = 32.01 kJ/mol.
`Fig. 4 shows that the activation energy ofviscous flow
`of solution !::.Es is a monotonically increasing fonction
`of concentration. How can one explain this fact? In a
`streamline flow of a solution molecules of both water
`and dissolved proteins take part. Therefore, the activa(cid:173)
`tion energy !::.Es should be a superposition of the activa(cid:173)
`tion energy of water !::.Ew and protein t:.EP molecules.
`So, it is quite natural to assume, that:
`
`(7)
`
`:::; 3. Temperature dependence of the viscosity of BSA aqueous solu(cid:173)
`D~--for the concentration c = 291 kg/m 3 in a log-normal plot. (e)
`~rimental points; straight lines show difTerent slopes at difTerent
`""""l'Cratures.
`
`where XP and Xw are molar fractions of the dissolved
`proteins and water, respectively. The molar fractions are
`defined as follows:
`
`Page 6 of 11
`
`

`

`64
`
`K. Monkos / International Journal of Biologica/ Macromolecu/es 18 ( 1995) 61-68
`
`(8)
`
`where NP and Nw denotes mole numbers of the dissolv(cid:173)
`ed proteins and water, respectively. The mole numbers
`NP= mrJMP and Nw = m.)Mw, where mp and mw are
`masses of the dissolved pro teins and water in a solution,
`respectively, and MP and Mw denote their molecular
`masses. A simple calculation shows that the molar frac(cid:173)
`tion of the dissolved proteins can be rewritten in the
`form:
`
`C
`Xp=------(cid:173)
`M
`C + (p - c) ....:.::.e._
`Mw
`where c and p denote the solute concentration and solu(cid:173)
`tion density in kg/m3, respectively.
`The solution density can be expressed in the form:
`
`(9)
`
`P = Pw + c(I - VPw)
`
`(10)
`
`where Pw and v are the water density and the effective
`specific volume of a protein, respectively. The insertion
`of equation (10) into equation (9) yields:
`
`C
`Xp=-----------
`
`(11)
`
`P ~ - c(p ~ V - 1)
`
`WM
`w
`
`w M
`w
`
`Because the molar fraction of water Xw = 1 - XP
`therefore the substitution of equation (11) into equation
`(7) gives the final form of the activation energy of a
`solution:
`
`150
`
`~110
`E
`' -,
`.,.
`~ w
`<)
`
`70
`
`100
`
`200
`c( kg/m 3
`
`]
`
`300
`
`400
`
`Fig. 4. Plot of the solution activation energy t:.E, versus concentra(cid:173)
`tion.<•) experimental points are obtained by using equation (A4); the
`curve shows the fit according to relation (12) with a = 3.667 X 106
`kg/m3, v = 1.417 x 10-3 m 3/kg,
`t:.Ew = 32.01 kJ/mol and t:.EP
`= 5.374 x 105 kJ/mol.
`
`where
`
`M
`a= Pw ....:.::.e._
`Mw
`
`and
`
`(3 = av - 1
`
`02)
`
`(13)
`
`(14)
`
`The molecular weight of hydrated BSA is 66 000
`a.m.u. [13). So, in this case a = 3.667 x 106 kg!m3.
`Relation (12) shows how the activation energy of a solu(cid:173)
`tion depends on concentration. In the relation the acti(cid:173)
`vation energy of dissolved proteins AEP and the
`effective specific volume of a protein v must be taken
`into account as two adjustable parameters. As seen in
`Fig. 4 relation (12) gives good approximation to the
`experimental values for AEP = 5.374 x 105 kJ/mol and
`V= J.417 X 10-3 m 3/kg.
`In Fig. 5 the experimental values of the coefficients B
`and D from equation (6) are shown. As is seen, they de(cid:173)
`pend on concentration exactly in the same way as ll.E1•
`It suggests that the reasoning presented above for t.E1
`can be repeated in this case, too. Therefore, one can ob(cid:173)
`tain the following relations:
`
`(15)
`
`and
`
`C
`D = - - (Dp - Dw) + Dw
`a - (3c
`where v, Bp and v, Dp are adjustable paramete~s. The
`above relations give a good fit to the expenmental
`values when BP = 3.891 x 105, DP = 648.8 K- 1 and
`
`(16)
`
`B ~ - - - - - - - - - - - - - - - - - : - - - lo [K " " :
`o.i
`
`100
`
`0.1
`
`0
`
`OL._ ____ 1..JOc....0 ___ --::2-c-0-::--0-------;:3::'.00;:-----
`c[~g/m3]
`
`tratioO·
`J).
`Fig. 5. Plots of the coefficients B <•) and D ( x) versus concen
`Experimental points are obtained by using relations (A2) and (A d
`respectively; the curves show the fit according to equations (15) an
`(16), respectively.
`
`Page 7 of 11
`
`

`

`K. Monkos / International Journal of Biological Macromolecules 18 ( 1995) 61-68
`
`65
`
`1417 X 10-3 m3/kg (Fig. 5). The insertion of equa(cid:173)
`= · 2) (1 5) and ( 16) into formula ( 6) yield~ the follow(cid:173)
`:: ~lation for relative viscosity of a solution:
`
`1i-:: _! = exp [ex ~ ~c (- É + DT+ ~;)]
`
`'lo
`
`,rbere 'lo denotes viscosity of water:
`
`(17)
`
`(18)
`
`.....a B. - B - B D = D - D and tJ.Ë = !J.EP - flEw .
`w,
`w
`.....
`-
`p
`p
`Equation (l 7) shows how relative vi~cosity of a s?lu(cid:173)
`UOD depends on temperature and protem concent_ration.
`Because the parameters Bw, Dw and flEw are umversal,
`tbe relative viscosity depends only on four parameters
`fl.E and v and these parameters have to be
`• D
`"A l t
`-,.p,
`P
`mluated for each type of dissolved protem.
`t eas
`two of the discussed parameters (!J.E and v) have _some
`physical meaning. An activation_ene_rgy tJ.E can be mter(cid:173)
`pmed within the frame of applications of the absolute
`rate theory to the process of flow [l]. In this theory tJ.E
`identified as the activation energy for the jump of a
`molecule from one equilibrium position in the liquid to
`tbe next. Differences in the values of activation energy
`of water (ll.Ew = 32.0l kJ/mol) and BSA
`(!J.EP =
`5.374 x 105 kJ/mol) reflect the differences in molecular
`·ghts and dimensions of these molecules. In the ab(cid:173)
`leace of a theoretical guide to the appropriate choice of
`ID effective particle size in solution, it is commonly
`1aUmed that the effective molar volume is proportional
`to the molar mass of a macrosolute; the constant of pro(cid:173)
`ponionality is referred to as the effective specific volume
`'(14).
`
`100
`
`10
`
`3.3. Mooney's approximation
`Despite substantial efforts (15], a useful theory for the
`viscosity of moderately concentrated and concentrated
`solutions does not yet exist. Much effort bas therefore
`been devoted to a search for empirical functional
`representations incorporating a wide concentration
`range ((16] and references therein). However, as bas
`been shown in our earlier work [l 7], in the case of aque(cid:173)
`ous solutions of globular proteins the most useful func(cid:173)
`tional form describing the dependence of relative
`viscosity on concentration is that of Mooney (10].
`
`(19)
`
`where 4> 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 fraction 4> = NA Vhc/ M where
`NA, Vh and M are Avogadro's number, the hydro(cid:173)
`dynarnic volume of one dissolved particle and the mo(cid:173)
`lecular weight, respectively. In his original work (10),
`the author obtained equation (l 9) for bard spherical
`particles for which S = 2.5, so that, in the lirnit 4>-:-0, t?e
`equation yields the expression developed by Emstem:
`'l'/r = l + 2.54>. In the case of particles of arbitrary
`shape, S should exceed 2.5.
`Surprisingly enough, the Mooney relationship may be
`obtained in the natural way from the generalized Ar(cid:173)
`rhenius formula (17). To do it, let us insert expressions
`(13) and (14) into equation (17). After simple transfor(cid:173)
`mations formula (17) can be rewritten in the form:
`
`'l'/r = exp
`[
`
`Mw
`PwNAVh
`
`(- Jj + DT+ tJ.Ë)4> ]
`RT
`) ~ A.
`'*'
`NAVh
`
`M w
`(
`l v----
`PwMp
`
`(20)
`
`This relation is identical with Mooney's formula (19), if
`the parameter
`
`and the self-crowding factor
`
`(21)
`
`(22)
`
`0 - - -- - -1~5 - - - - - -3L0 - - - - - - - :45
`,[oc]
`
`6
`: :
`· lempcrature dependence of the viscosity of BSA aqueous solu(cid:173)
`for the concentration c = 335.6 kg!m 3• (e ) Experimental points;
`shows the fit obtained by using equation (6) with B = 92.11,
`13 K-• and AE, = 124.23 kJ/mol.
`
`As is seen both coefficients can be calculated when
`the hydrody~amic volume of the dissolved proteins is
`known. BSA molecules in aqueous solutions can be
`treated as prolate ellipsoids of revolution with the main
`
`Page 8 of 11
`
`

`

`66
`
`K. Monkos I International Journal of Biologica/ Macromolecules 18 ( 1995) 61-68
`
`5.9
`
`s
`
`5 .6
`
`5 .3
`
`5 .0
`
`curve, so that linear extrapolation gives a serious err
`.
`[
`or
`m 77) and k 1•
`The problem can be treated in another way [20]. I
`our case the generalized Arrhenius formula (17) can ~
`expanded in the power series of concentration. Limitin
`to the second-order term, an expansion identical to tha~
`in equation (23) can be obtained from:
`
`l
`[77) = -
`a
`
`(
`
`-
`-
`- B + DT+ -
`
`t:i.É)
`RT
`
`(24)
`
`(25)
`
`(26)
`
`15
`
`,[oc]
`
`30
`
`45
`
`Fig. 7. The parameter S, given by equation (21), as a fonction oftem(cid:173)
`perature.
`
`k1 = -
`1 [
`2
`
`J
`
`+ 1
`
`2/j
`-
`-
`t:i.E
`-
`-B+DT+ - -
`RT
`
`axes a= 14.1 nm and b = 4.1 nm [13,18). It gives the
`hydrodynamic volume of BSA Vh = 1.241 x 10-25 m3.
`Takeda et al.
`[11) have shown, on the basis of
`measurements of fluorescence anisotropy of labeled
`BSA, that the protein volume remains almost unchang(cid:173)
`ed up to 45°C. In that case the expression (21) indicates
`that the parameter S depends on temperature. In Fig. 7
`numerical values of the factor S, obtained from relation
`(21), are presented.
`The self-crowding factor K (equation (22)), in turn,
`appears to be independent of temperature. Substitution
`of the hydrodynamic volume Vh = 1.241 x 10-25 m3
`in to relation (22) gives the numerical value K = 1.25. It
`is worth noting that in his original work [10) Mooney
`showed, on the basis of purely geometric arguments,
`that for rigid spherical particles values of K should lie
`between 1.35 and 1.91.
`
`3.4. Intrinsic viscosity and the Huggins coefficient
`At low concentrations, the relation between the solu(cid:173)
`tion viscosity and the concentration may be expressed
`by the polynominal (19):
`
`(23)
`
`where
`
`[77) = lim 77 5p/C
`c-0
`
`is the intrinsic viscosity, 77sp = 77r - 1 is the specific vis(cid:173)
`cosity and the dimensionless parameter k 1 is the Hug(cid:173)
`gins coefficient. The simplest procedure for treating
`viscosity data consists of plotting the 77 5pf c against con(cid:173)
`centration, extrapolating it to the intercept (equal to [77))
`and obtaining the coefficient k 1 from the corresponding
`slope. However, as was pointed out [19] even if 77sp <
`0.7, the concentration dependence of 77 5p/c forms a
`
`J
`
`+ 1
`
`+
`
`6/j
`t:i.É
`-
`-
`-B+DT+--
`RT
`
`So, both intrinsic viscosity and the coefficient of ex(cid:173)
`temperature. The
`pansion k 1 and k 2 depend on
`numerical values of [77) and k 1 for BSA are presented in
`Fig. 8. It is interesting that the value of the Huggins co(cid:173)
`efficient k 1 in the high temperature limit (k 1 = 0.753)
`agrees very well with the precise result obtained by
`Freed and Edwards [21). The authors calculated the
`Huggins coefficient for the Gaussian random coi! chaio
`and obtained a value k 1 = 0.7574. There are no theoret(cid:173)
`ical estimations of the second coefficient k 2 in equation
`
`6 .6
`
`6 .4
`
`~
`.... , 6 .2
`E
`"o
`
`76 .0
`
`5.8
`
`5 6
`
`0
`
`k,
`
`0 .76
`
`k,
`
`0 .75
`
`0 .14
`
`o.73
`
`0 .72
`
`0 .71
`
`15
`
`t[ 0c]
`
`30
`
`45
`
`Fig. 8. The intrinsic viscosity [71) and the Huggins coefficient k 1, given
`by equations (24) and (25), respectively, as a fonction of temperature.
`
`Page 9 of 11
`
`

`

`K. Monkos I International Journal of Biological Macromolecu/es 18 ( /995) 61-68
`
`67
`
`(23), However, relations (25) and (26) show that k2
`connected with the Huggins coefficient k 1 in a very
`11
`IÏJDple way:
`
`2 _1_
`k2 = k, -
`12
`
`(27)
`
`independently of temperature. Exactly the same relation
`was found by Maron and Reznik (22) on the basis of
`experiments with polystyrene and poly(methylmeth(cid:173)
`acrylate), and quite recently for some mammalian
`baemoglobins by Monkos (20). This suggests that rela(cid:173)
`tion (27) has quite a general character and is correct for
`difîerent sorts ofmolecules. Consequently, any property
`of a solution which determines the magnitude of k I will
`also determine the magnitude of k2• This conclusion is
`important, for it provides a test for any theoretical treat(cid:173)
`ment of dilute solution viscosity.
`
`4. Conclusions
`
`The viscosity of BSA solutions at temperatures up to
`45°C and in a wide range of concentrations at pH values
`near the isoelectric point may be quantitatively describ(cid:173)
`ed by the generalized Arrhenius formula (17). This for-
`
`n
`
`n
`
`n
`
`ÂE (
`;~ T; i~l z; - n ;~ z;T; - ---j--
`)2
`
`n
`;~i T;
`
`(
`
`n
`- n ;~ Tf
`
`n
`
`n
`
`J
`;~ T; ;~ T; - n
`
`D=
`
`flEs -
`R -
`
`asymptotic form of the generalized Arrhenius formula
`for small concentrations, the intrinsic viscosity and the
`Huggins coefficient may be calculated. Both quantities
`depend on temperature and the Huggins coefficient, in
`the high temperature limit, agrees well with the theoreti(cid:173)
`cal values obtained for the Gaussian random coi! chain.
`The Huggins coefficient k1 and the second coefficient of
`expansion k2 are connected by relation (27), which
`seems to have quite a general form.
`
`Appendix
`
`To find the coefficients B, D and t:.Es in equation (6),
`for a given concentration, we have minimized the square
`form:
`
`X= ~ (z· + B - DT: -
`
`i.J
`i= l
`
`,
`
`2
`t:.Es)
`, RT:
`
`1
`
`(Al)
`
`where zi = ln 7/i, with respect to B, D and t:.E5 • A simple
`calculation shows that:
`
`)
`2
`
`(A2)
`
`(A3)
`
`(A4)
`
`lllula may be transformed into Mooney's relation and
`Cllables the calculation of the parameter S and a self(cid:173)
`crowding facto